1230845

Etude des rayons cosmiques ultra-énergétiques avec
l’Observatoire Pierre Auger: de l’acceptance du
détecteur à la nature des particules primaires et aux
mesures d’anisotropies.
Julien Aublin
To cite this version:
Julien Aublin. Etude des rayons cosmiques ultra-énergétiques avec l’Observatoire Pierre Auger: de
l’acceptance du détecteur à la nature des particules primaires et aux mesures d’anisotropies.. Cosmologie et astrophysique extra-galactique [astro-ph.CO]. Université Paris Sud - Paris XI, 2006. Français.
�tel-00105431�
HAL Id: tel-00105431
https://tel.archives-ouvertes.fr/tel-00105431
Submitted on 11 Oct 2006
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
!"
#%$'&)(+* ,
-/.10325476
98;:<8;=>?
@ ABC;D D
EGFIHKJ'LNM'OPEQLSRUTVM'W'XCYILNO[ZX\J[]_^a`bO[XcZedIXBf
gihkj)l"monpomrqjtsvuxwyq\z\{|hvn}zyqm~l"€p~"w
:;‚ƒ „
†ˆ‡ G
$ ‰'*P‰'* ŠŒ‹ Žtt,Š/‘’tŠ”“ ()•–$'* Š/$'& ‡ ‹aU—9˜ ,*™‹šN˜ ‡ ("•›$'* Šœ K* ‘
&žŸ¡GŠ”*™‹¢£ ‡ N(+‹* ¤¥(+*™‹¢‹*¦-œ$'š§*™‹©¨Q‰*P&žN‘™‘*ª ‡ §,‘*«‰$
‰'˜ ‡ * ‘ ‡ * $[‹œ¬ &) ,G ‡ $[‹*«‰'* Š­ªG®‹ ‡ ()‘™$'&)* Š¯ª'‹("“ ®(+‹* Š¯* ‡ §$±°
“ * Š”$[‹* ŠŒ‰5žN,()Š ‡ ‹NªG(+* Š²
8;:}:³x´ @¶µ¸·º¹”» D µ C¼ @ ½|¾ C¿ =x½BÀBÀBÁ®DÂ;"} @ ‚º:ƒ¼KÃ8;¿£€8B D£„
Ä ÅoÇÉÈ9Ê~Ë
Æ
ÓÔÕcÇÉÔ³Ê
ÓÔ|Þcß"ÅoÇÉË
ÍÕcÔà”ËÊoÊoË
áKÅoÈiß)ä9Å~È
Ý ÅVÅVçkÔ
ÌÎÍϢϢÍ+Ð
Öx×ÆØNÙtÚ±Û®ÄÆÄ%ܙÝ
×%ÜÏÏaÛ
átÙtÜÄ%ÍÜÙ
åæØ™Ý Ýx×%ÍØNÜ
Ѐè±Û£åéÍ\êìë§ÙítÍ
¹ ƒ>ÑD} D|:£‚Ò:ƒ¼
"€8;ƒ:
"€8;ƒ:
ãâ "¿£>?!"ƒƒ>ÑÃ
ãâ "¿£>?!":
>?Ã>Ñà D™ƒî³ï
ð "³ @ ñÂÉ8;>Ñ D|:ƃî³
8;› µ >?¿£ïAB³ !" @ Ãò™:
³8;›ÆA"D³"}’8;:›ÆÃ8;:C"}
"ƒ}ƒ>?ÂÉ¿®có ð
ô
õæö
õæ÷Qøeùãõæö
õûú¡üQý
þ £ƒ>ѳñ´t¿®Ã>Ñx8;: D µ "=€8;D @¶µ ³¿x= @ ND ¿®¿ =ñD£¿®8;%‚Ò:ƒ¼Éÿ:>¢›;ÃÃND
@ >?ñÃUƒCcÂ;"> @ D£ µ ¼Æ>?Ccó þ Uƒ>ѳ ´KC @ :³!"ƒƒ>ÑÃ: @ >Ñï¿®} ¿®C"€8;ƒ:}:>ã8;}
ð @ :³Ãî³ ð Ã
@ 8;A ¿N"}:³Ãƒ>?cÿãñ8;: @ ÂÉ}:!"ƒ>?KD}:³ƒƒ>Ñ8;³ñ€ƒƒ>?³U8;:£": @ 8;A'D
@ : @ Ã:Bó
þ ƒ>ѳkA @ ¿®}¢´¿®Ã>Ñ @¶µ ³¿ = @ D|:x9C8;³ @ D @¶µ¸· ¹¢» k!"ƒƒ>ÑÃ: @ >Ñï¿®} @¶µ :>?€
D @ [D|>?Ã>Ñ8; 98;: @ :N;ÃÃ:³> @ ÃCî! @ :: â ": C>?ID|: @ "=€8;C"8;>?Bÿ”U€8;:£¿ µ ÂÉ8;>?®€ƒ¿£>Ñ
D µ €Ãƒ:³Ã ƒCÂB"> @ D™ƒî³ï D³"³<D ƒï=98;³’Ã8;³D|>?ƒ>Ñ8;³có
þ U¿®Ã>Ñ @¶µ ³C¿ = @ ®D @ KÃ8 @[email protected] "=€8;C"ƒ>Ñ8; :AÉ C"³ÃN":ïxD @ :³ @[email protected] ‚ µ ">=€c":³Ã8;:
"ƒ>ÑcÿC"³<8;:= @ >Ñ @ < â à @[email protected] }<¿®8;¿®}!;C< ƒAÉ}ƒ>?³Bó?ó?ó
@¶µ >?ƒ>Ñ:aDÃÃ8 @[email protected] "=€8;C"ƒ>Ñ8;ìÿ+‚Ò<C8;:î!">?<¿®Ã>Ñ @ :³”!"ƒƒ>ÑÃ: @ >Ñï¿®¿®8;t :>?€
D µ ;ÃÃ:³> @ „ @¶µ :>?€ :AÉ¢D @¶µ¸· ¹¢» ÿÉ":®>?®D @ }:³ @[email protected] ゠µ ">!;aƒ8;>Ñ¢"³ V8;ƒ”ƒ>ÑÃCî>ÑC"
Ã>у> }:³¿®›Kî}:¿N">?³¿®c
ó ±Ã> ´ >?>?!|ÿ · C"=€ @[email protected] Bÿ ƒ>ѳ' @ >?Â>Ñ%€8;:§¿ µ cÂÉ8;>?
³Ãc;D|x":›Ã8;:D ÃCcÂ;"> @ D ƒî³ï®„³ÂÉ8;ƒ ³ÃBÿiC @ Â;"ƒƒ>Ñà @ 8;’D¿®8;¿®D|> NÃ> @ cÿ
¿ µ –Ã8;³C"¿£¿®N³Ã8;:C"Aɛ´±D8;³ @ t¿®> @[email protected] :®Dt¿®8;>a
¿ ¿®Bó þ KÂÉ8;:³£ƒ:>ÑñÃ8;!">ÑC"}
98;:8;:³ @ ’ v8;ƒ
:³xÂÉ8;:³ÂÉ V8;:ƒ> €8;:’¿®x9ƒ¿®ƒƒUDx¿®³ ´®=>Ñ[Ãñ8"‚Ò™D³"³
D§Ã8;³D|>?ƒ>Ñ8;³Æ":³>"ABc"= @ có ±Ã>’A @ ¿®›": ⠃:³D|>r"%DÂÉ:³KD8Ã:%´}cÿ
@ â "³D|Bÿ >Ñ "¿£>Ñìÿv98;: cÂÉ8;>?!"ƒC"AɮÃC"³ÃCî³ DUÂ>Ñ®ÂÉã¿®8;>”D³"³ @ t=€8;³
î:¿®:cÿ³:>ÑC<³8B<8;:Ã8;>Ñ:®‚Ò8;:´U³8;:ÂÉc":ìó?ó?ó
!#"%$ õ
)
_
H
&
õæý
ö
! üQù('æ÷Gõûý
*+-,/.103234546+78+4:9;<2:=>[email protected]?BADCE+
FG
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
¾Bó?¾ P ãî>у8;ƒQ> :³ ó ó óó óó óó ó óó óó ó óó óó óó
¾Bó ½ Tì’8;=³ƒÂB"ƒ>Ñ8;³ ó óó óó óó ó óó óó ó óó óó óó
¾Bó ½|ó?¾
ÃCîQ> :³D DÃ>Ñ8; óó óó ó óó óó óó
¾Bó ½|ó ½ Tì 9à ƳƒAB>Ñ'ó ó óó óó ó óó óó óó
¾Bó ½|Wó V
8;¿£€8Bƒ>?ƒ>Ñ8; ó óó óó ó óó óó ó óó óó óó
¾Bó ½|ó X
>уƒƒ>?=:ƒ>Ñ8;±"AB: @ ">?œó óó óó ó óó óó óó
¾BWó V
ƒ>?AB>?³xD|:%C¼É8;³¿®™Ã8Bƒ¿£Q> }:³ó óó ó óó óó óó
¾Bó X TìC¼É8;³’Ã8Bƒ¿£Q> :³’":]:³8;ƒ>ÑD|>Ñ óó ó óó óó óó
¾Bó X³ó?¾ ãâ @ 8;C"ƒ>Ñ8;–ƒ!"ƒ>r @ <C¼É8;³’Ã8Bƒ¿£>Q:³§óó
¾Bó X³ó ½ :C":A"¿£¿N£<ABC"³D ⠃>?³Ãƒ>Ñ8;³ó óó
¾Bó X³Wó V à @ ">?’<C¼É8;³’Ã8Bƒ¿£Q> :³/óó ó óó óó óó
HO
ϔËÕ§ä!ß"Ô³ç3` ËÕ£äiËß"à”ËÕNÔ|ÞKJÈ!ÕKaDb:cß;ÅMLN7Ë}Õ
½|ó?¾ T µ "ƒ¿®8Bî³ïN„:ÆÃc @ 8;ƒ>?¿®ïƒ !"ƒ: @ ó óó óó
½|ó ½ Tì DÂÉ @ 8;9¿®}DAɃ=€"ƒ¿®8Bƒî³ƒ>Q}:³1ó
½|ó ½|ó?¾ f™ƒ=€< @ Ã8;¿N"AB³ƒQ> }:³ ó óó ó óó óó
½|Wó V f™ƒ=€î!;D|8;Q> }:³ óó óó óó ó óó óó ó óó óó
½|Wó V|ó?¾ [ â >?¿x:¿ÎDDÂÉ @ 8;€¿® @ 8;AB>?ƒ:³D|>?! @
½|Wó V|ó ½ ¹ >8 @ @ "C": â @ 8;AB>?ƒ:³D|>?!": â óó óó
½|ó X >?¿x: @ "ƒ>Ñ8;³ ó óó ó óó óó óó ó óó óó ó óó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
óó
ó ó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
ó ó
óó
óó
óó
óó
óó
óó
óó
óó
óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
óó
ó ó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
ó ó
óó
óó
óó
óó
óó
óó
óó
óó
óó
ó ó S¾ R
ó ó ¾SU
ó ó S¾ U
ó óP½BÀ
ó óP½>X
ó óPZ½ Y
ó óP\½ [
ó óP½ZU
ó óP½ZU
ó ^
ó V|¾
ó ^
ó V|¾
ded
óó
ó ó
ó ó
ó ó
ó óó
ó óó
ó óó
ó óó
ó ó óó
ó ó ó óó
ó ó ó óó
óó
ó ó
ó ó
ó ó
óó
óó
óó
óó
ó ó
ó ó
ó ó
óó
óó
óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
ó óó
óó
ó ó
ó ó
ó ó
óó
óó
óó
óó
ó ó
ó ó
ó ó
óó
óó
óó
ó ó^V>X
ó ^
ó V>X
ó ^
ó VBÁ
ó ^
ó VZU
ó g
ó XÀ
ó g
ó X³¾
ó g
ó XZX
d
l
)Z)
ÏihoÛ£àkÕcË}ßKjÔ|ÞcÈ9Å~ß"˛ÓxÅ~Ë}ß)ß"˖ØkNkä!Ëß
V|ó?¾ ;‚Ò: â Ã>Ñ}ƒ> :³óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
V|ó ½ T7 Ãî³8;> â D<ƒ>? ⠀ƒ>?¿®C": â ó óó ó óó óó óó ó óó óó óó ó óó óó
V|Wó V T7c":›Dƒ: ¶;à ó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
V|Wó V|ó?¾
Ã>?ƒ>Ñ8;±D µ :³ Ã:ÂÉ ´U v ÃCî³1oÉ8c ó ó óó óó óó ó óó óó
V|Wó V|ó ½ C @ 8;!"AÉñD<ƒC"ƒ>Ñ8;³D|:Æc":›Dƒ: ¶;à óó óó óó ó óó óó
V|Wó V|Wó V à }:>у>?ƒ>Ñ8;›D’D8;³Œóó óó ó óó óó óó ó óó óó óó ó óó óó
V|Wó V|ó X Ã8;³ƒƒƒ:³Ãƒ>Ñ8;–D<Âɳ¿®}cÂÉà @ ™c":–Dƒ: ;à ó óó óó
V|ó X T7 DÃ:’Dq
p³:³8;óÃeó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
V|ó X³ó?¾
Ã>?ƒ>Ñ8; óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
V|ó X³ó ½ à }:>у>?ƒ>Ñ8;›D’D8;³Œóó óó ó óó óó óó ó óó óó óó ó óó óó
V|ó X³Wó V Ã8;³ƒƒƒ:³Ãƒ>Ñ8;–D<Âɳ¿®}cÂÉà @ DÃ:Dq
p³:³8;óÃxó
V|Wó Y Ã8;³ƒƒƒ:³Ãƒ>Ñ8;–D @¶µ ³ƒAB>Ñ ó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
lnm
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
rs4E.utv06=>+xwE+=ywE234E4Ez{+=w6+|t~}€y‚=ƒ+,Z„n.n9Z23?,\+†k?‡+{,\,/+ˆrsC6‰3+{,
Ø ÇÉÇÉËa7ÞcÔ³çkÇÉˋ`DN ߃c}ÕcËÔ{NŒ` ˛ÕKNk߃ ÔÇÉË
U
X³ó?¾
³>?ƒ>Ñ8;³xóó óó ó óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
X³ó ½ ±8;ƒ>?ÂB"ƒ>Ñ8;³ˆó óó ó óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
X³Wó V @ AÉ8;ƒ>?ƒî¿®D Dà @ ³Ã®D™î!":<>?ÂÉc": XN Y óó óó ó óó óó
X³ó X T7 Ã8;³ÃD
8;³Ãƒ>Ñ8;]
T ¹ ó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
X³ó X³ó?¾
³>?ƒ>Ñ8;Só óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
X³ó X³ó ½ ¹ "C"¿®ƒƒ>ÑC"ƒ>Ñ8;'D
T ¹ ó óó ó óó óó óó ó óó óó óó ó óó óó
X³Wó Y NÃc;Ã>? D DÃ>Ñ8; óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó
@ Ã: @ D @¶µ NÃc;Ã>? D DÃ>Ñ8;›D³"³:³ à @[email protected] : @ î³ â "AÉ8;! @ ­óó
X³Wó Y|ó?¾
X³ó Á ÃÃC"³Ã >?³ƒC"C"³x’;ÃÃC"³Ã >?}AB D|:%c":–D ƒ: ¶;Ãéó óó
X³ó [ » 8;:ÂÉ @[email protected] 8;=³CƒÂ;"= @ @ >Ñ´ @¶µ ;ÃÃC"³ÃUóó óó óó ó óó óó óó ó óó óó
X³ó [ó?¾ ±ƒ: â 9ƒ>?¿®}C @ ¿® @ 8;³Ãƒ>Ñ8;³
T ¹ _ó óó óó ó óó óó
X³ó [ó ½ ‘’³³8;:ÂÉ @[email protected] 8;=³ƒÂB"= @ ®„ @ ’Tk;D|>?=98; óó óó ó óó óó óó ó óó óó
X³ó [Wó V ±ƒ: @¶µ NÃc;Ã>?xDDÃ>Ñ8;–< D8;:³’D @ £C"ƒ:C"ƒ>Ñ8; ó óó
X³ó [ó X ¹ >Ñ<€8;: @¶µ >ÑDƒ> !Ãc"ƒ>Ñ8;[D @ ñ!"ƒ: DCc¼É8;³Ã8Bƒ¿£Q> }:³Póó
ó ^
ó YBÀ
ó ó^YBÀ
ó ^
ó Y|¾
ó ó^Y>X
ó ó^YZY
ó ó^Y\[
ó ó^YZU
ó óPÁ|¾
ó óPÁB½
ó óPÁB½
ó óPÁZY
ó óPÁBÁ
ŠDF
O d
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
ó Ž
ó [ƒX
ó óŽ[Y
ó Ž
ó [;Á
ó ^
ó R|¾
ó ^
ó R|¾
ó ^
ó RB½
ó ó^RZV
ó ó^R>X
ó ó^R\[
ó ó^RZU
ó ó^RZU
ó ó^UZV
ó ó^U\[
ó ó^UZU
åéË}ÕKN7ß"Ë`DN ÕSa”ËÇÉÞcß)˖Ë}çŒc}çkËß"äiÅoË]` ËÕ£ß"Ô\I9ÈiçkÕ®ÇÉÈ!ՃJGÅMLN7Ë}Õ£Ô\j9ËǖØkNkä!Ëß
Y|ó?¾ @ Ã>Ñ8;±Ã8;³ƒƒƒ:³Ãƒ>Ñ8;–D’Âɳ¿®D|:%c":›D ƒ: ¶;Ãûó óó
Y|ó ½ Ã8;³ƒƒƒ:³Ãƒ>Ñ8;–D @¶µ Âɳ¿®} óó óó ó óó óó óó ó óó óó óó ó óó
Y|Wó V ƒ>?¿N"ƒ>Ñ8;–D @¶µ ³ƒAB>ѐóó óó ó óó óó ó óó óó óó ó óó óó óó ó óó
Y|Wó V|ó?¾
ƒ¿£>?!"ƒ>Ñ8;±D @¶µ ³ƒAB>ÑñÂÉà @ –8;} " @ 8®ó óó óó ó óó
Y|Wó V|ó ½ T7U¿®ƒî³8D D|: · —– ó óó ó óó óó óó ó óó óó óó ó óó
Y|Wó V|Wó V
8;:ƒ=€D µ "ƒ:!"ƒ>Ñ8;‹
±8; " @ 8Nó óó óó ó óó óó óó ó óó
Y|Wó V|ó X ±ƒî³8}D @ ƒ!"ƒ>?ÂÉ ó óó óó ó óó óó óó ó óó óó óó ó óó
@ Ã: @ D|:–ƒ9Ö³ƒAB>Ñbó ó óó óó ó óó óó óó ó óó óó óó ó óó
Y|ó X
“
)Z)Z)
H•”uO
ó ó
óó
óó
óó
óó
óó
óó
óó
ó ó
óó
óó
óó
óó
óó
óó
óó
rs45?=>2u9Z,\2˜‚?‡+=™w5Cš,\.103234546+78+4:9›;<2:=>[email protected]?BADCE+
ó Æ
ó ¾cÀZR
ó óƾB¾cÀ
ó óƾBS¾ Y
ó óƾB¾cÁ
ó óƾc½BÀ
ó óÆS¾ V|¾
ó óÆS¾ VBÁ
ó óƾSVZR
F{œ:G
åéÔß"Ç>bk˱Գʞc}Ô|ÞcÈiÅoß"˖ËÞ` ŞŸ5NkÕ+ÅoÈiç
H<“uH
Á|ó?¾
> €:³ƒ>Ñ8;› @ >?=!"Ã8;:¿®8c¼É ó óó ó ó ó óó óó ó óó ó ó ó ó ó ó ó ó ó ó ó ó Æ
ó ¾SYB½
Á|ó ½
> €:³ƒ>Ñ8;›;D|ÂÉÃ>Ñ8;Q„|¿N"ÃCî³x @ c"8;>? ³;8 %>Ñ8;ƒ8;9Uóó ó ó ó ó ó ó ó ó ó ó ó ó Æ
ó ¾SYZY

O
Ù§Ë}ÇÉÈiçkÕÞcßN7ÇBÞ+ÅoÈ9ç ` ËÕ®Ô³ç ÅoÕcȳÞcß"ÈaãÅoËÕ¡'äiß)Ô³ç:`kË]cÇb7ËÊoÊo˱Ô\j9ËǖÊoÔ¢J†cÉÞKbkÈu`kË[ÐìاÓ
[ó?¾ Ã8;³ƒƒƒ:³Ãƒ>Ñ8;–D µ :›D|>?£¤ @ ÂÉÃxÃ8;:ÂɃƒ: 8;C @ :> 8;ƒ¿® D|:›Ã>Ñ @ óó óó
[ó ½ Tì Ãc;<D µ :³ Ã8;:ÂɃƒ: 8;C @ ³8;Æ:>V8;ƒ¿® óó óó ó óó óó óó ó óó óó óó
[Wó V f™³C @ >Ñy"ƒ>Ñ8;±":ÆÃc;DÃ8;:ÂɃƒ: !"ƒƒ>Ñ @[email protected] D|:ÆÃ>Ñ @ ó óó óó óó ó óó óó óó
[ó X ¹ :>ÑCC"³Ã DÃ8;³ƒƒƒ:³Ãƒ>Ñ8;±< NÃc;Ã>? D @ ñ¿®ƒî³8D ó óó óó ó óó óó óó
[ó X³ó?¾ ¹ ƒ>?³Ã>?€ D<>?¿ : @ "ƒ>Ñ8;³’:¿®ƒQ> }:³ â 98B>?ƒ>Ñ8;% @ "ƒ>?ÂÉ óó óó óó
[ó X³ó ½ Ã8;³ƒƒ:³Ãƒ>Ñ8;–D!"C"¿®ïƒD|:ÆD|>?n ¤ @ §óó ó óó óó óó ó óó óó óó
[Wó Y
C">?¿®"! @ ¼}ƒQ> }:³ D|:›Ãc;<D @ ®Ã8;:ÂɃƒ: 8;C @ :> 8;ƒ¿®[ó ó óó óó óó
[ó Á f™³C @ >Ñy"ƒ>Ñ8;±":ÆÃc;D µ :³x">Ñ8;ƒ8;>Ñ
}:!;D|ƒ>?98 @ ">? óó óó óó ó óó óó óó
[ó Á|ó?¾ » Ã>? D µ >?³Ã @ : @ ¥
:!;D|ƒ>?£ ¤ @ óó óó óó ó óó óó óó ó óó óó óó
[ó Á|ó ½ ¹ "C"¿®ƒƒ>ÑC"ƒ>Ñ8;'D|
: p³: â óó óó ó óó óó óó ó óó óó óó ó óó óó óó
[ó Á|Wó V ¹ 8ÃD|: DÃ8;³ƒƒ:³Ãƒ>Ñ8;–D³"³ @ Ãc;D µ :³ Ã8;:ÂɃƒ: 8;C @ ó óó
[ó Á|ó X
>?n ¤ @ 
:!;D|ƒ>?£ ¤ @ ÂÉà :³ Ã8;:ÂɃƒ:!"ƒƒ>Ñ @[email protected] óó óó ó óó óó óó
[ó Á|Wó Y ¹ :>Ñy"³Ã DÃ8;³ƒƒƒ:³Ãƒ>Ñ8; óó ó óó óó óó ó óó óó óó ó óó óó óó
[ó [
ƒ>?¿£>ÑC"ƒ>Ñ8;±D @ ñ¿®ƒî³8Dæó ó óó óó ó óó óó óó ó óó óó óó ó óó óó óó
[Wó R
ƒ @¶µ >Ñ8;ƒ8;>Ñ«ó óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó óó óó
H<“£O
ó Æ
ó ¾SYZR
ó óƾcÁ|¾
ó óƾcÁB½
ó óƾcÁZY
ó óƾcÁBÁ
ó óƾcÁZR
ó óƾK[ƒX
ó óÆS¾ RBÀ
ó óÆS¾ RBÀ
ó óÆS¾ RZV
ó óƾSR>X
ó óƾSRZY
ó óƾSR\[
ó óÆS¾ UB½
ó óƾSUZY
H•”
8;¿£!"C">ÑC8;±ÂÉà @¶µ "! @ ¼ D c¼ @ >?ABî óó
[óWU|ó?¾
Ã>?ƒ>Ñ8;±D @ U¿®ƒî³8}Dbó óó ó óó
[Wó U|ó ½ @ "ƒ>Ñ8;›ƒ r1h α ó óó óó ó ó ó
[Wó U|Wó V ¹ :>ÑCC"³Ã DÃ8;³ƒƒƒ:³Ãƒ>Ñ8; óó ó ó ó
[óWU
óó
ó ó
óó
óó
óó
ó ó
óó
óó
ó óó
ó óó
ó óó
ó óó
óó
ó ó
óó
óó
óó
ó ó
óó
óó
ó óó
ó óó
ó óó
ó óó
óó
ó ó
óó
óó
óó
ó ó
óó
óó
ó ó®½BÀBÀ
ó ó®½BÀBÀ
ó ó®½BÀ|¾
ó ó®½BÀB½
Ù§Ë}Ç>bkËß"Çb7˛`Eh?Ô³ç Å~Õ+ȳÞcß"ÈaãÅ~Ë}Õ£Ô\j9ËǖÊoË}Õ`kÈ9çkç:cË}Õ`5h?ØkNkä!Ëß
R|ó?¾
"ƒ’DÃ8;:ÂɃƒ: ó óó óó ó óó óó ó óó óó óó ó óó óó óó ó ó ó ó ó ó ó ó
R|ó ½ ƒ: @ C"": â ABC"³DÃCî³ @[email protected] x„³" @ >ÑÃc"ƒ>Ñ8;±D @ ñ¿®ƒî³8}D ¹ ó ó ó ó ó ó ó ó
R|ó ½|ó?¾ ƒ:¿® Dƒ: @ C"D @¶µ "! @ ¼C ÂÉà @ ñ¿®ƒî³8D ¹ ó ó ó ó ó ó ó ó
¦
)S§
m
¨+{,=ƒ˜i+e;•9Z?‡„3+=
ØaDa
U|ó?¾
U|ó ½
ß)ÈvÇb7˛b7È9ÊoÅoÕÞ+ÅMLNkË]`(N ß"Ô\I9Èiç ç7˕JË}ç9Þ£ÇÉÈiÕKJGÅMLNkË
· ƒ8D|:³Ãƒ>Ñ8;Ió óó ó óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó
» 8;ƒ¿N @ >ÑC"ƒ>Ñ8;– @ "ƒ>?ÂÉ D<Ã8;¿£€8BC"’A @ ;Ã>Q:³ ⠃Cƒ¶A @ ;Ã>Q}:³
U|ó ½|ó?¾
³ƒ>? D<C¼É8;³Ã8Bƒ¿£Q> :³<A @ ;Ã>Q:³ ó ó óó óó óó ó óó
U|ó ½|ó ½
³ƒ>? ⠃Cƒ¶A @ ;ÃQ> }:³ óó óó ó óó óó óó ó óó óó óó ó óó
U|ó ½|Wó V » 8;ƒ¿N @ >ÑC"ƒ>Ñ8;›A @ ;ÃQ> }:³K «" ⠃Cƒ¶A @ ;ÃQ> :³ ó ó óó óó óó ó óó
U|óWV
>ÑÃ:³Cƒ>Ñ8;ÆD<ƒ: @ C"¦ó óó ó óó óó ó óó óó óó ó óó óó óó ó óó
|U ó X
8;³Ã @ :³>Ñ8;Æóó óó ó óó óó óó ó óó óó ó óó óó óó ó óó óó óó ó óó
_{”uO
ó®½BÀZR
ó®½|¾B¾
ó®½|¾X
©ªª
_ d “
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
óó
ó ®
ó ½ZVBÁ
ó ó®½ZV\[
ó ó®½ZVZR
ó ó®Z½ VZU
ó ó®>½ X½
ó ó®>½ X/V
ó ó®>½ XZX
¬Kú¡üG÷†­®&g¯
ø üQù(­ ú
:>Ñ @ [DÃ8;:ÂɃ±D @¶µ â >ѳÃÆD£C¼É8;³tÃ8Bƒ¿£>Q}:³§": D=:§D|: Â}>?ABƒ>Ñï¿®–ƒ>Ñïà @ Bÿ
@¶µ ƒ:³D D Ã<¿x¼ƒ>Ñ:³<!"ƒƒ>ÑÃ: @ ’ÂÉ:³D|:›Ã8Bƒ¿®8B£9ƒ¿£>Ñ": â î}¼>ÑÃ>ѳD µ â @ 8; @
ƒƒƒ:³Ãƒ:£D @ N¿N"ƒ>Ñï®": â @ :³™9ƒ>?ÃCî³ @[email protected] cÿ€=>ѐÂB" @ tÃ8;³ƒƒƒ:³Ãƒ>Ñ8;D™AB>?A"}~}:³
;Ãà @ C"::ƒ> @ >Ñ%î¼|ƒQ> :³ D!"ƒƒ>ÑÃ: @ ’D™³8B‚Ò8;:có
T7bÃc"C;Ã>уƒQ> }:³ @ Q @ :³K¿N"~:!"= @ D|: Cc¼É8;³¿®–Ã8B¿£Q> :³'ƒKÃC">?³¿®±8;
ƒ9ÃKD µ ³ƒAB>ÑB6ÿ }:> µ ³DIƒ:U @ :³£Dy
VB½±8;D|®DtABC"³D:®x
p³: â £D8;1: c%8;D|®D
ABC"³D:xQ³ƒAB>ÑBÿkD:>Ñ @ ÆÃC">?³tD
– °û‚º:³± : µ ´ 1020 ‡ ° €:¥
Ĩ
¿ ¿®t":< D @ ´|ó
TìKC¼É8;³–Ã8Bƒ¿£Q> :³ @ K @ :³Æ³ƒAɃQ> :³›DÛ´IÃt‚Ò8;:K€8BD}Æ:³'³ƒAB>Ñ[¿£> @[email protected] V8;>љƒ:€ƒ>Ñ:£´§Ã @[email protected] ¨
}:>C§;ÃÃCƒ>?= @ ®ÂÉà @ ²
o:ƒ:Ã8 @[email protected] >у>Ñ8;³:
T ™ £ó T µ ƒ:³D£D™Cc¼É8;³
Ã8Bƒ¿£Q> :³ @ ’ @ :³³ƒAɃQ> }:³9ƒ¿®™D µ â @ 8;™Dx¿N">Ñïx>?³D|>?Ãx:[D8;¿N">?³UD µ ³ƒAB>Ñ
³Ã8;>?³Ã8;:±%î¼|ƒQ> :³ D!"ƒƒ>ÑÃ: @ cÿ!CƒÂÉ9:³
ƒ´
:³ @ }:³<ƒ:ƒƒ>Ñ+ó
T µ â >у³ÃN
¿ ¿®KDN!"ƒƒ>ÑÃ: @ £ @ ¿®C">?U€8BD³"ñ:³t³ƒAB>Ñt:9ƒ>Ñ:K´ 1020 ‡ °
ƒ<:›D ±98;: @ <î¼|ƒ>ÑÃ>ѳñ„³› vcÿ @ 8;’D @ :8;!"A"ƒ>Ñ8;[D:>Ñ @ :8;:Ԃº:³~ }: µ ´
@ ƒBÿ @ <C¼É8;³Ã8B¿£Q> :³’ƒ:=>Ñ}<DD8;:C"= @ <€ƒD µ ³ƒAB>ђ%>?C"AB>ÑC"}™cÂÉÃ
@ î³8;8;³D|k
: 8;³D§D|> €:³aÃ8Bƒ¿®8 @ 8;ABQ> }:³B1ÿ :>9¿£ @ >Ñ}:> V8;ƒ¿®¿®} @¶µ :>?ÂÉD @ :5 V8;>ÑD
@ :¿£>ÑïBó –î³³8;¿®ï³Bÿ<"€ @ ±Ã8;::¢
fµn¶£ÿ>?¿£€8B›:³]
8;ƒ±Ã8;}ƒC">?'ƒ: @ QD|>уC"³Ã
D£8;:ã€8;ƒ>Ñ @[email protected] cÿ”D @¶µ 8;D|›D @ –Ã}C">?³›D™
–9ÃBÿ €8BC"£: Cƒ>Ñ: â 8;= @ ï¿®›": â
;ƒƒ8;î¼|ƒ>ÑÃ>ѳcó
’³8;¿x=: â 8;=|‚Ò<;ƒƒ8;î}¼Q> :³cÿ| @ @ aD µ â @ 8Bƒ>Ñ8;ÆD:9ƒ³8+Â;;Bÿ @ ‚ƒ¿£>Ñ
!" @ ’³8c¼": â ;Ã> VD A @ â >Ñ8;:±³Ã8; @ ƒ:C":’A"¿£¿N|ÿ9C8;™Ã8;:³™€8;:"=ƒ>?D
î³³8;¿®ï³ ⠃~ ¿®¿®}”Â}>Ñ8 @ }cÿ;ƒ:³Ãƒ>?= @ D µ ;Ãà @ ”D !"ƒƒ>ÑÃ: @ Ãî!"ƒAÉ+ó 9³D³"}cÿ
@
">?y
}: µ : ¼€'D µ 8;=|‚Ò–;ƒ8;î¼|ƒQ> :³:>ÑG;Ãà @ ›D›!"ƒƒ>ÑÃ: @ Ƴ'ƒ<: ®›!;›´ @ :>
Dó @ ›ƒ>?ƒ›D ð C8;:ÛD|:5C¼É8;³¿®%Ã8Bƒ¿£Q> }:³ ð „> @ ¶":§5 €§98;:ÂÉ8;>?K â @ Q> }:³
H<_
8;: @ 8;=³ƒÂB"ƒ>Ñ8;³aD8;a³8;:³aD|>у€8B8;³cÿɳ8;C"¿£¿®} @ ¢ÂB @ :a8;=³ƒÂÉaD|:p³: â ÿ @ €
D|:ƃ€ÃƒBÿ @ UÃ8;¿£98B>?ƒ>Ñ8;–">?³ƒ>3}:³ @ £D|>уƒƒ>?=:ƒ>Ñ8;–"AB: @ ">?Bó
T7D|> NÃ: @ a€8;: >ÑD}ƒ>! @ 8;:Ãã€8;ƒ>Ñ @[email protected] ”DãCc¼É8;³Ã8Bƒ¿£>Q}:³ãÂ}>Ñ}”A @ ¿®¢D|:
">?:
:³ @ Ãî!"¿£³k¿N"AB³ƒQ> }:³k>?} ¶A @ ;ÃQ> :³ãDÂ>Ñ @ ì!"ƒƒ>ÑÃ: @ cÿ)³D³" @ :7ƒC)‚ƒÃ8;>?
Ãî!;8;ƒQ> }:³£8;C @ ¿® >?¿£Â>у>?= @ Bÿv¿x= @ "= @ N´tà @[email protected] Uƒ: @ C"xD µ :±8Ã:³DUD|> €:³ƒ>Ñ8;ìó
T µ >?< 8;ƒ¿N"ƒ>Ñ8; ƒ: @ –D|>уƒ>?=:ƒ>Ñ8; ƒ!"ƒ>Ñ>8 ¶¿£98; @[email protected] ›D£8;:ãƒñ9D|:³% @ ±D|>уƒƒ>?=:ƒ>Ñ8;
"AB: @ ">? DC¼É8;³Ã8Bƒ¿£Q> :³"!"Cƒ·Ñ´ @¶µ 8;=³CƒÂ;":’ƒƒƒ Ã8;¿£¿® >Ñ8;ƒ8;9Bÿ!>?D|>ÑC"}
:§ÂÉ}ƒ:³ @ €8;>?C"Aə;ƒƒ8;³8;¿£Q> }:³ÂÉ @ C8;:Ãcó : @ @ ¢Cc¼É8;³Ã8B¿£Q> :³ @ ¢ @ :³³ƒAÉ
ƒQ> :³CC">Ñ ƒ:³CÃ>?= @ D µ ƒ£ƒ<: NC"¿£¿®} 9:DÂ>љ€8;:9ƒ¿®ƒƒU:³U>ÑD}ƒ> !Ãc"ƒ>Ñ8;
D¢8;:Ã+\ó ± @ î³::³C¿®cÿÔC¼É8;³Ã8Bƒ¿£Q> }:³C8;”C"’„ @ p³: â "ƒ³D|:t":8;:D 1020
‡ ° ƒD @¶µ 8;D|£D µ :³U!"ƒƒ>ÑÃ: @ ®!"qo> @ 8;¿®ïƒ£Ãc"ƒ£™!" ƒ>Ñïà @ B£ó T7t¿®ƒ:UD|:ƒ€Ãƒ£": â
@ :³î!":³ƒAB>ѳÃ>?D8;³Ã:³ ABC"³D ƒ: ¶;à DDÃ>Ñ8;ìó
T µ â 9ƒ>ѳà > c5
ÿ :>’:ƒ> @ >Ñ @ ÃîQ> }:³[D @ ¿®ƒ:–D @ @ :¿£>Ñï[Dy
p³:³8;óÃ
8;=³ƒÂÉ£:³§Ã8;::§D³"³ @ ®ƒ€Ãƒ§":< Dƒ:³xD 5.1019 ‡ °Uÿ7 @ 8;²
}:³ @¶µ â 9ƒ>ѳà f ÿ
:>:ƒ> @ >Ñt:QCc":bDKDÃ:U!"ƒƒ>Ñ£": 8 @ ƒ:ñ:³KÃC">?³KDk
o}> @ 8;¿®ïƒUÃc"ƒcÿ ³
Ã8;< ³ƒ¿®®!;ñÃN8;=³ƒÂB"ƒ>Ñ8;ìó ³GDN¿®ƒ:xñD µ ƒ:³D|>Ñ @ C¼É8;³UÃ8Bƒ¿£Q> }:³‚Ò:³~ }: µ ": â
@ :³î!":™³ƒAB>Ñcÿ @¶µ =³ƒÂB"8;>? ¹ >у :AÉcnÿ :>ベ;Ã:³ @[email protected] ¿® ›
³±DxÃ8;³ƒƒƒ:³Ãƒ>Ñ8;
ƒAɃ>?³Bÿi;ÃÃ:¿ : @ DD8;³D:>Ña @ :³D™D: â "³a¿N">?!"}có T µ ⠀ƒ>ѳà :AÉcÿ
:>”Ã8;¿x=>?³ @ %ÃîQ> }:³ND£¿®ƒ:ND @ kp³:³8;óÃtÂÉÃNà @[email protected] §D|:c":GDNDÃ:ñ":
8 @ ÿ€ƒ¿®ƒƒCNDxDÃ:³ ƒC">?³UD µ Âɳ¿®D µ ³ƒAB>ÑñD @¶µ 8;D|ñD 1020 °ˆ!""ìÿ
AB± ¸;à ´UCUƒ: ¶;ÃD DÃ>Ñ8;±D µ Â>?8;›
VBÀBÀB¨
À o¿ 2 ó
"³:³ ¿£>ÑïU!"ƒƒ>ÑBÿv"ï™:±= ¢î>у8;ƒQ> }:³Bÿi³8;:³™DÃ>?ÂÉ8;³ @ Nî³³8;¿®³8 @ 8;AB>Ñ®D|:
C¼É8;³¿®}§Ã8Bƒ¿£Q> }:³K @[email protected] y
}:³§³8;:³ @ [Ã8;!">Ñ8;³N":;‚ƒ8;:D µ î:>¶ÿbC"€ @[email protected] " @ £Dƒ>Ñ
ƒ: @ C"” â 9ƒ>?¿®}C": â ó T µ =³ƒÂB"8;>? ¹ >у :AÉ¢8;=³CƒÂÉ @ C¼É8;³Ã8Bƒ¿£Q> }:³”D¿N">Ñï
>?³D|>?ÃK!" @¶µ >?ƒ¿®D|>r">?ÆDñAɃ=€®"ƒ¿®8Bƒî³ƒQ> :³+ÿãAB>?A"~ }:³£Ãc;Ãc;D£Dt!"ƒƒ>ÑÃ: @ ƒ: @ C"ñD @¶µ >?C;Ã>Ñ8; D C¼É8;³ñÃ8Bƒ¿£Q> :³xD³"³ @¶µ "ƒ¿®8Bƒî³ïNƒƒƒB3ó Tk%î¼|ƒQ> }:³®D
ABC"³DAɃ=9™"ƒ¿®8Bƒî³ƒQ> }:³’>?}ƒ8}D|:>?Bÿ€›>?³>уC"ƒ: @ ’D|> NÃ: @ ™³Ã8;}ƒ™98;:
¿®8}D @ >ÑC @ 8}ÃCƒ:³<D€³D³"’D @ ñî¼|ƒQ> :³ î!;D|8;Q> :³ ´ñî!":x³ƒAB>ÑBó » 8;:³<DÃ>?ÂÉ8;³
³ƒ:>? @¶µ =³ƒÂB"8;>? ¹ >у :AÉcÿì">?³ƒ6> }:³ @ ¿®ƒî³8}D :ƒ> @ >р8;:ƒC">? @ D8;³
%8cÂÉ!"³ÃñD|:%c":›D : ;à ">?³ƒ> }:³ D|:ÆDÃ:Dq
p³:³8;óÃBó
"³¢:³Ã8;³D’!"ƒƒ>ÑBÿ³8;:³¢}8;³ @ ¢¿®ƒî³8DaDÃc @ Ã: @ D @¶µ ;ÃÃC"³ÃD|:§c":
H d
D±: ;Ã:>8;}ƱDÂÉ @ 8;€Æ": Ã8;:%D|: ƒCÂB"> @ D–ƒî³ïCBó‚TkQDƒ¿£>?!"ƒ>Ñ8;eÃ>Ñ
D @¶µ ;ÃÃC"³ÃNƒÃƒ:³Ã>r @ U98;: @¶µ C"= @ >Ñ¿®} D|:'ƒ€Ãƒ£[³ƒAB>Ñ£D™Cc¼É8;³ Ã8Bƒ¿£>Q}:³có
T µ NÃc;Ã>?™D’DÃ>Ñ8;%D€³D³"<D @ !"ƒ:DC¼É8;³aÃ8Bƒ¿£>Q}:³cÿ³8;:³¿®8;}ƒ8;³i: µ > @ ƒ
98B>?= @ ’D™Ãc"C;Ã>Ñ @ :Ã8;¿£€8Bƒ>?ƒ>Ñ8;KAB±¸;à ": â D8;³<D|:tc":ÆD’: ;ÃBó<Tì™Ãc @ Ã: @ D|:
ƒ9ÃD µ ³ƒAB>ÑaD7Cc¼É8;³Ã8Bƒ¿£Q> :³ ƒ ³ƒ:>?DÃ>?cÿ;U â @ >ÑÃ>?C"} @ kD|> v ¿®ƒî³8D
:ƒ> @ >р8;:ƒƒ>?¿® @¶µ ³ƒAB>Ñ DÂɳ¿®có
"³ã:³aƒ8;>у>Ñï¿®!"ƒƒ>ÑBÿ;³8;:³}8;³ @ ’8;= @ ¿N"ƒQ> }:³D”">Ñ8;ƒ8;>єD¢D|>?Ã>Ñ8;³
D µ "ƒƒ>?ÂÉxDxC¼É8;³UÃ8Bƒ¿£Q> }:³§„ @¶µ ƒ:³DNDU">Ñ8;ƒ8;>Ññ9ƒ¿®ñD µ 8;=>?ñDx>?< 8;ƒ¿N"ƒ>Ñ8;³
Ã>Ñ:³®ƒ: @¶µ 8;ƒ>?AB>?³% @ tƒC"³ƒ€8;ƒ®DUC¼É8;³NÃ8B¿£Q> :³£D:>Ñ @ :£C8;:قº:³± : µ ´ @
ƒBó » 8;:³U8;³£:³K¿®ƒî³8DKD µ "! @ ¼C™
:³t³8;:³®ÂÉ8;³NDÂÉ @ 8;9K€8;:£ƒƒ>?¿® @ !"C"¿®ïƒD µ :³ ">Ñ8;ƒ8;>Ñk
VD|>?n ¤ @ 
:!;D|ƒ:£ ¤ @ K a8;:³ ?‚;ÃxD³"³ @ <D8;³có ƒ¿®
ƒî³8}Dƒ" @ Q> }:³x": â ¿£>ÑïD8;³’D @¶µ =³ƒÂB"8;>? ¹ >у :AÉcó
®AB:>ÑDa9ƒ€Ãƒ>?ÂÉ<¢D µ 8;:ÂɃƒ:Bÿ³8;:³”}8;³ @ ”ƒ: @ C"¢D µ :®¿®8Dï @ ƒî³8;ƒQ> }:³
ƒïƒ>?¿£ @ xD³"³ @ }:³ @[email protected] £8;C @ >?xD|:–ƒ€Ãƒ D’C¼É8;³Ã8Bƒ¿£Q> }:³’ƒ’ â @ Q> }:³x´ @¶µ ">ÑDxD µ :
: @ ¼}€™D™8;:ÃBÿ>?;‚ÒÃC"} @ C¼É8;³<Ã8Bƒ¿£Q> }:³<ÂÉÃ:%ƒ€Ãƒ™ @ 8;>vD:>ÑC"³Ã:Q> }:³Bÿ
D8; @ UÂB @ :<ƒ<Ã8;¿£!"ƒ>?= @ ñÂÉà à @[email protected] ¿®ƒ: ´ñ=!;³ƒAB>ÑBó
H l
Ä
¹»º¼s½
õ
¾1¿›º¨¼
÷ !šÅ ­ ú ú õûö
ÀÂÁ-º²Ã²¾1¼
õæú¡ü
ø—­ ý'ö
ùÆǯ õ
È
ô
É Á|Àʾ•Ã¥º¨¼
Ë
!ÌÅ ­ ú
ø—­ ý'ö
ý
ùÆͯ õûý
Î ÏÑÐ ÒÓÏÔÒ›Õ Ö<×ÑØeـڀÛÝܒÞ

ß ØeÛ~àÑВáÙvÜ âeڇãZÚÒâ¢ÐÑهàÝÕäQÜÔ/ܒÛÝÜÝÕÖ<Ü ÛÝÕ Ö<ÜÑÐÑåäÝæ çÛ ÙèZٞânÜÔ/ÜÑвÛÝ×é\çâ1ВÕ~ç>Ð ÓÙvêÒ{ÜÑÐë™Ú‡çÒ<ÚÜÑвäQÜÑВÏânÜ ÛvèÙvÜ ÐÜ Ð Ú
ڇçÒÑì çҕ۱ÐÙMâeÕ~çZâ{â{Ò{ÜSí ß ØnÛÝàÑÐҕâîá Û~ÜMïÛÝ×ÑØ\Ø1Ü äEÖ/ٞРڇçÛ Ù€ê Ò1ÜÑå³âeçÒÐðÔ/ÏÝÕÛ ÙžÛÝçâ1ÐðäñÜÑÐØnÛÑÙMâeÕِØ<×äQÜÑÐðÕ~×ÛÝ×\Õ ÚÏÛÑٞÐÑò
ڀـêÒ{ÜÑÐ]Ô/ÜÑЙÛÝ×é\çâ1Ð]Õ~ç>РӨـêÒ{ÜÑÐÞäQÜ ÒÛÐØ1ÜÝÕ Ú€Û~܋Ô3æ Ï ânÜ ÛvèهÜêÒÙ²Ü Ð Ú³èZäç/á~×äñÜÓÜâÚÜ âŒäñçÙÔ/ܨØeҕÙóÐÑÐ×âeÕÑÜ
Ô/ÜØeҕÙóÐqä×Õ±Ü âڇ×ÙMânÜ¥Ô/ܳô‹Ü1õìÑÒÐê Òæ ë 20 Ü1õuåDäQÜ Ò•ÛÔZÙóÐڀÛÑهá Ò<ڀـçâ×âèҕä×ZٞÛ~ÜêҕÙ:Ð Ü Óá äQÜ¥öڀÛ~ÜٞÐç\Ú€Û çÑØ{Ü
10
ë¢Ú‡çÒ<ÚÜÑЙäñÜ ÐyÏ â£Ü ۀèZÙvÜÑÐÑå¨ÜÚ´äQÜ Ò•Û]Õ~çÓ³Ø<ç>Ð ÙMڀÙvçâ÷ÞqÕ~çâ{â{Ò{ܛëˆáÝ×>ÐÑÐ Ü‹Ï ânÜۀèهÜÑå´Ð ׆Ô/ÏÚÜÛÑÓ¨ÙMâe×ZڀÙvçâ8ë†Ú€ÛÝàÑÐ
֕×Ò<ÚÜÏânÜ ÛvèÙv܏Ø1çÒÛ ÛÝי×ÑØ\Ø<çÛÚÜ Û²ÔÜÑÐٞâKïçÛ Ó×ZڀÙvçâ1Ð5ØeÛÝÏ~ÕÙvÜÒ/ÐSÜÑХРÒÛ´ä~æ çZÛÑÙñèÙMânÜÔ/Ü¥ÕÑÜÑÐiØ<×ÛڀÙvÕÒäQÜÑÐSí
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
H•¦
øù±ø
ú û£üýsþ²ÿiû£ÿ ü
ï @ §D=:£D|:b½BÀ;ï¿®Kƒ>Ñïà @ Bÿ @ î}¼ƒ>ÑÃ>ѳ£cÂ;">Ñ}ñ¿N"~}:³k:³ @ :x @ Ã8BCÃ8;9U
DÃî!"ƒA">Ñ} @ }¿®cÿ€Ã8;³ }:³³ÃxD @¶µ >Ñ8;>ÑC"ƒ>Ñ8;–8+ÂÉ8}:³ !" @¶µ ;Ã>Ñ8;±D µ :–C¼É8;³¿®}
³ƒAɃQ> :³B:ó T µ 8;ƒ>?AB>?³t ⠃Cƒ¶ƒƒƒtDNîC¼É8;³¿®V:xDÃ8;:ÂɃ§!"°Uó x ¾SU|¾c½|ÿ
:>¿®ƒ:C›:³N>?³D>r"= @ K":AB¿®C"ƒ>Ñ8; D|:bC": â D µ >Ñ8;>ÑC"ƒ>Ñ8;ID³"³ @¶µ ">?UcÂÉà @¶µ @ ƒ>?ƒ:³DBÿ @ 8;
D µ :³’Cƒ>ђD’¿®ƒ:a§=! @[email protected] 8;ìó ’C¼É8;³¿®}<>Ñ8;>ÑC"‚ V:¢=!"ƒ>Ñ ð Cc¼É8;³Ã8B¿£Q> :³ ð !"
›> @[email protected] ñ> oB"ÆGS¾ UB½BÁ|{ÿ }:>ìC">?’ @ 8;<Ã8;}Â;">?³Ã:±D @ ñ!"ƒ: @ Ã8;¿N"AB³ƒQ> }:³xD Ã!"ƒƒ>ÑÃ: @ có
T µ ƒ:³D<D¢C¼É8;³Ã8Bƒ¿£>Q:³:®8;:ƒ> @ >?³D|>у€³C"= @ ":§DÂÉ @ 8;9¿®}D @ ™î}¼>Q:³
D!"ƒƒ>ÑÃ: @ 8;:ƒ>ÑC"}™": â î}¼>ÑÃ>ѳ™:]¶">ÑÃc": ð !"ƒ: @ ð D !"ƒƒ>ÑÃ: @ ³ƒAɃ>Q}:³có
T µ "ƒ> @ Ã8;xV :ñ">?³>DÃ8;:ÂɃñ!" ³D8;I ¾SUZVB½|ÿD:>aDÃ8;:ƒ>?ñƒ> â "³ñ @ :³xC"D @ ¿ :³8;ìó{7T <>Ñ8;³<ÃCî!"ƒAɏV :A @ ¿®}>ÑD}ƒ>! ’G¾SU>Xx
[ !" ó ¹ 8 @[email protected] ó
ï @ Ã8;³Dyf:³ƒ–8;³D|>r @ Bÿ @ x¿£>Ñ®;Ãà @ C":£D§!"ƒƒ>ÑÃ: @ U}ƒïN
ƒÂ>ÑÃBÿ{ 8;:ƒ>ÑC"}™": â î}¼>ÑÃ>ѳD¶">ÑÃc": â Dx!"ƒƒ>ÑÃ: @ ¼"} D’Ãc"C;Ã>уƒ>Q}:³™=>Ñ
¿£>Ñ: â Ã8;}ƒ~ ¤ @ "= @ :³Uà @[email protected] Ã8;¿£€8BC" @ ™Cc¼É8;³Ã8Bƒ¿£>Q:³ < ™!" @ §ƒ:>?K\󣑳ñ!"ƒƒ>Ñ
D @ –Ã8;¿£¿ :!":%Ã>Ñ}ƒ> :³KÃ8;}ƒ>?:![À³D³"®´Æƒ:³D|>Ñ @ xC¼É8;³£Ã8Bƒ¿£>Q}:³cÿkƒ³ID µ Ã>ÑD³cÂ;"}C"AÉ @ :<Ãc"C;Ã>уƒQ> }:³cÿ @ :<!"ƒ: <=>ÑÆ @ :<8;ƒ>?AB>?³Bó
¾SUZVZR|ÿ @ ™î}¼>ÑÃ>љoC"c">Ñ ¹ >у :AÉ¿£>Ñ<›Â>ÑD³Ã @¶µ â >у³ÃDABC"³D<AɃ=€
"ƒ¿®8Bƒî³ƒQ> }:³<ÆDÃC"ÆÃ8 ³Ã>ÑD³Ã @¶µ "ƒƒ>?ÂÉD!"ƒƒ>ÑÃ: @ <ƒ:<DDÃ:’ƒ!;Ã
D]
}:³ @ :³tD|ñ> +">?³KD›¿®ïƒK": 8 @ Eó TìK â 9ƒ>ѳÃtD±DÃ>Ñ8;5DtC¼É8;³KÃ8Bƒ¿£Q> }:³cÿ
´ @ ›V8;>Ñ£": 8 @ ƒ:³D|>r" @ UAɃ=9N"ƒ¿®8BQ> :³®®D|>?ÿ®}§´ @¶µ ">ÑD%DK=! @[email protected] 8;³®
C" @[email protected] >?cÿãK8;}ñ¿x: @ ƒ>? @ >ÑND:>Ñ @ £"³NÁBÀ|ó £8;=³ƒÂB"ƒ>Ñ8;³®8;}£³8;C"¿£¿®®¿£>Ñ£
Â}>ÑD³Ã @¶µ â >у³Ã'D±!"ƒƒ>ÑÃ: @ ÆD8; @¶µ ³ƒAB>ÑGD!; @ 1020 ‡ °UÿD8; @ bÃ8;¿£!"ƒ>?=> @ >?
ÂÉà @ KD|>ÑÃ>Ñ8;b S¾ UBÁB†
Á "³8;³Ã§!" @ î}¼ƒ>ÑÃ>ѳ’
f>ÑìDÿ µ€">?Q
¶™1: ¿£>?e D µ :³
D|:³Ãƒ>Ñ8;£D|C;ƒƒQ> :³D|²
: p³: â D7Cc¼É8;³Ã8Bƒ¿£Q> }:³ã": â ³ƒAB>Ñ @ k @ :³ã @ ÂÉ: ">?ã³Ã8; @¶µ 8;=|‚ƒ
D µ ƒ:³D’">8 8;³D|>Ñcó
¶@ µ î³: ;Ã:³ @[email protected] Bÿ @¶µ ƒ:³D™D|:KC¼É8;³¿®}’Ã8Bƒ¿£>Q}:³™D³"³y A @ 8;=! @ >? @ 8;>?%D µ ƒÃ @ 8BBÿ
@¶µ 8;=³ƒÂB"ƒ>Ñ8; D|: Ã8Bƒ¿®8B%cÂÉà @ ± @ :³KD–¿®C"AÉ%D|>vK€8Bƒ>?= @ soî³8;8;³cÿ³:ƒƒ>?³8Bcÿ
!"ƒƒ>ÑÃ: @ ãÃî!"ƒAÉcÿ)ABCÂ>?8;ƒì€ƒ¿®ƒƒCDÃ8;!ƒ·Ñƒ @ k¿®Ãc">у¿®´ @¶µ 8:Â}D³"³ @¶µ ‘’>?ÂÉ
98;:8D|:>? Ã>?}ƒƒ>?AB:!"!"ƒƒ>ÑÃ: @ có
Hó_ ϔË}Õ®È!à ÕcË}ßKjÔ|Þ+ÅoÈiçkÕ
H m
B¾ ó?¾ ÃCN%>?³D|>Q}:!" @ £D8;¿N">?³§D µ " @ >ÑÃc"ƒ>Ñ8;QD D|>v ¿®ƒî³8}D D®DÃ>Ñ8;
DC¼É8;³’Ã8Bƒ¿£>Q:³+ó{‘Æ8;D| D™ABC"³D:’D|:™p³: ⠃>?³D|>Q}:³ V8;³Ãƒ>Ñ8;±D @¶µ ³ƒAB>ÑBó
øù!
F,+ž©-+F
"ü#$&%'eü(û()eÿ&*'
.q+;0/54E?BADCE+=ywE+xwEz19Z+;•9\?‡234
Tì ¿®ƒî³8D ⠀ƒ>?¿®C @ ñDÂÉ @ 8;9x!" @ î}¼ƒ>ÑÃ>ѳñD:>Ñ ï D µ :Qƒ>Ñïà @ £€8;:
ƒ:³D|>Ñ @ ’Cc¼É8;³¿®Ã8Bƒ¿£>Q:³C8;Ã8;³D|>?ƒ>Ñ8;³<!" @ ™ ¼€D!"ƒƒ>ÑÃ: @ }:³ @¶µ 8;ÆÃî³ÃCî³
´xDÃcÿ³¿N">Ñ<ƒ:ƒ8;:!" @ ñÂB @ :aD|:³p : â ´ @¶µ ³ƒAB>Ñ Ã8;³ƒ>ÑDVà ƒó³AB:Ò¾Bó?¾KÝ\ó ¹ 8;:D
³ƒAB>є>?< ƒ>Ñ:´™:³<Ã}C">?³’D ‡U
° ÿ @ ‚³p : ⠃ƒ:<N
C"¿£¿® @ Âɀ8;:E: µ :³’DÃ>Ñ8;
D|>?çDñ!"ƒƒ>ÑÃ: @ £8;>?ñ98BCƒ>?= @ Bÿ @ t: @ 8;= @ ï¿®tC"}£D µ @[email protected] @ ñ8;=³ƒÂÉ®ÂB"’}: µ @[email protected] µ >?C"AB>Ñ}cÿ!":%Dƒ:³aD @¶µ "ƒ¿®8Bƒî³ï#1{ìT DÃ:8;}< @ 8;a>?³ƒC @[email protected] D³"³D=! @[email protected] 8;³
"ƒ¿®8Bƒî³ƒ>Q:³+ÿ³8;:›ƒ:<D<y" @[email protected] >?có
¹ 8;: @ ³ƒAB>Ñ ƒ:9ƒ>Ñ: ´k}:³ @ :³Ã}C">?³ D °ñÿ @ ¶">?= @ UD|:p³: â D™Cc¼É8;³
Ã8Bƒ¿£>Q:³U³ÃCƒ>?t:³K>?¿£98;ƒC"}ƃ: ¶;ÃtDKDÃ>Ñ8;ìÿ”Ã}:>ƒU8;= @ ¿N"ƒ>Q:³%€8;: @ ⠀ƒ>ѳÃa¿ =!"~}:³c<ó T7 DÃ>Ñ8;%D|>?ÙD @ !"ƒƒ>ÑÃ: @ ƒ>?¿N">? µ ƒ @ 8;¢ @ :³€8Bƒ>?= @ Bó
T µ "ƒ¿®8Bƒî³ïƒƒƒBÿ+D8;} @ C³Ã”>?D|>Ñy">? @ <DÃ>Ñ8;ñD|>?ÃBÿ)k @ 8;ì:ƒ> @ >ÑÃ8;¿£¿®
:5AB>?A"}~ }:³'Ãc @ 8;ƒ>?¿®ïƒD³"³ @ :³ @™@ ±C¼É8;éÃ8Bƒ¿£Q> :³±ƒ>?¿N">?'D|>Ñ>?9[8; ³ƒAB>ÑBÿ
Ãc"a:³¿x: @ ƒ>?ƒ:³D™D!"ƒƒ>ÑÃ: @ Ã8;³D³">?+•ó Tk î}¼ƒQ> }:³D¢AɃ=€"ƒ¿®8Bƒî³ƒQ> :³ @ :
DÃ>Ñ8;–ÂÉà @¶µ =³ƒÂ;"8;>? ¹ >у :AÉ<8;’’D³"³ @ <ÃCî!">?ƒ’ƒ:>?ÂB"có
TìD: ⠃>ÑÃ>?! @ ÃîQ> }:³ DUDÃ>Ñ8;GD™AɃ=€ "ƒ¿®8Bƒî³ƒQ> }:³ 8;} @¶µ Ãî!"ƒ> @[email protected] 8;<
!"AÉ D!"ƒƒ>ÑÃ: @ ™":›8 @ ÿ|AB± ¸;Ãx´ñ:›c":›D DÃ:’!"ƒƒ>у::³ ABC"³D ƒ: ¶;ÃBÿ³
@ ¿®ƒ:™D @ @ :¿£>Ñï™D
p³:³8;Cóÿ£>ÑC!"¿®8 @ Ã: @ aD µ >c8; â Ã>?!" @ !;C"AÉD
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
_{”
!"ƒƒ>ÑÃ: @ Ã8;³D³">?có ¹ "ƒ¿£> @ ’³8;¿ =:³C ⠀ƒ>ѳÙD|:– ¼}€xc":±Dñƒ: ¶;ÃBÿ @ ñDÃ
:D µ f ™32 R54 ÿ<:>vƒa:³™ â ³ƒ>Ñ8;ÆD|:%DÃ: oɳ8 2 [64 ÿ€8BïD’:³ƒ: ¶;à vÃ>?ÂÉ
DNDÃ>Ñ8; D µ Â>?8; 100 o¿ 2 £%€ƒ¿£>ÑñD µ ƒ:³D|>Ñ @ xC¼É8;³£Ã8B¿£>Q:³ñD:>Ñ 1016.5 °
‚º:³± : µ ": â @ :³”î!":³ƒAB>Ñcó T µ ⠀ƒ>ѳà > 2 ¾cÀ54 ÿ/}:>!ƒ @¶µ "¿® @ >Ñ8;C"ƒ>Ñ8;KD @ ¼ µ ¼É 2 U64 ÿ
®A @ ¿®}9ƒ¿£>Ñcÿ9±¿®ƒ:C" @ @ :¿£>ÑïxD¥
p³:³8;óÃU¿£>Ñx!" @ AɃ=€cÿiD µ ƒ:³D|>Ñ @ ƒ€ÃƒDC¼É8;³Ã8Bƒ¿£Q> }:³: @ ƒCƒ ³ƒAɃQ> :³+•ó T µ =³ƒÂ;"8;>? ¹ >у :AÉaƒ¢:KDÃ:
î¼=ƒ>ÑD
}:>Ã8;¿x=>?³§ÃxD: â ÃîQ> }:³ñD®¿®ƒ:BÿkcÂÉÃN:³§ƒ: ;ÃN €Ãƒ>?ÂÉtD 3000 o¿ 2
98;:x8;Gc":bD®ƒ: ¶;ÃNƒ>?ƒ:³§ ƒAɃ>?³Bó ƒNƒ: ¶;ÃNDNÃ8 @[email protected] Ã>Ñ8;b³8;ƒ¿®tƒ b ž
V³ÃC">?ñ€8;:Ã8;¿£9³ @¶µ ⠃~ ¿®ñC"£D’Cc¼É8;³ Ã8Bƒ¿£Q> }:³ @ ™ @ :³³ƒAɃQ> }:³ D
@¶µ :>?ÂÉcó
F,+ž©-+M©
*+x=>˜i+;•9Z,\+-+4šz4E+{,\‰3?+
¹ "ƒ¿£>8;: @ Ãc"C;Ã>уƒ>Q}:³DÃ>?Â;"} @ Cc¼É8;³ Ã8B¿£>Q:³cÿ @ @ :³7>?¿£>Ñ8;!"ƒ
ÃC">?³¿® @ :¢ƒ€Ãƒ®³ƒAB>Ñ „;> @ µ ³D®ƒ: @ :³¢DVB½™8;D|¢DaABC"³D:¨p³: â !;
¿®8;>?³™Dt¾c½tDÃc;D[³ƒAB>Ñ#1 8;'ÂÉ8 @ :ƒ>Ñ8;'9:ƒUDÃ>?£Ã8;¿£¿®x:³ @ 8;>Dx:>ÑC"³Ã
~ }:³–!" ¶">? dN ∝ E −α ÿ<ÂÉÃ': >?³D|>ÑÃ'9ÃC @ α ÂÉ8;>у>? D 2.7 ó ƒ8;!"
dE
AB: @ "ƒ>? D|k
: p³: â ::³™A"¿£¿®D µ ³ƒAB>Ñx":³ƒ>€Â;;ƒa:%î³³8;¿®ï³:>Q:³Kî}¼ƒ>Q}:³Bÿ
:>a>?}Â}>?Æ´±Ã8;³ƒ>ÑD @ ±î³³8;¿®³8 @ 8;AB>ÑÆD|: Cc¼É8;³¿®tÃ8Bƒ¿£Q> :³%C8;:³x: ;9ÃUA @ 8;=! @ ó
» 8;:³ ÂɃ8;³ D µ "> @[email protected] :xD³"³ ãƒî³ï’
}: µ :³N"8}ÃCî³®î³8 @ >уƒQ> }:³BÿìD³"³ @ }:³ @[email protected] @ Cc¼É8;³
Ã8Bƒ¿£Q> }:³8;}:98B‚ ƒ8}D|:>?a´ 8;: @ ³ƒAB>Ña!":t: @ ¼9D8;:ÃBÿ}¢³8;
: @ ¿®xÃ8;¿£!"ƒ>?= @ tcÂÉà @ 8;=³ƒÂB"ƒ>Ñ8;³cÿv¿N">Ññ"98;ƒ®A @ ¿®} :Ã @ ">?C"AÉN>?C"}
ƒ: @ ¨}:³ƒƒ>Ñ8;›D @¶µ 8;ƒ>?AB>?³ D<C¼É8;³Ã8B¿£Q> :³có
Tì8;~ }:³ @¶µ 8;K8;=³CƒÂɒD’ @ :³ï @ ƒ€Ãƒ’K³ƒAB>ÑBÿ|> @[email protected] :³ƒƒƒ: @ ´³AB:’
Ò¾Bó \½ \ÿ @ :³ƒ>Ñ:
=ƒ>у:D €ñ8;™Â>у>?= @ ñ„³ÂÉ 3 − 5 1015 ‡ °Uÿi8 7 @ N9} @ 8;A"ƒ>?ƒî¿£Q> }:³ñ!; D ∼ 2.7
´ ∼ 3.0 ÿvÃ8;¿£¿x:³¿®ñ"€ @[email protected] @ èÜ âeçÒ98 ÂÉ 5 1017 °ñÿ€8 7 @ §9}ñDÂ>Ñ ∼ 3.3 ÿv"€ @ @ Ð ÜÝÕ~çZâeÔðèÜ âeçÒ98 < ³ @ Õ Ö<Ü;:SٞäMäQÜ ÿ€ÂÉ 3 1018 ‡ °Uÿ€8 7 @ §9}£D|>?¿£>?:³N´§³8;:ÂÉc":'Âə:³
Â; @ : ∼ 2.7 − 3.0 ó
TìAB>Ñ8;³!"ƒƒ>ÑÃ: @ >Ñï¿®™>?CC"’D|:›ƒ€Ãƒ8; @ ƒ:>?Â;"} „
UÔÕcÕcËc}çkËß"äiÅoË>= 7T <C¼É8;³Ã8Bƒ¿£>Q}:³’D µ ³ƒAB>ÑñÃ8;¿£ƒ>Ñxƒ¥}:³ @ }:³’Ã}C">?³’D²–°
²
}:³ @ :³f™° 8;} ñD§ƒï @ 8;>? @ x @ :³ñ³8;¿ =: â ó · @ ñÃ8;³D|>?ƒ>Ñ8;³ @¶µ }:> @ >?=tD|:b¿£> @ >Ñ:
<
Hó_ ϔË}Õ®È!à ÕcË}ßKjÔ|Þ+ÅoÈiçkÕ
B¾ ó ½? 9ÃU'³ƒAB>Ñ£DC¼É8;³ Ã8Bƒ¿£>Q}:³£„p³: â D|>vƒ>Ñ @ [¿
V8;³Ãƒ>Ñ8;›D @¶µ ³ƒAB>ÑBó
_uH
ó¸
−2 −1
ó¸ƒ −1 óf™‡°
−1
>?ƒ @[email protected] ">?K">?³ƒ>E}:³ @ %}:³Ã @ 8Bƒ¼ƒî³ïtD @ ¿®} @ AÉñD³"³ @¶µ :>?ÂÉcó:T µ ƒ:³DtD®Ãƒ
!"ƒƒ>Ñ<D|:®ƒ€Ãƒƒ”Ã9³D³"}¢³D|:³<D|>NÃ> @ <!" @ ³Ã<D|:®ÂÉ}¢8 @ ">?BÿÉD8;} @ ”ÃCî!"¿£³
¿N"AB³ƒQ> :³98;:³C @ <!"ƒƒ>ÑÃ: @ >?³ƒ<: Ny"¿£¿®’³ƒAɃ>Q}:³có ›8;=³ƒÂÉ:³ ƒï³ƒ
¿®8}D|: @ "ƒ>Ñ8;tD|’
: p³: â :”:³<Ãî³ @[email protected] D¾B¾"³Â>?8;ìÿÃ8;ƒƒ€8;³D³"":Nü|à @ <D µ ;Ã>?Â}>?8 @ ">?Bó
kç Ëß"äiÅoËÕ%ÅVç9ÞcËßJ¢c` ÅoÔ³Åoß"ËÕA=S : â ³ƒAB>Ñ @ AÉï¿®} ƒ:9ƒ>Ñ:+ÿ‚Ò:³~}:³ ÂÉ 100 f™‡°Uÿ @ Ã>Ñ8;³x NÃc;ÃUD µ >?}C;Ã>Ñ8; D Cc¼É8;³ñÃ8B¿£>Q:³UÂÉà @ ®¿£> @ >Ñ: "¿ =>r"}®8;ñ³Ã8;§ƒ:<M
!C"¿£¿®ñABC"³D €8;:²
}:³ @ :ñÃ8;¿£98Bƒ>?ƒ>Ñ8;bQ8;>?Uƒ€ÃBó:Tìñƒ:³D³³xD C"€8;ƒ
D µ "=98;³D³"³Ã–D®D|> €}N³8+¼": â @ :®ÂB"ƒ>r"ƒ>Ñ8; ÂÉà @¶µ ³ƒAB>Ñƀƒ¿®ƒ§D µ 8;=>?tD
>?< V8;ƒ¿N"ƒ>Ñ8;³’ƒ: @ U8;!"A"ƒ>Ñ8;–D<C¼É8;³Ã8B¿£Q> :³’D³"³ @ UA @ â >ÑBó
T7GAB>Ñ8; D|: ƒ€Ãƒ±D³"³ @ GA"¿£¿® E ∼ 1013 − 1014 ‡ °ŒC">?%A @ ¿®}ƃ:³Ãƒ>?= @ D µ "=ƒ>?Dƒƒƒ:³Ãƒ:ƒ9Ã> }:³„ÉÒD8;¿N">?³D µ ³ƒAB>ÑÃ8;ƒƒ98;³Dt§ v´ @¶µ ³ƒAB>ђ¿N â >
@
__
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
¿N @ ´:>v€:ƒ Ã8;< ": â Cc¼É8;³Ã8Bƒ¿£>Q}:³!" @ <Dƒ:€ƒ³8cÂB;Bÿ8;:<D|:Æ¿®8;>?³
D³"³ @ ÂɃ>Ñ8;§ƒC"³D³"D§D”¿®8Dï @ có €³D³"cÿ @ ƒ: @ C"¢ â 9ƒ>?¿®}C": â ³C¿ = @ !;
>?³D|Q> :³ @ ñ³Ã D @[email protected] <ƒƒ:³Ãƒ:có
ϔˣäiË}çkÈeNB=S µ "ï @ ãÃ: @ C"”D @¶µ ⠀ƒ>ѳÃi¶ ™™C2 ¾SY54 ÿ @ Aɳ8;:®aƒ>?ƒ:³C">?
´ :³³ƒAB>Ñ ' 5 1015 ‡°Uó•Tìaî¼98;ƒî³ïCAɳC @ ¿®}’ÂB"³Ãa€8;: â @ >Q:³Ãƒƒ:ƒ:
D€>?}ÂÉ8 }:³8;>?<:³¿®8D|>!Ãc"ƒ>Ñ8;–D’Ã8;³D|>?ƒ>Ñ8;³’D8;!"A"ƒ>Ñ8;[DCc¼É8;³’Ã8Bƒ¿£>Q}:³
o³8;C"¿£¿® @ :”¿£³¢DÃ8;< ³³¿®Ñ \ÿ|8;>? @¶µ 8;:ÂɃƒ:D µ :®³8;:ÂÉc":§Ãc"! @ D µ >?C;Ã>Ñ8;ìÿ}Â}>r
@ :³ @ :³a!"ƒƒ>Ñ<D @¶µ ³ƒAB>Ñ<D|:£Cc¼É8;®Ã8Bƒ¿£Q> }:³aƒ>?¿N">?<Ãî!"€C">?": â DÃ:8;=³ƒÂB"
@ ãAɃ=9"ƒ¿®8Bƒî³ƒQ> :³ 8 8;:®³Ã8; @ ³ÃD µ :³<C8;:Ã<;ƒƒ8;î¼|ƒQ> }:³D³"³³8;ƒ8Ãî³
ÂÉ8;>у>?!"AÉB•ÿ :>v)‚Ò8;:C">?a:³Ã8;ƒƒ>?=:ƒ>Ñ8;ÆKóD|8;>?aÃ>ÑaD|:Kƒ9ÃB•ó ‘’³’ƒ:³D™DC"> @[email protected] D @ ›€8Bƒ>?ƒ>Ñ8;Q @ "ƒ>?ÂÉKD|:bAɳ8;:I98;: @ UD|> €}U ¼9UD§³8c¼": ⠀ƒ¿®ƒƒC–9: ƒKD
ƒC"³Ãî³ @ }:³ƒƒ>Ñ8;b„!D ¿N">ÑïxAɳC @ Bÿ @ ’ â @ >ÑÃc"ƒ>Ñ8;³™D !"ƒ:U;ƒƒ8;î¼|ƒQ> }:³ D|>Ñ}
:³ ÂÉ8 @ :ƒ>Ñ8;[D @ £€8Bƒ>?ƒ>Ñ8;›D|:–Aɳ8;:›8;€8;ƒƒ>Ñ8;³ @[email protected] ¿® ´ @ NÃî!"ƒAÉ Z D|:Ƴ8c¼":ìÿ€ @ 8;³
}:³
à @[email protected] <D!"ƒ:î}¼Q> :³ÂÉ8;>Ñ}:³ ÂÉ8 @ :ƒ>Ñ8;›8;98;ƒƒ>Ñ8;³ @[email protected] U´ @ U¿N; A ó
ϔÔEDBÇb7Ë/j7ÅoÊVÊoËFDG= : â }Â}>?8;³ñD E ∼ 3 1018 °ñÿ @ NÃî!"AÉ¿®}£D£9}ND|:bƒ9î9:
µ â @ Q> :³U!"ƒ: @[email protected] ¿®§Ã8;¿£¿® @ §!;CC"AÉKD µ :³%Ã8;¿£98BC"}%D8;¿£>?!"›´›=!;K³ƒAB>ÑÆ´
:³®³8;:ÂÉ @[email protected] KÃ8;¿£€8BC"Bÿ DN€§ @ :³¥ ">?= @ Bÿ3³>ÑC"}x!"ñD8;¿£>?³£´Æî!":§³ƒAB>ÑBó ƒ
ƒC"³ƒ>?ƒ>Ñ8;®ƒAɳC @ ¿®}¢>?ƒ’Ã8;¿£¿® @ a!;C"AÉD µ :³8;ƒ>?AB>?³aA @ ;Ã>Q}:³<D < ãÂÉ
:³’8;ƒ>?AB>?³ ⠃Cƒ¶A @ ;ÃQ> }:³¨
Và ³Ãc;D|K \ó 9³D³"}cÿƒ> @¶µ 8;%ƒ:€8Bq
:³ @ C¼É8;³Ã8Bƒ¿£>Q}:³
8;Ã8;¿£98B:Q> }:³¿®aD8;8;³cÿ @ Ãî³Â> @[email protected] ’ƒ¢>?ƒ 2 ¾B;¾ 4iÃ8;¿£¿®<:³
V8;ƒ¿®D³"³ @ ƒ€Ãƒ’Ãc"C;Ã>уƒQ> }:³™Da9ƒaD µ ³ƒAB>Ñ!"Ãc"ƒ>Ñ8;%D’!">? e± }:³’:=>Ñ @ 8;8;³
@ 8;xD @ :x8;!"A"ƒ>Ñ8;ìó · @ ƒñD8;³Ã§ ⠃~ ¿®¿®U>?¿£€8;ƒC"UDN¿®: @ ›Ã8;¿£98Bƒ>?ƒ>Ñ8;ID
C¼É8;³’Ã8Bƒ¿£Q> :³D³"³ @ ñAB>Ñ8;›D @ UÃCî³Â}> @[email protected] ñƒ³›D µ ÆÃ8;¿£³D| @¶µ 8;ƒ>?AB>?³Bó
”Ï Ë}Õkcç7Ë}ß)ä9ÅoËÕ%ËIHvÞcß5JJGËÕK= TìC¼É8;³Ã8B¿£>Q:³™D³"³™ÃƒUDƒ>ÑïU!"ƒƒ>ÑñD|:[ƒ€Ãƒ €8B
ïDx:³N³ƒAB>ѧÃ8 @ 8BC @ Bÿk @[email protected] "}<‚Ò:³~}: µ ´ @ ÆD|>ñ+">?³§D þ 8;: @ x98;: @ C" !"ƒƒ>ÑÃ: @ UD
º:³~ }: µ ´£Cc£ó T7N @ :!"ƒD}:³ƒƒ>Ñ8;³’C8;: @ Âɒ!" @¶µ â >у³ÃxDx @[email protected] !"ƒƒ>ÑÃ: @ D¿®: ³Ã8;£y"³™98;³Ã @ ">?N„€¼±ƒ¶ ¶> @ :³²
³G":ƒ€ÃƒL NM’:³ @[email protected] 8;} @ 8;:Ã
8}D|:>Ñy" @ ¢Cc¼É8;³Ã8B¿£Q> :³a´ D @[email protected] ³ƒAB>ÑF 8;¿£¿® â @ Q> }:³ @¶µ >Ñ8;ƒ8;>Ñ"!"}
D™D|>?Ã>Ñ8;³D µ "ƒƒ>?ÂÉ :[³Ã8;£Ã8;¿£¿® Ã!"ƒƒ>ÑÃ: @ 8; @[email protected] ™:'x8;!"Aɂº:³± : µ ´
Hó_ ϔË}Õ®È!à ÕcË}ßKjÔ|Þ+ÅoÈiçkÕ
_ d
átÔ³ÊoÔÇBÞ+ÅMLNkËÕ§ÈeN ÜHvÞcß"ÔPOCä!Ô³ÊoÔÇBÞ+ÅMLN7Ë}ÕRQ
STVULWYX;Z [6\0U,][5U_^ `ba6c0TdULeITgf0X6UhUhTji\0TW!k `lTgU_[6\5monp][ cIq
U_r0Wes [ W!`tk6`t\0T,k6XvubX ]mh`ba5cIT wxy[6cIWUds Td\&][6\5z{X `t\0]WhT6|us [6f~}
U_TdWhz{Xvmh`l[ \>e~Tus id^€`bU!U_`l[ \k5Xv^ ^‚XI|-][6\0U_iVa5cIT\0]Tƒe~TVU
][6utul`lUh`l[ \0UqIW![vmh[6\~}„qIWh[ mh[6\|dTd\pqIW![{z Td\0Xv\ ]TLe~TVUy\c0Xvk6TdU
e~T?†?X k TulubXv\>‡lˆV‰;ŠgX‹q TdWh^ `bU‚es TVUŒm!`t^ TdWa6c0T'ubXŽe~Td\~}
U_`tmhiRe~TgWYX;Z [6\0U,][5U_^ `ba6c0TdU,Z€TdU_moqIulc0U‘Xv`lfIutTa5cITgeIX \0U
\I[vm!WhT’k5XvubX{“~`tT6w&”`us [6Wh`lk `l\IT•e~c–W!X;Z6[ \I\0T^ T\5m][6U_}
^ `la5cITjim!X `—mLT“mhWYX{}„k6XvubX ]mh`ba5cIT |;ubX&e~T\ U_`tmhigeITg˜j™šUU_T}
W!X `—m#ubX€^€›d^ T&eIXv\0U#]Yœ X a5cITk6X ulXv“`lT w
 œ0X c~mhT?id\ITW!k `lT |LutTVUWYX;Z [ \ U€][6Uh^ `la5cITVUU_[6\5mU_c0q~}
q [5U_iVU-es [6Wh`lk `l\ITT“mhWYX{}„k6X ulX6] m!`la5cITg]dXvW-`lulU-\ITqPTcIz6T\5m
qIulc0Uy›m!WhT-][6\~ž0\IiVUeIX \0UFulXRk6X ulXv“~`tT6wdŸ uTdU_m,T\T¡¢Tme~`t£}
ž ]`lulT9e~T9][ \0]Tz [6`tW¤][ ^ ^ T\5mpulTdU¤WYX;Z [6\0U][6Uh^ `la5cITVU
q [6cIW!W!X `tTd\6m-›m!WhTqIW![~e~cI`tm!Uje0Xv\0U-ulT¤e~`lU!a5cIT¤k6X ulX6] m!`la5cIT
Tm#›m!WhT¤Uhc~¥UhX ^ ^€Td\5m-e~idz`tiVUoqP[ c0W#\I[ c0Ušq0X Whz6T\I`lW-e~T
³8;:³
^‚Xv\I`l¦W!TR`bUh[vmhW![ qPT |~][ ^ ^ TulTulX `lU!U_Td\5mšqPT\0UhTW#ulTdUš[6f~}
U_TdWhz{Xvmh`l[ \0Udw  mh`tmhW!T§es T“~T^ qIulT |,cI\AqIWh[ mh[6\¨e~T
19
T ©AeIX \0Uc0\&]Yœ0Xv^ q¤^‚Xvk6\Iim!`la5cITLeIT
]dXvWYX ] m!10
iW!`lU_}
3
µ
¬
ª
«
mh`ba5cIT#e~c ^€`lul`tTdc`l\5mhTWYU_mhTulubXv`lWhT;­ |;q [5UhUh¦deITocI\ WYX;Z [6\pe~T
k `lW!Xvmh`l[ \9eITRus [6W!eIWhTgeIT®p¯5`lul[ q0X W!UhTd]dU|6fI`tTd\UhcIqPiW!`tTdcIW
n§u°s iq Xv`bUhUhTcIW&eIc±e~`lU!a5cIT‚k6X ulX6] m!`la5cIT6w™oTqPT\0eIX \5md|ubX
q TdW_m!TšeITš][ \~ž0\0T^ T\5mFa5cI`qP[ cIW!WYXv`tmFUhTLqIW![~e~cI`lWhToq [6cIW
e~TdULqIW![vmh[6\0UX ce~TdU!U_c ULe~T
18 T ©p|UhTjqIW![eIcI`tWYXv`tmLn
cI\IT#id\ITW!k `lTLqIulc0U,idutTdz idToqP[ 3cI10
W,ulTdUF\I[{Z5Xvc~“w{SXRk6X ulXv“~`tT
q [6cIW!W!X `—mXvul[ WYU ][6\~ž0\ITdW‚utTVU \I[{Z6X c~“¨e~T§²TW&³Œc0Uha5cs n
19 T ©±´SX'e~imhTW!^ `t\ X{mh`l[ \e~T ulX'][6^ q [5U_`tmh`l[ \
∼
e~c‚8WY10
X;Z [ \0\IT^ T\5m,][5U_^ `ba6c0TjXvc~“ XvulT\5mh[6cIW!U,eIT#ulX¤]YœIT}
z`tulutTRTdU_moe~[6\0]#m!Wh¦VU`l^€qP[ Whm!X \5mhTjqP[ cIWoT\9][ ^ qIW!T\0eIWhT
u°s [ W!`lk `l\IT w
;8 ³Ãƒ!" ÃUDƒ>Ñï’}:³ƒƒ>Ñ8;ìÿ€Dï @ "³ ÁBÀ|ÿ @ ™î}¼>ÑÃ>ѳ¥f>Ñìÿµ€">?
¶™:1¿£>? 8; ">?§¿N"~:³k:³ @ '8;!"A"ƒ>Ñ8; D®C¼É8;³tÃ8Bƒ¿£>Q}:³N: @ ƒCƒ ³ƒAɃ>Q}:³K
C">?a}ƒCÂÉ!" @ x³Ã™D|k
: V8;³DKDC¼É8;³¿®}’Ã8Bƒ¿®8 @ 8;AB>Q}:³ 8Bƒ¿£>ÑÖ>ÑÃ8vcÂÉP;Ã~o
AB8;:³D \ÿÉ¿£>Ñ!" @¶µ :>?ÂÉ @ 8;± : µ > @ C">?"AÉD: @ ¿®5 VBÀBÀ¿£> @[email protected] >ÑD µ "³cÿÉ6}:>¿£ @ >?
:> V8;ƒ¿®¿®}8;: @¶µ :>?ÂÉc•ó Tì¢î³8;8;³D|: P™ÿD8;} @ @ 8;AB:³:D µ 8;³Dƒ”¿N">?}!"<D
@¶µ 8;D|D|:®¿£> @[email protected] >?¿®ïƒBÿ¿ = @ }=>ѧ>?³>8 €³> V¢¿N">ÑcÿD³"³ @ V}ƒ>Ñ @ D µ :§C¼É8;KÃ8Bƒ¿£Q> }:³<
8;!"AÉc"ÂÉÃ:ð
;Ã:Dq
Tì8;Ý D @¶µ 8;D|™D 1011 ÿ> @ 98BCïDƒ<: NC"¿£¿®}<D µ ³ƒAB>Ñ
98;:U>?C"AB>?KÂÉà @ ±!"ƒƒ>ÑÃ: @ ›Ã8Bƒ¿£Q> :³% 8D|:>ÑC"}NDU>Ñ8;³có · @ ƒ: @ K:³ @ >?¿£>?Cƒ
ƒ>Ñ8;–D|C;ƒƒQ> }:³xD @ ®D|>ÑC"³Ã²
}:³ €:Âə!"Ã8;:ƒ>? @ < : @ ƒCƒ ³ƒAɃQ> :³£„iD @¶µ 8;D|UD
:³ @ }:³’D|ñ> +">?³™D²
›€Ã €8;: @ ³ƒAB>љƒ:9ƒ>Ñ:´ 1020 °ñnó ‘³ D|:³Ãƒ>Ñ8;'ƒ>?AB> !Ãc"ƒ>?ÂÉ
D|]
: p³: â D < ƒ™D8;³Ã£"ƒ³D|:³®":'Dƒ:³™D ∼ 7 1019 ‡ °Uÿ9ÃUÃ8;::£ƒ³
V :³¿£¿®}
"9 @[email protected] ‹
fµn¶£ÿ¢ @¶µ î³8;³:tD§ƒ8;>ÑNî}¼>ÑÃ>ѳð
}:> @¶µ 8;%"³8;³Ã–> @ ¼ ':³›
:!"C"}C">?³
D µ "³có
T7Kƒ>?ƒ:!"ƒ>Ñ8; â 9ƒ>?¿®}C @ ®ƒ ;Ã:³ @[email protected] ¿®ñ; £Ã8;< o:³C®":[:;‚ÒD @ §³Ã£8;:[³8;
D µ :³xÃ8;::UD³"³ @ xƒ€ÃƒBnó T7’³AB:
Ò¾BWó V\< @ ™ƒƒ>?¿N"ƒ>Ñ8;³;Ã:³ @[email protected] ™D|]
: p³: â D|> v
ƒ>Ñ @ ":±DCƒ:³™D 1017.5 €Ãƒ:³™!"™D™ ⠀ƒ>ѳÙD ¼}€xc":'Dxƒ: ¶;Ã
f ™ ÿ
oɳ/8 \ÿ8;:N!" @ ¿®:’D @ @ :¿£>Ñï™Di
p³:³8;Cóè
€ @ ¼ µ ¼ÉBÿ > Ñ \ó T µ â 9ƒ>ѳà > Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
_ l
B¾ óWV9 €ÃƒDaCc¼É8;³<Ã8Bƒ¿£>Q}:³<": â @ :³aî!":<³ƒAB>ÑD µ "ï 2 ¾cÁ64 ÿ|D8;!"} @ ³p³: â
D|>€}ƒ>Ñ @ ¿x: @ ƒ>? @ >Ñ!" E 3 ¿®:<!" @ ⠀ƒ>ѳà > +ÿ @ ¼ µ ¼ÉBÿ oɳ8 f ó\Tì
³8;¿ =ñD µ Âɳ¿®™DÃ":›D @ ´§D 1020 ƒ’ @[email protected] ¿®³ ">?= @ ²:³ @ £Dƒ¿£>?!"ƒ>Ñ8;'D @
V8;ƒ¿®D|:–ƒ9ÃDÂ>ђ>?¿£98BCƒ>?= @ Bó
°
¿ = @ §8;=³ƒÂÉ :³§D|>?¿£>?:ƒ>Ñ8; D|:sp³: â ": â @ :³ î!":ñ³ƒAB>ÑcÿkÃ}:³£³NÃ8;<³ƒ¿®N!; @
Ã8 @[email protected] "=98;C"ƒ>Ñ8; f ó · @ Ã8;}Â}>Ñ}aÀ³D³"<D µ ƒ’ƒ:³D}:!"´ @¶µ >?ƒC"ƒ>Ñ8;›Dƒ: @
C"< â 9ƒ>?¿®}C": ⠄>ž @ UƒC"ƒ>уƒ>Q}:³ ;ÃÃ:¿x: @ <‚Ò:³~}: µ ´ñƒ @ 8;>?›D µ ƒ :<NC" 98;:
ƒ>?:³Ã8;³Ã @ :³ƒ>Ñ8;%D ³>?ƒ>?ÂÉ 8 >?>ž @ >?³Ãƒƒ>?ƒ:³Dƒ¼|ƒ¿N"ƒ>Q}:³ @ >Ñ´ @¶µ C @ 8;!"AÉK³ƒAB>Ñ
D< â 9ƒ>ѳó 8;<!;³Ã8;8;C @ ¿®™Ã8;¿£ƒ>Ñcó
T µ =³ƒÂB"8;>? ¹ >у :AÉ<€ƒ¿®ƒƒC£D™¿®ƒ:ÂÉÃÃ>у>Ñ8;Æà !"ƒƒ>Ña‚º:³± : µ ´x
9:§Ã8;:³D|:§ƒ€ÃƒBÿÉ¢€ƒ¿®ƒƒCAB± ¸;ô ¢D: â ¿®ƒî³8}DD<DÃ>Ñ8;s
žp³:³8;óÃ
– ÃCî!"ƒ> @[email protected] 8;!"Aɛ":I8 @ DNÃ8;¿£³D| @¶µ 8;ƒ>?AB>?³KD|:QDC;ÃÃ8;DI}ƒ @ ñ â 9ƒ>ѳÃñÃ
Dcó >¢:³tÃ8;::tD|:Q ¼}€k
fµn¶ÎDÂ;">?¨
ƒNDÃBÿ @ Æ¿®ƒ:NÃ>ÑC§D @ 8;ƒ¿®§D|:
ƒ€ÃƒU9ƒ¿®ƒƒC">? D µ 8;=>?ñDUÃ>Ñ:³ >?< 8;ƒ¿N"ƒ>Ñ8;³ : @ %D|>уƒƒ>?=:ƒ>Ñ8;GD C8;:Ãx">?³ƒ>
:³: @ <Ãî!"¿£³’¿N"AB³ƒQ> :³³
:>7ƒvà @ ñ8;!"A"ƒ>Ñ8;–D!"ƒƒ>ÑÃ: @ có
F,+ž©-+Mª
µð2:7Œ˜i2:=>?‡9\?‡234
!= ;Cx³ƒAB>љ ‚Ò:³~: µ ´k}:³ @ :³™Ã}C">?³ D ‡°\ÿ @ C¼É8;³ Ã8Bƒ¿£>Q}:³™8;Ã8;¿£€8B
D µ Â>?8; 1% D µ @ Ã8;³+ÿvD8; @¶µ "=€8;³D³"³ÃñDÃ8>·Ñ cÂÉà @¶µ ³ƒAB>ÑUD|:›¶">?D @ :ƒ€Ãƒñ @ :³
9}ƒ:ìÿ}D 99% D³8+¼": â ó ¹ "ƒ¿£>vó8c¼": â ÿ @ 8;8;³}<Â>?8; 89% D|:N8;C @ ÿ
Hó_ ϔË}Õ®È!à ÕcË}ßKjÔ|Þ+ÅoÈiçkÕ
_e“
@ x³8+¼": â D µ î³ @ >?:¿ 10% @ ñ³8c¼": â @ :³ @ 8;:D @ N€8;:ãƒC"}cóT7]oC;Ã>Ñ8; @ "ƒ>?ÂÉ
D à @ ¿® @ 8;:D™ƒ´N€:›ï¿ = @ "= @ £´§Ã @[email protected] x8;=³ƒÂÉñD³"³ @ xƒ¼|ƒï¿®x8 @ ">?Bÿ9cÂÉÃ
Ã9³D³"}:³ @ AÉïC8;:³ ¶C"ƒ>Ñ8;–D @ ¿® ™:³³ 8;ƒ™ƒ: ¶C"ƒ>Ñ8;
D< @ ¿®³
Tv>¶{ÿ Pa
P™ÿ8cÂÉ!"}’Dc;Ã>Ñ8;³’D ƒ! @[email protected] "ƒ>Ñ8;]}:³ƒ:=>Ñ} @ ³8+¼": ⛠ÿ »
@ 8;<D @ :<8;!"A"ƒ>Ñ8;ìó
T7£Dƒ¿£>?!"ƒ>Ñ8;'D @ U!"ƒ:ñD<Cc¼É8;³’Ã8B¿£Q> :³´U @ :³<î!":x³ƒAB>Ñ ƒ<:–8;= @ ï¿®
"D|:Q„!± €cÿ @ <¿®ƒ: µ vÃ:³™D ¿N">Ñïx>?³D|>?Ãx8;D9³D³"}™D¿®8Dï @ Dƒ>?¿ : @ "ƒ>Ñ8;tDAɃ=9"ƒ¿®8Bƒî³ƒQ> }:³”:ƒ> @ >Ñc/ó T µ ⠀ƒ>ѳÏ
¶ ’™ fC"³DB\ÿ }:>¿®ƒ:
@ GÃ8;¿£€8Bƒ>?ƒ>Ñ8; D|:5Cc¼É8;³¿®ÆÃ8Bƒ¿£Q> }:³[ƒ 1015 1017 ‡ °Uÿa¿ = @ –>?³D|Q> :³K:³[":A
¿®C"ƒ>Ñ8;–D @ oC;Ã>Ñ8;±D µ @ ¿® @ 8;:D’cÂÉà @¶µ ³ƒAB>Ñ 2 S¾ Y47D|>уƒ>?AB:³:³ ƒ:³ÃÃCƒ>Ñ8;›D
ð Aɳ8;: â ð 98;: @ xD|> vñ³8+¼": â ó ñƒ: @ C"xC8;UÃ9³D³"}U³Ã8;§ƒ8;G @ >?¿£>?!">?
98;:aDƒ¿£>?³ƒ> @¶µ ÂÉ8 @ :ƒ>Ñ8;ÆD @ €8Bƒ>?ƒ>Ñ8;K @ "ƒ>?ÂəDD|> €}aAɳ8;: â ÂB"ƒ>Ñ Z 8;:K A ÿ
ô
:>7":C">?D<Ã8;³ }:³³Ã<ƒï<D|> v’ƒ: @¶µ >?ƒC"ƒ>Ñ8;[Dà ƒƒƒ:³Ãƒ:Bó
@ :³”î!":³ƒAB>ÑBÿ @¶µ â 9ƒ>ѳà > ”ƒ:³D|>Ñ @ Ã8;¿£€8Bƒ>?ƒ>Ñ8;ND < ”ƒ 1017 1019
‡ °Uÿ§Ã8;¿£!"C"} @¶µ ÂÉ8 @ :ƒ>Ñ8;tD @ >8 8;³D:D|:§¿N â >?¿x:¿ D’DÂÉ @ 8;9¿®}D¢AɃ=€cÂÉÃ
@ ãD|>ÑÃ>Ñ8;³”Dƒ>?¿ : @ "ƒ>Ñ8;³+Zó Tì”8;=³ƒÂB"ƒ>Ñ8;³¿ = @ } ¶ÂÉ8;ƒ>Ñ:³aÃ8;¿£€8Bƒ>?ƒ>Ñ8; @ AÉï 2 S¾ V54 ÿ
2 ¾ X~4”ƒ:³D|>r" @ ðVC;Ã>Ñ8;DU¿ :³8;³¿®:®D³"³ @ Ã
}:³£Ã8;< ³ƒ¿® @¶µ â 9ƒ>ѳà f ™¶
AɃ=9<D µ ³ƒAB>Ñxƒ:€ƒ>Ñ: ´ 1019 ‡ °Uó
F,+ž©-+Qœ
·¢?B=K9Z,/?5C69Z?234÷.u4E‰3CEt.£?,\+
Tì<Cc¼É8;³Ã8Bƒ¿£>Q}:³C"}ƒ>?³Ã>?! @ ¿®}™Ã8;¿£€8B<D !"ƒƒ>ÑÃ: @ ’ÃCî!"ƒAÉcÿ³> @ C8;DÂ>Ñ
! " @ KÃCî!"¿£³%¿N"AB³ƒ>Q:³–": Ã8;:%D @ :K8;!"A"ƒ>Ñ8;e%D8;³Ã @ :%D|>?Ã>Ñ8;eD µ "ƒƒ>?ÂÉ
¿®ƒ:ƒ: ƒ ³™98;>?}!;ÂÉ @ :<C8;:Ãcó "³<ÃÃc;cÿ: @ @ 8;ƒ>Ñ´ñABC"³D
Ãî³ @[email protected] UD @ §D|>уƒƒ>?=:ƒ>Ñ8;G"AB: @ ">?ñ€:ÂÉ "98;ƒD’>?<V8;ƒ¿N"ƒ>Ñ8;³: @ N8;!"A"ƒ>Ñ8;D
!"ƒƒ>ÑÃ: @ @ KD|>уƒƒ>?=:ƒ>Ñ8;G!"ƒ>r @ ®D @ : 8;:Ãcó :‹¶">?D @ t98B>?ƒ>Ñ8;'!"ƒƒ>ÑÃ: @ >ÑïND @
ƒñD³"³ @ kf @ â >ÑBÿi8;–9: µ "ƒ³D|£´®8;=³ƒÂÉ: @ AÉ â Ãï[D|>?Ã>Ñ8;[D|:±ÃƒU8;:
D|: @ "A @ ;ÃQ> }:³Bÿ78;:G³Ã8;§D|:î! @ 8ÿìƒ:>?ÂB" @ î¼98;ƒî³ïC Ã8;³Ãƒ!" @ %D|>уƒƒ>?=:ƒ>Ñ8;bD
8;:Ãcó
9³D³"}cÿ7´§ƒï™î!":£³ƒAB>Ñ E & 1019 °ñÿ @ DÂ>r"ƒ>Ñ8;³98;:ƒC">Ñ}¥
ƒUƒ<: NC"¿£¿®}
">?= @ €8;:<€ƒ¿®ƒƒ DÂÉ8;>? @ £Ã8;ƒƒ>?=:ƒ>Ñ8;[D 8;:Ã>?³D|>?Â>ÑD|:³ @[email protected] ceó T µ ƒ:³D D @ ®D|>уƒƒ>?=<:
ƒ>Ñ8;b"AB: @ ">?K": â 9ƒ>?ñÃî³ @[email protected] ñ€ƒ¿®ñDNÃî³Ãî³®Dx8;:Ã
ð ;ƒƒ8;³8;¿£>Ñt!"ƒƒ>ÑÃ: @ _{
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
¾Bó Xƒ >?Ã>Ñ8;³ D µ "ƒƒ>?ÂÉ£D™C¼É8;³ Ã8Bƒ¿£>Q:³D µ ³ƒAB>Ñ ƒ:9ƒ>Ñ:
@
?
@
Ã8 "=98;C"ƒ>Ñ8; f ™¶2 ¾K[64 ó ¹ "ƒ¿£> @ Âɳ¿®x !" D™9ƒ>?
Ãc"ƒ™8;:Aə>?³D|>Q:³} Ã: â }:>ã98BCïD E > 1020 °ñóuT7™ABC"³DÃà @ @ AB8;:€¿®D µ Âɳ¿®D³"³<:%C¼É8;±D 2.5◦ ó
°
´
!" @
@
Ãà ÂɃcÿ @ Ã8 @ 8;™>?³D|>Q:³}
4 1019 °
à î!"ƒAÉ ð 98;}ƒ>Ñ @[email protected] aD < cÿÉ¢D’Ã8;ƒC">?³D| @ ™Â; @ :D¢ÃCî!"¿£§¿N"AB³ƒ>Q}:³¢>?}ƒA @ ;Ã
ƒ>Q:³+ó ¿N">Ñï AɳC @ Bÿ @ 8;=³ƒÂB"ƒ>Ñ8;³C¿ = @ ™Ã€³D³">?³D|>Q:³q:³ @ ND|>уƒƒ>?=:ƒ>Ñ8;
"AB: @ ">?D < kƒkÃ8;¿£!"ƒ>?= @ cÂÉà @¶µ >Ñ8;ƒ8;>ÑBÿ"kÃv‚º:³~}: µ ": â @ :³7î!": ³ƒAB>Ñk¿®ƒ:có
: ⠀ƒ>? ÃCî³ @[email protected] cÿ @¶µ ⠀ƒ>ѳà f ™ %³8;C"¿£¿®}U:= @ >ÑN ¾SUZUZU 2 ¾K[64 @ xƒ: @ C"
D µ :³ ÃCî³Ãî³£D µ ;ÃÃ:¿x: @ "ƒ>Ñ8;D µ Âɳ¿® D³"³:›Cc¼É8;'D 2.5◦ ÿi98;:™D³ƒAB>Ñ:9
ƒ>Ñ:£´ 4 1019 °ñ(ó T7›³ÃKD µ :Iƒƒ>? @ ñ£DtD: â D8;:= @ U!"ƒ¿£> X[ÆÂɳ¿®}£¼ƒ
"³8;³ÃBÿ7ÂÉî:³£8;=!"=> @ >?£ƒï
¶">?= @ ®D µ 8;=>? ÃUƒ: @ C" !"î!;C"Dvó ƒ: @ C" 8;
®Ã€³D³"ñ¿£>Ñ †
}:³ƒƒ>Ñ8;!"ñDx"! @ ¼|: @ ƒ>Ñ:
Và 2 ½ 4M\ó T7%D|>?Ã>Ñ8;ID|:ƒƒ>? @ ƒ>?ƒ:³ƒ8;›î!":D³"³ @¶µ £>уî³ï » 8;D›Ã @ ƒ
Và D³AB:N¾Bó X€8;:
ƒ 8;=³ƒÂÉ !" @ ƒ>?
:³D–D @¶µ =³ƒÂB"8;>? ¹ >у :AÉcÿi> @ ¶":³D|C§"ƒ³D| @ §Ã8;³ƒƒƒ:³Ãƒ>Ñ8;'D|:±ƒ>? » 8;D–€8;::³
ÂɃ:³ @[email protected] xÃ8;< ³ƒ¿N"ƒ>Ñ8;–D à â Ãïcó
T7[€8Bƒ>?= @ %Ã8;ƒ @ "ƒ>Ñ8;5ƒ @ ®D|>?Ã>Ñ8;³ND µ "ƒƒ>?ÂɖD < N @ [€8Bƒ>?ƒ>Ñ8; D µ 8;=|‚ƒ§;
ƒ8;î¼|ƒQ> }:³ƒ:³Ãƒ>?= @ <D µ ƒDa8;:Ãa€8;ƒ>Ñ @[email protected] ’ñA @ ¿®}>?³ƒ>?ÂÉ¿®}’ƒ:³D|>ÑBó
‘³–Â:³›AɳC @ [ ƒtD8;³[D³"³ 2 S¾ U54 ‚
ó T7bD|> NÃ: @ [D–Ã8;¿x=>?³ @ @ 8;KD±D8;³
D µ ⠀ƒ>ѳԀ8BD³"}¢D”Ãc"C;Ã>уƒQ> }:³D|> €}q
VÃCî³ @[email protected] D µ ³ƒAB>ÑBÿÉC8 @ :ƒ>Ñ8;§"AB: @ ">?Bÿ
Ãc"ƒ®D®Ã8;:ÂɃƒ:§D|:GÃ>Ñ @ ó?ó?ó ;8Ã>Ñ®´ @ Kƒï´
">?= @ ®C"ƒ>уQ> :³£D|>Ñ98;>?= @ ">?
}:³£Ã£AÉ
D µ "! @ ¼CtUÃ8;¿£ @ â KU9:£8;<: ®£Dt³8;¿ =: â =>r">Ñcó · @ "!"C">?
}: µ ":³Ã:³%Ã8;ƒ @ "ƒ>Ñ8;
"³8;³Ã›ÂÉÛD®8;:ÃN;ƒƒ8;î}¼Q> :³N µ ±î
: ƒ%Ã8;< ³ƒ¿®›ƒ:£: @ 8;®D›D8;³£>у:
D µ :³ : @ â 9ƒ>ѳÃBó
@ :³ABC"³D’ÃCî³ @[email protected] Bÿ f A @ ¿®}<8;=³ƒÂÉ 2 S¾ R49D³"³:³A"¿£¿®D µ ³ƒAB>ђÃ}ƒ
H dbÛ£ß"ÅoäiÅVçkËy`(N ß"Ô\IiÈ9çkçkËJGË}ç9Þ®ÇÉÈiÕKJGÅMLNkË
_£O
B¾ óWY¨ "ƒ @ >ÑCÆ´ 20◦ Ã88;D8;³N }:!"8;ƒ>r @ ®D @ [ƒ>?AB>!Ãc"ƒ>?Â>?†V ⠃>?¿®Æ
³8;¿ =®D σ ’€8;: @ Âɳ¿®ñD µ f ¶2 ¾SR64¢D³"³ @ tA"¿£¿®ND µ ³ƒAB>Ñ 1017.9 ≤ E1018.3
‡°Uó1Tì @ "› @ Ã}ƒ A @ ;Ã>Q}:³¼K8;<Ccó
":8;:KD 1018 ‡°Uÿ:5 â Ãï%´¢X σ D³"³t:³±D|>?Ã>Ñ8; 8}ÃCî³[D|:5Ã}ƒ–A @ ;Ã>Q}:³Bÿ;8Ã>Ñ
´': D !Ã>?KD³"³ @ QD|>?Ã>Ñ8; 8;€8BBó 9³D³"}cÿ @¶µ >?ƒC"ƒ>Ñ8; D±Ã›: @ C"tD8;>?ƒ
ƒ:³DñÃc"Ã8;¿£¿® 8;ƀ: @ ÂÉ8;>?ƒ: @ ’³AB:N¾BóWY|ÿ @¶µ â Ãƒ>?ƒ:³ ›=€8;D|: D @ UAB>Ñ8;
D µ 8;=³ƒÂ;"ƒ>Ñ8;ìÿ @ ´§8 7 @ §Ã8;:ÂɃƒ:£D|:[Ã>Ñ @ ƒƒï³ ">?= @ BónT7§Ãc"ƒUD µ â Ãï™C" @ >ÑCxƒ:’:³
Ãî³ @[email protected] ®D®½BÀ ◦ ÿ @ £Ãà @ ®Ã}ƒ®ƒ: @¶µ â Ãï @ U @ :³>?¿£€8;ƒC" Ã8;:®DAB>Ñ8;³¥:>³£8;}
!;<8;=³ƒÂÉcó
:>?": ⠃: @ C"D µ f ÿ @ aD8;³D @¶µ ⠀ƒ>ѳà ‘f ’ ƒ>?ƒ:³™D³"³ @¶µ £>у€ï
:³D%8;<™c"! @ ¼| 2 ½B5À 4 ÿ|:Æ â Ãï<´£½|ó ½ σ ñ™ƒ8;:ÂÉ D³"³<:³D|>?Ã>Ñ8;Æ8}ÃCî³
´ 7◦ D|:›ÃƒxA @ ;ÃQ> :³Bÿ9ÂÉÃñÃ9³D³"}:³xƒC"ƒ>уƒQ> }:³ ƒïi
¶">?= @ Bó T7 ƒ>? :³D–D @¶µ =³ƒÂB"8;>?
¹ >у :AÉkDº‚´9ƒ¿£>ÑkDãƒ7Ãì8;=³ƒÂB"ƒ>Ñ8;³ 2 ½|{¾ 4ÉAB± ¸;ôaCABC"³Dƒ: ;ÃD¢DÃ>Ñ8;§„
":³Ã:³ ">Ñ8;ƒ8;>Ñ µ <Ã8;< ³ƒ¿® €8;: @¶µ >?³ƒC"có
øù_¸
¹—û£ÿºÿ*ü¼»'
ûy)j½&&*'*²üj¾ ü#*À¿š&¾ ÿ€ü
± @ AB @ kD|>ñ+">?³ D µ "³ãD µ €8;ƒ Ã8;;‚ƒ8;>?ãÃ8;³C;ô @¶µ ƒ:³DD7Cc¼É8;³ Ã8B¿£>Q:³cÿ)ƒ:
@ a @ "® ⠀ƒ>?¿®C @ ƒî³8;ƒ>Q:³Bÿ @ }:³ƒƒ>Ñ8;ND @ :”8;ƒ>?AB>?³´8;: @ ”³ƒAB>Ñ¢ƒ8;:ÂɃBó
¹@ :³ƒ>Ñ: >ÑDx;£
Ã>Ñxƒ: @ K¿N">ÑïND8;} @ !"ƒƒ>ÑÃ: @ x9:ÂÉ}¥ƒN;Ãà @ ñD³"³
@¶µ :>?ÂÉ8;’Ã9³D³"}’ cÂ;"³Ãcó
‘tD¿®Ãc">у¿® @ ¢ @ :³a
NÃc;à @ ¢ @ :³aƒ:³D|>Ñaƒ @¶µ ;Ãà @ C"ƒ>Ñ8;ÆD|>9:³>?ÂÉ!"a8;³D
_{¦
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
DxÃCî³8}ÃBó @[email protected] £ @ >Ñ:":±ÂÉ8;>Ñ>?!"AÉñD™8;³D™DUÃî³8è:³x8}D|:>ÑC™³8;C"¿£¿® @ ™ â @ 8Bƒ>Ñ8;³
Dƒ:€ƒ³8cÂBxD³"³ @ ’¿£> @ >Ñ:t>?ƒ @[email protected] ">?£„ @ ¿N"ƒ>Ñ®ƒaº‚ÒÙ´ DÂ>?¢:98;>Q}:³
¢€ƒƒ:ƒ=9 @ ’¿£> @ >Ñ:K"¿x=>r"!" @¶µ >?}ƒ¿®D|>r">? DÃî!"¿£³¿N"AB³ƒ>Q:³+ÿ}8D|:>ÑC"}aDD|>Ñ
Ã8;ƒ>?}:>? D ¿£€C"ƒ:BÿvDñD³ƒ>?UDxÂ}>?Cx}ƒ @ ¿£> @ >Ñ: ð "¿®8; ð D|:[Ãî³8ÃU @ ¿£> @ >Ñ: ð –ÂB @ ð ó ¹ >ÑAɒD³"³ @ Ãî!"¿£›¿N"AB³ƒQ> :³ ;ÃÃ8;¿£!"AB!"}à ƒƒƒ:³Ãƒ: %¿®8;:ÂÉ
¿®´
V‚ }: µ @[email protected] aÃ8;ƒƒ>?=:³} @[email protected] ¶¿¿®´ ÃÑ \ÿÃC">?³a!"ƒƒ>ÑÃ: @ !"ƒÂ}>ѳ}´ ƒCcÂÉ
@ £D|>ÑÃ8;}ƒ>?:>?ñD µ "¿®8;™–ÂB @ :ÆABC"³D›³8;¿ = D´
V8;>Ñcÿ|A"AB!"}’:›9:›D µ ³ƒAB>Ññ´£Ãî!}:³
üà @ Bó 8Ã:³ 98BCïD:³ @ >?¿£>?a!"ƒ: @[email protected] Ãc"ã @ :³ @¶µ ³ƒAB>ÑDã!"ƒƒ>ÑÃ: @ ¢":AB¿®}BÿB @ :³
> @@ :
¶": D£¿£³x98;:A"AB³x:³8;:ÂÉc":Qƒ: @ ¿®}ñD µ ³ƒAB>ѧ @ :³> @ D|> NÃ> @ ND
@ Ã8;< ³³":›ÂÉ8;>у>?!"AÉxD @¶µ 8;³DxD Ãî³8ÃBnó T µ ³ƒAB>Ñ ¿N â >?¿N @ x"ƒ>?}ñ!" @ <!"ƒƒ>ÑÃ: @ ™D
9³D§ @ 8;¢D¢Ãc"C;Ã>уQ> :³Ã>Ñ¢D @¶µ Â>?8;³¿®<;ƒƒ8;î¼|ƒQ> }:³Ã8;³ƒ>ÑDBó ‘³<Â}:³
DC"> @[email protected] D¿®Ãc">у¿®D µ ;Ãà @ C"ƒ>Ñ8;›€:
ƒ™ƒ8;:ÂÉxD³"³ 2 X~4 ó
· @ ¶":xÀ³D³"UA"DU´ @¶µ ƒƒ>?
: µ > @ ³§<: ® !;ñD µ ;Ãà @ UDñ!"ƒƒ>ÑÃ: @ ñ€8;:²
ƒ
:³<8;:Ã<D<C¼É8;³Ã8Bƒ¿£Q> :³+\ó T7 @ :!"ƒD”8;:ÃÃ8;:³³€:ÂÉ"€8;ƒ":§¿£>Ñ: â
: µ :³™Ã8;}ƒƒ>?=:ƒ>Ñ8;›³A @ >?AÉc"= @ ":kp³: â 8;=³CƒÂÉBÿà µ ƒa:³DaC">Ñ8;³a98;: @ ~ }:³ @[email protected] @ ñ8;:Ã
ÂɃ>?C"= @ D ÃCc¼É8;³Ã8Bƒ¿£Q> }:³ÃCî!"9 ³Ã8; ": â ;ƒƒ8;î¼|ƒ>ÑÃ>ѳcó
· @ ƒ8;=!"= @ ´
}: µ :t ¼}€™D™8;:Ã!"ƒƒ>ÑÃ: @ >Ñ<D8;¿£>?³’8;:³ @ ":ƒa!"Cx:>ÑC"³ÃBÿ|
8}D|:>ÑC @¶µ }ƒ>Ñ @ DCc¼É8;³Ã8B¿£Q> :³™;Ãà @ ™D³"³ @ ®A @ â >ÑBnó [">Ñ´®ƒï’î!":ñ³ƒAB>ÑBÿ
D’C8;:Ò @ :³’:>ÑC"}™ ⠃C"A @ ;ÃQ> }:³D µ :– ¼}€xƒï’D|> v™€8;:ƒC">у§‚Ò:ìó
¹ "ƒ¿£> @ Ãc"³D|>ÑD³" @ @ :³ñƒ>Ñ: â ÿì8;G€: Ã>? @ ³8+¼": â ;Ã> VxD®A @ â >Ñcÿk"=ƒ>?C"}ñ
@ :<Ã8}:’Dƒ8;:³<³8;>?<:9ƒ¿N;> V<8;:ƳÃ8; @ <ƒ:C":A"¿£¿N|ó
» 8;:³a¿®8;ƒ8;³A @ ¿®!" @ U:>?´
: µ > @ ƒ Ø1ç>бРÙvá äQÜ {ÿ }: µ :% ¼€D8;:Ù!"ƒƒ>ÑÃ: @ >Ñ
8}D|:>ÑC @¶µ ƒ>Ñ @ D™C¼É8;³ Ã8B¿£Q> :³ 8;=³CƒÂÉ ƒ:™8;: @ tA"¿£¿®®D µ ³ƒAB>ÑBó T7 8;:Ã
C">Ñ @ 8;ì}kD³"³78;: @ vA @ â >Ñcÿ>?;‚ƒÃC" @ ì!"ƒƒ>ÑÃ: @ 7:>?Â;"}ì: ¿ ¿®¢9Ã
@ 8;>7D™:>ÑC"³ÃBó
ƒ%>?³Ã8;¿£!"ƒ>?= @ –cÂÉà @
T7'DÃ>Ñ8; D < ®D µ ³ƒAB>ћƒ:€ƒ>Ñ:Æ´ 1020 ° C¿ = @ "k
³Ã%D µ :³ÆÃ8;::›
f¥µn¶£ÿD£¿®8}Dï @ ® @ :³ ð â 8;ƒQ> }:³ ð V:®8;€8BU€8;:® â @ Q> }:³
@ D8;³ â 9ƒ>?¿®}C @ N„v @ 8;'ÿ®8Dï @ cÿ @ C¼É8;³ Ã8Bƒ¿£Q> :³ C">Ñ @ ™8D|:>? D
Dƒ>?ABC"ƒ>Ñ8;bDñ!"ƒƒ>ÑÃ: @ ƒï¿N;>?ÂÉcÿ€D8; @¶µ â >у³ÃUƒ®´KD¿®8;}ƒcó '!" @ N @ 8;
D™¿®8}Dï @ ð 8; v8 ð !"<8;98Bƒ>?ƒ>Ñ8;±": â ¿®8Dï @ ;ƒƒ8;î}¼ƒQ> }:³<D|>? ð Pa8;ƒ8;¿ ‘’ ð ÿ|!"Ã
: µ ":³Ã:³’;Ãà @ C"ƒ>Ñ8;N µ ƒ @ :³³ÃCC">?a98;:8D|:>?<D!"ƒƒ>ÑÃ: @ 삺:³± : µ ": â @ :³î!":
H lGϔË}Õ®ß"Ô\IiÈ9ç7ÕNÇÉÈ!ÕKJQÅMLN7Ë}Õ£Ô{NîLNkȳÞ+ÅM`ãÅ~Ëç
_ m
³ƒAB>Ñ „|> @ µ "AB>?’ @ 8;D µ :³ Ãc;Ãc;D ´£³ƒAB>Ñ DÃ8;>ÑC"Bó
Tì™!"ƒƒ>ÑÃ: @ ™8D|:>?D³"³ @ ’¿®8Dï @ 8; 8 [C8;[¿N)‚Ò8;ƒ>?£Dî³8;8;³D
³:ƒƒ>?³8BN>Ñ:³®D @ 'Dƒ>?}ABC"ƒ>Ñ8; D®>Ñ8;³có6T7®¿®ƒ:tD|:| 8;³D D|>9:³NA"¿£¿N ⠃C"Aƒ
@ ;ÃQ> :³KD @¶µ ⠀ƒ>ѳà f ’” D³"³ @ KD8;¿N">?³%D µ ³ƒAB>Ñ%Ã8;¿£ƒ>Ñ£ƒ @ yf™‡° @ ‡°
¿®ƒ’–D|> NÃ: @ @ U @ :!"ƒ’Dÿ®8}Dï @ 2 ½B6½ 4 ó q
V8;ƒÃ8;ƒC">?€8;:ƒC">Ñqƒ Aƒ
@ ¿®U>?¿£98BCxb¿®ƒ:C" @ ™oC;Ã>Ñ8; D§î³8;8;³UÃ8;¿£98BC"} @ xCc¼É8;³UÃ8Bƒ¿£Q> :³£´›ƒï
î!":ñ³ƒAB>ÑBÿv">?³ƒ> }: µ 'ƒ:³D|>r" @ : ">Ñ8;ƒ8;>ÑBÿvÃ8;¿£!"ƒ>?= @ £8;:±³8;'cÂÉÃU:³UD|>уƒƒ>?=:ƒ>Ñ8;
D<8;:Ã<D³"³ @ ™î! @ 8£A @ ;ÃQ> :³Bó
øùŒÁ
"ü#—ûy)j½&p*$¿š&F¾
ÿ ²
ü#$)pÂ
¤eÿ»ÿ~ü#*
TìñC¼É8;³®Ã8B¿£>Q:³U8;}ñ!"ƒ8;:có?ó?ó Ã:£D|:Iƒ:=ƒ> @ }:> @ >?=tD|:I¿£> @ >Ñ:I>? @[email protected] ">?Bÿ
@ :³ÃD³"³³8;ƒ}Â}>?8;³¿®}:³8;ƒ>ÑD|>Ñ%xDÃ8;³ }:³³Ã¿N"~:!"= @ +ó ¹ "ƒ¿£> @ ³8;¿ =:³ 8;ƒ>Ñ ;CÃ>?!" D|:'Cc¼É8;³¿®UÃ8Bƒ¿£>Q:³Bÿ€³8;:³ ÂɃ8;³ ":Ã8;:D£Ãƒ
=ïÂɛ!"}ƒî³ïs:³ @ §C¼É8;³%Ã8Bƒ¿£>Q}:³t98;:ƒC">Ñ}%=>Ñ €8Bt:5ƒ>Ñ: â 8;= @ ï¿®[": â
;ƒƒ8;!":x-:{ ND µ â @ 8;C"ƒ>Ñ8;Iƒ!"ƒ>r @ N: µ > @ 98;:ƒC">Ñ}UcÂÉ8;>?ñt´ @¶µ 8;ƒ>?AB>?³tD @¶µ :³
DU @ :³£>?¿£98;ƒC"}N ⠃>?³Ãƒ>Ñ8;³ND @ ±Â>љ: µ ±Ã8;}:³Æ³8;ƒK @ "³ïBó6ìT ›Dà @ ³Ã®KD
@ V8;:³D| @ 8;D8;C"Aɒ98;:ƒC">?q ƒ ³Ã8; @¶µ 8}: DC¼É8;³™Ã8Bƒ¿£>Q}:³cÿ!D8;} @¶µ ÂÉ}ƒ:³ @[email protected] >?<³p :³³Ã ƒ: @ à @ >?¿N"<ƒƒ ƒ<D|>ÑÃ:ñ´ @¶µ î³: ;Ã:³ @[email protected] Bó
F,+Qœj+F
Ã'Ę5t‡23,\.n9Z?234Ì=ƒ˜5.n9Z?.ut‡+†+{9],\.1032:4E=];<2:=>[email protected]?BADCE+=
· @ ƒ<C">Ñ8;!"= @ xD µ ƒ9q:³ @ U @ :!"ƒ’DD !<ÃCî>Q:³q:³ 98B @¶µ â @ 8;C"ƒ>Ñ8;[ƒ!ƒ
ƒ>r @ 8;ƒ:ƒ¿®8;’:£‚ƒ8;:có €³D³"cÿi:›8;=³ƒC;à @ D C"> @[email protected] 5Ń>Ñ~}:³ D @ >?¿£>? @ ®D|>уC"³Ã
D’ÂÉ8+¼"AÉ>?>ÑDC": â D³"³™:V:ƒ:™8}ÃCî³%„ @ Cc¼É8;³ Ã8Bƒ¿£>Q}:³1 ' €cÿ€ƒ>?ÂÉD @
8;Ã>Ñ8;›
}:³ @ :<>8 9 @¶µ "ƒ¿®8Bî³ï ƒƒƒBÿ @ ’;ƒƒ8;!":8;}8;:¿£>Ñ´ñ:³ >?ƒƒ>r;D|>r"ƒ>Ñ8;
9ƒ¿N"³} D™!"ƒƒ>ÑÃ: @ ³ƒAɃQ> :³có
Tì³p³: â ¿®8+¼É›DC¼É8;³<Ã8Bƒ¿£>Q}:³D³"³ @¶µ ƒ!;ÃC"}<D µ Â>?8;Æ:K8;8;†V8;:t:K³8+¼":e
!"ñÃ}ƒ>?¿®ïƒ%Ãc"ƒ§ñ!"ñCÃ8;³DBÿ }Â}>?8; BY ÀBÀBÀ›>Ñ8;³ñƒCÂÉ}£">?³> @ tÃ8;ƒ³ñî:¿N">? !"
Ã8;³DBÿ @ ">ÑC"}<ƒ: @ :<!;y"Aə:³ƒC">?³ D @ >r">Ñ8;³Ãî>?¿£>Q}:³<=ƒ>Ñ!" @¶µ "¿®8;à D µ :³
ƌÇÉÈ6ÊÈ6ÇÌˌÍYÎ ÇÉÏ&ÐdÑÉÒÓ
d ”
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
AɃ=9–DÆ!"ƒƒ>ÑÃ: @ KÃ8;³D³">?K¿x= @ "= @ [´Gà @[email protected] §ÃtD³"³ @¶µ "ƒ¿®8Bƒî³ïBóT µ :>?ÂB @ %D
D8B{ Ԓ{ :›":KÃ8;:D µ :%ÂÉ8+¼"AɃ!"ƒ>r @ D µ :›"%ƒƒƒ>?¿®´¨RBÀBÀñ¿ Â9ÿÂB @ :´ñÃ8;¿£!"’´U½BÀ
¿ ]
 }:>ãÃ8;ƒ98;³D´ @¶µ :>?ÂB @ DUD8Bñ¿N â >?¿N @ ":8;ƒ>Ññ98;:™:'¿£ @ 8c¼ÉND µ :³ñÃ}ƒC @ :³Ã @ c">?
oC"c">ÑBó
Tì’Â}>Ñ8 @ }™ƒ:ƒ>Ñ8;³8 @ ">?€:ÂÉ´ƒñ":³ƒ>7D8;:C"= @ +ÿi€8;:ÂB"™ ⠀8B:';ƒƒ8>
!":›³8; 8;Aɱ´G:³›D8BC›D– @ :³ƒ>Ñ:§>ÑÂɃt8:³ @ }:³§î³:có @¶µ î³:';Ã:³ @[email protected] Bÿ
@ N vND µ :³]V 8;ƒÆ>?ƒC;D|>r"ƒ>Ñ8;5ƒ:§:³ @ 8;AB:³›9ƒ>Ñ8D›D›¿£³§8;}N>?³Ã8;}:³cÿ @ §: @ >?<V 8;ƒ¿N"ƒ>Ñ8;³´§Ãx:;‚Ò8cÂÉ!"}¿N @ î³::³¿® :>Q}:³¿® D’Â>ÑÃ>?¿®D µ â @ 8Bƒ>Ñ8;³8;:
D µ ;ÃÃ>ÑD’:³Ã @ c">?cÿi8;:ƳÃ8; D µ ;ÃÃ>ÑD™D @ "=€8;C"8;>?Bó
¹ 8;:C8;:³D| @ ’8;= @ ï¿®™D|:§Cc¼É8;³¿®Ã8B¿£>Q:³Bÿ @ :³ƒ>Ñ:8"‚ƒ8;}’ƒ:³D|>Ñ 2 ½\[64
Ã8;¿£¿®™!"< â ¿£ @ @¶µ ÃC"%¿N"ƒ>Ñ @ 8;: @ ™=98;:³Ã @ >Ñ¿N"AB³ƒ>Q:³Bó{T µ ÃC"Æ¿N"ƒ>Ñ @ Ã8;³ƒ>? ´U<
8;: @ ñ¿®8}D|: @ Uî!"=>?C"= @ £!":³U!">ÑC: ƒ<: NC"NDx¿N"ƒ>Ñi€8;:™:ÃC"Ã8;³ƒƒ>?ƒ:³
D µ c":ìÿ;> @ ":³D|C">?:³!">ÑC: ¿£>?>?¿N @ D Y¿Æÿ;à }:>³Ãƒ>?C">?:³¿N;D µ c":£D @¶µ 8;D|
D
YBÀBÀU8;³cÿ³C8;>?<ƒ> â 8;>Ñ @ £ÃCî!"ƒAÉ :ƒ> @ D @ U!cÂɃ ƒ!"ƒ>r @ x"¿®ƒ>ÑÃc">?³# 1
Tì’=98;:³Ã @ >Ñ¿N"AB³ƒ>Q}:³³}:!"´ @ :>9Ã8;³ƒ>?C">?<´8D|:>?:KÃî!"¿£K¿N"AB³ƒ>Q}:³":8;:D|:
Â;">ÑCc":›ƒ:<NC"¿£¿®}:>Ñy"98;:DÂ>Ñ @ !"ƒƒ>ÑÃ: @ ’ÃCî!"ƒAÉcón± @ î³::³¿®}cÿ @¶µ >?}<
ƒ>?³ÃC">?€8;:€8;:³”Dã8;8;³¢Da @ :³>Ñ:5™
f ‡° ƒ:6}:³ @ :³ã¿®ïƒ¢D<D|>уC"³Ãƒ
D @¶µ 8;D|D µ :³Â>?ABC">?³D™ @ ;cÿ8;>?<ÁBÀBÀ£ÀBÀBÀ 8;>Ña @ :³i:³ @ ÃCî!"¿£Æ¿N"AB³ƒ>Q}:³ƒƒƒ#1
· ³D€³D³"¿£¿®™D|:™¶ ">?: µ > @ 8;>?’;Ã:³ @[email protected] ¿®Ãî>Q}:³¿®™>?ƒc @ >ÑC"= @ xD8D|:>? :Æ @
Ãî!"¿£b¿N"AB³ƒ>Q:³K":8;:xD µ :QÂ;">ÑCc":Qƒ!"ƒ>r @ ÿ @ £8;= @ ï¿®tDN8;Q>?³8}Ã:>?N98;: @ NÃ8;ƒ³
î:¿N">?t€8B<A @ ¿®có•’
‘ ³<8 @ :ƒ>Ñ8;§CC">? @ 8;D<8}D|:>?:NÃ8;³DNÃCî!"¿£N¿N"AB³ƒ>Q}:³Bÿ
": @ "} @ NÃD®´ @¶µ >?}ƒ>Ñ:®D|:GÂB">Ñc":ìÿkÃ}:>Ã8;¿£ @ >Q}:³C">?UU @ 8;:D|>?C">?£³Ã8; @ ƒ¼|ƒï¿®Bó
Tìx8;= @ ï¿®x’D8;³Ã @ 8;>?±D µ ƒx8 @ ì: ÿ9’> @ ƒ>Ñ~:³xD ³D| @¶µ â @ ;8 C"ƒ>Ñ8;±î}:¿N">?³UD @
@ "³ï[
"=9c":³Ã8;:›¿®8;>?³’"ƒƒC¼"} € 8;: @ V:ƒ:’;ƒƒ8;!":+ó
Õ!їÒÖ ÇÉÒY×;ÒhØ ËÙ‘Ö ×6Ú,ÒY΄Ë#Û6È5ÒgÛ5È5ÇÌˌÍRÜ0Ý Í Þ{Û5ÇÉ×dÐdÑÉÒYÈ{Ë#Ü ÒÜ ßVÎ ÒVàIádÝ Ò!Î„Ë âãâ°Ü6ÇÌ،ÒÛ6È5ÒgϤÒYÎ Û6،ÒRÜ6ÒgїÐÞ{ÛÐÈ{ˌÇÉˌÍRÜ6ÒjØ_ÐYä{ßVÈ6È5ÒYϤÒ!ÈvË
Ð å5Î ßd،å~Í!ҚæÐØyÛ6È5ÇÌËŒÍšÜ ÒšÏ&ÐÎ Î ÒRÙ
Úhàvæ~ßVÈ5Ü6Í!،Í!Òoæ5ÐØ,Û6È&çbÐdá!ˌÒ!Û6ØÜ ßVÎ ÇÉϤÍ!Ë ØŒÇ—Þ{Û5ҐÇÉÈvË ØŒßvÜ Û5ÇÌË,æ~ßdÛ6Øyæ ،ÒYÈÜv،ÒoÒ!Èpá!ßVϤæ6ˌÒ
−1
ÑÉÒYΐÒ!èIÒ!ˌΐå6ǗßdÑÉßVéVǗÞ{Û5Ò!ÎšÜ Ò-Ñ£Ý Í!È5Ò!،édJ.kg
ǗÒgÐå5Î ßd،å~Í!ÒgÎ Ò!їßdÈ ÑÉÒ-Ëä æ~Ò#Ü ÒšØ_ÐYä{ßVÈ6È5ÒYϤÒYÈ{Ë ê
H lGϔË}Õ®ß"Ô\IiÈ9ç7ÕNÇÉÈ!ÕKJQÅMLN7Ë}Õ£Ô{NîLNkȳÞ+ÅM`ãÅ~Ëç
F,+Qœj+ž©
ë(CE,/=>.uC9\=y‰[email protected]
d H
+19]‰3,\.u4Ew6+e=]+Ä(9\?45;•9\?‡2345=
Tìƒ:C": A "¿£¿Ns‡f"¿£¿N c¼›P:ƒcÿf Pe!" @ Kƒ:>?K’8;} Dî³³8;¿®ï³ ⠃~
¿®¿®¢Â>Ñ8 @ 5:> @ >?=9ï}¢:³}:!"ƒ>?’D µ ³ƒAB>ÑÃ8 @ 8BC @ Bó/T µ â @ 8B>Ñ8;§D µ :Nƒ:y":”A"¿£¿N
D³"³ @ ñ8Ãî³ñÂÉ8;>у>?!"AÉUD @ ƒBÿì´}:³ @ }:³o}€ÃBÿ!€8;:ƒC">?Ãc":³™DD8;¿£¿N"AÉÃ8;³ƒ>ÑD
C"= @ ñ´K8;: @ %=>Ñ8Bƒî³ïNƒƒƒBó NÃñƒƒ>?¿N"ƒ>Ñ8;³ 2 ½BÁ64D|:C": â Df aP D³"³ @
°”8;>Ñ7
T ;î¿x= @ ñ>?³D|>Q:³²: µ :GÂÉﳿ®}xD³"AÉ: ⠀8;:ƒC">? U8D|:>?®¿®8+¼É³
D: â 8;>Ñ!" ¿£> @[email protected] >r"DQD µ "³có3kT ƒK¼"}xÃ8;:I":¿®8;>?³ñÃ>?{–ABC"³Dx ⠃>?³Ãƒ>Ñ8;³U":
Ã8;:xD @¶µ î>у8;>?KD @ %Â>ÑBÿì> @ ƒx98B>?= @ }: µ :Qƒ:C":xA"¿£¿N[">?xÃ8;ƒƒ>?=:³%´›:QDñÃ
Ãc"C;à @ ¼ƒ¿®+ó ¹”@ :³a!"ƒƒ>ÑÃ: @ >Ñï¿®cÿ³> @ ¼KXZX Àñ¿£> @[email protected] >Ñ8;³D µ "³cÿ @ x¿£>Ñï™ABC"³D ⠃>?³Ãƒ>Ñ8;
D @¶µ D8cÂ>ÑÃ>ћ€8;:ƒC">?ÂÉ8;>?’>?>?ƒ>Ñ!" @¶µ â @ 8Bƒ>Ñ8;›D µ :f P 2 ½>~X 4 ó
Tì v DÂB;ƒC":D µ :£ƒ:C":ãA"¿£¿NÂ>ѳ”D|:¶">?D}:³ @¶µ ¿£>у>Ñ8;£D|:UC¼É8;³¿®
ƒa=ïÂÉ â ƒ~ ¿®¿®’>?³Bó · ¿N"AB>?³8;³ @ ™Ã!"ƒ>Ñ8£Ãc"C;ƒƒ8;î³ ƒ:>?ÂB"„:]f P5ƒ>?ƒ:³´
[;ÀBÀU9à >?ƒC;D|>Ñ €³D³"U¾cÀ£Ã8;³D<:³ ;à D @ ƒ D µ :y³p : â D¥RBÀo{ì !" m2 ÿ³Ã8;¿£€8B
ƒ>?³Ã>?! @ ¿®D î³8;8;³í γ ó î³8;8;³’>?C"AB>ÑCD³"³ @ NƒƒC"8Bƒî³ïñ›>Ñ8;>ÑC"}
®D|>Ñ8Ã>r" @ £¿®8 @ Ã: @ +ÿ¢ƒC">?!"} @ ‹V8;ƒ¿N"ƒ>Ñ8; D µ 8 â ¼|DND µ >c8;–": D9³D D @¶µ 8c8;³Bó
‘³
VC;Ã>Ñ8;–D Ã<î³8;8;³’ƒ>?¿N">?’€:’"ƒ>?³D| @ 8 @ C8;:³i 8;ƒ¿® D µ :›Cc¼É8;³¿®‘³°
ƒ<: NC"¿£¿®N>?³%€8;:
ƒ%¿®8;ƒ @ €8;:£8;:K:³%ÂB"ƒ>Ñ%D µ 8;ƒA">у¿®có €³D³"cÿ @ ƒ>?³Ã>?!": ⠀DÂ;;C":8+Â}>ѳ}D @ xD @ ƒ>Ñ8;tD @ Ã8;:³ÃD µ 8 c8;³ƒ:¢:³ABC"³D
ƒ: ;ÃBÿ€ â 98By" ">?³ƒ> @¶µ ’ » D¿N @ î³::³´
V8;ƒ¿®DñÂ>Ñ } ":[C¼É8;³¿®}²
‘³°
D|:KC8 @ > @ ó ¹ 8;:;ÃCî³ÂÉ @ ƒCÂB"> @ ÿ @ D|>Ñ8 â ¼DD µ >c8;q
8;ƒ¿®™D³"³ @ UƒƒC"8Bƒî³ï£ÿ|C"<8;!}:³
´ @ @ :¿£>Ñï Â>у>?= @ Bÿ|8cÂɕ8 :³ :³ Ãî}: A @ 8;=! @ D<¿£9C"ƒ:´ @ Uƒ: ¶;ÃD|:ÆA @ 8;=9Bó
TìC¼É8;³™Ã8Bƒ¿£>Q}:³98;:ƒC">Ñ}":³Cƒ> cÂÉ8;>?:±>?¿£!;Ã: @ N=>Ñ8Bƒî³ïñƒƒƒ 2 ½ZY54„9
€cÿ}ƒ> @ ‚f aP ;Ãà @ }DC¼É8;³Ã8Bƒ¿£>Q}:³ì‚Ò:³~}: µ ´™ƒï¢î!":’³ƒAB>ÑBÿ @ Ep³: â D<¿x:³8;³
Ã8;³D³">?®"ƒƒ>?Â;"}§": 8 @ 9:ÂÉ}®³D8;¿£¿N"AÉ @¶µ » D£8;ƒA">у¿®£Â>?Â;"}U8cÂÉ8•:!"}
D¿ :C"ƒ>Ñ8;³có
F,+Qœj+žª
Ò;1t‡.u?‡,=ð+{9],\.1032:4E=];<2:=>[email protected]?BADCE+=
!î :³’‚ƒ8;:cÿ€Â>?8;sX§¿£> @[email protected] >Ñ8;³ D µ à @ ">?qVC"€ @ §ƒ: ;ÃUD|:[A @ 8;=€Bÿ9 @ ñî³³8;¿®ï³®
Dº‚ƒ§x8;=³CƒÂÉñ:’D µ ":ƒ™ @ "³ïcó € ³D³"cÿvÃ8;¿£¿® @ ™8;C"Aə8D|:>ÑDà @ ">?
d _
Ù§Ô/IiÈiçkÕ®ÇÉÈiÕKJGÅMLNkËÕ
ƒx:³²}:³ƒƒ>Ñ8;[8;:ÂɃx=>ѐ"ï @ ™DÃ8;:ÂɃD²P™ó C"1o @ >?'b¾K[YB½|󣑒³x â @ >ÑÃc"ƒ>Ñ8;
>?y"a8;€8B!" @ ãî¼|ƒ>ÑÃ>ѳ<„ @ à @ ">?”C">Ñ}>?ƒ>?¿®¿®¿® @ >є": â Cc¼É8;³
Ã8Bƒ¿£Q> }:³có
Tì›8}Ã:³§}Â}>ÑC"AɱC">? @ ›ƒ:>?ÂB"[„¢D³"³ @ 'î!":["ƒ¿®8Bƒî³ïBÿ: Cc¼É8; Ã8B¿£>Q:³
ƒ>?¿N">? }ƒ –Ã8 @[email protected] >Ñ>Ñ8;±ÂÉà :³¿®8 @ Ã: @ ñD @¶µ ">?8D|:>ÑC"™">?³ƒ>7:³ AɃ=9xD!"ƒƒ>ÑÃ: @ Ã8;³D³">?™D î!":U³ƒAB>ÑBó£Tì!"ƒƒ>ÑÃ: @ ™D @ NAɃ=€Bÿi±!"ƒƒ>ÑÃ: @ >Ñ @ ’ @ Ã8;³+ÿVC"€
@ ¿®8 @ Ã: @ D µ ">?ñD³"³ @ £}:!"AÉN8;C"AÉ: â º‚ƒÃxD µ ":ƒx @ Ã8;³x´Kƒï î!":N³ƒAB>ÑBó
Ãà @ t!" @ tÃî!"¿£³% @ Ã>Q}:³k}:>ïAB³}t}ƒ @ N!"ƒƒ>ÑKÃî!"ƒAÉt€8Bƒ>?ƒ>?ÂÉ¿®%
³A"ƒ>?ÂÉ¿® D|:±:!"AÉBÿ€Ã’!"ƒƒ>ÑÃ: @ AɳD|™:³ñcÂ; @ "³ÃCî³£D µ @ Ã8;³D î!":U³ƒAB>Ñ
@ >?=9C"}a!"=¿®CƒƒC"î @ :A£Dî³8;8;³ íéA"¿£¿N|ó ¿®ƒ:à 2 ½ZV64v8;¿®8;}ƒ
:³Nà =€8;<: €ñD£C¼É8;³ñA"¿£¿NÆ9:ÂÉ}
ƒ§ƒ<: NC"¿£¿®ñ>?³C €8;:ñ"ƒ>?³D| @ N8 @ ÿ
@¶µ ³ƒAB>Ñ D ÃC">?³î³8;8;³<€8;:Â;"}D!; @ £D|ñ> +">?³ D
– °ñó
‘’³´
8;>Ñ @¶µ à @ ">? "¿®8;ÃBÿ @ ’ @ Ã8;³C 8;!"AÉ!"=€8;³D @ @ 8;A%D µ :±Ãc"! @ >Ñ8;>ÑBó Ãî!}:³KC"9BÿãÃ: â Ã> µ ;ÃÃ:¿x: @ §": â ³D|8;>?£8;: @ tÃc"! @ Ãî!"AÉ%DKD|>?Ã>Ñ8;ìÿÃc"£:
Ãî!"¿£ @ 8Ãc @ >?³¥
:>7;Ãà @ ïD µ cÂ;"}C"AÉ D µ @ Ã8;³c{ó T µ "¿£ @ > !Ãc"ƒ>Ñ8;–™€8;:ƒ:>?ã‚Ò:³~ : µ ñÃ
:³ @ Ãc"! @ "ƒ>?AB³ @ 8 @ ÿ€ƒ¿®ƒC"">?³ƒ> @ !;y"AÉ D @¶µ >?¿£: @ ƒ>Ñ8;±D Ã8;:C"
:>v8D|:>?
@ @ :¿£>ÑïKÂ}>Ñ>?= @ Bó T µ ƒ:³DÆD£Cc¼É8;³?
í 8D|:>?£!" @ ®Ã @ ">?Nƒ®D8;³ÃK:³K³8;:ÂÉ @[email protected] %ÂÉ8;>Ñ
DNÃCî³Ãî³y
}:>9ƒ¿®ƒƒC–€:
ƒKD§9à @ §¿ ¼|ƒïKDtÃñ>?¿£ƒ>Ñ8;!"}£î³³8;¿®ï³
!"ƒ: @ ó
È
Ä
É Á|Àʾ•Ã¥º¨¼
õæý
ï
î
÷ ! úg& õûý
ï
õé÷ " õûý
! üQö
­ ý’ðòñôóæ÷QùÆǯ
õæý
Î ÏÑÐ ÒÓÏÔÒ›Õ Ö<×ÑØeـڀÛÝܒÞ

õ3Ü´ÚÜÛÑÓÜÔ܏è/Ü ÛSá~ܨ×ZڀÓç>ÐØÖ<Ï Û ÙvêÒ1ܲÔ/ÏÑÐ ÙèZânÜäñ×Õ~×ÐÕ~×\ÔܲÔÜ5Ø<×ÛڀÙvÕÒäQÜÑдÐSÜÝÕ~çâeÔ/×ٞÛ~ÜÑÐØeÛÝçSÔҕـÚÜÑдäçÛÑÐÔ/Ü
äÝæ ٞâeÚÜ Û ×/Õ Ú€Ù€çZâ-Ô3æ Òâ£Ü³Ø<×ÛڀÙvÕÒäQÜ´ØeÛ ÙžÓ×ÙMÛÝÜ×:>ÜÝÕðäQÜÑШӒçZäñÏÝÕ ÒäQÜÑÐðÔ/ÜäÝæ ×ZڀÓç>ÐØÖ<à Û~ÜSí€ö6ܳØeÛÝçSÕ±Ü Ð±ÐÑÒÐðڀÛÝ×â1Ð ò
ïçÛ ÓÜä~æ ÏânÜ ÛvèÙvܳÔ3æ Òâ£Ü‚Ð ÜÒäQÜDØ1×Û Ú€ÙvÕҕäñÜqÕ~çÐÑӨـê Ò1ÜqÜ âҕânܳêÒ1×âڀـÚϏÕ~çäñç>ÐÑÐ×äQÜqÔ/Ü:Ø<×ZÛ Ú€ÙvÕҕäñÜÑгÔ:æ Ï â£Ü ۀèZÙvÜÑÐ
Øeä ÒÐ(ï×هá äQÜÑÐ´Ü ÚEÔ/ÜÛÝ×é\çâ{â£Ü ÓÜ âÚyí õ×ÔÜÑÐÕÛ ÙØnڀـçZâÔ/ÜÑÐ5ØeÛÝçÑØeÛ ÙvÏ ÚÏÑдÔÜÑÐièÜ Û á±ÜÑв×ZڀÓç>ЇØeÖ<Ï Û Ù€êÒ{ÜÑдânÏÝÕÑÜÑÐÑò
ÐÑـÚÜ´ä~æ Ò<ڀÙMä ٞÐ×ZڀÙvçâ›ÔܴРٞӨҕä×\ڀـçâ{Ðqâ{ҕÓÏ Û Ù€êÒ{ÜÑÐÑå‚Ô\çâÚ6äñÜÑÐÛ~ÏÑÐ Òäڇ×ZÚMвÔ/ÏØ{Ü âeÔ/Ü âeڂÔ/Ü¥âeçдÕ~çZâ{âe×ٞбР×âeÕ±ÜÑÐ
÷ çÒ²Øeä ҕڇã\ÚEÔ/ÜânçZڀÛÝܲÔ/ÜvèZÛÝÏÔ3æ ÙèZâeçÛÝ×âeÕ±o
Ü ø ù’Ô/ÜÑÐÐ ÜÝÕSڀـçâ{ÐÜ úîÕ±×\Õ±ÜÑвÔ3æ ÙMâÚÜ ÛÝ×\Õ Ú€Ùvçâ›ëڀÛ~àÑÐ֕×Ò<ÚÜ¥ÏânÜ ÛvèÙvÜ
Ô/ÜÑЏØ1×Û Ú€ÙvÕҕäñÜ Ð³ØnÛÑÙMÓ×ٞÛ~ÜÑÐSL
í ûçZÒ/ÐÔ/Ï ü6âٞÛÝçâ1ÐÔ/×â1ШÕÑÜÕ Ö•× ØeÙMڀÛ~Üäñ'
Ü :>çSÕ~×/á ҕäñ×ٞÛ~ܙÜÚäQÜÑвânçZڀـçZâ1ШÜÑÐÑÐ Ü â{ò
ڀÙvÜ äMäQÜÑÐ¥Õ~çâeÕ±ÜÛÑâe×Zâڂä×´ØÖ/é>ÐÑÙvêÒ{ܨÔ/ÜÑÐè/Ü ÛSá~ÜÑВ×ZڀÓç>ЇØeÖ<Ï Û Ù€êÒ{ÜÑÐSí
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
d{l
ù±ø
(û£ü
"’ý_)F¾þ&ÿþ
šû£ü
'*¿š) iû£ÿ¾
*)¤¥û£ü
T7›DÃ>Ñ8;bD|>?ÃtD Cc¼É8;³UÃ8Bƒ¿£>Q:³xƒ>?¿N">?ñD µ : @ ƒCÆî!":t³ƒAB>ѧ Ãî>Q}:³
¿®®>?ƒc @ >Ñy"= @ „ãD|:x ">?®D @¶µ ⠃~¿®¶">?= @ KD|:xp³: â ´[㳃AB>Ñ+ÿ> @ ":³D|C">?®D @ 8+¼É
:³ ƒ: ¶ ;ÃñD DÃ>Ñ8;[AB>?A"~}:³ñD³"³ @¶µ !;ÃBó?ó?óeT7§: @ x8 @ :ƒ>Ñ8;–€8;: @ ’î¼|ƒ>ÑÃ>ѳ™ƒ
D8;³ÃD µ ƒ:³D|>Ñ @ ¢!"ƒƒ>ÑÃ: @ aÃ8;³D³">?8D|:>? @ 8;D @¶µ >?}C;Ã>Ñ8;ÆD @ ™!"ƒƒ>ÑÃ: @ ƒ>?¿N">?
ÂÉà @ <"8;¿®<D @¶µ "ƒ¿®8Bƒî³ï ƒƒƒBó @[email protected] Ã>‚ƒ8;:³ @ ~¤ @ D µ :ÆÃc @ 8;ƒ>?¿®ïƒ D³"³ @ }:³ @[email protected] !"ƒƒ>ÑÃ: @ ÂÉ8;<D|>ÑCƒ>?9a8;: @ :³ƒAB>ÑBó €³D³"cÿ!Ã8;ƒC">?¿®": â Ãc @ 8;ƒ>?¿®ïƒ:ƒ> @ >Ñ
Ã8;:C"¿£¿®Nî}¼Q> :³D”!"ƒƒ>ÑÃ: @ cÿ @¶µ "ƒ¿®8Bƒî³ï’ µ ƒ!;¢:§¿£> @ >Ñ:tƒ>?¿£ @ ñ„à @[email protected] Ã>9 µ ƒ
î³8;¿®8;AÉï³ > D³"³ @¶µ ƒ!;à >ãD³"³ @ ¿£³ 1 @ :³ @ xÃî!"¿£±AÉ8;¿N"AB³ƒQ> :³ñƒƒƒñÃ8;¿¨
@ Q> :³ @ 8;= @ ï¿®ñD !"8;Æ>?< p³:³³Ãñƒ: @ î³8;8;³’ƒ>?¿N">?D µ : @ ƒCNî!": ³ƒAB>Ññ’ƒ:
@ ”!"ƒƒ>ÑÃ: @ Ãî!"ƒAÉaD @ AɃ=998;:¢D"A @ E c>?ƒî!": â >?¿£€8;ƒC"+ó · @ ƒ¢D8;³Ã>?¿£98;ƒC"}
D§Ãc"C;Ã>ÑCUÃ8;ƒÃ¿®}®8;: @ ñ8;ƒ>ÑUî¼|ƒQ> :³UD @¶µ "ƒ¿®8Bƒî³ï¢
oƒ>Ñ8;ìÿk¨
¿
9C"ƒ:Bó?ó?ó ™":< DCƒ:³™D|:[>?ñ ⠀ƒ>?¿®C @ ó ƒUÃ8;!">ÑC"³Ã£ƒD µ "> @[email protected] :Ã:³Ã>r @ £D³"³ @
¿®ƒî³8}DD<DÃ>Ñ8;N!"Ep³:³8;óò
Và ƒóÉÃî!">?ƒq
V\(}:>!ƒ”D @ :³³ƒ>?= @ ´D¢!"C"¿®ïƒ
@ 
}:³ @ C": â D µ ;8BC8 @ D³"³ @¶µ ">?cÿ @ ™C": â D µ î:¿£>ÑD|>?ñÃBó?ó?ó
"³ @ Ãc;D|D @ <DCÃ>?ƒ>Ñ8;ñD|:UDÂÉ @ 8;9¿®}D7AɃ=€cÿ @ ”!"C"¿®ïƒ @ ” @ :³k>?¿£98;ƒC"}
ƒ @ §¿N;C ÂÉ8 @ :¿£Q> }:³ñD @¶µ ">?cnó T7k:!"}ƒ>?¨
:>ãDÃ>?™>?ƒ:>?ƒ>?ÂÉ¿® @ ®ÂB"ƒ>r"ƒ>Ñ8;'D @ N¿N;
ÂÉ8 @ :¿£Q> :³®cÂÉà @¶µ @ ƒ>?ƒ:³D®ƒ @ §>8 8;³D:x"ƒ¿®8Bƒî³ƒQ> }:³£ÂɃƒ>ÑÃc @ Ƅ Xv (h) = R ∞ natm (z)dz
h
8;: natm ƒ @ ND³ƒ>?UD @¶µ ">? h @¶µ @ ƒ>?ƒ:³DBó 9³D³"}cÿ€98;::³ AɃ=€x>?³Ã @ >?³Bÿ @¶µ !">ÑC:
D¿N"ƒ>ÑCÂɒÂ;"ƒ>ђÂÉà @¶µ "A @ 
c>?ƒî! @ θ „Éƒ> @¶µ 8;§³A @ >?AÉ @ Ã8;:ƒ=:D @ ƒ Å ÿÉ8;t
@ ñ @ "ƒ>Ñ8;›ƒ>?¿£ @ X(h) = X (h)/cos(θ) {ó T7£>8 V8;³D:<ÂɃƒ>ÑÃc @ x"ƒ¿®8Bƒî³ƒQ> :³xÃc @ Ã: @ ">?³ƒ>
v
ƒ’D µ }Â}>?8; ¾cÀBÀBÀNA³ó¸Ã¿ −2 ":–>?ÂÉc":'D @ ®¿®cÿ9’± :³U’
Á 8;>ђ @ :³’€8;::["A @ c>?ƒî! @
D
RBÀ ◦ ó
ù!
"kü
-(ü¤ÿ'ÿðüj¾
»
ü#*$»²ü#Àºü(û%kü#
(û£ÿ
)F¾þ&ÿþ
²ü#
ìT ™D|>v ¼9™Dx!"ƒƒ>ÑÃ: @ ™Ã8;¿£98BC"}™:³ AɃ=€xC8;™Ã8;¿£¿ :³¿® AB8;:9
ƒ8;>Ѷ"¿£> @[email protected] „
@
@
@ @
@
∗ UÃ8;¿£€8BC"x Ã8;¿N"AB³ƒ>Q:³t„ < Ã8;³Ñ«)€8Bƒ>?8;³ î³8;8;³
@
@
@
∗ UÃ8;¿£€8BC" ¿ :³8;>Q}:³N„ µ± ÿ :³:ƒƒ>?³8B’;8Ã>Ñ νµ ÿ νµ
ƌæ~ßVÛ6ØLÜ ÒYÎLÐdÈ6éVÑÉÒYÎ YÍYÈ5ÇÌË 5ÐdÛ pÇÉÈ6çlÍ!،ÇÉÒYÛ6،ÎLã
◦
à5Ñ£Ý Ò!Ø ØŒÒYÛ ØLÎ Û6Ø
X
pê
Ü6Û6Ò#ãáYÒ!Ë ËŒÒ-Ðdæ5æ ØŒß 6ÇÉÏ&ÐˌǗßdȀÒY΄ËLÇÉÈ6çlÍ!،ÇÉÒYÛ6،Ògã
_yó_ ϔ˛`:c/jiËÊ~Èea:a¢ËJËç9Þ`kË}Õ®ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
°
d “
½|ó?¾ }C"ƒ>Ñ8;–D<Ãc"C;Ã>у>Q:³AÉ8;¿®ƒƒ>Q:³D µ :³AɃ=€ "ƒ¿®8Bƒî³ƒ>Q:³Bó
@ tÃ8;¿£98BC"}Uî!;D|8;>Q:³%„ @ D|>vî!;D|8;³™8D|:>? π± ÿ π0 ÿ K ± ÿ K 0 ÿ€ÃBó?ó™">?³ƒ>
:³ @ :³Ã @ 8;³’ @ VC"AB¿®<}:³Ã @ c">?có
P>Ñs}:³ñ8;:;‚Ò8;:™} D³"³™:³UAɃ=€Bÿ @ ™8;€8;ƒƒ>Ñ8;³ cÂÉà @ ~:³ @[email protected] C8;™
Ããƒ8;>єÃ8;¿£98By"”D€³D 8;ƒ¿®D @ !"ƒ:D @ !"ƒƒ>ÑÃ: @ <ƒ>?¿N">?BÿÉD8;£³ƒAB>Ñ
">?³ƒe> }:³8;K"A @ ’D µ >?³Ã>ÑD³Ã™D³"³ @¶µ "ƒ¿®8Bƒî³ïBó ¹ "a â ¿£ @ Bÿ:³AɃ=9’>?ƒ>?ƒ>ђ!":§î³>8
8;[8;:±:' @ Ã8;'³UÃ8;ƒ>ѳD|CKC"ƒQ> }:³¿® ":³Ã:³£Ã8;¿£€8BC"Uî!;D|8;Q> }:³U> ¿ :³8;Q> }:³Bó
¼}€ DxAɃ=9ñƒ™"9 @[email protected] ñAɃ=€ @ Ã8;¿N"AB³ƒQ> }:³®Ãc" @ xDÂÉ @ 8;9¿®} D @ §Ãc;Ãc;Dñƒ
D8;¿£>?³<!" @¶µ >?}C;Ã>Ñ8;t @ Ã8;¿N"AB³ƒQ> }:³B<ó TìAɃ=9¢>?>?ƒ>Ñ¢!"¢D:³Ã @ 8;³8;:§D¢³8c¼": â
D8;³ @ >Ñ:I´›D AɃ=€xD|>?ñî!;D|8;Q> }:³có @[email protected] Ã>€8BïD} :³tÃ8;¿£98By"t @ Ã8;¿Nƒ
AB³ƒQ> :³‹
}:>™C–DÂÉ @ 8;€±!"C @[email protected] ï @ ¿®cÿ¿£€8;ƒC"Æ}Â}>?8; UBÀ _D @¶µ ³ƒAB>Ñ'D|: ƒ>?¿N">?Bó
T µ ÂÉ8 @ :ƒ>Ñ8;–AɳC @ xD µ :³ Ãc;Ãc;D D !"ƒƒ>ÑÃ: @ ƒD8;³Ã D8;¿£>?³ !" @ Dƒƒ>?–D CUÃ8;¿£€>8
C"£ @ Ã8;¿N"AB³ƒQ> }:³Bó · @ ™D8;³ÃU€D³"AÉ8;ABQ> }:³¿® >?}C"xD µ ƒ:³D|>Ñ @ £Ã8;¿£98;ƒ¿®}
D µ :³AɃ=9 @ Ã8;¿N"AB³ƒQ> }:³ñD³"³<:%¿£>Ñ¿£³+ó
T7!"ƒ:ƒ8Ãî!;ƒƒQ> }:³D8Ã:³D µ >?}C;Ã>Ñ8;N¿£>љ‚ƒ:N€8;:”Ã8;³ }:³³ÃBÿÉ":£ABC"³D
DC"ƒ8;>Dãî¼|ƒ>ÑÃ>ѳcZÿ }:³ÃCî!:³aAɃ=€aƒã:Q> :³aã9:”cÂÉ8;>?”DÃc"C;Ã>уƒQ> :³¢ @ 8;>?AB³
∗
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
d 
Ú%ËÕcÇß"ÅMakÞ+Å~È9ç `5hQNkçkËÆä!Ë}ß)à¢Ë•=È!à ÕcË߃jÔà ÊoËÕaDbnIìÕ+ÅMLNkËÕ
STVU m!TW!^€TVUN][ cIWYXv^ ^ T\5mNc~m!`tul`bU_iVU?qP[ cIWe~id]Wh`lWhT¨ulX
k d
i [ ^ im!Wh`lTpe~TulX9k TdWhfPT ]ž0k6cIWhT ‰~wlˆV­gU_[6\5mRulTdUU_cI`t}
«
«
z{Xv\5m!U
I] s TVUŒm#cI\0TeIWh[6`—m!Ta5cI`][ W!WhTVU_qP[ \0e9nubXpmhWYX{}
∗
³ŒTV] m![ `lWhT&a5cs XvcIWYXv`tmgUhcI`tz`ubX q0X W_m!`l]cIutT¤q0Wh`l^‚Xv`lWhT`l\~}
]`leIT\5mhT&Uh`TululT&\s X;z;X `—m#q0X6U#`l\6m!iWYXvk6`°wISLs Xv“T&eIiž0\I`tm
eI[ \0]'ubXeI`tW!Td]mh`l[ \’es X WhW!`lz iT§e~T ubXk TW!fPT§X cKUh[ u|
ulTdU€q0X W_m!`l]cIulTdU U_TV][6\0eIXv`lW!TdUpUhT'qIWh[6q0Xvk6TdX \6m€e~T'‘X{}
[ \'k ul[ f0X utTd^ T\5mj][ ulul`t^ iT¤X c~mh[6cIW#e~T¤u°s X{“~T6w
šTVUŒm'utTƒq [6`t\5m§es `l\5mhTdW!UhTd]mh`l[ \>e~Tƒu°s X{“~T
∗
X;z6Td]ƒulX’UhcIWh‘X ]Tm!TW!WhTVUŒm!WhT6wšŸ u][ W!W!TdUhq [6\0e~WYXv`tm'Xvc
qP[ `l\5mes `t^ q0X6] m§XvcUh[ uReIc•W!X;Z6[ \’][6Uh^ `la5cITNqIWh`t}
^‚X `tW!T¤Us `tu\s X;z{Xv`tm#q0X U-`l\5mhTdW!X k `w
eIidUh`tk6\IT u°s T\0UhT^fIulT eITdU¤q0X W_m!`l]cIulTdU¤UhT}
∗
][ \0e0Xv`lWhTVUUhT‚qIW![ q0X k TVXv\5mz6TWYUutTUh[ uLn'ubX§z`—m!TdU!U_T
eIT ulX‹ulcI^ `t¦dWhT6w \’qIW!T^ `l¦W!T?XvqIqIW![;“~`t^‚Xvmh`l[ \|y[ \
qPTcIm^€[~e~idut`bUhTWyutTš²W![ \5m,e~T-k TW!fPT-][ ^ ^ Toc0\IT#U_cIWh}
‘X6]T?q0ulX \IT |š]TqPT\ eIXv\5md|outTVUq0X W_m!`l]cIulTdUUhcIfI`bUhU!Xv\5m
eITdU‚e~`t¡Pc U_`l[ \0U^pcIutmh`lqIulTdU‚ulT§ut[6\IkŽe~T§utTdcIW‚mhWYX{³ŒTV] }
! "
#%$'&(*) #,+')-&/.10 32
m![ `lWhT6|`tuLTdU_mq0utc0UW!idX ut`bU_mhT‚e~T‚][6\0U_`be~idWhTdWc0\IT‚U_cIWh}
‘X6]T?e~[6\5m ubX‹][6cIWhf0cIWhT§TVUŒm‚cI\IT [6f0UhTW!z;X fIulTqIœZ5}
Uh`ba6c0TRa5cI` eIiqPT\0ee~TjubX&\0XvmhcIW!T |5e~Tgus id\ITW!k `lTjTmoe~T
us Xv\Ik6utT¤es `t\ ]`be~T\0]T&e~T¤ubXq XvWhmh`b]cIulTqIWh`l^‚Xv`lWhT6w
Ne~iVU_`lk \IT ubXEqIW![v³ŒTV] mh`l[ \ e~c
∗
\0[ ^pf0WhT&e~T¤q XvWhmh`b]cIulTdUje0Xv\0U#c0\ qIulX \§q TdWhqPT\0eI`l]c~}
ubXv`lW!Tƒn±ubX¨eI`tW!Td]mh`l[ \ eITNqIW![ q0X k6Xvmh`l[ \w#™Rs TVUŒm'cI\IT
m!W!X \0]YœIT s¢e~T ulXk TdWhfPTe~iV][6cIq idT‚n9cI\^ [ ^ T\5m
eI[ \I\Ii?eIT?U_[6\’eIiz Tdut[6qIqPT^ T\5mdwST qIW![vž0ujubX{m!iWYXvu
TVUŒm#^ TdUhcIW!iX c'Uh[ u¢Td\e~`bU_qP[6U!Xv\5m-cI\§WhiVU_TVXvc9eITeIi}
m!Td]mhTc0W!U€W!ik c0ut`l¦W!T^ T\5mpTVU_q X ]iVUwySXƒa5c0Xv\5mh`tmhi'e~T
Uh`lk \0X u^ TdUhcIWhi¤Td\'²[ \0]mh`l[ \?e~T&ubX‚e~`bUŒmYXv\0]T&n u°s X{“~T
W!Tq0WhiVU_Td\6m!TRulT¤qIW![vž0uubX{m!iWYXvue~T¤ubX€k TW!fPT w
geIidUh`tk6\ITŽu°s iz6[ ulc~mh`l[ \¬e~c
∗
\0[ ^pf0WhTNeITNq0X W_m!`l]cIutTVUulTNul[ \IkKe~TNubX‹m!W!Xv³ŒTd] m![ `lWhT
eITNubXŽk TdWhfPT?eIXv\ Uus Xvmh^ [6UhqIœI¦dWhT6woSX±qIWh[ ²[ \0eITcIW
z6TWhmh`b]X utTX{mh^ [5U_qIœ0iW!`la5cITnpubX a5cITdutulTRulT\I[ ^fIWhTe~T
q XvWhmh`b]cIulTdUpTdU_m^‚X{“~`l^ X uLTVUŒm XvqIqPTuli
Tm€TVUŒm
Xmax
^ TVU_cIW!iT\§k0w ]^ −2 w
#546)7& 8 9:9;/0=<>)-/9 ?/@
?
#46)7& 8 99;&A.CBEDF0G(1HADF.I/9 D|:'Ã8;¿£98;ƒ¿®} ¿®8c¼Éì󑒳ñAɃ=9£D µ :³£³ƒAB>Ñ£D8;³U€: ">?³ƒ>¿x= @ ´§:³UAɃ=9
D µ :³–":ƒ–³ƒAB>ÑBÿ¢8;: 8+ÂÉ!"%D µ :³Æ!"ƒƒ>ÑÃ: @ –ƒ>?¿N">?›D|>vBó ƒ–8;ƒ>ÑƳD
:!;>?¿®™>?¿£98BCƒ>?= @ @¶µ >?³D}ƒ>!Ãc"ƒ>Ñ8;GD|>?ÃUD @ N!"ƒ:UD|:–Cc¼É8;'Ã8Bƒ¿£>Q:³xƒ>?¿N">?Bÿv: @ :³!"C"ƒ>Ñ8;–C"ƒ>уƒQ> :³<Â>ÑC"AÉc"= @ Bó
©-+ž©-+BF
Js+{,\i+=]zt‡+e;•9,/2378.n‰:46z19\?A(CE+=
TìtAɃ=€% @ Ã8;¿N"AB³ƒ>Q:³† ‹§Dƒ>?AB³} @ %Ãc;Ãc;DKD–!"ƒƒ>ÑÃ: @ Æ8}D|:>?K!"
@¶µ >?C;Ã>Ñ8;'D µ :›î³8;8;[8;:[D µ :– @ Ã8; Ô D î!": ³ƒAB>ÑUÂÉà @ £¿N"ƒ>ÑïBónìT ’AɃ=9 @ Ã
ƒ8;¿N"AB³ƒ>Q:³U8;} :QÃ8;¿£€8;ƒ¿®U¿®8;>?³¥³p :³Ãƒ:!"’}:³ @ AɃ=€xî!;D|8;>Q:³t„7b €cÿ
@ î³8;8;³ @ @ Ã8;³}:>7Ã8;³ƒƒ>?ƒ:³} @ AɃ=9 ­
8;}:³ÂB"ƒ>Ñ D¿®8DD µ >?}C;Ã
ƒ>Ñ8;³¢8;=!"= @ =€c":³Ã8;:§¿®8;>?³ƒ>ÑÃî³q}:³<€8;: @ î!;D|8;³có · @ §ƒ: @ :³AB: @ "ƒ>?™D³"³ @ DÂÉ @ 8;9¿®}D @ Ãc;Ãc;Dq}:³ @¶µ 8;§€:” @ :³E ;Ã> @ ¿®}Ãc"C;Ã>ÑCcó•kT ƒî³8;ƒ>ÑD @¶µ @ Ã8>
D|¼}!"¿£>Q}:³q:!"}ƒ>Q:³™€ƒ¿®D µ vÃ:³<DaÃc @ Ã: @ ƒïÃ>Ña€8;:aƒ>?¿ : @ @ ™DÂÉ @ 8;9¿®}
-K
L
6L
Õ #ÐdÈ6Î,áYÒ!Ë ËŒÒ#æÐØ ËŒÇÉÒVà5ÑÉÒjϤßdË hÍYÑÉÒYá!Ë ØŒßdÈ #Ü ÍYÎ ÇÉéVÈ5ÒhØ_ФÐdÛ5Î Î Ç å5ÇÉÒYÈÛ5ȀÍ!їÒ!á!Ë ØŒßVÈ ßdÛ Û5Èæ~ßVÎ ÇÌˌßVÈ
_yó_ ϔ˛`:c/jiËÊ~Èea:a¢ËJËç9Þ`kË}Õ®ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
d O
DAɃ=9 @ Ã8;¿N"AB³ƒ>Q:³có
ڙc\jiËÊoÈaDa”ËJGË}ç9Þ`5hñN çkËÆä!Ë}ß)à”Ë]c}ÊoËÇBÞcß"ÈeJGÔäiç:cÉÞ+ÅMLNkË
T7®DCÃ>?ƒ>Ñ8;±DC"> @[email protected] xD|:–DÂÉ @ 8;€¿®D µ :³ AɃ=€x"ƒ¿®8Bƒî³ƒ>Q}:³x³Ãƒ>? @ Ã8;:
´ @ Kƒ>?¿ : @ "ƒ>Ñ8;}:¿®ƒ>Q:³Bó » c"¿®8;>?³+ÿì'¿£>Ñï§"8 â >?¿N"ƒ>Ñ8;ìÿ7> @ ƒ98BCƒ>?= @ £D£DÃ>?
"! @ ¼}ƒ>Q}:³¿® @ Ã8;¿£98;ƒ¿®}¿®8c¼É%D µ :³AɃ=€ @ Ã8;¿N"AB³ƒ>Q:³Bó1T7 ƒî³8;ƒ>љ"! @ ¼}ƒ>Q}:³
Dð
f >ѐ 8Bƒ> 2 ½Z6R 4 @ "=€8;®D³"³ @ "³¥X ÀKÃ8;³D|:>?U´%D 8;ƒ¿ : @ ñ¿£> ¿£>?ƒ>Q}:³
:>ì8;}< @ "ƒ>?ÂÉ¿®=>ÑÆ8D|:>?!" @ ’ƒ>?¿ : @ "ƒ>Ñ8;³+ó ³ ÆD Ã8;¿£³D| @ ƒ>?³Ã>?! @ Ãc"C;Ã>уƒ>Q:³›DKAɃ=9Æ @ Ã8;¿N"AB³ƒ>Q}:³›y"³Kƒ›D³"³ @ ÆDC"> @ +ÿÃ8;³ƒ>ÑD8;³ @ ¿®8}Dï @ ⠃~ ¿®¿®ƒ>?¿£ @ >у DÂÉ @ 8;€!" >? @ 2 BV À54 ó
|½ ó ½€’–8Dï @ <ƒïƒ>?¿£ @ >!D|:§DÂÉ @ 8;€¿®D µ :³AɃ=€ @ Ã8;¿N"AB³ƒ>Q}:³Bó :N=€8;:D
Ãî!}:³ @ 8;AB:³:’D™C;D|>r"ƒ>Ñ8; X0 ÿ @ ™³8;¿ = D !"ƒƒ>ÑÃ: @ D8;:= @ Bó
Tìƒ>?³Ã>?€ ƒ @ ƒ:>?ÂB"x„
∗
∗
OP
RQ
: @ @ 8Ã:³DÃc"ƒ>Ñ8;³D!">?’<D=¿®ƒƒC @ :ANC8;Ã8;³ƒ>ÑD
":%=98;:’D µ :³ @ 8;AB:³:DC;D|>r"ƒ>Ñ8;NM’€:³ @¶µ 8;%³8;C X0 ™„
Ð#æ~Ò!Ø ËŒÒØŒÒYїÐˌÇÉ×;ҚÜ0Ý ÍYÈ5Òh،éVÇÉҚÜIÝ Û5È&ÍYÑÉÒYá!Ë ØŒßdÈpæ5ÐØØ_Ð ä;ßVÈ5È6ÒYϤÒYÈ{ËÜ6Ґçl،ÒYÇÉÈ5ÐdéVҚїß،ΌÞ{Û Ý ÇÉÑË Ø_Ð×VÒ!،ΠÒoÛ6È5ҐÍYæ5ÐdÇÉÎ Î ÒYÛ6Ø
Ü6Ò
dx
Ï&ÐËŒÇ !،ÒyÒY΄ËÜ6ßdÈ5È6ÍYÒyæÐØ d<E>
V
ß
Û
Y
Ò
„
Î
Ë
t
Ñ
o
Ð
—
Ñ
d
ß
5
È
d
é
5
Û
Y
Ò
Û
F
Ø
Ü
F
Ò
_
Ø
V
Ð
Ü
—
Ç
Ð
Œ
Ë
É
Ç
V
ß
0
È
ê
o
Ð
6
æ
Œ
Ø
V
ß
5
å
d
Ð
6
å
—
Ç
É
Ñ
Ì
Ç
Œ
Ë

Í
6
Ü
F
Ò
!
á
Œ
Ø
Í
Ð
Œ
Ë
É
Ç
d
ß
&
È
6
Ü
y
Ò
5
æ
d
Ð
Ì
Ç
Œ
Ø
Ò
' dx
X
e+ e−
æ~ßdÛ6ØFÛ5Ȥæ 6ßdˌßdÈ
Ë Ø_Ð ×;Ò!،ΌÐEÈvËїÐgÏ X!0ϤÒLÍ!æÐd0 ÇÉÎ Î ÒYÛ Ø,ÒY΄ËyÐdÛ5Î Î Ç~Ò!Èpæ ،ÒYÏ¤Ç !،ÒLÐdæ6æ6ØŒß Ç—Ï&ÐˌÇÉßVÈ
ê ÐgÑÉßVÈ6éVÛ6ÒYÛ6Ø
dx
'æ6X
ÍhË_ÐdÈ{Ë#ÇÉÈv×;Ò!،ΠÒYϤÒYÈ{γ˚æ6،ßVæ~ßØ ËŒÇÉßVÈ5È6ÒYÑÉÑÉÒ¤ãїÐÏ&ÐÎ Î Ò×;ßdїÛ6ϤǗÞvÛ6ÒgÜ6Û9ϤǗÑÉÇÉÒYÛ0à ßdÈÛ ËŒÇÉїÇÉÎ ÒÎ ßdÛ5×;Ò!ÈvdP
Ë-їÒ
،ßv0Ü6Û6ÇÌË
Ï&ÐX
Î Î 0Ò
(X
×
0
×VßVÑÉÛ5ϤǗÞ{Û5ÒÚÒ æ6،ÇÉϤÍ-ÒYÈ
ê
g.cm−2
C
-
TS
RQ
P
P
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
d ¦
?
?
Ãî!}:³ @ Ã8;±ñ8}D|:>?<!"=¿®ƒƒC"î @ :A®:Æî³8;8;–:› @ Ã8;–¼"}’ÃCî!;Ã:
@ U¿®8;>?ƒ>Ñ D @¶µ ³ƒAB>Ñ >?>?ƒ>r @ Bó
Ãî!}:³ñî³8;8;®8D|:>?™:³x!">? e+ e− ÿ @¶µ ³ƒAB>ÑUC"} A @ ¿®!"ƒƒ>ÑUƒ @ D: â @ Ã8;³có
átË}ß)à”Ë}Ւc}ÊoËÇÉÞcß)ÈnJÔäiç:cÉÞ+ÅMLNkËÕ¨=ÊoË}Õakß"ÈvÇÉË}ÕcÕKNkÕa(bnIìÕ+ÅMLNkËÕ
STVUgq0Wh`l\0]`lq0X c~“?qIW![]TdU!U_c Uga5cI`y`t\5m!TW!z5`lT\0\IT\5me0Xv\0UulT€e~idz Tdut[6qIq Td^ T\5mes cI\IT€k TdWhfPTpidutTV] m!Wh[6^ X k \Iimh`ba5cIT
Uh[ \5m#utTVU#U_cI`lz{Xv\5m!U
yU_[6c0U&us T¡¢Tm€eIc¨]Yœ0Xv^ q±][ cIul[ ^fI`tTd\±e~TdU
?\I[{Z5Xvc~“|ulTdUpiulTd]mhW![ \0Upq TdW!e~Td\5m e~Tu°s i\0TW!k `lTT\±id^ Tm_mYXv\5m€eITdUqIœI[vm![ \0Uk6X ^ ^ X0wST9qIœIid\I[ ^ ¦d\IT9qPTc~m
›m!WhT€e~id]Wh`tm]ulX6UhUh`ba6c0T^ T\5mgTd\ƒ][6\0Uh`le~idW!X \5mRa5cITpus idutTV] m!Wh[6\?U_cIf0`Fc0\ITX6]]idutidW!Xvmh`l[ \?e~c0TnubXq0WhiVU_Td\0]TpeIc
]Yœ0X ^€q'iulTd]mhW!`la5cIT¤e~c§\I[{Z5Xvc|WYX;Z [6\I\0Xv\5m-Xv`l\0U_`cI\IT[ \0eIT¤iulTd] m!Wh[6^‚Xvk \0imh`ba5cIT w
UWVYX/Z[]\I^#^_V!`aV!^6bdc_VfeGX/V!gh^_ZiIVY\Ijlk#X/V!`amnmbXZ1o#pGj#^_i UWZaqE\I^6r]VCX/mgG\]^5c#V!mtsuo_\1bn\]^_mvV!^wsuZgxX/V!m
0TdutulTp\0id]TVUhUh`tmhT¤ik5XvulT^
T\5mgubX‚qIW!idUhT\ ]T&e~cN]Yœ Xv^ q
?][6cIul[ ^pf0`tTd\€e~TVU,\I[{Z5Xvc~“w ”TdcIulUutTVU,qIœI[ mh[ \ UU_cI¥‚eU!+Xve^ −^ T\5m,i\0TW!k im!`la5cITVUyq TdcIz Td\5mqIWh[~e~c0`tW!T\5mLe~TVU,q0Xv`lW!TdUd|
ubX€UhTd]mh`l[ \'T¥]X6]T¤e~Tdz T\ Xv\5mj][ \ U_iVa6c0T\5mhT¤n€q0XvWhmh`lW#e~T¤qIulc0Uh`tTdcIW!U#†?T ©pw
UWVCmsWVXbnVCmTcWy;z!^_VCXiIghVtc_V!mTz!phV!qAbX\I^_ms_Z*X{gh\I^#gGmZbgh\I^%\]j|V}~qgxbnZ*bgG\]^:c_V!mTZ*b\I`aV!mc#ja`|ghphgGVCj qP[ cIWšubX a5cITululTjutTVULqPTWhmhTVULq XvWL`l[ \0`lU!X{m!`t[6\e~Tz`lT\I\ITd\5moik5XvulTdUoXvc~“
WYX;Z [ \0\IT^ T\5mLe~Tg²WhTd`t\ Xvk T6w6SLs id\ITW!k `lT,2
?]TVUqPTWhmhTVULUh[ \5mLqIWYX{m!`la5cITd^€Td\5m`t\0eIiqPT\0eIX \5mhTdUše~Tjus i\ITdWhk6`tTje~Tgu°s iulTd]mhW![ \‚TmšU_[6\6moT\][6^ q imh`tmh`l[ \9X;z6Td]-ulT
c id\ITW!k `lTR]Wh`tmh`ba5cITTm-TdU_m-e~Tus [6W!e~W!TReIT
q TWhmhTdUšq0X Wof0WhTd^ U!U_mhWYXvœIulcI\IkTdU_m-X qIq TdutidTR
P
† T ©wS[ WYUha5cs cI\idutTV] }
mhW![ \ X{mhmhTd`t\5mL]Tm_m!Tšid\ITW!k `lT#]W!`—m!`la5cIT6|{`tu0z;XX ut[6W!UFqPTWYe~WhTšm![ c~m!T-Uh[ \€i\0TW!k `lT-UhcIWycI\I83
T#ul[ \0k cITdcIW,e~T#W!X6e~`bX{mh`l[ \w
U Z:c_gxWj#mngh\I^f`dj_pxbngxsuphV 0ulTdU#idutTV] m!Wh[6\0UjUhcIfI`bUhUhT\5mje~TdUgq
Tmh`tmhTVUge~idz5`bX{m!`t[6\0U#Xv\0k cIubXv`lWhTVU#eIXv\ U#utT&]Yœ Xv^ q
?e~TVU,\I[{Z6X c~“pTmoe~iz6Tul[ qIqPT\5mLX `t\0Uh`IubX¤k TdWhfPTjeIX \0U,ulT#qIulX \‚q TdWhqPT\ e~`l]cIubXv`lWhTjnubX¤e~`lWhTV] mh`l[ \‚e~TgqIWh[6q0Xvk5X{m!`t[6\w
€Xv\0U-ulT&]X6e~WhT¤es cI\IT&eITdU!]W!`tq~m!`t[6\q0utc0Ušž \IT&e~c e~iz6Tul[ qIqPT^ Td\6mje~T¤ubX€k TdWhfPT |~`lu‘Xvc e~W!X `—mjXv³Œ[ c~m!TW
u°s [ W!`tk6`t\0TKes cI\IT
\1bnœ0\1X6s_e~X/W!\~[ \Ic_`bj#a5qEcIbT ghwš\I™o^‚Tdq VTdbƒ\0eIp X y„zC\6pGmVV!|qAulbTdX/U9\]Uhs#TdX/]\~mh`lc_[ j_\0UqAbTgG¥\]^ ]X6]c#TdV†
Zc‡]TVX/U‚\I^#qIm W![~]pTda5U!UhcIc0`9UUhUh[ [ \5\5m±m‚n¬
?Uh[ c0U U_Z
}k6Ts_
W!f o_T§
U9e~o_TN
‘Xv`lfIulTdUe~Tdz{Xv\5m9ubX
]W!idXvmh`l[ \ e~T-q0Xv`lW!TšTmutT#f0WhTd^ U!U_mhWYXvulcI\Ik0w;Ÿ u~‘X c~m\IiVXv\I^ [ `l\0UyulTdU,q0WhTd\0e~W!TšTd\€][ ^ q~m!TjU_`Ius [ \€z TcImL][ \0\0Xbˆ mhW!T
ubXq0Wh[6q [6W_m!`t[6\e~T¤^cI[ \0U-q0Wh[~e~cI`tm!U-q XvW#ulTdU#k6TW!f TVUoidutTV] m!Wh[6^ X k \Iimh`ba5cITdUdw
U,y„VWVbv‰Š\]`as#bn\]^ I`l\5mhTW!z Td\0Xv\5m-q [6cIW#ulTdU#qIœ0[vmh[6\0U#e~Tf0X6UhUhTi\ITdWhk6`tT « e~T¤us [ WYe~WhT¤e~c † T ©­ w
?]TVUhU‹Uhc0
y;VCU9 iVCulbY
 Xvk \0ŽU imhZ`ba5^_cIc_TdZU‚j~e0'Œ‹Xv\0\IU9`aulXŽ
phVCX/^‚ZXv^_mhq`l¦o‘W!j‡T?’6q '“
iIulc#TdZ1Up•”qIœIš[ mha5[6cI\0`Ue~[6`l^€c•`l\iulcITdT] m!ulWhXA[6\0U_UTV] emh`ls cI[ ut\•mhWYTXŽ¥œ0]X X c~]mhTƒTƒe~iTV\IU9TdWhq0k6Wh`t[ T6} w
Td] m!UWWhŒT
[6^‚
[6cIghW9
STVU][ \0Uhida5cITd\0]TVUe~`lWhTV] mhTVUe~Tš]TmT¡¢TmFUh[ \5mcI\e~iz6Tul[ qIqPT^ T\5mq0utc0Um!X W!e~`t~TmFqIutc U‘0
– c0]mhc0X \5me~TLubX-k TdWhfPT w
VulTdUqIœI[vm![ \0Ues cIu—m!W!Xgœ0XvcImhTLid\ITW!k `lT « E > 5.10 T ©K­qPTc0z T\5mF`t\5mhTdW!X k `lWX;z TV]
? U‹y;VC VCbc#V—s#X/z-mno_\!˜—VCX
?
19
ulT]Yœ Xv^ q^‚X k \Iimh`ba6c0Tgm!TW!WhTVUŒm!WhTq [6cIW#]W!iTdW-eITdU-q0X `tW!TdU + − a5cI``l\I`tmh`lT\5m#cI\IT¤qIW!i}„k TdWhfPTRX;z{Xv\5m-u°s T\5m!WhidT
eIX \0U u°s X{m!^ [6UhqIœI¦W!T wo™oTVU q0XvWhmh`b]c0utTVU‚X;Z6Xv\5m€^€[6`t\ Ues i\IeTdWhk6e`tT§a5cIT utT§qIœ0[vmh[6\Aq0Wh`l^‚Xv`lWhT6|,TulutTVU‚U_cIf0`lU!U_Td\5m
^ [ `l\0Ušu°s T¡¢TmgSxL†3Tmje~[ \0\IT\5m-ut`lTc§npc0\ITk TW!fPTRidutTV] mhW![ ^‚X k \Iimh`ba6c0TRe~[6\6m#ulTeIiz Tdut[6qIqPT^ T\5m-TdU_m-^ [6`t\0U
0c0]mhc0X \5mdw
–
¹ ³8;³Ã8;¿£¿®U€8;>?DUD!"ƒ™:³x!"ƒƒ>ÑÃ: @ U
ƒ>?¿N">?Bÿ€î³8;8;'8;:' @ Ã8;ìÿvD µ ³ƒAB>Ñ E0
9³ƒC"} @¶µ "ƒ¿®8Bƒî³ï’ƒƒƒBó :®=€8;:¢D µ :³ @ ;8 AB:³:DC;D|>r"ƒ>Ñ8;ìÿD: â !"ƒƒ>ÑÃ: @ D µ ³
_y dbátË}ß)à¢ËÕbkÔ1`kß"Èiç ÅMLN7Ë}Õ
dem
AB>Ñ E0 « 2 8;<Ãx„:³!">? e+ e− D³"³ @ Ãc;<8;: @ ™ƒ>?¿N">? C">?:%î³8;8;–<:Æ @ Ã8;
;ÃÃ8;¿£!"AB³D µ :tî³8;8;K>Ñ:tD|:t=¿®ƒƒC"î @ :A®D³"³ @¶µ ":ƒ™Ãc;có :t=€8;:D 2X0 ÿ @ ³8;¿x=
D !"ƒƒ>ÑÃ: @ !;U´’Xt™">?³ƒ> Dx:>?Bó :›=€8;:D –C"€cÿ @ ®AɃ=9ñÃ8;ƒ>Ñ} 2n !"ƒƒ>ÑÃ: @ ¼"ÃCî!;Ã:³ @¶µ ³ƒAB>Ñ E = E0 /2n ó T7 AɃ=€Ã>8 ·Ñ">?³ƒ>‚Ò:³~}: µ ó:³ @ @ Ã8;³"ƒ>?AB³}
@¶µ ³ƒAB>ÑÆÃ>?ƒQ> :³ ´±!"ƒƒ>?ND @ }:³ @[email protected] %> @ ÂÉ8;
ƒÆ"ƒ~ t": =98;:®D µ :³ @ 8;AB:³:ND%Cƒ
c
D|>r"ƒ>Ñ8; ¿®8c¼É³BEó T7K¿N â >?¿x:¿_D%DÂÉ @ 8;€¿®§D @ ±AɃ=€%ƒ£D8;³ÃÆ"ƒ>?K"£:
³8;¿ =£D µ C"€ nmax @ :³ Ec = E0 /2n £ó T7§>8 V8;³D:Ã8;ƒƒ€8;³D³" ":[¿N â >?¿ :¿¡D
DÂÉ @ 8;9¿®} ƒD8;³ÃUD8;³ñ!"U„ Xmax = X0 ln(E /E ) ÿ9 @ ³8;¿x=U¿®8c¼ÉDx!"ƒƒ>ÑÃ: @ ln(2)
":%¿N â >?¿x:¿ !" @¶µ ⠃>Ñ8; Nmax = E0 /Ec ó
¿®8Dï @ ñ9ƒ¿®™D8;³ÃUDxDA"AÉ @ ;ƒ€Ãƒ>?³Ã>?!": â D|:'DÂÉ @ 8;9¿®} D µ :³xAɃ=€
@ Ã8;¿N"AB³ƒQ> :³Bó Tì%8Ã:³§ µ vÃ:³› ƒ8;>ѧî!;N:³ÃÃ>?Âɛ„”:³Æî!;ÆD–Ã8;>Ñ
C"³ÃU8;: @ ™!"ƒƒ>ÑÃ: @ ñ¿ : @ ƒ>? @ >Ñ} AB± ¸;î´ @¶µ @ ƒ!"³ÃNÃc"ƒ>Ñ8;DU!">?Ý ¶=¿®ƒƒC @ î:A 8
:x¿N â >?¿ :¿ DDÂÉ @ 8;9¿®}”< ³x:³î!;D µ ⠃>?³Ãƒ>Ñ8;®8;: @ !"ƒƒ>ÑÃ: @ D98BCã8;:
@ :³ƒAB>Ñ D³"³ @¶µ "ƒ¿®8Bƒî³ïBnó T7U>8 8;³D:":›¿N â >?¿ :¿ D DÂÉ @ 8;€¿®Â;"ƒ>Ñ @ 8;A"ƒ>?ƒ<î
¿£Q> :³¿®}£cÂÉà @¶µ ³ƒAB>ÑK @ 8;
}:³ @ §³8;¿ =KD§!"ƒƒ>ÑÃ: @ ®":Q¿N â >?¿ :¿¯ÂB"ƒ>Ñ @ >?³c">?¿®
ÂÉà „
max
0
c
Xmax ∝ ln(E0 /Ec )
Nmax ∝ E0
½|ó?¾K
½|ó ½\
£ :!"ƒ>Ñ8;³§ ⠃>?¿®} @ k ">?}:³t @ :³ @ ±!"ƒƒ>ÑÃ: @ %ƒ>?¿N">?›±:³%³ƒAB>ћ @ ÂÉBÿ @ :³ @ ³8;¿ = D µ C"€³Ãy">?<€8;:!"ƒƒ>? @¶µ ³ƒAB>Ñ }ƒ @ !"ƒƒ>ÑÃ: @ ’Ã8;³D³">?ƒABC"³Dvó
· @ ƒa>?}C"D™³8;:³™¿N @ ABC8;Æ â ƒ~¿®ƒ>?¿£ @ >ÑÃ>?Bÿ!Ù¿®8}Dï @ ™€ƒ¿®<D™D|>?
:³%ÂÉ8 @ :ƒ>Ñ8; D Nmax Xmax cÂÉà @¶µ ³ƒAB>Ñy:>ƒN;CK8}ÃCî³%D @ ›c @ >?BóETì @ Ã:
!;ƒ>Ñ8;³t98;:ƒC›=>Ñ ƒ:Uƒ8;:ÂÉ£Dx¿®8Dï @ ®"! @ ¼ƒQ> :³U @ :³UDC"> @[email protected] £D³"³ @ @ >?ƒC"ƒ:
2Z
½ U64 ó » c"¿®8;>?³+ÿ @ 'DÃ>?ƒ>Ñ8;î
³³›D%8;: @ ®8;ƒ>ÑtD @ 'AɃ=€%ƒ› ⠃~ ¿®¿®}
Ã8;¿£ @ â ³ÃCƒ>? @¶µ :ƒ> @ >ÑC"ƒ>Ñ8;±D™8;ABC"¿£¿®’D >?¿ : @ "ƒ>Ñ8;ìó
ù_¸
™
ü(ûF%ðü#÷þ')&»²ûF&*ÿ ü#
Tì8;~}:³ @ U!"ƒƒ>ÑÃ: @ xƒ>?¿N">? <:›³8+¼":±8;:Æ:–:³Ã @ 8;ìÿ @ £Ãc;Ãc;DxD¿N"ƒ ! " @ = >r">Ñ
D µ :³£>?C;Ã>Ñ8;†V 8;ƒ®cÂÉà @ £³8c¼":QD µ :Q"8;¿®£} D³"³ @¶µ "ƒ¿®8Bƒî³ï 2 VZV64 ó ƒ ð Ã8 @
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
l ”
@ >у>Ñ8; ð ⠃~¿®¿® Â>Ñ8 @ NÃ8;³D|:>?x":³ƒ>=>ÑG´ @ ðVC"AB¿®C"ƒ>Ñ8;bD|:³8c¼":GÃ>?= @ ’}: µ ´Kà @[email protected] D|:Q³8+¼": >?³Ã>ÑDcóTì²VC"AB¿®£}:³Ã @ c">?UÂÉ8;®Ã8;}ƒ>?:³N´–>?C"AB>?®ñÂÉ8;£Ã8;³ƒƒ>?ƒ:³ @ Ã8}:D @ tÃc;Ãc;D£î!;D|8;>Q}:³BóuTì8; D£ÃƒU¿£>ÑïU>?}C;Ã>Ñ8;ìÿ7D>Ñ8;³ π± ÿ π0 ’ D
o;;8;³³
K ± ÿ K 0 8;A @ ¿®¢8D|:>?có\T7>Ñ8;³”³:ƒ π0 c¼":®¿£³”DÂ>Ñaƒï”Ã8;:ƒ
D>?ïAB[D: â î³8;8;³cÿn:> ÂÉ8; @ 8;>?>?ƒ>љ:³x8;:³ ¶AɃ=€x @ Ã8;¿N"AB³ƒ>Q}:³BóuTì
>Ñ8;³Ãî!"ƒAÉ
}:!" ´t: â ÂÉ8;>?C"AB>? C"
}:³ @ : @ 8;AB:³: D µ >?}C;Ã>Ñ8;Gƒ™ @ :³Ã8;:ƒ
:³ @ : @ 8;AB:³:DDƒ>?ABC"ƒ>Ñ8;›
šB\ó Tì π± Daî!":<³ƒAB>Ñ>?C"AB>ÑC @ 8;¢”8}D|:>ÑC
D µ ":ƒ>Ñ8;³’³:ƒ™’ÃCî!"ƒAÉcó ÃCî!:³ñC"9BÿiÂ>?8;–:–ƒ>ђD @¶µ ³ƒAB>ÑñƒÃDU": â
@
@
@
@
UBÀ D @¶µ ³ƒAB>ћD|:
π 0 :> >?¿®} [Ã8;¿£€8BC"– Ã8;¿N"AB³ƒQ> :³Bó : 8;C ÿ¢Â>?8;8
ƒ>?¿N">?<ƒ8;:ÂÉ8;:³ @ 8;ƒ¿®<D µ @ Ã8;³Dî³8;8;³98BCD³"¢:³³ƒAB>Ñ¿®8+¼É³’D
@¶µ 8;D|£D µ :³®D|ñ> +">?³®D’
±‡ °Uuó Tì8;~ }:³ @¶µ ³ƒAB>Ñ£¿®8c¼É³§D π± "ƒ>? :³£³ƒAB>ѮÃ>?ƒQ> }:³
@ @ 8;AB:³:D µ >?C;Ã>Ñ8;[ @ @ 8;AB:³:DDƒ>?ABC"ƒ>Ñ8;[DÂ}>ѳ}A @ x„ @ Eπc ' 9 f™ °ñÿ
>Ñ8;³UÃ8;¿£¿®³ÃN @ 8;£´›CNDƒ>?}AB®Q¿x:³8;³£x³:ƒƒ>?³8Bcÿ @ >?¿®}C"N">?³ƒ> @ ›Ã8;¿£€8>
C"x¿ :³8;Q> }:³Buó Tì¿ :³8;³+ÿ9D|›
: ¶">?D @ tD|> @ "C"ƒ>Ñ8;± @ "ƒ>?Â>уUD @ :’¿£³DxÂ}>ÑBÿi€:ÂÉ
!"Ã8;:ƒ>?DABC"³D’D|>уC"³Ã’cÂ;"}D Dƒ>?}AB’%:› @ Ã8;–<:%³:ƒƒ>?³8ó
Tì
oB;8;³cÿ|D!"ƒ @ :ABC"³D™ÂB"ƒ>ÑD™Ãc"!": â D™Dƒ>?}ABC"ƒ>Ñ8;›€8Bƒ>?= @ @ >?¿®}’D
@ U¿¿®´
¶8;]
}:³ @ <>Ñ8;³ @ D|> vÃ8;¿£98By"D @ £AɃ=€ Æ8D|:>ÑC"D π± ÿ π0
">?³ƒ> }:³D<¿ :³8;³+ó
©-+žª-+BF
œ
.FÄ6?7îCE7
w6+|wEz1„3+t2˜E˜i+78+4:9›t2346‰:?v9\CEw5?46.ut
Tì8;~}:³ @ k>Ñ8;³ Ãî!"ƒAÉ"ƒ>?AB³ ¶@ µ ³ƒAB>ÑÃ>?ƒ>Q:³ Eπ ÿc> @ 7C¢Dƒ>?}ïAB7>?}ƒ8;¿£9}
c
@ 8; @ N8}ÃCƒ:³xDN¿ : @ ƒ>? @ >ÑÃc"ƒ>Ñ8; Dx!"ƒƒ>ÑÃ: @ có 9³D³"}cÿ @ ›Ã8;¿£98By"K @ Ã8;¿N"AB³
ƒ>Q:³NÃ8;ƒ>?:³KD®C®DÂÉ @ 8;9ñC":³ @ @ Ã8;³ñ8;x:³N³ƒAB>ÑN:9ƒ>Ñ:t´ @¶µ ³ƒAB>Ñ
Ã>?ƒQ> :³ Ec D ³>Ñ®":!"CcÂ;"}cóT왃: @ C" D|:[¿®8Dï @ Uƒ>?¿£ @ >уUDÃ>?Â;"} @ UDÂÉ @ 8;9¿®}
D µ :³¢AɃ=€a @ Ã8;¿N"AB³ƒQ> :³a€:ÂɔD8;³Ã µ " @ >Q}:³D³"³ @ aÃc; 8;: @ ¢ƒ>?¿N">?aƒ :U³8c¼":
8;:§:t:³Ã @ 8;'„t vcÿ @ ¿N â >?¿ :¿¥DDÂÉ @ 8;9¿®}<D µ :³ @[email protected] AɃ=€’Ã8;ƒ98;³D%":t¿N â >
¿ :¿¥DDÂÉ @ 8;€¿®™D™CUÃ8;¿£98By" @ Ã8;¿N"AB³ƒQ> }:³Bó %¿£>Ñïx"8 â >?¿N"ƒ>Ñ8;ìÿ!:
³8c¼":DU¿N;CU"8;¿£Q> }:³ D µ ³ƒAB>Ñ E0 9:
ƒ£Ã8;³ƒ>ÑD£Ã8;¿£¿®£C"} Ã8;¿£€8BUD A
:³Ã @ 8;³>?³D9³D³"}c/ó ‘’³AɃ=€8D|:>?<!"”:®³8+¼":N€: @ 8;E ƒ<¿®8}D @ >ђÃ8;¿£¿®:³

ÑtФÑÉßVÈ6éVÛ6ÒYÛ6ؚÜ0Ý ÇÉÈ{ˌÒ!Ø_Ðdá!ˌÇÉßdÈ9ÒhË-їФїßdÈ5édÛ5ÒYÛ Ø-Ü ÒRÜ ÍYÎ ÇÉÈ{ˌÍYédØ_ÐˌǗßdÈá!ßdØ ØŒÒYÎ æ~ßdÈÜ6Ò!Èv˚ã¤Ñ—ÐpÜ ÇÉ΄Ë_ÐdÈ5á!ÒgÏ¤ß ä{Ò!È5È5Ò#æ5Ð،áYßdÛ6،Û6Ò
æ5ÐؐїÐæÐØ ËŒÇ—á!Û5ÑÉÒjÐ ×VÐÈvËLÜIÝ Ç—È{ˌÒ!Ø_ÐéVÇÌØLßVۀ،Ò!Î æ~ÒYá!ˌÇÉ×;ÒYϤÒYÈ{ËÜ Ò#Î ÒjÜ6ÍYÎ ÇÉÈ{ˌÍYé،Ò!Ø ê
_y dbátË}ß)à¢ËÕbkÔ1`kß"Èiç ÅMLN7Ë}Õ
l H
ƒ:9ƒ€8Bƒ>?ƒ>Ñ8;ÆD A AɃ=9D µ ³ƒAB>Ñ E0 /A :>Ñ~}:³ @ a:³Ã @ 8;³8;’ƒ:98BC<ÂÉ8;>?D98;>?}
D¿£>Ñï >?}C;Ã>Ñ8;–>?³D€³D³" @ <:³’D":ƒcóT µ :!"ƒ>Ñ8;x ½|ó ½\a>?³D|>Q}:³ @ 8;}:³ @
>8 V8;³D:D|:›¿N â >?¿x:¿ÎDDÂÉ @ 8;€¿®Â;"ƒ>Ñ cÂÉà A C @ 8;„ Xmax ∝ ln(E0 /(A × Ec )) ó
Ö ÈeJ†a”È!Õ+Ô³ç9Þc˛J‹N7È9çkÅMLNkË%ËÉÞ§ÅM` Ë}ç9Þ+ÅGžãÇÉÔ|Þ+ÅoÈiç
STVU^cI[ \0UFqIW![{z Td\0Xv\5mTVUhUhT\5mh`lTululT^ T\5m,e~TLubXReIidUh`t\5mhi}
k YW X{m!`t[6\eITdUqI`t[6\0Ud|ubXjqIW![ qP[ Whmh`l[ \¤e~To^pcI[6\0UeIX \0Uc0\IT
k TdWhfPT TdU_mpU_Td\0U_`lfIulT‚nubX§\0XvmhcIW!T‚e~T ulX§q0X W_m!`l]cIulT€q0Wh`t}
^‚Xv`lWhT6w0x,XvWRT“~T^ qIulTpcI\0Tpk TdWhfPT&`l\I`tmh`liT€q0XvWRcI\?qIœI[ }
mh[6\'][ \5mh`lT\0eIW!X&mhW!¦dUšq Tdc'eITR^cI[ \0U-][ ^ q0XvW!iTnpc0\IT
k TdWhfPT?`l\I`—m!`tidTNq0X W9cI\>qIWh[ mh[6\|oa5cI`TulutTƒ^ ›^ TNqP[6U_}
U_¦Ve~TcI\IT¤][6^ q [5UhX \6m!TR^cI[ \0`la5cIT^ [ `l\0U-`l^€qP[ Whm!X \5mhT
a5cs c0\IT‚k TdWhfPTq0Wh[~e~cI`tmhT‚q XvWcI\‹\I[{Z6X cƒe~T ²TdWdwxy[6cIW
][6^€q0WhTd\0e~W!TLus [6Wh`lk `l\IToeITš]Tm_m!T#e~`—¡¢iW!T\ ]T-e~cp\I[6^pfIW!T
e~T
Td\5mhW!Tšc0\IT#k TdWhfPT#e~T-qIWh[ mh[6\Tmyc0\IT#k TdWhfPT#e~Tš²TWV|
µ
WhTdqIW!T\I[6\0U'\I[vm!WhTŽqPTm!`—mN^ [~e~¦dutT±e~T±eIiz Tdut[6qIqPT^ T\5m
e~TdU-k6TW!f TVU#eIXv\ U-utT¤]X6U-[ c'ulX€q0X W_m!`l]cIulTq0Wh`l^‚Xv`lWhTTdU_m
cI\§\I[{Z6X c9[6ccI\§\c0]utid[ \w
™oTmhmhTo²[ `bU]`°|;us cI\0`—m!iše~Tšul[ \Ik6cITcIWy]X W!X6] m!iW!`lU_mh`ba5cIT,TdU_m
ulX¨ut[6\Ik c0TcIW9es `l\5mhTWYX ]mh`l[ \e~TVU9q0`t[6\0Ue0Xv\0U9u°s X{m!^€[ }
U_q0œI¦W!T
|]X WR]vs TdU_mTulutT a5cI`FeIimhTdWh^ `l\IT€u°s iz6[ ulc~mh`l[ \
λ
e~c’\I[ ^fIπW!T'e~T
wLxWhTd\I[ \ U u°s T“~T^ qIulT es cI\IT k TdWhfPT
π
`t\0`—m!`tidTjq0XvW,c0\‚qIWh[ mh[6\ 5XvqIW!TdU,]Yœ0X6a5cIT
|6U_cIq0q [5U_[6\0U
λπ `t\5mhTdW!X6] m!`t[6\|
a5cIT
q0X W_m!`l]cIutTVU-U_[6`tTd\6mj]WhidTdU-n]Yœ X a5cIT
n
u°s i\0TW!k `lT `t\I`tmh`bXvulTNUhT?W!iq0X W_m!`lU!UhX \5m Td\Cˆ v®‹e~T
Tm
π0
‰ v®pe~T
w
j
c
f
6
[
~
c
j
m
~
e
T
i
Y
m
v
X
P
q
d
T
d
U
|
l
u
p
X
k
T
!
W
P
f
T
]
[
5
\
h
m
l
`
T
5
\
m
±
Ÿ
π
¡
k
¢
e [ \ ]
~
es i\ITdWhk6`tT E0 [
TVUŒmus id\ITW!k `lT
2n k
E
π±
e~c€qIWh`l^‚3Xv`lWhT6w{SX^pcIutmh`lqIul`l]dX{m!n
`t[6k\€e~c€\I[60^pfIW!TšeITšq0`t[6\0U
]TVUhUhT#a6c Xv\0eutTdcIW,i\0TW!k `lTšXvm_mhTd`t\5m
π Uh[ `tm,X cfP[ cImLe~T
E
ul[ \0k cITdcIW!URes `t\5mhTdW!X6] cm!`t[6\wPSTp\I[6^pfIW!T
ln(E0 /Eπ
c)
e~Tg^cI[ ln(n)
\ ULqIW![~e~cI`tm!Uoq0XvWocI\ITk TdWhfPTge~TqIW![vmh[6\Us id]Wh`tm
Nµp± =
2n
3
π
0 /Ec )
ln(E
ln(n)
∝
E0
Ecπ
ln(2n/3)
ln n
∝ (E0 )
α
ST\I[ ^fIW!Tƒe~Tq0XvWhmh`b]c0utTVU9qIW![~e~cI`tmhTVU'nK]Yœ0X6a6c0TN`l\~}
mhTdW!X6] mh`l[ \
X cIk ^ Td\6m!TŽX;z Td]ƒus id\ITW!k `lT RXvc~“id\ITWh}
n
k `lTdUŽT“mhW!›^ TdUŽa5cI`\I[ c0U±`t\5mhidWhTVUhUhT\5mV|
TVUŒm¨][ ^€}
n
qIW!`lU9T\5mhW!Tƒ‰ ‹Tm
Ž]TNa5cI`][ \ e~cI`tmnŽcI\ITƒiz6[v}
utcImh`l[ \ e~c \0[ ^pf0WhTCe~TG^pcI[6\0U>X;z6Td]Bu°s i\0TW!k `lTCT\
[ c
WhTVU_qPTd]mh`lz Td^€Td\5mdwy™o[6^ ^€T9qIWhi}
(E0 )0.92
(E )0.94
]iVe~T^ ^ T\5md|~c0\I0Tk TW!fPTRqIW![~e~cI`tmhTq0X Wšc0\\I[{Z5Xvc Z
X
q Tdc~my›mhW!TL^ [~e~idut`bU_idTo][ ^ ^ TocI\0ToUhcIq TdWhqP[6Uh`tmh`l[ \pe~A
T
A
k TdWhfPTdUFes i\ITdWhk6`tT
w{STš\I[ ^fIW!Toe~Tš^pcI[6\0UFUs id]Wh`tm
E0 /A
][6^€^ T
‰~w ®6­
NµA± = A1−α Nµp±
«
j`l\0Uh`°| q [6cIWjcI\IT¤k6TW!f T`l\I`tmh`liT&q0X W#cI\ \I[{Z5XvceIT¤²TWV|
utTR\I[ ^fIW!Tge~Tjq0X W_m!`l]cIulTdUidutTV] m!Wh[6^ X k \Iimh`ba5cITdU
|
­
« γ e±
\IT]Yœ Xv\Ik6T‚q0X UXvul[ WYU&a5cITulT9\I[6^pfIW!Te~T9^cI[ \ U&TdU_m
XvcIk6^ T\5mhies c0\‘X ]mhTdcIWš][ ^ qIW!`lULT\5mhW!Tpˆ6| ® ¤Tmˆ | ®pUh`
u°s [ \NqIWhTd\0e
Tm
[6c
WhTVU_qPTd]}
A = 56
n = 200
n = 600
mh`lz Td^€Td\5mdw
Y
A££ ¥¤E£E£
t
¡
¤
™oTdUz{XvulTcIWYUUh[ \5m'T\fP[ \BX ]][ WYe•X;z TV]
utTVU9q0WhiVe~`l]}
mh`l[ \0Ue~TdU9qIW![ k WYXv^ ^ TdU‚eITNU_`l^pc0ulXvmh`l[ \0Ue~TdU9k TW!fPTdU
X{m!^€[5U_q0œIiW!`la5cITVUw5Ÿ uP\ITg‘X c~m#]TqPT\ eIXv\5m-q0X UšX{mhm!X ]Yœ0TW
mhW![ q•es `t^ qP[ Whm!X \0]T
n‹]Tm_mhT
z{X utTdcIW‚\cI^ iW!`la5cIT6|La5cI`
'2
WhTVUŒm!T,cI\ITTVUŒm!`t^‚Xvmh`l[ \mhW!¦dUk W![6U!U_`l¦W!T w \¤T¡PTmd|c0\¤^€[ }
k=
e~¦ulT‚Xvc0U!Uh`yUh`l^€q0utT a5cIT‚]TulcI`,a5cIT \I[ c UX;z6[ \0Ugc~mh`lul`lUhi
e~[ \ ]
][6^€^ T
Š
©-+žª-+ž©
e~c
T“Td^ qIutT
\0T?UhX cIW!X `—me~iV]W!`tW!T§ulXŽ][ ^ qIulT“~`tmhi
e~iz6Tul[ qIqPT^ Td\6mjes cI\IT¤k6TW!f Tœ0X6e~W![ \I`ba5cIT w
,/2§¦‚tB=™t‡.n9Zz{,\.uC#ÄÌ+19›t‡2:46‰3?‡9ZC5wE?B46.uC#Ä
Tì®Ãc"C;Ã>уƒ>Q}:³tƒ>Ñ [email protected] @ §D µ :³%AɃ=€›"ƒ¿®8Bƒî³ƒ>Q}:³%9:ÂÉ}ðƒÆDƒ¿£>?³t
¿®ƒ:C"cÿ9D µ :³ !"ƒ @ ®D|>уƒƒ>?=:ƒ>Ñ8; @ "C @ D’!"ƒƒ>ÑÃ: @ ":±8 @ ÿ!D µ ":ƒ !"ƒ ¶@ µ ÂÉ8 @ :ƒ>Ñ8;'D|:
³8;¿ = D™!"ƒƒ>ÑÃ: @ ÂÉà @¶µ !">Ñ:D µ "ƒ¿®8Bƒî³ï ƒCÂÉBó
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
l _
|½ óWV ¹ 8> @ì@ "C @ 8;=:–!" @ Uƒ>?¿ : @ "ƒ>Ñ8;±D µ :Æ8;8;–DN¾cÀ 19 ‡° VD µ " 2 Y64M€8;: @
Ã8;¿£98By"ñ¿ :³8;>Q}:³£ @ Ã8;¿N"AB³ƒ>Q}:³BóT7³AB:UDUD|8;>?xC @ Uƒ>?AB! @ Ã8;}ÂÉ8 @ :³
ÂÉà @ ñ€8;³ D µ :–DÃ:’D|:%c":›D: ;à D @¶µ =³CƒÂ;"8;>? ¹ >у :AÉcó
°
Ó "ß È~žÊ¢ÊoÔ|ÞScß"Ô³Ê T7UD|>уƒƒ>?=:ƒ>Ñ8;±D|:%³8;¿ =xD™!"ƒƒ>ÑÃ: @ ™ 8;³Ãƒ>Ñ8;›D @ UD|>уC"³Ã ´ @¶µ â D
@ –AɃ=9KD€³D D @ –!"ƒ:%D @ –!"ƒƒ>ÑÃ: @ Kƒ>?¿N">?Bÿ”D @¶µ ³ƒAB>ÑBÿ”D @¶µ "A @ c>?ƒî! @ ó C @ ¿® @ "C @ D @ NAɃ=€xƒƒ>?³Ã>?! @ ¿® D¢'": â D|>€:³ƒ>Ñ8;³¿ : @ ƒ>? @ ´}:³x:=>Ñ @ !"ƒƒ>ÑÃ: @ ¢ÃCî!"ƒAÉ£>?}C"AB>ÑC"ÂÉà @ <Ãî!"¿£tÃ8;: @ 8;¿ =>ѧD¢³8c¼": â D @¶µ "ƒ¿®8Bƒî³ïBóTì
@ Ã8;³aC" @ :³³8;¿x=: â =9c":³Ã8;:t @ :³aDÂ}>т:³ @ ¢¿x:³8;³cÿ @ 8> @[email protected] "C @ D8;³Ã
D8;¿£>?³™!" @ Ã8;¿£98;ƒ¿®}’D @ UÃ8;¿£€8BC" @ Ã8;¿N"AB³ƒ>Q}:³BóT78;~:³ @ UD|>ÑC"³Ã ´ @¶µ â ":AB¿®Bÿ @ ³8;¿ =x¿®8c¼É±D !"ƒƒ>ÑÃ: @ D|>?¿£>?:³xC">ÑD¿®U„³Ãƒ´
8;³Ãƒ>Ñ8;–Dƒ¿£>?³
D ¿N">ÑïU¿£>?ƒQ> }:³U’€:q
ƒx!"C"¿®ƒƒ>ÑUDñD|> €}™¿N">Ñïc£ó T7’³AB:£½|Wó VN> @[email protected] :³ƒƒ
:§ â ¿£ @ ’D>8 @[email protected] "C @ 8;=:t!" @ ƒ>?¿x: @ "ƒ>Ñ8;tD µ :t8;8;tD µ ³ƒAB>ÑA @ ™´®¾cÀ 19 °ñó T7
D³ƒ>? D™!"ƒƒ>ÑÃ: @ V8;³Ãƒ>Ñ8;–D @ £D|>уC"³Ã 98;: @ <¿ :³8;³cÿ @ î³8;8;³
@ @ Ã8;³ceó T7U€8;³ ƒ>?¿x: @ xD µ :–DÃ:D|:ÆCc":–D ƒ: ¶;à D @¶µ =³CƒÂ;"8;>? ¹ >у
:AÉxƒ }§ƒ: @ k³AB:ND§D|8;>?Bÿì¿®ƒC"Uà @ ">?¿®UGÂ>ÑD³Ã @ Æ @ :³ ABC"³D
³ƒ>?=> @ >? D|:ÆDÃ:™": â ¿x:³8;³có
¹@ :³ƒ>Ñ:V8;³Ãƒ>Ñ8;³8;<:ƒ> @ >ÑC<!" @ UÃ8 @[email protected] "=€8;C"ƒ>Ñ8; :AÉcÿ³8;C"¿£¿®ñ„
_y dbátË}ß)à¢ËÕbkÔ1`kß"Èiç ÅMLN7Ë}Õ
∗
lnd
@ ²V8;³Ãƒ>Ñ8;–D ¼9 » ¶qft¨£„
S(r) = A
∗
r
rs
1+
r
rs
−β
87 @ ™
!"C"¿®ïƒ @ >?= ƒ A ÿ β C" ⠙!" ¶@ µ "A @ ´c>?ƒî! @ @ ²V8;³Ãƒ>Ñ8;–D|>?!"C"=98 @ @ ;8 A @ 8;A›„
S(r) = 10
„
A+B
“
r
r0
”
+C
“
r
r0
½|ó X
rs = 1000¿Æó
”2 «
½|óWY\
8;: @ ’!"C"¿®ïƒ B C €:ÂɃ´ â 8;: @ ">Ñ @ >?=cÿ!]V8;³Ãƒ>Ñ8;[D @ :!"}ƒ>?
D µ >?< V8;ƒ¿N"ƒ>Ñ8;±D|>у€8;>?= @ D³"³ @ <D8;³98;: @¶µ )‚Ò:³ƒ¿®}’D @ UÃ8;:ƒ=€Bó
· @ Ã8;Â>Ñ DñC"€ @ }:³UóV8;ƒ¿x: @ DÃ>?ÂÉ @ ñÃ8;¿£98;ƒ¿®} Óçé/Ü â D™AɃ=€™}:³
Ãî!}:³ Ãc;Ãc;DxD!"ƒƒ>ÑÃ: @ ™Dà ƒ<:>Q}:³BóTk£ƒ>Ñx›Ã8;¿£xDp³:³Ãƒ:!"ƒ>Ñ8;³’D³"³ @ DÂÉ @ 8;9¿®} DAɃ=€™ƒ’>?¿£€8;ƒC"ñ€8;:Dƒ¿£>?³™:³ñ8ÃD|:xDxÃ8;³ƒƒƒ:ƒ>Ñ8;
D µ "! @ ¼ @ £ @ :³Ã>р8Bƒ>?= @ Bó
Ó ß)ȇžãÊaÊoÈiçkäiÅ~ÞKN:` ÅVçkÔ³Ê Tì 8> @ @ 8;AB>?ƒ:³D|>?! @ Ãc"C;Ã>Ñ @ DÂÉ @ 8;€¿®D @ ®AɃ=9UÂÉà @
8>V8;³D:¢D µ "ƒ¿®8Bî³ï<ƒCÂÉCBó/T7q³AB:’½|ó X <:³<ƒ>?¿ : @ "ƒ>Ñ8;tDAɃ=€a>?>?ƒ>Ñ<!"
:%8;8;–D£¾cÀ 19 ‡ °ˆ @ >8 @[email protected] 8;AB>?ƒ:³D|>?! @ ;8Ã>ÑBó
Tì<³8;¿x=¿®8c¼ÉKD!"ƒƒ>ÑÃ: @ ´:³8>V8;³D:<"ƒ¿®8Bƒî³ƒ>Q}:³D8;³€:5ƒ’!"C"¿®
ƒƒ>љ!" @ ² 8;ƒ¿ : @ D¥
f ">Ñ > @[email protected] ; „
N (X) = Nmax
X − X0
Xmax − X0
Xmax −X0
λ
exp
Xmax − X0
λ
½|ó Á\
ó Nmax ƒ @ <³8;¿ =¿N â >?¿ :¿ D<!"ƒƒ>ÑÃ: @ "ƒ>?} @ 8;±:³ @ AɃ=€’
@
@
@
â
Xmax X0 ƒ <8> 8;³D:”"ƒ¿®8Bƒî³ƒ>Q}:³D ¿£>Ñï>?}C;Ã>Ñ8;ìó\T7Â; :ãD|:ñ¿N >?¿x:¿
DDÂÉ @ 8;9¿®}’ÂB"ƒ>Ñ ÂÉà @¶µ ³ƒAB>Ñ @ ñ!"ƒ: D|:%Cc¼É8;–Ã8Bƒ¿£>Q}:³™ƒ>?¿N">?Bó
ÂÉÃ
λ = 70A³ó¸Ã¿
−2
©!ÒYÈ،ÍhçbÍ!،ÒYÈ6áYÒg㪚ÇÉÎ=6ǗÏRÛ6Ø_Ðvà«jÐÏ&ÐË_Ь ­n®¯ Ò!˱°o،ÒYÇÉÎ ÒYÈ]¬ ­E²¯
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
ll
|½ ó X >?¿ : @ "ƒ>Ñ8;®D µ :³aAɃ=€´ 1019 ‡°5D µ " 2 XÀ64ì„)U= @ :£ã8}ÃCî³D|:£Ã8:8;ã
@ ƒC)‚ÒÃ8;>?aD¿ :³8;³cÿN8;:AÉ @ e± §ÂɃ @ î³8;8;³có•Tì<8> @[email protected] 8;AB>?ƒ:³D|>?! @
ƒ>?³D|Q> }:³ ´UD|8;>? ">?³ƒ3> }:³ @ ñ8> 8;³D:’D|:%¿N â >?¿ :¿ÎD DÂÉ @ 8;9¿®} Xmax ó
°
ùŒÁ
³ÿ¾ Ÿ )¤eÿ&*'
Tì ÃñDÂÉ @ 8;9¿®}UD @¶µ >?<V8;ƒ¿N"ƒ>Q}:³N8;} € ƒ¿£>Ñx": â î¼|ƒ>ÑÃ>ѳñD£¿®8}D @ >ÑCxD
î³³8;¿®ï³aD< @ :³N @ :³¢Ã8;¿£ @ â AB±;¸ Ò´ @ ƒ>?¿x: @ "ƒ>Ñ8;§}:¿®ƒ>Q:³Bó @[email protected] Ã>9ƒ¢>?³Ã8;}8;:
!"= @ £€8;: ƒ:³D|>Ñ Ãc"C;Ã>ÑxÃ>Ñ¿® @ £DÂÉ @ 8;9¿®}ñD µ :³£AɃ=€®"ƒ¿®8Bƒî³ƒ>Q}:³Bó
¹ @ :³ƒ>Ñ:U8;ABC"¿£¿®§Dt>?¿ : @ "ƒ>Ñ8;³®8;}®tDÂÉ @ 8;€® µ ":¼"}Nƒ:£D® @ AÉ8;ƒ>?ƒî¿®
:ƒ> @ >Ñ î}¼>Q:³£D !"ƒƒ>ÑÃ: @ cÿìƒ:>?ÂB" @ =98;>?³ @ £>?ÂÉc":GD£Cƒ®³¿®}x â >?AÉ !"
@ ’D|> v ⠀ƒ>ѳÃceó T7’8;ABC"¿£¿®Dñƒ>?¿x: @ "ƒ>Ñ8;³D AɃ=€"ƒ¿®8Bƒî³ƒ>Q:³ @ @ :³
2 X³;¾ 4E · ƒî³8 ãâ ³DD >?¿x: @ "ƒ>Ñ8;ea · ¶ 2 X½54E 8B
:ƒ> @ >Ñ<´ñÂÒ8;:8;} · ¨
¿£>Ñà c¼ >?¿x: @ "ƒ>Ñ8;³
8;
¶ ;Ãc;DK \ó Dƒ>Ñ<Ã8;³ƒ>ÑD Ã8;¿£¿® @ :³
i"= @ < @ :³<DC"> @[email protected] :³ · Uƒ :ƒ> @ >ÑCK;Ã:³ @[email protected] ¿®}£!"x:IABC"³DQ³8;¿ =%D µ â 9ƒ>ѳÃñ @[email protected] ¨
}:³ ÿ
¶ ;Ãc;D 8;: :AÉcó
T왃>?³Ã>?€ D @ Uƒ>?¿x: @ "ƒ>Ñ8;›Ã8;³ƒ>у ´Uƒ:>?Â!;´ñ!; @ ñ8;!"A"ƒ>Ñ8;›D<!"ƒƒ>ÑÃ: @ D³"³
p-air
σ
l “
[email protected]
650
inel
(mb)
_y lIÐìÅMJ‹N ÊoÔ|Þ+Å~È9ç7Õ
600
550
500
450
Mielke et al.
Yodh et al.
Baltrusaitis et al.
Honda et al.
Aglietta et al.
Belov et al.
400
QGSJET 01
QGSJET II
SIBYLL 2.1
350
300
250
10
11
10
12
10
13
inel(exp)
10
14
10
15
10
16
10
17
10
18
10
19
20
10
Elab (eV)
σ inel = 241
½|óWY Ã>Ñ8;b NÃc;ÃN8;C @ §D µ >?C;Ã>Ñ8;I>?³ @ ;ƒƒ>Q}:³®8;8;< ">?ñ¿®ƒ:®!"ñD|>€}
⠀ƒ>ѳòVD µ " 2 XÀ54M„ þɔ 2 V>X~4 ÿ{ @ ¼ µ ¼É 2 VZY 4ÿ 2 VZ5U 4M\ ÿ Éo ³8 2 BV Á64ÿ ” ¹ 2 V\[ 4k
D µ :xÃc @ 8;ƒ>?¿®ïƒî!;D|8;>Q:³:ƒ> @ >Ñ¢!": · T3¶ B2 VZ6R 4 óƒìT kD|>ÑÃ>Ñ8;³ D7¿®8}Dï @ D µ >?C;Ã>Ñ8;
î!;D|8;>Q:³M´f þɔ ¾ ½®">?³ƒ>3}:³ >?=¼ @[email protected] 8;}<’!" @ <Ã8;:ƒ=€›ƒC">?< @ >?›
σ
= 237 ± 2
98;>?}ƒ> @[email protected] có
@¶µ "ƒ¿®8Bƒî³ï™t>?³Ã @ :!"}8;: @ >?}C;Ã>Ñ8;³8;:KDƒ>?ABC"ƒ>Ñ8;³<98B>?= @ cóT µ "ƒ¿®8Bƒî³ïƒ
@[email protected] £
¿ ¿®N¿®8D @ >Ñ£!"x:³® @ "ƒ>Ñ8;bƒ @ ÂB"ƒ>r"= @ >Ñ8;ìÿì¿£€C"ƒ:NxD³>?N
V8;³Ãƒ>Ñ8; D @¶µ @ ƒ>?ƒ:³DB‚
ó T7–³8;¿x=[D›!"ƒƒ>ÑÃ: @ Æ´GƒC">?%C"%Ã8;³ƒ>ÑDC"= @ Bÿa: @ AÉ8;ƒ>?ƒî¿®
D µ Ãî!"ƒ> @[email protected] 8;!"AÉ[ƒt" @ Q> :³%€8;:®D|:>? @ %¿£³NDÆÃc @ Ã: @ „ @ ®!"ƒƒ>ÑÃ: @ §³Æ8;}® @ :³
ƒ:>?Â}>Ñ>?³D|>?Â>ÑD|:³ @[email protected] ¿® Dï @ 8;
}:³ @ :³ƒAB>Ññ!;Cx[D8;:³™D µ :':> @ â Bó : @ ñ:³
oC;Ã>Ñ8;›DÃ<!"ƒƒ>ÑÃ: @ ’ƒ<Ã8;³CƒÂÉ ÂÉà :³™8;=!"=> @ >? D€³D³"’D @ :³ƒAB>ÑBó
TìU>?C;Ã>Ñ8;³§î!;D|8;Q> :³t´–=!;%³ƒAB>ц
o€8;:£:³%³ƒAB>Ñ%D³"³ @ %ÛDK¿N;K>?<
Vƒ>Ñ:%
´ :³ @ }:³£D|ñ> +">?³®Dy
f™ °´ñ8;}£ƒ>?¿ : @ £!"®DU¿®8Dï @ ®)‚Ò:³ƒUƒ: @ £D8;³
⠀ƒ>?¿®C @ a>Ñ:³D @ xî¼|ƒQ> :³ƒ:;Ãà @ C":c<ó Tì8;ABC"¿£¿®
DT3‘³¶ 2 X/V4€(T‘Ã:!ƒ
ƒ>?²
A ¶ Ãc;DK ãƒa;Ã:³ @[email protected] ¿® @ @ :³:ƒ> @ >Ñ< @ < @ :³§;ÃÃ8;DKÂÉà @ ¢D8;³aD|>у€8;>?= @ có
¹ 8;: @ ã>?}C;Ã>Ñ8;³”î!;D|8;Q> }:³´’î!":<³ƒAB>ÑBÿ @¶µ "=³³ÃDD8;³” ⠀ƒ>?¿®C @ Ã8;³D|:>?
´ ⠃C"€8 @ @ 7Ã>Ñ8;³ NÃc;Ããƒ:7 @ :³ƒ>Ñ: 8;D| D”ABC"³D:ã8;:ñ´<)‚Ò:³ƒ @ 7¿®8Dï @ 7€8;:
³D|’Ã8;¿£™DD8;³<´ @ :³=!;’³ƒAB>ÑBó ¹ "ƒ¿£>vÿ®8}Dï @ cÿ @ ƒ8;>Ñ @ :³aÃ8;:C"¿£¿®}
:ƒ> @ >ÑU8; >?=¼ @[email protected] 2 XZX4 -ÿ M´f þ 2 X/Y64´ŒM’:!"Ýo|f @ :³8; ƒƒ>?A[¿®8}D @ >?ƒî þ Ñ x ¹ þ 2 X5Á 4
:! @v¹ "ƒ8;y
–8D @ >?ƒî þ Ñ \ó a¿®8}Dï @ a}<D³8;¿ =:³C<D|> €³ÃaÂÉ8;>Ñ}
DxAɃ=€UD8; @ tDÂÉ @ 8;9¿®}®9:¨
ƒK³ƒ>?= @ ¿®}£¿®8}D|> !Bó ¹ "® â ¿£ @ Bÿ @ UÃ>Ñ8;³
Ï¢ËÕNäiß)Ô³ç:` ËÕ£ä!Ë}ß)à¢ËÕ®Ô|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
Fly´s Eye
HiRes-MIA
HiRes 2004
Yakutsk 2001
Yakutsk 2005
CASA-BLANCA
HEGRA-AIROBICC
SPASE-VULCAN
TUNKA
DICE
2
Xmax (g/cm )
l 
800
700
600
[email protected]
proton
iron
QGSJET 01c
500
QGSJET II
SIBYLL 2.1
400
´
10
14
10
15
10
16
10
K inel
17
10
18
10
19
10
20
Elab (eV)
10
21
σ inel
|½ ó Á°¢"ƒ>r"ƒ>Ñ8;ÆD Xmax cÂÉà @¶µ ³ƒAB>љ¿®ƒ:!"<D|>€}< ⠀ƒ>ѳÃVD µ " 2 XÀ54M\ó
Tì Ã8;:ƒ=€ C @ D|>ÑÃ>Ñ8;³ñDND|>€} ¿®8Dï @ xD µ >?}C;Ã>Ñ8;Qî!;D|8;>Q}:³§98;:
Dƒ>?¿N">?<8;8;³’8;™
: có
°
NÃc;ÃxD µ >?}C;Ã>Ñ8;I>?³ @ ;ƒƒ>Q:³N8;8;< ">?ñÃc @ Ã: @ x!" >?=}¼ @[email protected] 8;} @ :³ ABC"³D}:³Nà @[email protected] 8;=:³<cÂÉÂM´f þÉ,µ „|´ î!":™³ƒAB>ђҾcÀ 20 ‡° @ xD|>€³Ã™€:"ƒ>?³D|™½BÀ¶#Và ³AB:
½|ó X\£ó ‘'Ãî!"AÉ¿®}xD @ §Â; @ : DUãÃ>Ñ8;' NÃc;ÃU>?<p³:³£ƒ: @¶µ @ ƒ>?ƒ:³D£D @ t¿£>Ñï
>?C;Ã>Ñ8;±<!"<Ã8;³ }:³’: @ Xmax @ ƒD|:›DÂÉ @ 8;€¿®D @ ñAɃ=9Bó
T7¥³AB: |½ ó XU¿®8;ƒ @ D|>vD|>ÑÃ>Ñ8;³D Xmax !" @ a¿®8}Dï @ pM´f þ,µ oÂɃ>Ñ8;
¾%N½\U
>?=¼ [email protected] @ ¢
ÿ Ã8;¿£!"t": â D8;³§ ⠀ƒ>?¿®C @ +óETìND|>ÑÃ>Ñ8;³NDAM´f þ,µ ¾Æ
D >?=¼ @[email protected] 8;}Nƒ>?AB>! Ãc"ƒ>?ÂÉ¿®}tD|>v§98;: @ N8;8;³§":< D:³®D¾cÀ 17 ° @ 8;ð:³
M´f þÉ,µ ½›¿ = @ ƒtb;ÃÃ8;D ÂÉà >?=¼ @[email protected] ó(7
T Ƴ8;:ÂÉ @[email protected] tÂɃ>Ñ8;IDM´f þÉ,µþ2 XÀ548D|:>?
D UAɃ=9§¼"§: Xmax @ :³N @ ÂÉÆD µ :³KÂ}>?ABC">?³›DKA³ó¸Ã¿ −2 :³ @ ±ÂɃ>Ñ8; ÃD}Bÿ
ƒ>?Â}> @ AB>r"}ñD8;³ÃN:³§Ã8;¿£€8Bƒ>?ƒ>Ñ8;Q @ :³ @ 8;:DBóDTìxƒ: @ C"xD @¶µ ⠀ƒ>ѳö 2 X[64
¿ = @ }a>?³D|Q> :³i
}:³ @ ¿®8Dï @ ¤
M´f þÉ,µ ÿ ¹ þÉ,µ >?=}¼ @[email protected] }<DaD|>v³Ãi:>
D|:>р8;: @ ã³8;:ÂÉ @[email protected] ÂɃ>Ñ8;³+ÿ"£C"”Ã8;¿£!"ƒ>?= @ cÂÉà @ D8;³”D³"³ @ A"¿£¿®
D µ ³ƒAB>Ñ 5 1014 eV , 5 1016 eV ó Tì >?¿ : @ "ƒ>Ñ8;³x:¿®ƒQ> }:³ 8;} :ƒ> @ >Ñ !" @ î¼|ƒ>ÑÃ>ѳ
98;:DÂÉ @ 8;€ @ a¿®ƒî³8D<D™Ã8;³ƒƒƒ:³Ãƒ>Ñ8;›<D µ >ÑD}ƒ> !Ãc"ƒ>Ñ8;±DaAɃ=€có :>?Â;"} @ a¿®
ƒî³8}D:ƒ> @ >Ñ+ÿ @ ƒ: @ C"a€:ÂÉi ƒ @ :³a8;:t¿®8;>?³a³>?= @ a´ :%Ãî!"AÉ¿®}D¿®8Dï @ D ƒ>?¿x: @ "ƒ>Ñ8;ìó · @ ƒ’D8;³Ãñ ⠃~ ¿®¿®>?¿£€8;ƒC"D ƒ™™Ã8;ƒC">?³D|£Ãƒ>?¿ : @ "ƒ>Ñ8;³
_y lIÐìÅMJ‹N ʕ·|Þ+Å~È9ç7Õ
l O
¿ : @ ƒ >? @ >r" @ £8;=³ƒÂB"ƒ>Ñ8;³£ ⠀ƒ>?¿®C @ §U Dƒ¿£>?!"}ND£8;=³ƒÂB"= @ £î¼|ƒ>Q}:³ @ @ :³<³ƒ>?= @ ": â 8Ã:³î}¼ƒ>Q}:³’´Uƒ:³D|>Ñcó
"³Ãƒ99Ã>?ÂÉBÿ @ (V:ƒ:ãƒ: @ C"”D|:T ¸ ":8;}D8;³ÃD8;¿x=9¢>?¿£98;ƒC"}
98;: @ î}¼Q> :³±D|: C¼É8;³¿®–Ã8Bƒ¿£>Q:³±t9ƒ¿®ƒƒ8;}ÆD µ "¿® @ >Ñ8;%³8;ƒ±Ã8;!">ÑCC"³Ã
D>?C;Ã>Ñ8;³ñî!;D|8;Q> }:³x´Kî!":®³ƒAB>ÑBó 8;³Ã>ÑñD 8;= @ ï¿® @ >Ñx´ @ ki"=> @ >?§D
ƒ>?¿ : @ "ƒ>Ñ8;³cÿ @ ™Ã8 @[email protected] "=98;C"ƒ>Ñ8; :AÉ¢™D8;³Ã<Ã8;³y;Ã=9c":³Ã8;:§D µ €8;ƒ¢´ @ ¿£>ÑC£ @ ;Ã<D µ :³
8}ÃD|:aDÃ8;³ƒƒƒ:³Ãƒ>Ñ8;®D AɃ=9 ƒï 9:£D€³D³"Dãƒ>?¿x: @ "ƒ>Ñ8;³có ƒ¿®ƒî³8Daƒ
⠀8BD³"³ @ ¢Ãî!">?ƒ¢Ã8;³y;ô @ aÃ8;³ƒƒƒ:³Ãƒ>Ñ8;UD|: ƒ€Ãƒ”D µ ³ƒAB>ÑDvC¼É8;³ Ã8Bƒ¿£Q> }:³có
l ¦
Ï¢ËÕNäiß·³ç:` ËÕ£ä!Ë}ß)à¢ËÕ¹·|ÞKJGÈ!ÕSa(b3c}ß"ÅMLNkËÕ
È
Ä
É Á|Àʾ•Ã¥º¨¼
»u¼
º
½¿¾fÀÂÁfÃÅÄÅÆÈǂÉÊÁ5À
Ë
ÉÊÀÂÁÈÁ5À
Ì
ÍÏÎÀÂÁ
Ð ÑÒnÓ]ÔÕÑÖ!ÓY×nØ6ÙÚ#ÛÝÜÝÞ߶à
ß~íîìÕê>á Ù!ÛÝÖ]ߥÖ1ß
õâáRãänÒ/ßnÞnåÙ*Ü æ!ەÞ>ßtçÛ=ßÞÞ>ßtèŠÓ!é]ßnÞ¥Ö]Ñ܏ß×/܏ßêxßҊé1ßÞ/ä>ßnÒ¶Ù*ÜÝÔ¹æCҏڇØ6ÑnÞnÛ=ëӑßnÒìvÚ6Ù!ÞÜÝÛGÞ¥Ö1ß
1018
Ö1ßnÓ*ïd܏ß>×nØ1ð~Û=ëÓ~ßÒàWÓ]ðYÞÑnÒ/ßÙ!ÓfÖ1ßvÒÓIÞGñ/Ù×ß×>æ*ԗڑæCÒ/ÑÖ1ßòóôôdÖ1Ñ܏ß>×/܏ßnÓIÞÒì%ߕõŠßÜtöÊ×nØ6ßÞßnð~÷Cæ!å¶Þ>яÚ6Ù!ÞÜÝÛhÒ
ÒÓIÞ¹Ó]ðußÕÒÓIÞGñ/Ù×ß:Ö]ߛø]ôôô|÷EÔ 2 Ñ×nØIÙ*ð‡ÜÝÛGê•êFæ!ð~ð_ßdêxßÒùÚ6Ù!ÞÜÝÛ=×Ó]êxßÒ:Ö1ßdêúÙ:é]ßnÞ/äß:Ù!ÓwÒ/æ!ê;ûùßÜ{üEýþ܏ÑnêxßÒ×>ænڑßÒ
ÒÛÝÜÝӑÑҗÙ!Ó6Ü æ!ÓIÞ{Ö!ÓÕÞÑÒAßÙ!ӛÞß>×ӑßÛGê•êFßð‡Ü±êúٟê„Ó]Ô¥Û ÿnÞßùÖ]ß Ó‘æ!ÞßnÒ×ßnð_×ßâÑnÔ¥ÛGÒ/ßÚ6Ù!ޗêFßÒÔ¹æ!êxÑ×ÓIêxßÒùÖ1ß{Ö!Û=Ù Eæ*܏ß
ßï×ÛÝ܏Ñ>ßÒڑÙ!ÞêxßÊڑÙCÒÒ/ÙAé1ßùÖ1ßÒÊڑÙ!ÞnÜÝÛ=×ÓIêFßnҊÖ1ߗêúÙùé1ßnÞAä>ß væ!Ó]ҊÖ1Ñ×/ÞەÞ'æ*ð‘ÒêFßnÒÊÚ#Þەð#×Û Ú6Ù!êxßҊ×>Ù*Þ'Ù×A܏ÑnÞÛGÒnÜÝÛ=ëÓ~ßÒ
Ö!Ó|Ö1Ñ܏ß×A܏ßnÓ]ÞÕß܋êúÙvÚ#ÞæA×Ñ>Ö!Ó]Þ>ß¹Ö]ßvÞ>ß×>æ*ð‘ÒnÜÝÞnÓ6×AÜÝÛÝæ!ð Ö1ßҟÑnåCÑnðußnÔÕßnð‡Ü•Ò
!"#%$'&()*
+-,/.
02143 576"8
)9;:<571:>[email protected]?A6B5#
C D'EGF<H#[email protected]@R%SLK<S'THVU WLNXR#NZYGSL[GNGK\YGD]R#UGU SD]^`_aY R#U ]\b µ D]>[email protected]<TDLFBbgD'h ¼ ]>K<iLF<D
YGD]fF/R ¼ HVU ]'TH`]>hdgj*N D]kY µ NGbeK>F/R2l)R#NGK<DmSLU DLF>P`dgD`noCD]kYGHVUGU SD]pRVTTLNGh"NGbgSD]BIJN ]<jN µ R#bgHVF<]']<DLh"[GbqR#dgDLU*K
deU YdgjN DLFrb µ Dtsdg]>K<DLU TDAYGDuF/R ¼ HVU ]TH`]>hdgjN D]vEcHVNGw`R#UKR#K>K<DLdeU YF<DxYGD]rSLU DLF>P`dgD]vDtsK>F<yLhfD]azb µ DtszEcSLF>dgDLU TD
{;b ¼ µ ]o| ¼ DOR ¼ R#UKvhfyLhfDOHV[ ]/DLF>wQS}NGU'SLwQSLU DLhfDLU*KY µ SLU DLF>P`dgDOD]>K>dehfSD ´"~ n ½ _ 20 D€ƒ‚ „^†n‡rDLEDLU Y R#U*Ka
×
bgDˆU HVhB[GF<D%YGD%F/R ¼ HVU ]pTH`]>hdgj*N D]'Y µ NGbeK>F/R‰l)R#NGK<D!SLU DLF>P`dgD%HV[ ]<DLF>wQS]'F<D]>K/R#deKŠ[DR#N THVNGE‹K>F<HVEŒŽR#de[GbgD
EHVNGF7EcHVNGwQHVdeF4SLK>N YdgDLFob µ HVF>deP`deU DuDLK7bqRAU)R#K>NGF<D}YNkF/R ¼ HVUGU DLhfDLUKTH`]>hdgj*N D}Y R#U ]TDLK>K<D}PR#hhfDuY µ SLU DLFJ
P`dgD`n C‘RTHVU TDLEGK>dgHVU!Y µ NGUmP`dePR#U*K<D]<jN DBYGSLK<DTLK<DLNGF}Œ’NGKuR#bgHVF<]}bqR#U WSD`aTDLbeNGd“†TLdYGDLw`R#deK}NGK>debedg]<DLF}bgD]OYGDLNs
K<DT<lGUGdgj*N D]BYGDkhfD]>NGF<D]\YGD]-PQDLF>[cD]\R#K>hfH`]>EGl SLF>dgj*N D]\THVUGUN D] ´ b µ SLEcH*j*N Dˆ”‘bqRŠ• N HVF<D]<TDLU TDŠDLK"bgD]
F<S]<DR#NsˆYGD-YGSLK<DTLK<DLNGF<]AYGD-]>NGFJŒRVTD`n CD"EGF<H#IJDLK–MuNGPQDLFuD]<KORVTLK>N DLbebgDLhfDLUKANGU D-THVbebqR#[cHVF/R#K>dgHVU%deU*K<DLF>U)R
K>dgHVU)R#bgDF<SLNGUGdg]<]/R#U*K4EGbeN ]oYGD ~ _`_AEGl ¼ ]>dgTLdgDLU ]rDLKdeUGPQSLUGdgDLNGF<]rYGDE)R#F>KbgDhfHVU YGDOY R#U ]NGUkDt—˜HVF>K4THVUVI>HVdeUK
EHVNGF}YGSLK<DLF>hdeU DLFObgDA• Ns˜abqR\U)R#K>NGF<D"DLKb µ HVF>deP`deU D"YGD]}F/R ¼ HVU ]uTH`]<hdgjN D]OY µ NGbeK>F/[email protected])R#NGK<D-SLU DLF>P`dgD`n
š ¹ Mx
™OUkYGD]4HV[I>DTLK>d“Œ’]4EGF>deU TLdeE)R#Ns'YGDb µ›š [ ]<DLF>w`R#K<HVdeF<D ¹ dgDLF>F<DuMONGPQDLF–œ˜j*N Db µ HVU'U HVhhfDLF/RBR#N ]<]<d D]>KAYGD"hfD]>NGF<DLF-RwQDTBEGF<STLdg]<dgHVU!bgD\]>EcDTLK>F<DBY µ SLU DLF>P`dgDYGD]–F/R ¼ HVU ]xTH`]>hdgjN D]AYGD 19 D€žIJN ]<j*N µ R#Ns
10
SLU DLF>P`dgD]oDtsK>F<yLhfD]nzCD]rYGHVUGU SD]rDtsEcSLF>dehfDLUK/R#bgD]rF<STDLUK<D]v]<DLhB[GbgDLUKyLK>F<DODLUŠYGS]/RVTTHVF<YpR#N']>NVI>DLKoYGD
b µ Dtszdg]<K<DLU TDOY µ NGU DATHVNGEGNGF<D ´ l)R#NGK<D–SLU DLF>P`dgDAY R#U ]rbgDA]>EcDTLK>F<DB”zŸudeF<D]AazjNGdNGK>debedg]<DObqRBK<DT<lGUGdgj*N D–YGD–bqR
• N HVF<D]<TDLU TD`aHV[ ]<DLF>[email protected] [email protected]/lNGK<D[GF>NGK/R#bgDfYN!• NsX‚ Y _oR#bgHVF<]xjN [email protected]¡zM¢aj*NGd£NGK>debedg]<[email protected]¤F<S]<DR#N
š ¹ M¨EcDLF>hfDLK>K>F/R
YGDuYGSLK<DTLK<DLNGF<]R#N']/HVba*R¥fF>hfD–HV[ ]<DLF>wQDLFrYGD]rSLwQSLU DLhfDLUK<]}R#NŠYGDLb ´ YGD 20 D†€ƒ‚§¦†nC µ›
10
YGDOTHVhEGF<DLU YF<D–b µ HVF>deP`deU DAYGD–TDOYGS]/RVTTHVF<YpDLUpSLK/R#[Gbedg]<]/R#U*KrNGU DOEGF<HzTSYNGF<DOY µ D]>K>dehkR#K>dgHVUˆYGDOb µ SLU DLF>P`dgD
NGK>debedg]/R#UK–R#NˆhkRszdehBNGh©bgD]deUŒŽHVF>hkR#K>dgHVU ]uYGD]uYGDLNspK ¼ EcD]}YGDxYGSLK<DTLK<DLNGF<]n
C‘RŠYGSLK<DLF>hdeU)R#K>dgHVU‰YNª]>EcDTLK>F<[email protected] µ SLU DLF>P`dgDYGD]AF/R ¼ HVU ]-TH`]>hdgj*N D] ´ NGbeK>F/RŠl)R#NGK<DSLU DLF>P`dgDBF<DLK>dgDLU*K
]<HVNGwQDLUK'b µ R#K>K<DLUK>dgHVU«YGD¬bqRªTHVhhBNGU)R#NGK<S%]<TLdgDLU*K>d“­)jN D`aoTDLEDLU Y R#U*KkbgD¬EcHVK<DLUK>dgDLbAYGD¬b µ›š [ ]<DLF>w`R#K<HVdeF<D
¹ dgDLF>F<D"MONGPQDLFOD]<K}[GdgDLUmEGbeN ]uwVRV]>K<D"DLKuEcDLF>hfDLK>K>F/RfY µ SLK>N YdgDLF–K<HVNGK<D-NGU D-YdewQDLF<]>deK<SBYGD"YGHVhkR#deU D]–YGD-bqR
EGl ¼ ]>dgj*N D`aYGDLEGNGdg]-bgD]"deUK<DLF/RVTLK>dgHVU ]BŒ’HVU Y R#hfDLU*K/R#bgD]BYGD]"E)R#F>K>dgTLNGbgD]}IJN ]<j*N µ R#NsªEGF<H*TD]<]<N ]"RV]>K>F<HVEGl ¼ 
]>dgj*N D]n
+-,¯®
°Š5±9c²'}:J8´³B5# :Y5#¨5;8xµŠ¶£·:/¸¹571ºu6-8
C D'EGF<H#IJDLKMONGPQDLFf»mSLK<SŠTHVU WLNZEHVNGFSLK>N [email protected]]\F/R¼QHVU ]TH`]>hdgj*N D]@Y˜½ NGbeK>F/Rml)R#NGK<DpSLU DLF>P`dgD¬DLU
EGF<HwQDLU)R#U TD}YGDrK<HVNGK<DvbqRA]>EGl iLF<DvTSLbgD]>K<Dx”#bqRATHVU ]>K>F>N TLK>dgHVUfYNf]<deK<Dr]<N Y\DLUMOF>PQDLUK>deU D}]/DLF/ROYGHVU T]<NGdew*dgD
E)R#FxTDLbebgDfYNª]>deK<[email protected] HVF<Yª]>deK>N [email protected]Y R#U ]xbgDk‡rHVbgHVF/RVYGHGnC½¾HV[ ]/DLF>wVR#K>dgHVU‰YGD]xYGDLNs!l SLhdg]>EGl iLF<D]-EcDLF>hfDLK>K>F/R
¿˜Àg¿2%Á* & ` m&#†GÃQ
˜Ä
˜Y ½¾SLK>N YdgDLF}YGD]rHV[I>DLK<]RV]>K>F<HVEGl¼]>dgj*N D]THVhEGbgiLK<DLhfDLUK<]OYd“—cSLF<DLU*K<]aGTHVhhfD–E)R#FvDtsGDLhEGbgD–bgDATDLUK>F<DxYGD
bqRx€HVdgDuCRVTLK<SDOwzdg]>de[GbgDOYGDLEGNGdg]4b½ l SLhdg]>EGl iLF<DA]>N Y'HVNkb½§R#hkRV]rYGDuPR#bqRsdgD]rYGD}€–deF>PQHGa[cDR#N THVNGE'hdgDLNs
w*dg]>de[GbgDuYGDLEGNGdg]4b½ l SLhdg]>EGl iLF<D–U HVF<Y!œ’THVhEGK<DuK<DLU*NŠYGDubqRxbqR#K>deK>N YGDuYNŠ]>deK<DuMONGPQDLFo]<N Y)tnC‘R"THVNGwQDLF>K>NGF<D
K<HVK/R#bgDYN"TLdgDLb]<DLF/RuR#N ]<]>dQY˜½ NGU-deU*K<SLF<yLK‘hkRI>DLNGFEHVNGF‘b½¾SLK>N YGD4YGD]‘R#UGdg]<HVK>F<HVEGdgD]YGD]‘YdeF<DTLK>dgHVU ];Y˜½§R#F>F>dewQSD
YGD]F/R¼QHVU ]OTH`]>hdgjN D]a)R#NsŠP`F/R#U YGD]OTHVhhfD"R#NspEDLK>deK<D]OST<l DLbebgD]n
‡ol)RVTLNGU‰YGD]"]<deK<D]-YGHVdeK-EcDLF>hfDLK>K>F<DfbqRpYGSLK<DTLK>dgHVUÅY˜½ NGUÅU HVh"[GF<D']>N¥k]/R#UKBY˜½¾SLwQiLU DLhfDLUK<]»ŠNGbeK>F/R
l)R#NGK<D\SLU DLF>P`dgDY R#U ]–bqR'wzdeUGP`K/R#deU D\Y˜½§R#UGU SDYGDBŒŽHVU TLK>dgHVUGU DLhfDLUK-dehE)R#F>K>dgDR#N¤EGF<H#I>DLKnCRkK>F<i]–ŒR#de[GbgD
wVR#bgDLNGFYNZ• NsÆYGD]@F/R¼QHVU ]kTH`]>hdgj*N D]f»!TD]SLU DLF>P`dgD]fDtsK>F<yLhfD]a7YGDpb½¾HVF<YF<D¬Y˜½ NGU DpE)R#F>K>dgTLNGbgD¬E)R#F
Ç debgHVhfiLK>F<DrTR#F>F<S4DLKE)R#F;]>dgiTLbgDr» 20 D†€\adehEH`]/D4YGHVU T4YGDrTHVU ]>K>F>NGdeF<DoNGU DdehEcH`]/R#UK<Dr]>NGFJŒRVTD4Dt—˜DTLK>dewQD
10
YGDYGSLK<DTLK>dgHVUnCD]"TR#F/RVTLK<SLF>dg]>K>dgjN D]\YGD]-]>deK<D]BHVÈ2]<HVUKpœ’HVȉ]<DLF<HVU*K-EHVNGF"bgD]>deK<DU HVF<Y)–deU ]>K/R#bebgS]"bgD]
YGSLK<DTLK<DLNGF<]4]<HVU*K4YdgTLK<SD]E)R#F7bqRAEGl*¼z]<dgjN Dj*N Db½¾HVUwQDLNGK4SLK>N YdgDLFrDLK7E)R#FbgD]THVU YdeK>dgHVU ]K<DT/lGUGdgjN D]rYGD
F<SR#bedg]/R#K>dgHVUfYNEGF<H#IJDLKn`É;HVNGFb½ deU ]>K/R#bebqR#K>dgHVU'YNF<S]<DR#NfYGDvYGSLK<DTLK<DLNGF<]4YGD]>NGFJŒRVTD`a#bgDvEGF<HV[GbgiLhfDF<DLw*dgDLU*K
»K>F<HVNGwQDLFANGU DBSLK<DLU YN [email protected]˜½§R#N¤hfHVdeU ] ~ _`_`_ Ç h 2 a)]<deK>N SD\»kNGU D\R#beK>deK>N YGD\bqRkEGbeN ]ATHVU ]>K/R#UK<D\EcH`]<]>de[GbgD`a
THVhEGF>dg]<DpDLUK>F<D _`_`_mDLK Y _`_#hˆa7R­ UZYGDkEcHVNGwQHVdeFHV[ ]<DLF>[email protected]"NGhÊYGDpYGSLwQDLbgHVEGEDLhfDLU*KfYGD]


PQDLF>[D]uR#K>hfH`]<EGl SLF>dgjN D]n
C‘R¤bqR#K>deK>N YGD%YNË]>deK<DˆYGHVdeKkyLK>F<DˆTHVhEGF>dg]/DˆDLUK>F<D ~ZY DLKf„_¤YGDLP`F<S]fEcHVNGFfEcDLF>hfDLK>K>F<DˆNGU D¬[cHVUGU D
THVNGwQDLF>K>NGF<D"YN%TLdgDLba)DLKub½¾DLU ]<HVbgDLdebebgDLhfDLU*K–YGHVdeK–yLK>F<D"]>N¥k]/R#U*K}EcHVNGF–R#bedehfDLUK<DLF–bgD]uTLNGwQD]OYGHVUKObgDxE)R#U
U DR#Nˆ]<HVbqR#deF<D–Œ’HVNGF>[email protected]]/DLNGbgDAF<D]<]<HVNGF<TDxSLU DLF>PQSLK>dgj*N D`n MÌTDLbqR\debcŒR#NGKuRIJHVNGK<DLFbgD]THVU*K>F/R#deUK<D]ObedgSD]u»
b½¾HV[ ]<DLF>wVR#K>dgHVU!YGDxbqR\• N HVF<D]<TDLU TD'”GNGU%TLdgDLb‘K>F/R#U ]>E)R#F<DLUKADLK}EcDLNˆUN)R#PQDLNsmTHVU*K<DLU)R#UK–EDLNmY˜½§RVSLF<H`]/HVbg]
DLU!]>N ]<EDLU ]>dgHVU-ÍNGU [email protected]#[ ]<DLU TDBYGD\EcHVbebeNGK>dgHVU%beNGhdeU DLN ]/DpœbgHVdeUªYGD]–wzdebebgD]Ν}R#deU ]>d£j*N½ NGU!K<DLhE ]–bgDBEGbeN ]
]<DT}DLKo]<K/R#[GbgD}EcH`]<]>de[GbgD`nQÉ£R#F>hdbgD]4K>F<HVdg]o]<deK<D]F<SLEcHVU Y R#UK»-TD]4TLF>deK<iLF<D]rY R#U ]ob½ l SLhdg]>EGl iLF<DO]<N Y˜a`b½¾DLU
PR#PQDLhfDLUKAYNˆPQHVNGwQDLF>U DLhfDLUKADLKOYGD]EGl*¼z]<dgTLdgDLU ]–R#F>PQDLU*K>deU ]–R\ŒR#deK}EcDLU T<l DLFObqR[)R#bqR#U TD"DLUˆŒŽRwQDLNGFOYN
]>deK<DYGDÏ!R#bqR#F>P`Ð DODLUfMuF>PQDLU*K>deU D`nCDu]>deK<D]>N YfYGDvb½ š [ ]<DLF>w`R#K<HVdeF<DuɏdgDLF>F<D}MuNGPQDLF4¼\D]>KDLUfTHVU ]<K>F>N TLK>dgHVU
»\b½ l DLNGF<D"RVTLK>N DLbebgD`a)DLK}YGDLwzF/R#deKOR#K>K<DLdeU YF<D"]Î[email protected]]>NGFJŒŽRVTD–­ U)R#bgD"YGD ~ _`_`_ Ç h 2 »\bqRB­ UmYGDAb½§R#UGU SDBÑ`_`_ [ n
+-,†+
°p5Ò·c¶#)57º}6´³\5 )6B·Óºu95
C R!hkRI>DLNGF<D'E)R#F>K>dgDmYGD]SLwQiLU DLhfDLUK<]kYGSLK<DTLK<S]fE)R#F\b½ š [ ]/DLF>wVR#K<HVdeF<DˆÉdgDLF>F<D¬MuNGPQDLFEGF<Hw*dgDLUGU DLU*K
‘
YN‰]<DLNGb4F<S]/DR#NÅYGDk]<NGFJŒŽRVTD`a˜bgD]BSLwQiLU DLhfDLUK<]@l*¼*[GF>dgYGD]BF<DLEGF<S]<DLUK/R#U*[email protected] DŒ’F/RVTLK>dgHVUÅYGDfb½¾HVF<YF<D'Y˜½ NGU
YdeÔdgiLhfDxYNŠU HVhB[GF<DxK<HVK/R#bn ‡–½¾D]>K}YGHVU T–bgDAYGSLK<DTLK<DLNGFuYGDA]>NGFJŒŽRVTDœ’j*N D–b½¾HVU¬U HVhhfDLF/RR#N ]<]>d‘¡zÕÌEcHVNGF
¡*NGFJŒŽRVTDÕADLK<DTLK<HVFΝxj*NGd7w`RŠEDLF>hfDLK>K>F<DfY˜½¾HV[GK<DLUGdeFBNGU D]>K/R#K>dg]>K>dgj*N DdehEcHVF>K/R#UK<DfP`F/ÖVTD'»p]<HVU2SLU HVF>hfD
]>NGFJŒŽRVTDxYGD"THVbebgDTLK>dgHVUn
_
!"#%$'&()*
C Df¡zÕ×D]<K–THVU ]>K>deK>N [email protected]˜½ NGU¤F<S]<DR#NªYGD _`_']<K/R#K>dgHVU ]AdeU YGSLEDLU Y R#U*K<D]xF<SLE)R#F>K>dgD]x]>NGF–NGU D\]<NGFJŒŽRVTD
Ø
Y˜½¾DLUwzdeF<HVU ~ _`_`_ Ç h 2 abqR\hkR#debebgD-SLbgSLhfDLUK/R#deF<D\YN¬F<S]<DR#NmSLK/R#U*K}NGUˆK>F>dqR#UGP`bgD-Sj*NGdebqR#K<SLF/R#bYGD a Y Ç hÙYGD

TÚVK<S`nC‘RBYdg]>K/R#U TD–DLUK>F<D–YGDLNsk]>K/R#K>dgHVU ]R-SLK<SOT<l HVdg]>dgD–YGD}hkR#UGdgiLF<Dx»-TDOjN½ NGU DOPQDLF>[cD}deUGdeK>dgSDOE)R#F4NGU D
E)R#F>K>dgTLNGbgDuEGF>dehkR#deF<DAY˜½¾SLU DLF>P`dgDA]>NGEcSLF>dgDLNGF<DOHVNpSLPR#bgDA» |D†€ÌK<HVN T<l DxR#N'hfHVdeU ]oK>F<HVdg]vTLNGwQD]a*THVU YdeK>dgHVU

F<DjNGdg]/DfEcHVNGFBEcHVNGwQHVdeFBF<DTHVU ]<K>F>NGdeF<D'b½¾SLwQiLU DLhfDLU*Kn7C‘R%]>NGFJŒŽRVTD'YGDkb½¾DLhEGF<DLdeU*K<D¬R#NÆ]<HVbrY˜½ NGU DŠPQDLF>[D
R#NGP`hfDLUK/R#U*K"RwQDTBb½¾SLU DLF>P`dgD`aHVU%EDLNGKOF<DLhkR#F<jN DLFxj*N D"T/l)R#UGPQDLF–bqRkYdg]>K/R#U TD\DLUK>F<DBbgD]OTLNGwQD]–F<DLwzdgDLUK
YGHVU T¬»mYGSTR#[email protected]½¾SLU DLF>P`dgDˆ»mE)R#F>K>[email protected]ŠbqRVj*N DLbebgDkb½¾Dt¥kTRVTLdeK<S¬YGDpYGSLK<DTLK>dgHVUZw`R#NGK\NGUn£CD]@YGSLK/R#debg]YN
TR#bgTLNGb‘YGDAb½§RVTTDLEGK/R#U TD\YN¬F<S]<DR#NmYGDx]>NGFJŒRVTD-]<HVU*KuDtszEcH`]<S]Y R#U ]ubgDxT<l)R#EGdeK>F<D-„ n
C DBEGF>deU TLdeED\YGD-bqR'YGSLK<DTLK>dgHVUªYGD]uF/R¼QHVU ]ATH`]>hdgjN D]xRwQDT"bgDBF<S]<DR#N%YGD\]>NGFJŒRVTD"D]>K–bgD"]<NGdewVR#U*K\”
bgHVF<]<j*N½ NGU DBPQDLF>[[email protected]#K>hfH`]>EGl SLF>dgjN [email protected]<HVhB[[email protected]]>NGFAbgDBF<S]<DR#Nª[email protected]]>NGFJŒŽRVTD`aNGU DBŒF/RVTLK>dgHVUªYGD]–hdebebedqR#F<YG]-YGD
E)R#F>K>dgTLNGbgD]-j*NGd7bqR'THVhEcH`]<DLU*K-K>F/RwQDLF<]<DLU*KBj*N DLbgjN D]xTLNGwQD]B»pDt—cDLKB‡ol DLF<DLU Ç Hwa˜¼!YGSLEH`]ÎR#UK-K<HVNGK-HVN
E)R#F>K>dgDkYGDbgDLNGF\SLU DLF>P`dgD`nC‘D]"]>K/R#K>dgHVU ]\jNGdoHVUKBTHVbebgDTLK<SŠ]>N¥k]/R#hhfDLU*K\YGDk]>deP`U)R#boDLUÅdeUŒ’HVF>hfDLU*KBE)R#F
HVU YGD];F/RVYdgH`]bgDrTDLUK>F<DY˜½§RVTj*NGdg]>deK>dgHVUcÛ7jNGdzw`RuYGSLK<DLF>hdeU DLF7DLUR#EGEGbedgjN)R#U*[email protected]#bePQHVF>deK>lGhfDYGDr]<SLbgDTLK>dgHVU
jN D4U HVN ]£YGSLK/R#debebgDLF<HVU ]£E)R#F‘bqRO]>NGdeK<Do]<db½¾SLwQSLU DLhfDLUKD]<KTHVU ]<DLF>wQSrHVNBU HVUn#ÕxR#U ];bgD4TRV]£HVNBb½¾SLwQSLU DLhfDLU*K
D]>Ku]<SLbgDTLK>dgHVUGU S`a)bgDB‡rÕ}Mx¡pdeU*K<DLF>F<HVPQD\R#bgHVF<]ubgD]uTLNGwQD]OEHVNGFuj*N½¾DLbebgD]}K>F/R#U ]<hfDLK>K<DLUKAbgDLNGFuYGHVUGU SD]–jNGd
]<DLF<HVUK"R#bgHVF<]"]>K<H*T Ç SD]n˜[email protected]<S]<DR#N2THVhEGbgDLKBYGD _`_p]>K/R#K>dgHVU ]-YGDLwzF/R#deK"R#deU ]>dYGSLK<DTLK<DLF"DLUwzdeF<HVU „_`_
Ø

SLwQSLU DLhfDLUK<]OE)R#F£IJHVNGFn
ÜÞÝßJàáŽâ†ãÎäå7ã/á†ãvæ˜ç>èéê›ëŽê¾áê§ìtàxíîë’áßJïAðìtéxÝå;æ;í}ñ`ìÎéâ‘ä§ßJëê›àáê§ïßJë
¿˜Àg¿2%Á* & ` m&#†GÃQ
)¿
!ò ` móô õG(ö‘
CD]YGSLK<DTLK<DLNGF<]}YGDAb½ š [ ]/DLF>wVR#K<HVdeF<DxÉ7dgDLF>F<DxMONGPQDLF]<HVUK}YGS]/HVF>hkR#dg]vw*dg]>de[GbgD]YGDLEGNGdg]vbgD–[cHVF<YpYGD
bqRBF<HVNGK<D`abgHVF<]<jN D–b½¾HVUmR#EGEGF<HzT<l DxYGD–bqRBw*debebgDAYGDxϤR#bqR#F>P`Ð Dfœ’DLUwzdeF<HVU ¦@_`_`_"l)R#[GdeK/R#U*K<]Νta)R#Np[cDR#N
hdebedgDLN¬YGD}bqRŠ÷#ø§É;R#hE)[email protected]#F>debebqRG÷V½ea*DLUŠF<StŒŽSLF<DLU TDA»-bqR"THVNGbgDLNGF;I<R#NGU Du YGDObqR-wQSLPQSLK/R#K>dgHVUpbgHzTR#bgD`n*C‘D
]>deK<DuDtsEcSLF>dehfDLUK/R#b˜D]>KoTHVhEGF>dg]vDLU*K>F<D} ~ZY ◦ DLK ~ZY n ~ ◦ YGDObqR#K>deK>N YGD`a»xNGU DAR#beK>deK>N YGDOTHVhEGF>dg]<D–DLUK>F<D
_`_DLK „_`[email protected]>F<D]n
~

C‘R¤­ P`NGF<D%TLd“†YGD]<]<HVN ]kdeU YdgjN Dmb½¾SLK/R#K'YGDˆYGSLEGbgHVdgDLhfDLU*KŠYGDmb½ š [ ]<DLF>wVR#K<HVdeF<D%É7dgDLF>F<DmMONGPQDLFŠR#N‹Ñ`^
ù`NGdeU'Ñ`_`_ ” Y ]>K/R#K>dgHVU ]]<HVU*KdeU ]>K/R#bebgSD]oY R#U ]7bqRAE)R#hE)RaQYGHVU*K ~ZY F<DLhEGbedgD]oY˜½¾DR#NkDLKo^ ¦xj*NGd
EH`]/]<iYGDLUØ KxR#N` ]<]>d;Ø bgDLNGF–SLbgDTLK>F<HVUGdgj*N D`ncCD]AjN)R#K>F<D\EcDLK>deK<]u[)R#K>dehfDLU*K<`]x U HVdeF<]A]>deK>N S]–]>NGFObgDBEcHVNGØ F>K<HVNGF
YN2F<S]<DR#N2THVF>F<D]>EcHVU [email protected]#Ns![)R#K>dehfDLU*K<]BYGD\• N HVF<D]/TDLU TD`n‘CD]xK>F/R#dg]BDLUªEHVdeU*K>debebgS]"HVF>deP`deU)R#deF<D]
YGD][)R#K>dehfDLUK<]–YGDA• N HVF<D]<TDLU TD-deU YdgjN DLU*KubgD-T/l)R#hEmYGDxwzN DxYGD-T<l)RVTLNGUmYGD]O]>d“s'K<SLbgD]/THVED]n
Gú
!"#%$'&()*
û£üòû£üÞý
þªÿ 4ÿ‘ÿ ÿ}ÿ!"$#4ÿQÿ%&'%
C D];TLNGwQD]7»}Dt—˜DLK)(T<l DLF<DLU Ç [email protected]]<HVU*KbgD];SLbgSLhfDLUK<]7YGDo[)RV]<D4YN\F<S]<DR#[email protected]]>NGFJŒRVTDrYGD4b½ š [ ]<DLF>w`R#K<HVdeF<D
É7dgDLF>F<DªMONGPQDLFn}CD¤EGF>deU TLdeEcDªYGD¤bqRZYGSLK<DTLK>dgHVUÌNGK>debedg]<D¤b½¾Dt—˜DLK*(‘T/l DLF<DLU Ç HwžEcHVNGF¬hfD]>NGF<DLFˆb½¾SLU DLF>P`dgD
YGSLEH`]/SDAE)R#FbgD]E)R#F>K>dgTLNGbgD]}YGDAbqR\PQDLF>[cDf”zbgHVF<]<jN½ NGU D-E)R#F>K>dgTLNGbgD-T<l)R#F>PQSD-]/DxYGSLEGbqRVTD-Y R#U ]NGU¬hdebedgDLN
hkR#K<SLF>dgDLb–RwQDTmNGU Dmw*deK<D]<]/Dˆ]>NGEcSLF>dgDLNGF<D%»ªTDLbebgD%YGDˆbqR¤beNGhdgiLF<D!YGDˆTDˆhdebedgDLNaoNGU D‰÷LHVU YGD!YGDˆT<l HzT
beNGhdeU DLN ]<DQ÷ŠD]>[email protected]]<DŠE)R#FBbqR%E)R#F>K>dgTLNGbgDŠbgDŠbgHVUGP!YGDp]/RmK>F/RIJDTLK<HVdeF<D`n‡rDŠF/R¼QHVUGU DLhfDLU*KkD]>[email protected][ ]<DLF>wQS
E)R#FK>F<HVdg]oEGl HVK<HVh"NGbeK>deEGbedgTR#K<DLNGF<]j*NGd)THVUwQDLF>K>dg]<]/DLUKvbgD]EGl HVK<HVU +
] (‘T<l DLF<DLU Ç Hw'DLU'NGUk]>deP`U)R#bSLbgDTLK>F>dgj*N D
DtszEGbgHVdeK/R#[GbgD`n
,.-0/.1 ~ n
32
ÕAD]<TLF>deEGK>dgHVU!Y˜½ NGU D-]>K/R#K>dgHVUˆYNˆF<S]/DR#NmYGDx]>NGFJŒRVTD-Y˜½¾MuNGPQDLFn
CD]£]>K/R#K>dgHVU ]7YN\F<S]<DR#NYGDo]<NGFJŒŽRVTDvYGDob½ š [ ]<DLF>wVR#K<HVdeF<DɏdgDLF>F<DrMONGPQDLF£]<HVUK7THVU ]>K>deK>N SD]7Y˜½ NGU DvTLNGwQD
TL¼*bedeU YF>dgj*N DuYGD ~ a hÒYGDuYdqR#hfiLK>F<D–DLK a Y hÒYGD}l)R#NGK<DLNGFa*THVUK<DLU)R#UKNGUŠ]/RVTF<DLhEGbedcY˜½¾DR#NkEGNGF<D`nzCD]
Ø

E)R#F<HVdoYN‰]ÎRVTf]<HVUKBF<DTHVNGwQDLF>K<D]BY˜½ NGUÅhkR#K<SLF>dqR#NÅF<St•)ST/lGdg]<]/R#U*Kpœ4(4¼*wQD Ç xj*NGdEcDLF>hfDLK"YGDfF<St•)ST/lGdeF\DLK
Yd“—N ]/DLFrbqRBbeNGhdgiLF<"
D (‘T/l DLF<DLU Ç HwˆY R#U ]vK<HVNGK<DAbqRBTLNGwQD-R­ U¬Y˜½¾HV[GK<DLUGdeFNGU DAF<SLEHVU ]<D–bqRBEGbeN ]rl HVhfHVPQiLU D
EH`]/]>de[GbgD`n`C‘R–beNGhdgiLF<DuD]<KTHVbebgDTLK<SDE)R#FK>F<HVdg]7EGl HVK<HVhBNGbeK>deEGbedgTR#K<DLNGF<6
] 5.7AÉ ¦`_ Y YGD}^–EcHVN TD]EGF<HzYNGdeK<]

E)R#F-bqRˆ]/H*TLdgSLK<[email protected]Œ’F/R#U WR#dg]<D'É7l HVK<HVUGdg]n‘C‘D]BɏÏ!]"]/HVUK"]>deK>N S]B]>NGF"bqR¬E)R#F>K>dgDk]>NGEcSLF>dgDLNGF<DkYGDbqRmTLNGwQDkDLK
THVbebgS]}]>NGF}NGUmlNG[GbgHVK}K>F/R#U ]>E)R#F<DLU*Ka)bgDLNGFuEGl HVK<H*TR#K>l HzYGD"HVF>dgDLU*K<SD-wQDLF<]bgDAŒŽHVU YˆYGDAbqRTLNGwQD`n)C‘R\­ P`NGF<D
~ n  F<DLEGF<S]<DLUK<D"NGU D-]>K/R#K>dgHVUmYN¬F<S]/DR#NmYGD-]>NGFJŒRVTDxDLK}]<D]}YdewQDLF<]OSLbgSLhfDLUK<]n
CD}]<deP`U)R#b D]>K4DtsK>F/R#deK4YGDvb½§R#U H*YGDuDLK4YGDvbqRxYGDLF>UGdgiLF<DuY¼zU HzYGDYGD}T<l)RVj*N D}ɏÏÆaQTDLK>K<D}YGDLF>UGdgiLF<DOSLK/R#UK
R#hEGbed“­)SD'Y˜½ NGUƌRVTLK<DLNGF\DLU*w*deF<HVU ~ _!R­ UÆY˜½¾HV[GK<[email protected] D'bqR#F>PQDkPR#hhfDŠY¼*U)R#hdgj*N D`a£]½¾SLK<DLU Y R#UKYGD
8JãÎé#ëŽëŽê*ãÎññ`ßJä:9Jë<;>=–ëñVã<â‘ä ãoëŽé#ê¾áß
¿˜Àg¿2%Á* & ` m&#†GÃQ
j N DLbgj*N D]‘EGl HVK<HzSLbgDTLK>F<HVU ]IJN ]<jN½§»ODLUwzdeF<HVU _ 5 nCD]£]>deP`U)R#Ns\]<HVUK7DLU ]>NGdeK<D4­ beK>F<S];EGNGdg];ST/l)R#UK>debebgHVUGU S]

]>NGF _"[GdeK<]r»"NGU DuŒF<Sj*N DLU TDOYGDO„_BÏ%ŸOÔuE)R#FrNGUŠTHVU*wQDLF>K>dg]<]<DLNGF}R#U)R#bgHVP`dgj*N DtUNGhfSLF>dgj*N D`n š UŠHV[GK>dgDLU*K

R#deU ]>d;]<NGFubgD] wQHVdgD]œrR#U H*[email protected]?–Y¼zU HzYGD
É4Ï%]Ν}NGU¤]>deP`U)R#bU*NGhfSLF>dgjN DTHVhEGF>dg]ADLU*K>F<[email protected]_'DLK _`Ñ ~
Ø
×~

E)R#F–deU*K<DLF>wVR#bebgD%œ[GdeU–YGDfÑ Y U ]n˜¡*d7bqR'wQHVdgDDLU2]<HVF>K>[email protected]@bqRŠY¼zU H*YGDD]>Kx]/R#K>NGF<SD`a˜T`½¾D]<Kx»pYdeF<Dj*N½ NGU
[GdeU¬EH`]<]/[email protected]#bgDLNGF _`Ñ ~ a)bqR\wQHVdgD"R#U HzYGDxD]>K}NGK>debedg]<SD`n

™uU¨TLF>deK<iLF<D‰YGD2YGSTLbgDLU T<l DLhfDLU*K¤YGD¤EGF<DLhdgDLF%UGdewQDR#N 4œ ( pD]>KmR#EGEGbedgj*N S2U*NGhfSLF>dgjN DLhfDLU*K¤R#Ns

]>deP`U)R#Ns2EHVNGF\YGSLK<DTLK<DLFBbgD]\]>deP`U)R#Ns2EGl*¼z]>dgj*N D])
n (‘HVNGK<D]@TD]"Œ’HVU TLK>dgHVU ]\]<HVUKRV]<]>NGF<SD]BE)R#F"bqRmTR#F>K<D
SLbgDTLK>F<HVUGdgjN D–YGDO{ F<HVUK|7U Y˜nzCD]rhfD]>NGF<D]oYGDOK<DLhE ]r]<HVUKoŒ’HVNGF>UGdgD]rE)R#FobqR"TR#F>K<DxAÉo¡kjNGdEDLF>hfDLKrbqR
]>¼*U T/lGF<HVUGdg]/R#K>dgHVUªK<DLhEcHVF<DLbebgDYGDBb½§RVTj*NGdg]>deK>dgHVU2DLKxYNªK>F/R#U ]JŒŽDLF>KxYGD\YGHVUGU SD]n˜C‘RŠTR#F>K<[email protected]“­)SDˆœ’B
™ A}
PQiLF<DubgD]4Œ’HVU TLK>dgHVU ]rK<DLbebgD]vj*N DubgD–YGSTLbgDLU T<l DLhfDLU*KuYGDO]/DTHVU YkUGdewQDR#N24œ (}ÑQta*b½¾SLK/R#bgHVUGU)R#PQDxYGDubqR"]>K/R#K>dgHVU
DLKxbgD]-THVhh"NGUGdgTR#K>dgHVU ]@RwQDTkbgDk‡rÕ}MA¡n‘C‘RpTR#F>K<Df3
™ A RV]/]>NGF<DSLPR#bgDLhfDLUKBbgDTHVUK>F<ÚVbgDkYGD]-K<DLU ]>dgHVU ]
R#deU ]>d7jN [email protected]Š]>NGF>wQDLdebebqR#U [email protected]ŠK<DLhEcSLF/R#K>NGF<DfYGD]xYd“—˜SLF<DLUK<]BTHVhEH`]/R#U*K<][email protected]½¾SLbgDTLK>F<HVUGdgj*N Dmœ’TR#F>K<D
{GF<HVUKu|U Y˜a)ɏÏ!]nenen›tn
‡ l)RVjN Dk]>K/R#K>dgHVUÅYGDLw`R#UK"yLK>F<DkdeU YGSLEDLU Y R#U*K<D'SLU DLF>PQSLK>dgj*N DLhfDLUKa;YGDLNsªE)R#UGU DR#Ns‰]<HVbqR#deF<D]"T/l)R#FJ
o
PQDR#UKT<l)RVTLNGUfYGDLNs\[)R#K>K<DLF>dgD]YGD Ñ#€ EcDLF>hfDLK>K<DLUK4YGDoŒŽHVNGF>UGdeF7bgD] _C U STD]<]/R#deF<D]»Ob½¾SLbgDTLK>F<HVUGdgj*N D`n


C‘RA]>K/R#K>dgHVUkTHVhhBNGUGdgjN DuE)R#F4HVU YGD]7l DLF>K>ÔdgDLUGU D]vRwQDTbgDO‡rÕuMA¡BP`F/ÖVTDu»A]ÎR–EGF<HVEGF<DOR#UK<DLUGU DuDLKo»A]<HVU
F<STDLEGK<DLNGFOF/RVYdgHŠR#deU ]>d‘j*N½§R#NsˆF<DLbqR#dg]}EGbqRVTS]–DLU%Yd“—˜SLF<DLUK<]–DLU YF<HVdeK<]–YNmF<S]<DR#Nn%DÞb‘D]>KAR#deU ]>dEcH`]<]>de[GbgD
Y˜½¾DLUwQH¼QDLFAYGD]}deU ]<K>F>N TLK>dgHVU ]OR#NsˆTLNGwQD]uYGDLEGNGdg]ObgDB‡rÕ}Mx¡aGE)R#FuDtsGDLhEGbgD-EcHVNGF}F<SLdeUGdeK>dqR#bedg]<DLFONGU D-TLNGwQD
HVN¬deU ]>K/R#bebgDLF}NGUˆU HVNGwQDLbR#bePQHVF>deK>lGhfDBYGD-YGSTLbgDLU T<l DLhfDLU*Kn
û£üòû£üFE
GHJILKILMÿ4ÿ%NOJI> +*P Qÿ%ILR4ÿS TUIVÿ
C ½ NGUGdeK<SAYGDuhfD]<NGF<DOYNp]>deP`U)R#bcYGSLK<DTLK<S–Y R#U ]rbgD]rTLNGwQD]vYN¬¡zÕ D]>KobgD–]>deP`U)R#bcYGSLEH`]/SuE)R#FrNGU'hBN HVU

K>F/RwQDLF<]/R#U*K\wQDLF>K>dgTR#bgDLhfDLU*KbqRˆTLNGwQDŠDLUÅ]<HVUÆTDLUK>F<D`n£CDkh"N HVUXYGSLEH`]/DfbqR¬hfyLhfDŠj*N)R#UK>deK<SpY˜½¾SLU DLF>P`dgD
Y R#U ]xb½¾DR#N‰YGDbqRpTLNGwQD`a˜EGF<HVEcHVF>K>dgHVUGU DLbebgDLhfDLUKf»ŠbqRŠbgHVUGP`N DLNGF\YGDf]<HVUªE)R#F<THVNGF<]aTDTLd4EGF/R#K>dgjN DLhfDLU*K
deU YGSLEDLU Y R#hhfDLU*KoYGD]<HVUfSLU DLF>P`dgDdeUGdeK>dqR#bgD`aTDrj*NGdGEcDLF>hfDLKY˜½ NGK>debedg]<DLF4TDvEGF<HzTD]<]>N ]7THVhhfDF<StŒŽSLF<DLU TD`n
‡rDLK>K<D-NGUGdeK<S\YGD"]>deP`U)R#b;D]>K–R#EGEcDLbgSD"€A|Ï EcHVNGFu€DLF>K>dgTR#b£|j*NGdewVR#bgDLU*K–Ï%N HVU%DLKOEDLNGK–]<D-hfD]>NGF<DLFADLU
THVNGF/R#UK
HVNŠDLU¬T<l)R#F>PQDxTHVbebgDTLK<SD
œdeU*K<SLP`F/R#bgDxYN¬THVNGF/R#UKΝrDLU¬]<HVF>K>dgDAYNŠÉ4ÏÅaDLU¬SLU DLF>P`dgD
IVEM
QVEM
œŽÑn§ÑmÏ!D€ÙTLh −1 Ñ`_mTLh'-HVNÆDLU THVF<DŠDLUÅNGUGdeK<S]@SLbgDTLK>F<HVUGdgj*N D]¬œ’TR#U)R#NsÅY˜½¾MOÕ-‡tn;CD'F<ÚVbgD'YGD'bqR
× 
EGF<H*TSYNGF<DOY˜½¾SLK/R#bgHVUGU)R#PQDOD]>KrYGD}YGSLK<DLF>hdeU DLFrbqRxw`R#bgDLNGF4Y˜½ NGUk€A|Ï¹DLUkNGUGdeK<S]rSLbgDTLK>F<HVUGdgjN D]vRwQDTuNGU D
[HVUGU DAEGF<STLdg]>dgHVUn
C‘R¬hfSLK>l HzYGDpRVYGHVEGK<SDkE)R#FBbqRˆTHVbebqR#[cHVF/R#K>dgHVUÆMONGPQDLF'‚ Y oNGK>debedg]<DkbgDf• Ns2DLU
YGD]Bh"N HVU ]
Ø
cos2 (θ)
!"#%$'&()*
W
#R K>hfH`]>EGl SLF>dgj*N D]xK>F/RwQDLF<]/R#U*K"bgD]-]<K/R#K>dgHVU ]AEHVNGF-D]>K>dehfDLF"bqRkw`R#bgDLNGF-YNª€A|ÏÅn˜C4½ lGdg]>K<HVP`F/R#[email protected]
T<l)R#F>PQDŠF<STHVbeK<SD¬DLUX]<HVF>K>dgDŠY˜½ NGUZÉ4Ï DLUZF<SLEHVU ]/DŠR#NsÆh"N HVU ]fR#K>hfH`]>EGl SLF>dgj*N D]DLUZEGF<HwQDLU)R#U TDˆYGD
K<HVNGK<D]xbgD]xYdeF<DTLK>dgHVU ]-EGF<S]/DLUK<DNGU2EGdgTf»ŠNGU [email protected]`R#bgDLNGF-YGDT<l)R#F>PQD peak F<DLbedgSDf»[email protected]
QVEM
Ydg]>K>F>de[GNGK>dgHVU¤DLU!T/l)R#F>PQDBYGD]Oh"N HVU ]–wQDLF>K>dgTR#Ns
n)CD\]>EcDTLK>F<D"YGD"T/l)R#F>[email protected]]uhBN HVU ]OwQDLF>K>dgTR#Ns
QVEM
D]>KkHV[GK<DLU*N‹]>NGF'NGU DmTLNGwQDmK<D]>KkSj*NGdeESD%YGDmYGDLNs ]<TLdeU*K>debebqR#K<DLNGF<]Š]<deK>N S]kYGDˆE)R#F>KŠDLK'Y˜½§R#NGK>F<D!YGDˆbqR
TLNGwQD`aR­ UªYGD]<SLbgDTLK>dgHVUGU DLFxbgD]AK>F/RIJDTLK<HVdeF<D]-h"N HVUGdgj*N D]-wQDLF>K>dgTR#bgD]f‚ Y ¦†n˜C‘R'F<DLbqR#K>dgHVUÅR#deU ]<d£HV[GK<DLU*N D
D]>K peak
n
QVEM = 1.05 × QVEM
M ­ UkYGD}hkR#deU*K<DLUGdeFrb½¾Dt¥kTRVTLdeK<S–YGDuYGSLK<DTLK>dgHVUpYNkF<S]/DR#Nadeb)D]<K4U STD]<]/R#deF<DOYGD}hkR#deU*K<DLUGdeFrbgD]o]<DLNGdebg]
u
YGDAYGSTLbgDLU T<l DLhfDLUK<]ObgHzTR#Ns¤œ4( DLKX(}ÑQoY R#U ]T<l)RVj*N Dx]>K/R#K>dgHVUm»\YGD]vwVR#bgDLNGF<]]>K/R#[GbgD]DLKTHVUGU*N D]nGCD

EGdgT"YGDBb½ lGdg]>K<HVP`F/R#[email protected]!THVNGF/R#U*KxYGD]uhBN HVU ]AR#K>hfH`]<EGl SLF>dgjN D] peak D]<KuNGK>debedg]<S\THVhhfD\NGUGdeK<SBYGD
IVEM
F<StŒ’SLF<DLU TDkEHVNGF"­GsDLFBbgD]B]/DLNGdebg]"YGD]\YGSTLbgDLU T<l DLhfDLUK<]n peak D]>KBSLPR#bgDLhfDLU*[email protected]<DLbedgS'»
E)R#F"NGU D
IVEM
IVEM
F<DLbqR#K>dgHVUm]>dehdebqR#deF<D"»fTDLbebgD-]>NGFEcHVF>K/R#UKu]>NGFbqRfT<l)R#F>PQD`n
C RmEGF<H*TSYNGF<DpY˜½¾SLK/R#bgHVUGU)R#PQDpYGD]@]>K/R#K>dgHVU ]@D]>KBYGHVU TŠbqR%]>NGdew`R#UK<D!”‘bqRmEGF<DLhdgiLF<DpSLK/R#EDŠTHVU ]>dg]>K<D
‘
»'­GsDLF"bqRpwVR#bgDLNGFBYGD peak » Y _¬TR#U)R#Ns2MOÕ-‡ôEGNGdg]B»ˆRI¯N ]<K<[email protected]#deUÅYGD]-É4Ï%]-YGDfhkR#UGdgiLF<D'»¬TD
IVEM
jN DfbgDK/R#Ns‰YGDfTHVhEGK/R#PQDpR#N†YGD]<]>N ]"YGD Y _ˆTR#U)R#Ns‰M–Õ-‡ ]<HVdeK"SLPR#bv» _`_ˆŸOÔQn‡rD]BwVR#bgDLNGF<]BHVUK


SLK<S¬YGSLK<DLF>hdeU SD]kP`F/ÖVTD%»¤YGD]hfD]>NGF<D]fDt—˜DTLK>N SD]k]>NGFNGU DmTLNGwQDpK<D]>KkYGHVUKkbgDpPR#deU YGD]fɏÏ!]fD]>K
THVUGUNn*C‘RAF<SLEcHVU ]<D}YGD]4É4Ï%]7EcHVNGw`R#UKoSLwQHVbeN DLFvRwQDTubqRAK<DLhESLF/R#K>NGF<D`abqRAw`R#bgDLNGF4YGD peak D]>KoRIJN ]>K<SD
IVEM
bgH*TR#bgDLhfDLU*K–DLKAR#NGK<HVhkR#K>dgjN DLhfDLU*KxEHVNGFOT<l)RVj*N DBTLNGwQD\R­ U!YGDBTHVhEDLU ]<DLFONGU D"SLwQDLU*K>N DLbebgDBw`R#F>dqR#K>dgHVU
YGDxF<SLEcHVU ]<D”)»@K>deK>F<D-Y˜½¾DtsGDLhEGbgD`a bqRwVR#bgDLNGF}hfH¼QDLUGU DBYGD peak ]>NGFbgDxF<S]<DR#NmD]<KuYGDx„
„kTR#U)R#Ns˜a
IVEM
ر
TDxjNGdD]>KNGU Dxw`R#bgDLNGF}EDLN%SLbgHVdeP`U SD"YGD] Y _TR#U)R#Ns¬deUGdeK>dqR#Ns˜n‡rDLK>K<DxEGF<HzTSYNGF<D-EcH`]<]<iYGDAb½§RwVR#U*K/R#PQD
Y˜½¾yLK>F<D}]>dehEGbgD`aVF<HV[GN ]>K<D`a*R#deU ]>d)j*N DvK>F<i]7NGUGd“Œ’HVF>hfDu]>NGF7b½¾DLU ]<DLh"[GbgDOYNF<S]<DR#NnQC‘R-YGSLK<DLF>hdeU)R#K>dgHVU'YGDbqR
wVR#bgDLNGFrYN peak YGDuT/l)RVjN D–]>K/R#K>dgHVUpD]>K4Dt—˜DTLK>N SDOYGD}ŒŽRVWHVU'F<SLP`NGbedgiLF<DOE)R#FobgD–‡rÕ}Mx¡DLU'F<SR#bedg]/R#UKvNGU
QVEM
RI¯N ]<K<DLhfDLUKxYGD"b½ lGdg]<K<HVP`F/R#hhfDYGDBT/l)R#F>PQD]>K<H*T Ç S"K<HVNGK<D]AbgD]–hdeU*NGK<D]AE)R#FObqRŠ]>K/R#K>dgHVUnC‘R'EGF<STLdg]>dgHVU
HV[GK<DLUN DBP`F/ÖVTDB»kTDLK>K<D-EGF<HzTSYNGF<D"Y˜½¾SLK/R#bgHVUGU)R#[email protected]]>KuYGD"b½¾HVF<YF<D"YGD ~ZY EcHVNGFubgD]}w`R#bgDLNGF<]–YGD
QVEM
DLK
n
IVEM
C D]"R#NGK>F<D]-E)R#F/R#hfiLK>F<D]-U STD]<]/R#deF<D]"EHVNGF-DtsEGbgHVdeK<DLF-bgD]>deP`U)R#b]/HVUK-bgD]Aw`R#bgDLNGF<]-YGD]xEGdgSYGD]>K/R#Ns
YGDT<l)RVTLNGU D'YGD] wQHVdgD]"R#deU ]>dj*N [email protected]#deU2F<DLbqR#K>d“Œ}DLU*K>F<DbqRŠwQHVdgDfY¼zU H*YGDfDLKxbqR¬wQHVdgDfR#U H*YGDfj*NGdD]>K
Ø
YGSLK<DLF>hdeU S"»fÑ Y EGF<i]n
¿˜Àg¿2%Á* & ` m&#†GÃQ
û£üòû£üòû
L[
\N]^+_` <ÿ* + Q ÿ
C Df]>¼]>K<iLhfDYGDfYGSTLbgDLU T/l DLhfDLUKYNÆ¡zÕ R¬SLK<SfTHVU WLN‰EHVNGF-EcDLF>hfDLK>K>F<DfbqRpYGSLK<DTLK>dgHVUZYGD]xF/R¼QHVU ]
TH`]>hdgjN D]kY R#U ]fNGU D¬P`F/R#U YGDmPR#hhfDˆY˜½¾SLU DLF>P`dgDmDLKfEcHVNGFfK<HVNGKfbgD]'R#UGP`bgD]kY˜½ deU TLdgYGDLU TD]nv¡zHVUËF<ÚVbgD
D]>KkYGDˆ]<SLbgDTLK>dgHVUGU DLF'bgD]kSLwQSLU DLhfDLU*K<]'EGl*¼z]<dgjN D]'YNË[GF>NGdeK'YGDpŒŽHVU YËK<HVNGK'DLUËPR#F<Y R#UKŠNGU K/R#NsËYGD
YGSTLbgDLU T<l DLhfDLUK¤RVTTDLEGK/R#[GbgD`nCD!EGF<H*TD]/]>N ]pYGD!YGSTLbgDLU T/l DLhfDLUKmD]<KŠlGdgSLF/R#F<T/lGdgjN DX”rdebADtszdg]<K<D!YGDLNs
UGdewQDR#NsmYGD"YGSTLbgDLU T<l DLhfDLU*K'4œ ( DL3
K (}ÑQYdeK<]ObgH*TR#NsmTR#FAYGSTLbgDLU T<l S]xR#N!]<DLdeU%YGDBT<l)RVj*N DB]>K/R#K>dgHVU¤DLK

NGU'UGdewQDR#N¬YGDAYGSTLbgDLU T<l DLhfDLU*KuTDLUK>F/R#'b ( ~ YGSTLbgDLU T<l SAE)R#FrbgD-‡rÕ}Mx¡kj*NGdF<DT<l DLF<T<l DAE)R#F>hd˜bgD.
] (}ÑBYGD]
THVU­ P`NGF/R#K>dgHVU ]PQSHVhfSLK>F>dgj*N D]7EH`]/]<SY R#UK£NGU DTHVhE)RVTLdeK<ShdeUGdehBNGhˆnQCHVF<]7YGDvb½§R#U)R#be¼]<DvYGD]7YGHVUGU SD]a
YGDLNs\R#NGK>F<D];UGdewQDR#Ns\YGDrYGSTLbgDLU T<l DLhfDLUKo]<HVUK7DtsdePQS];R­ UYGDr]<SLbgDTLK>dgHVUGU DLF£bgD];SLwQSLU DLhfDLU*K<]7EGl¼]>dgj*N D]
4œ (v„*4DLKY˜½§RV]<]<NGF<DLFvj*N D–bqRBF<DTHVU ]>K>F>N TLK>dgHVUˆD]>KTHVF>F<DTLK<Df4œ ( Y tnCDA]>¼z]<K<iLhfD–YGDAYGSTLbgDLU T/l DLhfDLUKOYNm¡zÕ
D]>K}YGSTLF>deKuYGD–ŒRVWHVUmYGSLK/R#debebgSD"Y R#U ]-‚ Y\[ †n
,6-0/.1 ~ n§Ñ
2ba
DLEGF<S]<DLU*K/R#K>dgHVU%Y˜½ NGU DxEcHVF>K>dgHVUmYGDAF<S]<DR#NˆEcH`]<]<SY R#U*K}NGU D-]>¼zhfSLK>F>dgDAl Dts R#PQHVU)R#bgD`n
M ­ U¤YGD\EGF<STLdg]<DLFxbgD]ATHVUwQDLU*K>dgHVU ]-NGK>debedg]<SD]AEcHVNGFxYGSTLF>deF<DBbgD]xTHVU­ P`NGF/R#K>dgHVU ]-PQSHVhfSLK>F>dgjN D]-j*N D
}
Œ’HVF>hfDLU*K}bgD]]>K/R#K>dgHVU ]K<HVN T/l SD]a bqRB­ P`NGF<D ~ n§Ñ@debebeN ]<K>F<DANGU DAEHVF>K>dgHVUmYN¬F<S]/DR#NˆYGDx]>NGFJŒŽRVTDAEcH`]<]<SY R#U*K
NGU D}]>¼zhfSLK>F>dgDl DtsGR#PQHVU)R#bgD`nzÉ;HVNGF4bqRxTLNGwQD}TDLU*K>F/R#bgD`abqRxEGF<DLhdgiLF<DuTHVNGF<HVUGU D–D]>K4YGSt­ UGdgDE)R#FobgDEGF<DLhdgDLF
l DtsGR#PQHVU D-Œ’HVF>hfS"E)R#FubgD] ]>K/R#K>dgHVU ]ObgD]}EGbeN ]OEGF<H*T/l D]a bqRkYGDLNszdgiLhfD\THVNGF<HVUGU DBDLU*K<HVNGF<D"bqRfEGF<DLhdgiLF<D
Ø
DLKBR#deU ]>doYGD']>NGdeK<D`n‘C‘D]BTHVU­ P`NGF/R#K>dgHVU ]@YGDk]<K/R#K>dgHVU ]B]<HVU*KBYGSt­ UGdgD]BE)R#F"bqRˆU HVK/R#K>dgHVU
HVÈ D]>KBbgD
iCj
i
U HVh"[GF<[email protected]"]>K/R#K>dgHVU ]AY R#U ]ObqRfTHVNGF<HVUGU D DLKubgD]–THVNGF<HVUGU D]AdeUŒ’SLF>dgDLNGF<D]a)bqRk]>K/R#K>dgHVU!TDLU*K>F/R#bgD\deU TLbeN D`n
j
!"#%$'&()*
c
É;R#FBDtsDLhEGbgD`a˜bgD]"THVU­ P`NGF/R#K>dgHVU ]\ÑQ‡
‡vшTHVF>F<D]>EHVU YGDLU*K\»¬R#NªhfHVdeU B
] YGDLNs¤]>K/R#K>dgHVU ]"wQHVdg]>deU D]
ed ~
Y R#U ]ŠbqR‰EGF<DLhdgiLF<DªTHVNGF<HVUGU DªDLKpNGU D!K>F<HVdg]>dgiLhfDª]>K/R#K>dgHVU«EcHVNGwVR#U*Kˆ]<D!]>deK>N LD F\I¯N ]<j*N½§»‰bqRÅYGDLNsdgiLhfD
THVNGF<HVUGU D`n
CD]}TR#F/RVTLK<SLF>dg]<K>dgjN D]–YGD]}Yd“—˜SLF<DLUK<]uYGSTLbgDLU T/l DLhfDLUK<]A]<HVU*K}bgD]}]>NGdew`R#UK<D]B”
 ` ÁÃõgf‘Ã_h‘gi2gf ` a #iÅjfò &lknmfÄo EcDLNGK–yLK>F<DBF<SR#bedg]<SBE)R#F–YGDLNs!THVU YdeK>dgHVU ]xYd“—cSt
F<DLUK<D]\”GbgD-]>deP`U)R#b;YGD-T<l)RVTLNGU¤YGD]}K>F<HVdg]–ɏÏ!]uYGSLE)RV]<]/DxNGU%]<DLNGdebYGD a [Y €A|Ï DLUmTH_pŽU TLdgYGDLU TD\HVNmbgD

]>deP`U)R#b4Y˜½§R#NÅhfHVdeU ]BYGDLNs2ɏÏ!]"YGSLE)RV]<]/D'_a§Ñp€A|Ï EDLU Y R#U*KBEGbeN ]BYGD ѬTR#U)R#Ns˜n‘ÕxR#U ]"bgDfEGF<DLhdgDLF

TRV]abgb
D ( D]>[email protected]>dgjN DLK>K<“
S ö Ø1Þ>ßÒØ6æ!êúÖQa;Y R#U ]\bgDŠ]<DTHVU YZTRVf
] ö ãâö EcHVNGH
F (vdehfD š wQDLq
F (vlGF<D]>l HVbgY˜n£CD

K/R#Ns2YGDfYGSTLbgDLU T<l DLhfDLU*[email protected]
N ( D]>K-Y˜½¾DLUwzdeF<HVU _`_¬ŸuÔQa‘YGHVhdeU SE)R#F"bgs
D (vlGF<D]<l HVbgY œ
_`_¬ŸOԝ–bgD
'


( š (ÌR¼R#U*KuNGU D–Œ’F<SjN DLU TD"Y˜½¾DLUwzdeF<HVU a ŸOÔQn
 Ø
MON UGdewQDR#N«Yt
N ( aobqR2]>K/R#K>dgHVUU D%THVU*K>F>de[GN D%E)RV]p»¤ŒŽHVF>hfDLFkNGU«YGSTLbgDLU T<l DLhfDLU*KˆTDLUK>F/R#bAhkR#dg]

EHVNGF>F/R\EGF<DLU YF<D"E)R#F>KO»\NGUmSLwQSLU DLhfDLU*KO]>d˜NGUmYGSTLbgDLU T/l DLhfDLU"
K ( ~ D]>KvŒŽHVF>hfS`n
,.-F/.1 ~ n ~ |7sDLhEGbgDxY˜½ NGUˆ]>deP`U)R#b˜wQSLF>d“­R#U*KubqR\THVU YdeK>dgHVUN(}Ñu(vlGF<D]>l HVbgYŜŽR#N¬hfHVdeU ] [GdeUmR#N†YGD]<]>N ]
YGD n [Y €A2 
| ϤrDLUml)R#NGKa DLKX(}Ñq( š (¨DLUˆ[)RV]BœŽR#N¬hfHVdeU ] ~ [GdeU ]uR#N†YGD]/]>N ]}YGD"_n§Ñ\ €A|Ïªtn


¿˜Àg¿2%Á* & ` m&#†GÃQ
%v
 ` *Á ÃõfÃhi‰f ` ‹Ãf ` f;ò* &wknm_>o D]>KF<SR#bedg]<S¬]>dbqRª]<K/R#K>dgHVUË]/R#K>dg]JŒR#deKkR#N TLF>deK<iLF<D
( š ( œ’Y R#U ]fTDpTRV]bgDx( ( š (ÙD]>[email protected]<HVh"NR#NGK<HVhkR#K>dgj*N DLhfDLUKŠDLU(}Ñy( š (u\HVNË]>dvbgD¬]>deP`U)R#b}YGD

T<l)RVTLNGUªYGD]AK>F<HVdg]AɏÏ%]AYGSLE)RV]<]/D"NGUª]<DLNGdeb£YGD ~ a§Ñf€A|Ï´DLUªTH_pŽU TLdgYGDLU TDˆœ+(}Ñb(vlGF<D]<l HVbgY)tnC‘DBK/R#Ns
YGDxYGSTLbgDLU T<l DLhfDLUKxY*
N (}ÑD]<K}Y˜½¾DLUwzdeF<HVU!Ñ`[email protected]ŸOÔQa YGHVhdeU S-DLU THVF<DxNGU DAŒŽHVdg]E)R#FbgDz(vlGF<D]<l HVbgY˜n
 ` ÁÃõfÃ_h‘gi2f ` B#0{gi2sf;q &SknmB¿Lo D]>K£PQSLU SLF<SvE)R#F£bgDrF<S]<DR#NYGDv]>NGFJŒŽRVTDvHVNE)R#F£bqR
• N HVF<D]<TDLU TD`nCD|( ~ EDLNGKyLK>F<Dw`R#bedgYGS}YGD}YGDLNs\ŒRVWHVU ]}”`bqR–EGF<DLhdgiLF<DuD]<K4R#EGEDLbgSD ~ ( š ( DLK4YGDLhkR#U YGD
R#NŠhfHVdeU ] ~ ]>K/R#K>dgHVU ]}R¼R#UKuYGSTLbgDLU T/l SxNGUj( š («DLUpTH_pŽU TLdgYGDLU TD"Y R#U ]vNGU D–THVU­ P`NGF/R#K>dgHVU%THVhE)RVTLK<D
ÑQ‡
‡vÑmœŽÑ']>K/R#K>dgHVU ]OYGHVdewQDLU*KxyLK>F<DBY R#U ]ObqRfEGF<DLhdgiLF<D\THVNGF<HVUGU [email protected]>F<HVdg]>dgiLhfD\]>K/R#K>dgHVU%Y R#U ]–bqR
 d ~
}
YGDLNszdgiLhfDfTHVNGF<HVUGU DtnCDfYGDLNsdgiLhfDfF<Dj*NGdgDLF>K-„m]>K/R#K>dgHVU ]-E)RV]<]/R#UK"bgDs(}Ñ2œ4( š (ƒHVN~(vlGF<D]<l HVbgY)xDLU
T_H pŽU TLdgYGDLU [email protected]\NGU DBTHVU­ P`NGF/R#K>dgHVUªF<DLbqR#K>dewQDLhfDLU*K-THVhE)RVTLK<DpœbgD]A]>K/R#K>dgHVU ]AYGHVdewQDLU*KAyLK>F<D\Ydg]>K/R#U*K<D]
YGD"hfHVdeU ]AYGD Ç h©bgD]–NGU D]AYGD]xR#NGK>F<D]Νt'n ON)R#U Y!NG!
U ( ~ D]>KAYGSTLbgDLU T<l S`acK<HVNGK<D]–bgD]A]>K/R#K>dgHVU ]xR¼R#U*K
Ø
YGSTLbgDLU T<l S-NG€
U ( Y R#U ]}NGU D–ŒŽDLU yLK>F<DxK<DLhEHVF<DLbebgDB]<HVUKdeU TLbeN D]OY R#U ]}b½¾SLwQSLU DLhfDLU*Kn

SLwQSLU DLhfDLUK<]ŠE)R#F-I>HVNGF'DLK'E)R#FkK>F>dqR#UGP`bgD!SjNGdebqR#K<SLF/R#b
CDˆK/R#Ns YGN
D ( ~‰~ ( š (¢D]>KkY˜½¾DLUwzdeF<HVU
1.3
YGD ~ ]>K/R#K>dgHVU ]AHVESLF/R#K>dgHVUGU DLbebgD]n˜[email protected]>deK<iLF<D\YN ~ ( š ( D]>K–K>F<i]–Dt¥kTRVTD\EcHVNGFObgD]–PQDLF>[cD]OŒŽR#de[GbgDLhfDLU*K
deU TLbedeU SD]@œ
xa)deb;]/SLbgDTLK>dgHVUGU DBDLU*w*deF<HVU2^`_ Y Y˜½¾SLwQSLU DLhfDLUK<]xEGl*¼z]<dgjN D]nCD\YGDLNszdgiLhfD\TLF>deK<iLF<D
θ < 60◦
Y*
N ( ~ aGEGbeN ]EcDLF>hdg]<]>d“Œ>aGD]>KU STD]<]/R#deF<DxEcHVNGF}F<STLNGEcSLF<DLF}bgD]PQDLF>[cD]uR¼R#U*KuNGUˆP`F/R#U Y%R#UGP`bgDxÔSLUGdeK>l)R#bn
û£üòû£ü‚
ƒ¤ÿ%[email protected]$+g „ÿ%NQQ%4ÿ%‹ÿ%< jI‘ÿ%Kÿr Q%#ÿIVl4ÿ_ $T†IVÿ
C R¬YGSLK<DLF>hdeU)R#K>dgHVUZYGD]xE)R#F/R#hfiLK>F<D]\YGSt­ UGdg]<]/R#UKBNGU DPQDLF>[cDkR#K>hfH`]>EGl SLF>dgj*N D'»pb½§R#dgYGD'YN2F<S]<DR#N
‘
YGD]>NGFJŒŽRVTD]½¾Dt—cDTLK>N DDLU2EGbeN ]<dgDLNGF<]-SLK/R#ED]n˜CD]xK<DLhE ]"Y˜½§R#F>F>dewQSDkYGD]xE)R#F>K>dgTLNGbgD]"Y R#U ]xbgD]"]>K/R#K>dgHVU ]
EDLF>hfDLK>K<DLU*KrYGDF<DLhfHVU*K<DLFv»-bqRAPQSHVhfSLK>F>dgD–YGDbqRxPQDLF>[cDuR#bgHVF<]rjN Db½ deU*K<DLU ]>deK<S–YGD]4]>deP`U)R#NsF<DLU ]<DLdeP`U DLU*K
]>NGFOb½¾SLU DLF>P`dgDYGD\bqR'E)R#F>K>[email protected]>dehkR#deF<D`n˜C‘R'EGF<HzTSYNGF<DYGDBF<DTHVU ]<K>F>N TLK>dgHVU2YGD]xSLwQSLU DLhfDLUK<]B¡zÕÒD]>K
[GF>dgiLwQDLhfDLUKvYGSTLF>deK<DuTLd“†YGD]<]<HVN ]aDLbebgD}D]>KoYGSLK/R#debebgSDOY R#U ]4bqRAE)R#F>K>dgDuTHVU ]/RVTLF<SD–»xb½§R#U)R#be¼]<D}YGD]oYGHVUGU SD]
YGDAb½ š [ ]<DLF>wVR#K<HVdeF<DBÉ7dgDLF>F<D-MONGPQDLFn
*¡ NGdewVR#U*KAb½§R#UGP`bgDY˜½ deU TLdgYGDLU TD YGD\bqRfE)R#F>K>[email protected]>dehkR#deF<D`abgD]AE)R#F>K>dgTLNGbgD]x]<DTHVU Y R#deF<D]AwQHVU*KAK<HVN
θ
T<l DLFfbgD]fYGSLK<DTLK<DLNGF<]pRwQDTmYGD]fYGSTR#bqR#PQD]kK<DLhEcHVF<DLbg]fYd“—˜SLF<DLUK<]!”£bqRªYGSLK<DTLK>dgHVU ]<DLF/RªEGF/R#K>dgjN DLhfDLU*K
]>deh"NGbeK/R#U [email protected] D\PQDLF>[cD"wQDLF>K>dgTR#bgDHVNªR#NªTHVUK>F/R#deF<DYGSTR#bgSDYGDBbqRkYdg]<K/R#U TDBdeU*K<DLF-]>K/R#K>dgHVU ]œ n Y

Ç ˆ
h ‡
OEHVNGFBNGU DPQDLF>[D'Y˜½§R#UGP`bgDfÔSLUGdeK>l)R#b n˜ÕxR#U ]"NGU DfEGF<DLhdgiLF<DŠR#EGEGF<HszdehkR#K>dgHVUa;[email protected]Œ’F<HVUK
c cos θ
θ
YGDPQDLF>[DfEDLNGK"yLK>F<DŠRV]<]>dehdebgSk»¬NGU2EGbqR#UÅ]<DfEGF<HVE)R#PQDR#UK»pbqRpwzdeK<D]<]<DfYGDbqR¬beNGhdgiLF<D`a‘bgD]xK<DLhE ]\YGD
W)
!"#%$'&()*
YGSTLbgDLU T<l DLhfDLU*K–DLUGF<DLP`dg]>K>F<S]uE)R#F}bgD]}]>K/R#K>dgHVU ]OYGHVdewQDLUK–Y R#U ]}TRV]}HV[cSLdeFO»\bqR\F<DLbqR#K>dgHVUŔ
œ~ n 

cti = cT0 − (uxi + vyi )
HVÈ
DLK
]<HVU*KbgD]‘THzDt¥kTLdgDLUK<]YdeF<DTLK<DLNGF<]YN"EGbqR#U"DLK
D]>K˜bgD7K<DLhE ]
u = sin(θ) cos(φ) v = sin(θ) sin(φ)
T0
˜Y ½§R#F>F>dewQSDYNTH*DLNGF7YGDrbqROPQDLF>[D}R#N]<HVbnVC‘ROEGF<STLdg]>dgHVUfYGDrbqROhfD]>NGF<DYGD];K<DLhE ]YGDvYGSTLbgDLU T<l DLhfDLU*KoYGD]
]>K/R#K>dgHVU ]}PQHVNGwQDLF>U [email protected]<STLdg]>dgHVU!YGDxbqRF<DTHVU ]<K>F>N TLK>dgHVU¤R#UGP`NGbqR#deF<D'” deb‘D]>KOYGHVU T-U STD]<]/R#deF<DBYGD-YGSt­ UGdeF
THVF>F<DTLK<DLhfDLUKATDAK<DLhE ]uYGDxYGSLE)R#F>K"‚ YZY †n
C R"EH`]<deK>dgHVUŠYNpTH*DLNGFvD]>KrD]>K>dehfSDAY R#U ]oNGUŠEGF<DLhdgDLFrK<DLhE ]THVhhfDOSLK/R#U*Kv]<deK>N SD–R#NŠ[)R#F>¼zTDLU*K>F<D
‘
YGD]@]>K/R#K>dgHVU ]fYGHVUKbgD]@THzHVF<YGHVUGU SD]f]<HVU*KkR—cDTLK<SD]Y˜½ NGUXEcHVdgYG]\EGF<HVEHVF>K>dgHVUGU DLbO»%bqR%F/RVTLdeU D¬TR#F>F<SD
YGD-bgDLNGF–]>deP`U)R#b‘hfD]>NGF<S`nMôE)R#F>K>deF–YGDBTD]}deUŒŽHVF>hkR#K>dgHVU ]OPQSHVhfSLK>F>dgj*N D]abgD-EGF<H#­ bbqR#K<SLF/R#baT`½¾D]>KA»YdeF<D
bgDf]>deP`U)R#brDLU2ŒŽHVU TLK>dgHVUÅYGDfbqRmYdg]>K/R#U TD'»¬b½§RsD'YGDfbqRpPQDLF>[cD`a‘D]>[email protected]¯N ]<K<Sf]>NGF"bgD]BYGHVUGU SD]n‘ɏbeN ]>dgDLNGF<]
Œ’HVU TLK>dgHVU ]BEcDLNGwQDLUKByLK>F<DkNGK>debedg]<SD]BEHVNGFBE)R#F/R#hfSLK>F>dg]<DLFBbgDfEGF<H#­ b4bqR#K<SLF/R#bxœ’TtŒT/l)R#EGdeK>F<D']<NGF"bgD]-PQDLF>[cD]
R#K>hfH`]>EGl SLF>dgj*N D]Νtab½¾D]<]<DLUK>dgDLbSLK/R#U*K-YGDBYdg]<EH`]<DLFAY˜½ NGUªD]>K>dehkR#K<DLNGFxF<DLbedgS»'b½¾SLU DLF>P`dgD\YGD\bqR'E)R#F>K>dgTLNGbgD
EGF>dehkR#deF<D`nÉ£R#F>hd*TD]Œ’HVU TLK>dgHVU ]aR#EGEDLbgSD]CÕA{pEHVNGFC‘R#K<DLF/R#bzՖdg]>K>F>de[GNGK>dgHVUB{GNGU TLK>dgHVUaNGU DDtsEGF<D]<]>dgHVU
YGSTLF>dewVR#U*K}[GdgDLUˆbgD]}YGHVUGU SD]O]<DxYGSt­ UGdeKOYGDAbqRhkR#UGdgiLF<D-]>NGdew`R#UK<Df”
ln(S(r)) = ln(S1000 ) + βln
HVÈ
r 2
r + γ ln
1000m
1000m
œ ~ n§ÑQ
<F DLEGF<S]<DLU*K<D"bgDB]>deP`U)R#b‘hfD]>NGF<S\»NGU DBYdg]>K/R#U TD YGDxb½§RsGD`a)DLK
THVF>F<D]>EcHVU Y%R#N!]>deP`U)R#b;»
r
S1000
_`_`_ŠhˆncCD]AE)R#F/R#hfiLK>F<D] a EcDLNGwQDLUK-yLK>F<DfRIJN ]>K<S]xHVN¤bqR#dg]<]<S]–­GsGD]acDLU!Œ’HVU TLK>dgHVU2YNªU HVhB[GF<DfYGD

β γ
]>K/R#K>dgHVU ]xY R#U ]–b½¾SLwQSLU DLhfDLUKnC½§RI¯N ]<K<[email protected]>K<D\Œ’HVU TLK>dgHVU2R#Ns¤YGHVUGU SD]AEDLF>hfDLKxY˜½¾HV[GK<DLUGdeFxNGU D
wVR#bgDLNGF}YGD
R#deU ]>dj*N DxbgD]}THzHVF<YGHVUGU SD]uYNmTHzDLNGF"œ a vDLU¬hdeUGdehdg]/R#U*Kub½¾DtsEGF<D]<]>dgHVU‰”
S(r)
S1000
xc yc
χ2 =
Nstations
X
i=1
HVÈ
Siobs
D]>KobgDO]<deP`U)R#b)hfD]>NGF<S–Y R#U ]rbqR-]>K/R#K>dgHVU
i
DLK
Siobs − Si (r)
σS2
σS2
2
œ ~ n~ 
D]>KrbqRxw`R#F>dqR#U TDuK>l SHVF>dgj*N DAYN']>deP`U)R#b7œ’TtŒ4‚ YZY tn
C D]7U HVNGwQDLbebgD]4TH*HVF<YGHVUGU SD]oYNfTH*DLNGF]<HVU*K7NGK>debedg]<SD]THVhhfDw`R#bgDLNGF<]£deUGdeK>dqR#bgD]EcHVNGF7NGU DvU HVNGwQDLbebgD
deK<SLF/R#K>dgHVUmYGDxbqR\EGF<HzTSYNGF<D"YGDAF<DTHVU ]>K>F>N TLK>dgHVUn
™ U DxD]>K>dehkR#K>dgHVUmYGDAb½¾SLU DLF>P`dgD"YGDAbqR\PQDLF>[cDxD]>Kv­ U)R#bgDLhfDLU*K–HV[GK<DLU*N D-DLUˆNGK>debedg]/R#U*K}NGU DAF<DLbqR#K>dgHVU%YGD
O
THVUwQDLF<]>dgHVU!DLUK>F<DxbgD"]>deP`U)R#b‘» _`_`_\hfiLK>F<D]
DLK}b½¾SLU DLF>P`dgD"YGSYNGdeK<D]OYGD]}]>deh"NGbqR#K>dgHVU ]n

S1000
C‘R\­ P`NGF<D ~ n „debebeN ]>K>F<DxNGUmDtsGDLhEGbgD-YGDAF<DTHVU ]>K>F>N TLK>dgHVU¤Y˜½¾SLwQSLU DLhfDLUKARwQDT-bgDAF<S]/DR#NmYGDx]>NGFJŒRVTD`n
¿˜Àeú‰ ` ÁÃ`z&‘ ` *‰7&‘*ÃQgf‘ÃQ
WÄ
,6-0/.1 ~ n „
‡vR#EGK>NGF<DfY˜½¾STLF/R#U2YN¤bgHVP`dgTLdgDLb[email protected]*dg]<N)R#bedg]/R#K>dgHVUªYGD]xSLwQSLU DLhfDLUK<]nCRŠPQDLF>[D\F<DLEGF<S]<DLU*K<SD
œwQSLF>deK/R#[GbgDx2 LS wQSLU DLhfDLUKΝ}RBK<HVN T/l S „f]>K/R#K>dgHVU ]vF<DLEGF<S]<DLUK<SD]}E)R#FbgD]vbqR#F>PQD]TDLF<TLbgD]wQDLF>K<]n™uUˆRIJN ]>K<Dt
hfDLUKOYGD"[email protected]ŒŽHVU TLK>dgHVU!YGD"Ydg]>K>F>de[GNGK> dgHVU¤bqR#K<SLF/R#bgD"D]>KOF<DLEGF<S]<DLUK<S\]>[email protected]­ P`NGF<DBDLU%l)R#NGK>F–»fYF<HVdeK<D`nc‡rDLK
SLwQSLU DLhfDLUKD]>K4F<DTHVU ]<K>F>NGdeKvRwQDTuNGU¬R#UGP`bgD}ÔSLUGdeK>l)R#b˜YGD ◦ a*DLKr]/HVU'SLU DLF>P`dgD–D]>KoD]>K>dehfSD–»
19 D€@n
42
+-, Š
5 10
°p5±³\¶š579576"·ô³B5Œ‹@6·c5#)9;571\95
C RP`F/R#U YGD-Œ’HVF<TDBYNmEGF<H#I>DLKOMONGPQDLFOD]>KubqRfTHVU ]>K>F>N TLK>dgHVU¤Y˜½ NGU%YGSLK<DTLK<DLNGF–l¼z[GF>dgYGD`aTHVhB[GdeU)R#UKAbqR
‘
K<DT<lGUGdgj*N D"Y˜½¾ST<l)R#U*K>debebgHVUGU)R#PQDR#Nm]<HVb‘YGDxbqR\PQDLF>[cD"RwQDT-bqRYGSLK<DTLK>[email protected]<D"YGD–• N HVF<D]/TDLU TD`n
CHVF<]<j*N DbgD]E)R#F>K>dgTLNGbgD];YGD4bqRPQDLF>[cDK>F/RwQDLF<]<DLUK£b½§R#K>hfH`]>EGl iLF<D`a#DLbebgD]DtsGTLdeK<DLUK;bgD]hfHVbgSTLNGbgD];YGDoYdqR#ÔHVK<D
YGD-b½§R#deFOj*NGd‘F<DLK<HVNGF>U DLU*K-»bgDLNGFOSLK/R#KOŒŽHVU Y R#hfDLUK/R#b7DLU%SLhfDLK>K/R#U*KAYGD]uEGl HVK<HVU ]n˜‡rD-F/R¼QHVUGU DLhfDLU*K-D]>K
SLhdg]-YGD\ŒŽRVWHVU‰dg]<HVK>F<HVED`a˜Y R#U ]"bqRŠPR#[email protected]Œ’F<SjN DLU TD]"THVF>F<D]>EHVU Y R#U*K\R#Ns¤NGbeK>F/RwzdgHVbgDLK<]"DLK-EcDLNGK
yLK>F<DxHV[ ]<DLF>wQSxbgD]}U*NGdeK<]u]/R#U ]vbeNGU D`a»@EGbeN ]>dgDLNGF<] Ç debgHVhfiLK>F<D]uYGD-Ydg]<K/R#U TD`n
C RxYGSLK<DLF>hdeU)R#K>dgHVUpYNkU HVhB[GF<D}K<HVK/R#bYGD}EGl HVK<HVU ]oYGDv• N HVF<D]/TDLU TDuSLhdg]K<HVNGKvR#NfbgHVUGPBYNkE)RV]<]/R#PQD
‘
YGD£bqRvPQDLF>[DEDLF>hfDLK‘Y˜½¾D]<K>dehfDLFb½¾SLU DLF>P`dgD4YGD7bqRrE)R#F>K>dgTLNGbgD4EGF>dehkR#deF<DdeUGdeK>dqR#K>F>dgTDoYGD7bqRrPQDLF>[D`n#‡rDLEDLU Y R#U*Ka
EHVNGFvF<SR#bedg]<DLFvNGU DAhfD]>NGF<D–EGF<STLdg]<D`azdebŒŽR#NGKTHVUGU)ROgK>F<DxbgD]vYd“—˜SLF<DLUK<][GF>NGdeK<]uYGDuŒŽHVU YpDLKvbgD]r]/HVN ]>K>F/R#deF<D
THVF>F<DTLK<DLhfDLUKnCD]EGF>deU TLdeE)R#bgD]rTHVUK>F>de[GNGK>dgHVU ]vE)R#F/RV]<deK<D]oR#N']<deP`U)R#b)]<HVUKobqRxbeNGhdgiLF<DOYNkTLdgDLb)U HzTLK>NGF>U D
[email protected]<z
D (‘T/l DLF<DLU Ç HwmSLhdg]/DxE)R#FbgD]}E)R#F>K>dgTLNGbgD]OT<l)R#F>PQSD]uYGD-bqR\PQDLF>[D-Y R#U ]}b½§R#deFn
We_
û£ü‚üÞý
!"#%$'&()*
þªÿ C D–YGSLK<DTLK<DLNGFYGD}• N HVF<D]<TDLU [email protected]‚ Y œ’jN Dub½¾HVU'U HVhhfDLF/[email protected]#N ]<]<d{£ÕžEcHVNGFr{;beN HVF<D]<TDLU TDAÕADLK<DTLK<HVFΝ

YGDb½ š [ ]<DLF>w`R#K<HVdeF<DrÉ7dgDLF>F<DoMONGPQDLF£D]>KTHVU ]>K>deK>N SrYGD4j*N)R#K>F<D[)R#K>dehfDLU*K<]R#[GF>deK/R#UK7T<l)[email protected]]>d“s-K<SLbgD]<THVEcD]n
CD]p[)R#K>dehfDLUK<]m]<HVUKˆ]>deK>N S]ˆR#NGK<HVNGF¬YNžF<S]<DR#NžYGD¤]>NGFJŒRVTDXœ’TtŒJnvTR#F>K<DªYN«F<S]<DR#NžY R#U ]¬b½¾DLU TRVYF<S
EGF<STSYGDLUKΝtaR­ UmYGD-YGSLK<DTLK<DLF}bgD]PQDLF>[cD]}DLUmT_H pŽU TLdgYGDLU TD\RwQDTBTDLbeNGd“†TLdn
,.-F/.1 ~ n Y
É7l VH K<HVP`F/R#EGlGdgDÅY˜½ NGUžK<SLbgD]/THVED2YN¨YGSLK<DTLK<DLNGF%YGD¤• N HVF<D]<TDLU TD˔}»ÆPR#N T/l D2bgD¤hdeF<HVdeF
<] DLP`hfDLUK<Sp2 YGD ~ a Y h
a hÊDLKBbqR!TR#hfSLF/Ra»%YF<HVdeK<D'bgDpYdqR#EGlGF/R#P`hfD¬[email protected]½§R#UGU DR#NËYGDŠTHVF>F<DTLK>dgHVU
× ~ Y
HVEGK>dgjN D`n
C R}beNGhdgiLF<D4YGD7• N HVF<D]/TDLU TD4DLUK>F<DrY R#U ]bgD4K<SLbgD]<THVEcDo»vK>F/RwQDLF<]‘NGU\YdqR#EGlGF/R#P`hfDoYGD a hÌYGD4F/R¼QHVU
‘
 
DLKD]>KrŒ’HzTR#bedg]<SDOE)R#FbgDOhdeF<HVdeF}]<EGl SLF>dgjN Dx]<DLP`hfDLU*K<SxYGD ~ a Y h
a~ Y h ]>NGFvNGU DATR#hfSLF/RnGC‘[email protected]#hfSLF/R
×
D]>KTHVhEH`]/SD¬YGDˆÑ`_ Ñ`Ñ!EGl HVK<HVhBNGbeK>deEGbedgTR#K<DLNGF<]kl Dts R#PQHVU)R#NsËYGDŠ„ Y hh´YGD¬YdqR#hfiLK>F<DˆTHVNGw*F/R#U*K
×
T<l)RVTLNGU¬NGU DOEHVF>K>dgHVUˆR#UGP`NGbqR#deF<DxYN¬TLdgDLb˜YGD a Y ◦ YGDOYdqR#hfiLK>F<Dfœ’TtŒ‘­ P`NGF<D ~ n Y tnCD]]<deP`U)R#Nskdg]<]>N ]vYGD]

ɏÏ!];]/HVUKST<l)R#U*K>debebgHVUGU S]r» _–Ï%lGÔQn#C‘DT<l)R#hEfYGDrw*N DvY˜½ NGUK<SLbgD]<THVEcDvD]>K7YGD ~ _AYGDLP`F<S]7DLUR#ÔdehBNGK>l

DLKBYGDŠÑ`¦a YGDLP`F<S]\DLUÅSLbgSLw`R#K>dgHVUa‘NGUÆ]>deK<DkYGDf• N HVF<D]<TDLU TD'THVNGwzF<DkYGHVU T ¦`_ ◦ DLUZR#Ôdeh"NGK>lÆP`F/ÖVTDŠ»
Ø

]<D] K<SLbgD]<THVEcD]n7™OUXR#UGU DR#NYGD¬THVF>F<DTLK>dgHVUHVEGK>dgj*N DpEDLF>hfDLKfY˜½§R#NGP`hfDLU*K<DLF'b½¾HVNGwQDLF>K>NGF<DˆDt—˜DTLK>dewQD
Ø
YN2K<SLbgD]<THVEcD`n{;deU)R#bgDLhfDLUKaNGUª­ beK>F<DfHVEGK>dgj*N DkRVY R#EGK<S'R#NÅ]>EDTLK>F<DkYGDb½¾SLhdg]<]>dgHVU‰YGD• N HVF<D]<TDLU TD%œ
DLUwzdeF<HVU ~ _`_"UGhƒ‘„_`_-UGh'D]<KEGbqRVTSOYGDLw`R#UKobgDuYdqR#EGlGF/R#P`hfD–EcHVNGF4F<SYNGdeF<D}bgDu[GF>NGdeKrYGD}Œ’HVU YkYN'TLdgDLbn
CD]OK<SLbgD]<THVEcD]O]<HVU*KudeU ]>K/R#bebgS]ARwQDTBNGU D-EGF<STLdg]<dgHVU!YGD"EH`]>deK>dgHVUGU DLhfDLU*KxY˜½¾DLUwzdeF<HVU¤_a ◦ jNGdEDLNGK

yLK>F<D-R#hfSLbedgHVF<SD"Y˜½ NGU¬ŒRVTLK<DLNGFuYd“s¬DLUˆhfD]>NGF/R#U*K}[email protected]`]>deK>dgHVUˆY˜½¾SLK<HVdebgD]OYGDAF<StŒŽSLF<DLU TD`n
û£ü‚ü0E
\€])+_` ŽÿN < Q)ÿ%
C ½¾HV[ ]<DLF>w`R#K>dgHVUžYGDmbqRª• N HVF<D]<TDLU TD!U½¾D]>KŠEH`]<]<de[GbgDmj*N DmbgHVF<]pYGD]'UNGdeK<]p]/R#U ]ŠbeNGU DŔ4b½¾Dt¥kTRVTLdeK<S

DLU«K<DLhE ]¬YGDmŒŽHVU TLK>dgHVUGU DLhfDLUK%YN«{£Õ D]>K¬Y˜½¾DLUwzdeF<HVU _ Y nvÉ;DLU Y R#UKˆbgD]ŠEcSLF>dgH*YGD]¬YGD!EGF>dg]<D¤YGD

¿˜Àeú‰ ` ÁÃ`z&‘ ` *‰7&‘*ÃQgf‘ÃQ
W¿
YGHVUGU SD]ˆœ7ÒØ1Û ñnܕҐ#a;bgDpTHVUK>F<ÚVbgDˆYGD]YGSLK<DTLK<DLNGF<]k]<DfŒR#deKf»¤Ydg]>K/R#U TDpYGDLEGNGdg]bgDŠ[)R#K>dehfDLUK'R#[GF>deK/R#U*KfbgD
‡rÕ}MA¡n‘MuN‰THVNGF<]BYGDhfHVU2K>F/RwVR#debrYGDK>l i]<D`aI½§R#dE)R#F>K>dgTLdeEcS'»pNGU DEGF>dg]/DfYGDfYGHVUGU SD]"YNGF/R#U*K\YGDLNs
]<DLhkR#deU D]a`DLK7[GdgDLUkj*N DvK<HVNGK7bgDYGSLK<DTLK<DLNGFo]<HVdeK7THVU*K>F<ÚVbgSO»AYdg]>K/R#U TD`a`deb SLK/R#deK4E)R#FJŒ’HVdg]£U STD]<]ÎR#deF<D}Y˜½§R#bebgDLF
F<DLbqR#U TDLF4hkR#UN DLbebgDLhfDLU*KvbgD}]>¼]>K<iLhfD`a`U HVN ]EcDLF>hfDLK>K/R#UKvY˜½§RVYhdeF<DLF4NGU'hkR#P`UGd“­)jN DubgDLwQDLFoYGD}]/HVbgDLdeb)]>NGF
bqR¤E)R#hE)Rn4CD%]>¼]>K<iLhfDmR¼R#U*KŠSLK<SmYGDLEGNGdg]kP`F/R#U YGDLhfDLU*KpR#hfSLbedgHVF<S`aodeb}U½¾D]>KfEGbeN ]kU STD]<]/R#deF<D%YGDˆ]<D
YGSLEGbqRVTDLF}EGl¼]>dgj*N DLhfDLUKnenen
CHVF<]<j*N½ NGU DkPQDLF>[cDfK>F/RwQDLF<]<DŠb½§R#K>hfH`]>EGl iLF<DŠY R#U ]BbgD'T<l)R#hEXYGDfwzN DkY˜½ NGUÆK<SLbgD]<THVEcD`a‘bqRˆbeNGhdgiLF<D
THVbebgDTLK<SD"wzdgDLUK–Œ’F/R#EGEDLFObgD]OYd“—˜SLF<DLUK<]AEGd“sGDLbg]OYGDBbqRfTR#hfSLF/RpR#NmŒNGFADLK–»khfD]>NGF<D\YGD-bqRkEGF<HVE)R#PR#K>dgHVU
YGDbqRAPQDLF>[cD`nQCDu]<deP`U)R#b UNGhfSLF>dgj*N DuF<DTLN DLdebebedDLUk]<HVF>K>dgDOYGD]7Œ’F<HVUKJ†DLU YkD]<K4]<HVNGhdg]o»xEGbeN ]>dgDLNGF<]UGdewQDR#Ns
YGDxYGSTLbgDLU T<l DLhfDLUKfœ’YGSTLbgDLU T/l DLhfDLUKΝB”
-gi‰Sf;ò & ` ` ÁÃõfÃ_h‘gi2fmfÄ ”4D]<KkYGSTLbgDLU T<l S¤R#N UGdewQDR#N‹Y˜½ NGU EGd“sGDLbOj*N)R#U YËbgD
]>deP`U)R#bYGSLE)RV]<]<[email protected]]<DLNGdeb4R#N†YGD]/]>N ]-YNª[GF>NGdeK"YGD\ŒŽHVU Y˜n˜CD]<DLNGdebD]>K"RIJN ]>K<SYGD\ŒRVWHVU‰»pHV[GK<DLUGdeF-NGU
K/R#NspYGz
D ( THVhEGF>dg]ODLUK>F<D _`[email protected]Ñ`_`_ŸOÔQn


‘ *&’‘F{gi2zf;ò & ` ` ÁÃ*õògf‘Ã_hi2gfXm²_
”YGDLhkR#U YGD4Y˜½§RwQHVdeF Y EGd“sDLbg]R¼R#UK£YGSTLbgDLU T<l S4NGU“(

Y R#U ]£NGU\deU*K<DLF>wVR#bebgDuYGD » ~ Ñ ]7Ydg]>EcH`]<S]£Y R#U ]£NGU DYGD] _`¦ATHVh"[GdeU)R#dg]<HVU ]4PQSHVhfSLK>F>dgjN D]4RVTTDLEGK<SD]n


µ
CD];P`F<HVNGED]7YGDo„AEGd“sDLbg]£THVF>F<D]>EcHVU Y R#UK4»ONGU DvYGD]£THVh"[GdeU)R#dg]<HVU ]]/HVUK7SLPR#bgDLhfDLUKoRVTTSLEGK<S]n#C‘DoK/R#Ns
YGD”(}ÑD]>K}Y˜½¾DLUwzdeF<HVU ŸOÔxE)R#F}hdeF<HVdeFn

” F<DT<l DLF<T/l DpbgD]"TH_pŽU TLdgYGDLU TD]@K<DLhEcHVF<DLbebgD]\DLU*K>F<D
m4#0{i‰Sfò* & ` ` ÁÃ*õògf‘Ã_hi2gf•m"¿ ‘
bgD]}
( ÑYGDxK<HVN ]ObgD]}hdeF<HVdeF<]OY˜½ NGUm[)R#K>dehfDLU*K–YGDA• N HVF<D ]/TDLU TDBY R#U ]uNGUmdeUK<DLF>w`R#bebgD"YGD"„_`[email protected] ]O» _ ]n
 µ
‡ DLK>K<DOTHVU*K>F/R#deUK<DxEcDLF>hfDLKYGDuF<D¯I>DLK<DLFobgD]vSLwQSLU DLhfDLU*K<]uR¼R#U*KNGU DuP`F/R#U YGDAYNGF<SDATHVhhfD–bgD]oE)RV]/]/R#PQD]
r
Y˜½§RwzdgHVU ]uHVNˆYGDxEGbqR#U iLK<D]OY R#U ]bgD-T<l)R#hE%YGDxw*N D`nGC‘DxK/R#NsˆYGD”( ~ D]>KuR#deU ]>dY˜½¾DLU*w*deF<HVU¤_a§_`џuÔQn
W ú
!"#%$'&()*
– õ>f<fG()!Qf&*’õ’õf‘Ã ` %õÞ[email protected]) a h<{#
– õò>f<fG() — ˜y™Jšœ›3@ž Ÿ šœn¡†¢¤£›B¥ ¦J§H¨O© ª"«¬šu­L®
œš ›¡)™ š+¯"šœ›© ª†š°ª<«F± §°˜ šœª²O¢¬š6™Jšœ›<²@šœªžgšœ›)³µ´@š¶© ˜ š+¢¬˜·š°ª¸
¡¢¬¡©™Jš”¢¬˜ £‚§°ª†¢¬šœ© ªš"¹bº¶»O¼“½¾L@©$ª.³¿¡¡†š°¢À˜$™Jª†š3°š3˜$¢¬´Ošœ³¿©
™Jš|¨ ª†§œ°¢À›¢À@˜ÁJ¢À«šœ›¡ ˜ §¶·šœ›†›†³¿¢Àªš.™± §·¡†³@«¬O˜ ˜ šœª«¬šX™J§°¡š¶n¸
¡šœ© ª"³µ´@šœz© ˜ š“šœªª†š°©$ªX¢¬˜ £‚§°ª†¢¬šœ© ªš}@©*§œ²O³@«¬š“¹e¼ÙJš
£³@Ä°@˜"› ¡U³¿ž «Àš^š·¡'·O˜¡ª†@«À§°š@½Å—˜3§°¡†³@«¬O˜ ˜$³¿²Oš)³@ž$›@«À©šœ›¡
š·Æšœ·¡© §.¨g@© ªUÇ$³OÈ© š ¡†§°«Àšœ›†·@¨gš+¡ª†@¢É›O©}È©$³Å¡†ªš+£‚@¢É›
¨$³@ª^³@˜zš·¡© ˜H§·¡U³¿«À@˜ ˜³¿²@šªšœ«À³¿¡¢¬£%šœ›¡š·Æšœn¡†© §.UÇ$³@È© š
˜_© ¢¬¡ ¨šœ˜$™ ³¿˜¡ «Àšœ›^¨ ª†¢É›š¶›™Jš|™J@˜ ˜$§°šœ›XÊ »O˶Ì0½
Í ± §°¡†³¿«À@˜$˜$³¿²Oš}³¿ž›O«¬©€©J¡¢À«¬¢É›šH©$˜ šH›@©$ª†°šH™JšH«À©J¸
¯"¢¬Îœªšs™ šq«¬O˜ ²@© šœ© ªz™± @˜$™ še·šœ˜O¡†ª§œšq³¿©J¡†@© ªz™JšbÏп»
˜ ¯Ñ™JO˜¡B«Àšœ›B¨ ªO¨ ª†¢¬§°¡§œ›B›O˜¡B·O˜ ˜_© šœ›Bš·¡3¯3š¶›©$ª§œšœ›
š°˜r«À³@žOª†³¿¡O¢¬ª†š@½ Í ³*¨$ª§¶·¢É›O˜~³@·¡© šœ«¬«ÀšeOžJ¡šœ˜©$šZ›© ª
«0± §°¡†³¿«À@˜$˜$³¿²Oš“™Jšœ›B¾Òs›"™Jš}«À³b°³@¯3§œª†³sšœ›¡3™± š°˜_´_¢¬ª†@˜
º¶Ëz¼“½Ó<š·¡¡š¨$ªJ·§¶™J© ª†šX›šœª†³”³@¯"§°«À¢¬Oª§œš|¨$³¿ª «É³z›© ¢¬¡š
š·¡›š°ªU³B§·¡šœ˜$™J© š|¨g@©$ª ™± ³@©J¡ª†šœ›^«À@˜$²@© šœ© ª†›^™± @˜™Jš@½
Í ± §°¡†³¿«À@˜$˜$³¿²Ošª†š°«É³Å¡¢¬£<š¶› ¡.š·Æšœ·¡© §"šœ˜b¢À«¬«À© ¯"¢À˜$³¿˜¡
™J¢¬Æg§œªšœ˜¡†› šœ˜$™Jª†@¢¬¡†›+™Jš«É³z°³@¯"§°ªU³"²@ªUÔ@·šX¹“™Jšœ› ¥$ž ª†šœ›
@¨ ¡¢ÉÈO©$šœ›<š·¡<šœ˜“¯"šœ›© ªU³¿˜¡«É³UÇ$³@ª²Oš ·O«¬«Àšœ·¡§°š+¨$³¿ª)«Àšœ›
¾Òs›œ½@՘”©J¡¢À«¬¢É›†³¿˜¡'«F± §·¡U³¿«À@˜ ˜³¿²@š³@ž$›O«¬©”°@¯"¯"šª§°£‚§·¸
ª†š°˜$°š@Á¿·š°¡¡†š¨ ªJ°§œ™J© ª†š)šœ›¡V›©JÖz›†³¿˜¡š^¨g@© ª'°@˜¡ª†×@«Àš°ª
«É³3›¡†³@ž ¢À«¬¢¬¡§X™J©q›ØJ› ¡†Î°¯"š|¹3Ïz¼Ù¨ ªÎ¶›°½
™J© X
š ³@©J¦H™ ¢¤Æ§°ª†š°˜¡†›³¿§°ª†O›@«É›<¨ ª§¶›šœ˜¡†›^š°˜q›©$›¨šœ˜$›¢À@˜
™ ³@˜$›«0± ³¿¢Àªœ½ Í šª†š°˜™Jš°¯"š°˜¡.™JšXÞ$©$@ª†šœ›†·š°˜·š“æ‚£‚ª†³On¡¢À@˜
™Jš ™J§¶›š°¦ ·¢¬¡†³Å¡†¢¬O˜$›VÈ© ¢$°@˜$™J©$¢À›š°˜¡¹6© ˜ š§œ¯3¢É›†›¢À@˜"™Jš
Þ$© Oªš¶›°š°˜$°š¶é§°¡†³@˜O¡>™J§°¨gš°˜$™$³¿˜¡>™Jš<«À³¡š°¯"¨g§°ªU³Å¡†© ªšš·¡
™J©"¡U³¿©J¦"™± Ç_© ¯"¢É™J¢¬¡§ ™Jš«0± ³@¢¬ª¶Á¿¢À« šœ›¡'§°²³¿«Àš°¯"š°˜¡V˜ §¶·šœ›¸
›†³¿¢Àªš.™Jš.¯"šœ›© ªšœª^«F± §°´O@«À©J¡¢À@˜H³µ´@šœ«F± ³¿«¬¡¢¬¡©™Jš|™Jšœ›°šœ›
™Jšœ©J¦}¨$³@ª†³@¯"η¡ª†šœ›œ½
Í ³z·O«¬«É³¿žg@ªU³Å¡†¢¬O˜}Ý ©$²@š°ª6¹"¯"¢À›š°˜q_š°© ´_ª†š|™± ¢À¯3¸
¨g@ª¡†³¿˜¡U›x¯3ÅØOš°˜$›Z¨g@© ª*œ³¿ªU³@·¡§°ª†¢É›šœªb«0± ³¿¡¯"O›¨ Ç Îœªš
³¿© ¸4™Jš¶››©$›q™J©S›¢¤¡†š*™Jš*Òs³@«À³@ª²Oê š€ë+™Jšj˜ @¯Bž ª†š°©$›šœ›
°³@¯"¨$³¿²O˜ šœ›.™Jš"ª†š°«Àš°´@§¶›6¯3§°¡§œ@ª†@«À@²@¢ÉÈ© šœ›6¹H«F± ³¿¢É™Jš"™Jš
ªU³@™J¢ÀO›@˜$™ šœ›^¢À˜$› ¡U³¿«À«¬§¶› ™ ³@˜$› ™Jš¶› ž$³@«¬«À@˜$› @˜¡ §·¡†§X¯"š·¸
˜ §œšœ›Všœ˜”¨$³¿ªU³¿«À«¬Îœ«¬š³µ´Ošœ^™Jšœ›'¯3š¶›©$ªš¶›V³¿©”›O«F½_Ó<šœ›'™JO˜J¸
˜ §œšœ›+@˜¡¨gš°ª†¯3¢É›6™± §°¡†³@ž «¬¢Àª©$˜e¨ ª†¿¥$«>³Å¡†¯3›¨$Ç §°ª†¢ÀÈ© š
«À_œ³¿«'¯"ÅØOš°˜ ˜ §z›© ªX©$˜ š“³¿˜$˜ §°šzO©j›© ª© ˜j¯"@¢É›È© ¢
šœ›¡© ¡¢À«¬¢É›§)«À@ªU›%™Jš<«À³ªš¶·@˜› ¡†ª©$·¡¢À@˜|™Jš¶›%§œ´@§°˜$š°¯"š°˜¡†›œ½
Í ± ¢À˜$·šœª¡†¢¤¡†©$™Jšq™Jšu¯"šœ›© ªšu›© ª"«À³b¨ ª†¿£‚@˜™Jš°© ª"³Å¡†¯"¿¸
›¨ Ç §°ª†¢ÉÈO©$š+³¿©}›O«È© ¢š°˜“™ §œ·O© «Àššœ›¡<š¶› ¡†¢¬¯"§œš+¹Xšœ˜_´¢¬¸
ª†@˜q»²½ °¯ −2
Í š”ž$³Å¡†¢¬¯"š°˜¡X™J©NÓ<š°˜¡†ª†³@« Í ³@›š°ª|­³@·¢À«À¢¤¡ ØNæ4Ó Í ­Vé
Ê컿ϵ̚œ›¡<¢À˜$›¡†³¿«À«À§³¿©z°š°˜¡ª†š™J©zª†§œ›šœ³@©”™Jš+›© ª£³@·šš°¡)³
¨g@© ªVOžJŸ šœ·¡¢¬£™Jš¯"š¶›© ª†š°ªL«Àšœ›L¨ ª†@¨$ª¢À§·¡†§œ›>@¨J¡†¢ÀÈ© š¶›V™Jš
Ú &‘zõ’õfÃQrÛ;Xõޓ@i‰)¿Ü^h<{# Ý¥$˜j™± šœ›¡¢¬¸ «0± ³¿¡¯"O›¨ Ç Îœªš6³¿©J¸`™Jšœ›†›©$›<™ ©}›¢¤¡†š@½$Ó<š6°š°˜¡ª†š+°@˜¡¢Àš°˜¡
¯"š°ªB«À³sÈ©$³@˜¡¢¬¡§}™ šz«¬©$¯3¢Àઆš}™ š”Þ$© Oªš¶›°š°˜$°šz§°¯"¢É›š © ˜ š6› ¡U³Å¡†¢¬O˜”¯3§°¡§œ³¿¢À˜$›¢È©± © ˜“›Ø_›¡Îœ¯"š Íí ® Ý+èæ·£ ½
¨$³@ª6«É³H²@šœªžgš@Á¢À«Všœ›¡.˜$§œ·š¶››†³¿¢Àª†šB™± š·Æšœ·¡© šœª|© ˜ š”·Oª¸ ¨ Ç @¡e°¢¤¸`™Jš¶››@©$›Ué|È© ¢)š°˜_´@O¢¬šz™ šœ›X¢À¯"¨ © «É›¢À@˜›«É³@›š°ª
ª†šœn¡†¢¬O˜•™J©•˜ O¯ž ª†še™Jšb¨ Ç @¡@˜›bß@à°á`â†ã°á`ànäešœ˜å£‚@˜n¸ ™ ³@˜$›%«0± ³Å¡¯"›¨ Ç$આš'š·¡>¯3š¶›©$ªš'«É³È©$³@˜¡¢¬¡§<™ š«À© ¯"¢¬Îœªš
¡¢À@˜€™Jš“«É³q¨ ª†¿£‚O˜$™Jš°©$ªB³Å¡†¯"O›¨ Ç §°ª†¢ÉÈO©$š@½Õ'˜€š·Æš·¡¶Á¢À« ª†§·¡ª†J™J¢¬Æg©$›§°š"æ賵ثÀš°¢À²@Ǔš·¡Òq¢¬šµén½J®+š6¨$«¬©$›œÁ_«¬š.UÇ$³@¯3¨
šœ›¡˜ §¶·š¶››†³¿¢Àªš^™Jš ¨$ªšœ˜$™Jª†šš°˜”·O¯3¨ ¡š«É³X™J¢¬Æg©$›¢À@˜”™Jš ™Jš3´_© š3™ šœ›¡§°«Àšœ›†·O¨š¶›š¶› ¡.@ž›šœª´O§|¡O©J¡š¶›+«Àšœ›.»“¯3¢¬¸
«É³“«À© ¯"¢¬ÎœªšB¨$³¿ª6«Àšœ›6¯"@«À§œ·©$«¬š¶›+™Jš3«0± ³¿¡¯"O›¨ Ç ÎœªšHæ™J¢¬£ç¸ ˜_©J¡š¶›X¨$³@ª|© ˜jª†§œ›šœ³¿©x™Jšzœ³¿¯"§°ªU³@›.¢À˜J£‚ªU³¿ª†@© ²Ošœ›6¨O© ª
£‚©$›¢¬O˜N™JšHè ³µØ_«¬šœ¢¬²OÇé.³¿¢À˜$›¢)È© šz«É³e™ ¢¤Æ©$›¢¬O˜*™Jš}Òq¢Àš ›© ª´Oš°¢À«¬«Àš°ª«À³+·O© ´@šœª¡†© ªš˜©³¿²@šœ©$›š³¿©J¸`™Jš¶››©$›%™ ©›¢¬¡šO½
¿˜Àeú‰€ÛÁÃ`z&‘xÛ;*‰7&‘*ÃQgf‘ÃQ
û£ü‚üòû
W
ƒ¤ÿ%[email protected]$+g „ÿ%NQQ%4ÿ%‹ÿ%< jI‘ÿ%Kÿ Q`ÿ%g`ÿ%!4ÿî6 Qÿ%_ÿ%ÿ
ï *ÃQfV&‘Ã`>fËf(&;õò #
C‘R"EGF<DLhdgiLF<D–SLK/R#EcD–YN'EGF<HzTD]<]>N ]rYGDuF<DTHVU ]<K>F>N TLK>dgHVUˆY˜½ NGUŠSLwQSLU DLhfDLU*K}HV[ ]<DLF>wQSuE)R#FvbgDOYGSLK<DTLK<DLNGF
[email protected]• N HVF<D]<TDLU TDkD]<K-bqR¬YGSLK<DLF>hdeU)R#K>dgHVUZYGDbqRpPQSHVhfSLK>F>dgDkYGDfbqRpPQDLF>[D`nÕxR#U ]xNGU2EGF<DLhdgDLF"K<DLhE ]abgD
EGbqR#UmYGD-YGSLK<DTLK>dgHVU%[email protected]>[D"D]>K}YGSLK<DLF>hdeU Sk:‚ ð#„`EGNGdg]b½ deU TLbedeU)R#dg]<HVU%YGD-b½§[email protected]>[D"Y R#U ]uTD
EGbqR#UÅD]>K"D]<K>dehfSD`nC‘R¬­ P`NGF<D ~ n debebeN ]>K>F<DfbgD]BU HVK/R#K>dgHVU ]"NGK>debedg]/SD]"EcHVNGFBYGSTLF>deF<DkbqR¬PQSHVhfSLK>F>dgDŠYGDfbqR
Ø
EGF<HVE)R#PR#K>dgHVU%[email protected]>[cD`n
CDxEGbqR#U!YGD-YGSLK<DTLK>dgHVU%YGDxbqRfPQDLF>[D"D]>KuYGSLK<DLF>hdeU S-E)R#FuNGU!RIJN ]>K<DLhfDLUK–YGD]uYdeF<DTLK>dgHVU ]OhfD]>NGF<SD]
E)R#FbgD]EGd“sGDLbg]4YGDubqRxTR#hfSLF/RaT/l)RVjN DuEHVdeU*KoSLK/R#UKrEcHVU YGSLF<S}E)R#Fb½§R#hEGbedeK>N YGD–YN']>deP`U)R#bY R#U ]bgD}EGd“sGDLbn
C½ deU TDLF>K>deK>N YGDu]>NGF7b½ deU TLbedeU)R#dg]/HVUkYNEGbqR#UfYGSLEDLU YkYNfT<l)R#hEkYGDvwzN DYGDT<l)RVj*N D}ɏϢYGDvbqRATR#hfSLF/RaQYGD
bqR-K/R#debebgDxR#UGP`NGbqR#deF<Dx]<HVN ]rbqRVj*N DLbebgD–D]<KowzN DubqRBPQDLF>[cDxR#deU ]>dj*N DOb½¾SLE)R#dg]<]<DLNGFYGDA]/R-K>F/RVTDA]>NGFrbqR\TR#hfSLF/Rn
ÉHVNGF-NGU DfPQDLF>[DKÞ¼*EGdgj*N D`a˜bgD]-]<deh"NGbqR#K>dgHVU ]\Ï%HVU*K<D'‡vR#F>bgH¬EGF<SYdg]<DLU*K"NGU DdeU TDLF>K>deK>N YGD'[email protected]½¾HVF<YF<DkYGD
_a ~ ◦ ]>NGFbqRYdeF<DTLK>dgHVUmYNˆEGbqR#Un
CHVF<]vYGDubqR"]<DTHVU YGDASLK/R#EDOYGDubqR"F<DTHVU ]>K>F>N TLK>dgHVUˆR#UGP`NGbqR#deF<D`abqRxEcH`]>deK>dgHVUpDLKob½¾HVF>dgDLU*K/R#K>dgHVUˆYGDub½§RsGD
YGD-bqRkPQDLF>[D\Y R#U ]ubgDBEGbqR#U¤]<HVUKAYGSLK<DLF>hdeU SD]xDLUmNGK>debedg]/R#U*KAbgD]uK<DLhE ]AYGD\YGSTLbgDLU T<l DLhfDLUKBYGD]uEGd“sGDLbg]n
C½§RsDkD]>K-YGSt­ UGdE)R#F-bgDE)R#F/R#hfiLK>F<D'Y˜½ dehE)RVTLK
DLUK>F<DkbgDYGSLK<DTLK<[email protected]• N HVF<D]/TDLU TDkDLK-bqRpPQDLF>[cD`a
Rp
R#N jN DLbGdebzŒR#NGK4RIJHVNGK<DLF7bqRATHVUGU)R#dg]<]ÎR#U TDYGDvb½§R#UGP`bgD j*N DrŒR#deK;bqR–PQDLF>[cD}RwQDTvb½ l HVF>deÔHVU*K/R#bgD`n*C‘DvK<DLhE ]
Ψ
Y˜½§R#F>F>dewQSD YGDxbqR\beNGhdgiLF<DBY R#U ]}T<l)RVj*N D-EGd“sGDLbD]>K}YGHVUGU SxE)R#F}b½¾DtsEGF<D]<]>dgHVUŔ
ti
Rp
tan
ti = T 0 +
c
π − Ψ − χi
2
œ ~ n „*
D]>K7bgDK<DLhE ]4YGDvE)RV]<]/R#PQDYGDbqR–PQDLF>[cD}R#NfEGbeN ]7EGF<HzT<l D}YNfYGSLK<DTLK<DLNGFa D]>K7b½§R#UGP`bgD}Y˜½¾SLbgSLw`R#K>dgHVU
VH È
T0
χi
YNXEGd“sDLb DLKkTpbqR¤wzdeK<D]<]<D¬YGD¬bqR¤beNGhdgiLF<D`n4CD]fE)R#F/R#hfiLK>F<D]
a
DLK
]<HVUK'R#bgHVF<]'HV[GK<DLUN ]ŠDLU
i
R p T0
Ψ
hdeUGdehdg]/R#UKOb½¾DtszEGF<D]<]<dgHVUˆ]>NGdewVR#U*K<Df”
2
χ =
N X
ti − tobs 2
œ ~ n:ðQ
i
i=1
σi2
Y R#U ]7bqRVjN DLbebgD
F<DLEGF<S]<DLUK<DbgDvK<DLhE ]4hfD]>NGF<SYGD}YGSTLbgDLU T/l DLhfDLUKvYNEGd“sDLb DLK 2 NGU DuD]>K>dehkR#K>dgHVU
tobsi
i σi
YGDAb½ deU TDLF>K>deK>N YGD"YGDxhfD]>NGF<D"YGD n
ti
ÕxR#U ]bgD]uTRV]}HVÈ[email protected]/R#debebgD"R#UGP`NGbqR#deF<D"YGD-bqR\PQDLF>[D-D]<KŒŽR#de[GbgD`a)TDLK>K<DxEGF<H*TSYNGF<D-U DxEDLF>hfDLKuE)RV]uYGD
YGSLK<DLF>hdeU DLFb½¾HVF>dgDLUK/R#K>dgHVUmYGDOb½§RsDATR#FrNGUpP`F/R#U Y'U HVhB[GF<DxYGDATHVNGEGbgD]Aœ a EDLF>hfDLK>K<DLU*KuY˜½§RIJN ]>K<DLF
Rp Ψ
!"#%$'&()*
Ax
e
Rp
χi
ψ
FD
Point d’impact
détecttio
n
de
la
θi
Plan de
ge
rb
de la ger
b
e
e
W%W
plan horizontal
,.-F/.1 ~ n
‡rHVU*wQDLUK>dgHVU p
] PQSHVhfSLK>F>dgj*N D]kEcHVNGFfbqR¤F<DTHVU ]>K>F>N TLK>dgHVUžR#UGP`NGbqR#deF<D¤RwQDTˆbgDmYGSLK<DTLK<DLNGFŠYGD
Ø DLU 2 TDkœ’Y˜½§R#EGF<i]B‚ †n
• N HVF<D]<T€
~
bgD]uYGHVUGU SD]nc‡rDLK>K<D"YGSLPQSLU SLF<D]<TDLU TD\EcDLNGKuyLK>F<D-bgDLwQSD"E)R#FubqRkTHVUGU)R#dg]<]/R#U TD"Y˜½ NGU D"deUŒ’HVF>hkR#K>dgHVU!K<[email protected]
EHVF<[email protected]]>NGEGEGbgSLhfDLU*K/R#deF<D`a˜EcHVNGwVR#U*K-yLK>F<D"ŒŽHVNGF>[email protected])R#FANGU [email protected]]x]>K/R#K>dgHVU ]xYN¤F<S]<DR#NªYGD\]<NGFJŒŽRVTDR¼R#U*K
YGSTLbgDLU T<l S`nC½§RIJHVNGK}Y˜½ NGU D"]>K/R#K>dgHVUˆYN%¡zÕ R#hfSLbedgHVF<D-P`F/R#U YGDLhfDLUKObqR\F<DTHVU ]>K>F>N TLK>dgHVU%R#UGP`NGbqR#deF<D`a)EDLFJ
hfDLK>K/R#UKuR#deU ]>dcY˜½¾HV[GK<DLUGdeFNGU DOdeU TDLF>K>deK>N YGDxY˜½¾DLUwzdeF<HVN
U ð`_"h±]>NGFrbqRBEH`]>deK>dgHVUpYNpEHVdeU*KY˜½ dehE)RVTLKvDLKYGD
_:a ð ◦ ]>NGFb½¾HVF>dgDLUK/R#K>dgHVU¤YGDAb½§RsGDf:‚ ðJð†n
C½¾HV[ ]<DLF>w`R#K>dgHVUËY˜½ NGUZhfyLhfD¬SLwQSLU DLhfDLUKf]>dehBNGbeK/R#U SLhfDLUK'E)R#[email protected]<DTLK<DLNGF<]fYGDk• N HVF<D]<TDLU TD
EDLF>hfDLKOSLPR#bgDLhfDLUKAYGD"YGSLK<DLF>hdeU DLFub½¾HVF>dgDLU*K/R#K>dgHVUªYGDxb½§RsD-E)R#FudeUK<DLF<]/DTLK>dgHVU!YGD]OYGDLNspEGbqR#U ]uF<DTHVU ]J
K>F>NGdeK<]]/R#U ]7NGK>debedg]/DLFY˜½ deUŒ’HVF>hkR#K>dgHVU'K<DLhEHVF<DLbebgD`n*‡rDvKÞ¼*EcD}Y˜½¾SLwQSLU DLhfDLU*K<]a*R#EGEcDLbgS]u÷L]>K<SLF<SH ÷VaQ]/HVUK7[GdgDLU
hdgDLNsŠF<DTHVU ]<K>F>NGdeK<]uj*N DObgD]uSLwQSLU DLhfDLUK<]B÷thfHVU H ÷xP`F/ÖVTD"R#NspTHVhE)R#F/R#dg]/HVU ]uj*N½ debD]>KvEH`]<]<de[GbgDxY˜½¾DtŒò
Œ’DTLK>N DLFrRwQDTObgD]YGHVUGU SD]oYGD]oYGDLNsfK<SLbgD]<THVEcD]n*‡ol)RVj*N DuSLwQSLU DLhfDLUKvY˜½¾SLU DLF>P`dgD–]>NGESLF>dgDLNGF<DO» _ 19 D†€

]<DLF/RŠHV[ ]<DLF>[email protected])R#F-R#NªhfHVdeU ]ANGUªK<SLbgD]<THVEcD`a˜DLK–E)R#F"R#N¤hfHVdeU ]"Ñ'K<SLbgS]<THVEcD]xY R#U ]-DLU*w*deF<HVU _ Y YGD]
Ø
TRV]n
+-,†ñ
ò 579š}1')·c6B9L:u1
³\5ôó@õ¶71\5£·>öu:<5
™ U D-ŒŽHVdg]–bqRkPQSHVhfSLK>F>dgDYGD\bqR'PQDLF>[cD\YGSLK<DLF>hdeU SD`acbgD\EGF<H#­ b£bgHVUGP`deK>N YdeU)R#b4YGDBbqR'PQDLF>[[email protected]]>KAD]>K>dehfS
O
‚:ð#„Vc»"E)R#F>K>deFrYGD]odeUK<DLU ]<deK<S]vhfD]>NGF<SD]4E)R#FrbgD]oEGl HVK<HVh"NGbeK>deEGbedgTR#K<DLNGF<]}K<HVN T/l S]nC½ deUK<DLU ]<deK<S–YNp]>deP`U)R#b
DLUˆ]<HVF>K>dgD-Y˜½ NGUmÉ4Ï YGSLEcDLU YˆYGDxU HVhB[GF<DLNsŠŒRVTLK<DLNGF<]}K<DLbg]ujN D”
bqR'K/R#[email protected]'PQDLF>[D
a˜T`½¾D]>K-»ŠYdeF<[email protected]@U HVh"[GF<DfYGD\E)R#F>K>dgTLNGbgD]-T/l)R#F>PQSD]-EGF<S]<DLU*K<D]-Y R#U ]xbqR
∗
Ne
YdeF<DTLK>dgHVU%YNmEGd“sDLbTHVU ]>dgYGSLF<S
¿˜Àq ï ÃQ>f‘V&Ã`fÛ;%õÞeÁf#(
WV[
[email protected]]>K/R#U TD-Y˜½¾HV[ ]<DLF>w`R#K>dgHVUa b½ deU*K<DLU ]>deK<S-w`R#F>dqR#UKOTHVhhfDxb½ deUwQDLF<]<DBYNmTR#F>F<S-YGDxTDLK>K<D-Ydg]<K/R#U TD
b½¾Dt¥kTRVTLdeK<S'YNÆYGSLK<DTLK<DLNGF¬”˜bqRˆ]>NGFJŒRVTDkYGD'THVbebgDTLK>dgHVUXYN‰hdeF<HVdeFBDLKBb½¾Dt¥kTRVTLdeK<SpjN)R#U*K>dgjN DŠYN
∗
EGl HVK<HVh"NGbeK>deEGbedgTR#K<DLNGFkœ’YGSLEcDLU Y R#UK–[email protected]`N DLNGFuY˜½¾HVU YGD
YN¬F<DLU YGDLhfDLUKAYGD–• N HVF<D]<TDLU TD`a)YGSLEcDLU Y R#UK–YGD]}THVU YdeK>dgHVU ]AR#K>hfH`]>EGl SLF>dgjN D]
∗
YN¬K<DLhE ]uY˜½¾DtsEH`]<deK>dgHVUmYN¬EGd“sDLb;R#Np• NspbeNGhdeU DLNs
∗
‡rHVUGU)R#dg]<]/R#U*[email protected]<DLbqR#K>dgHVU¤DLUK>F<DBbgD-U HVhB[GF<DBYGD"EGl HVK<HVU ]OHV[ ]/DLF>wQS]ODLK}bqRfK/R#debebgD"YGD"bqRPQDLF>[cD
a bgD
Ne
EGF<H#­ bbgHVUGP`deK>N YdeU)R#bEDLNGKyLK>F<D4TR#bgTLNGbgS`n#C½¾DLF>F<DLNGF;]<¼z]>K<SLhkR#K>dgj*N D4]<NGF‘TDLK>K<DoTHVU*wQDLF<]>[email protected]]>KRVTLK>N DLbebgDLhfDLUK
YGD Ñ Y :‚ ð`^o»'bqRVj*N [email protected]ŒR#NGK"RI>HVNGK<DLF–NGU DDLF>F<DLNGF-]<¼z]>K<SLhkR#K>dgj*N [email protected] ~jY ‚ 4]>NGFAbqRkw`R#bgDLNGF-YN


Ø
F<DLU YGDLhfDLUK£YGD7• N HVF<D]<TDLU TD`aVR#deU ]>d*j*N½¾DLUwzdeF<HVq
U ð Y EGF<HwQDLU)R#U*K7YGD]‘hfD]>NGF<D];YGD7EGF<D]<]<dgHVUaK<DLhEcSLF/R#K>NGF<D
DLKlNGhdgYdeK<S`n
∗
,6-0/.1 ~ ¤n ÷ |£sGDLhEGbgD–Y˜½¾SLwQSLU DLhfDLU*K}HV[ ]<DLF>wQSOE)R#F4bgD–YGSLK<DTLK<DLNGFvYGD}• N HVF<D]<TDLU TD`azF<DTHVU ]>K>F>NGdeK}RwQDTONGU D
SLU DLF>P`dgDŠ]<2NGESLF>dgDLNGF<D¬» _ 20 D†€Ù[email protected]#UGP`bgDŠÔSLUGdeK>l)R#b}YGDk„÷ ◦ n;C½§RIJN ]>K<DLhfDLUKfYGD'bqRmŒ’HVU TLK>dgHVUXYGD
xR#dg]<]<DLF\ŸudebebqRV]\]>NGF\bgDfEGF<H#­ bvbgHVUGP`deK>N YdeU)R#bD]>K"F<DLEGF<S]<DLU*K<Sˆ»¬PR#N T/l DŠDLK\bgD]BEGd“sDLbg]\K<HVN T<l S]\bgHVF<]@YN
E)RV]<]/R#[email protected]>[cDf»¬YF<HVdeK<D`a˜[email protected]/R¼QHVUÅYGD]-TDLF<TLbgD]BSLK/R#UK-EGF<HVEcHVF>K>dgHVUGU DLbr»ŠbqR¬j*N)R#UK>deK<SkYGDf]>deP`U)R#b
YGSLK<DTLK<SD`n
DLKbgDˆU HVh"[GF<DmYGDpE)R#F>K>dgTLNGbgD]pR#N
M ­ UËYGDmYGSLK<DLF>hdeU DLFkbgDphkRsdeh"NGh YGDmYGSLwQDLbgHVEGEDLhfDLU*K
}
Xmax
hkRszdehBNGh
aNGU DAŒ’HVU TLK>dgHVU%YGDBxR#dg]<]<DLFJ†ŸOdebebqRV]}D]>KuRIJN ]>K<SDx]>NGF}TD-EGF<H#­ b4”
Nmax
Ne (X) = Nmax
X − Xint
Xmax − Xint
Xmax −Xint
λ
exp
Xmax − X
λ
œ~ n 
Ø
VH È
D]>[email protected]<H#ŒŽHVU YGDLNGFuYNˆEcHVdeUKOYGDAEGF<DLhdgiLF<D"deUK<DLF/RVTLK>dgHVU¤YGDxbqR\E)R#F>K>dgTLNGbgD-EGF>dehkR#deF<DBDLK D]<K}bqR
Xint
λ
bgHVUGP`N DLNGFY˜½§R#K>K<SLUN)R#K>dgHVUøaVYGDrb½¾HVF<YF<DYGDX÷V_OP n¾TLh −2 Y R#U ]7YGD]7THVU YdeK>dgHVU ]oR#K>hfH`]>EGl SLF>dgjN D]4]>K/R#U Y R#F<Y˜n
Jù ä›ã}ä§ìÎàtú éßJéâ)û$ü ã<áŽá 9JàéVã/áê§ìÎàzûO9Jñ`ßJàû#ã/àá<ûßJëç¯ìtà_ûê¾áê›ìÎà#ë;ã/áïìtëŽñOý9¯âê›èé#߯ëJðVßJä§ä§ß6ûìtê¾á<þ¯áŽâßïßJëŽéâ 9JßÿvçýVãÎèéߏñâê§ëŽß
ûßûìÎà#àO9JßJë
!"#%$'&()*
W%c
C‘RA­ P`NGF<D ~ n¤÷AhfHVU*K>F<DuNGUŠDtsDLhEGbgDOY˜½¾SLwQSLU DLhfDLUKvF<DTHVU ]>K>F>NGdeKrE)R#FbgDuYGSLK<DTLK<DLNGFvYGD• N HVF<D]<TDLU TDAR#deU ]>d
jN DAbqR\K>F/RVTD"YNˆE)RV]<]/R#PQDxYGDxbqR\PQDLF>[cD-]>NGFbgD]}EGd“sDLbg]}YGD]OTR#hfSLF/RV]n
450
400
350
300
250
200
150
100
50
0
0
200
400
600
800
1000
1200
,.-F/.1 ~ n§¦ –HVh"[GF<D\YGD"EGl HVK<HzSLbgDTLK>F<HVU ]AF<DWLN ]ADLUˆŒŽHVU TLK>dgHVUªYGD-bqRkEGF<H#Œ’HVU YGDLNGF-R#K>hfH`]>EGl SLF>dgj*N D\EHVNGF
GN U D7PQDLF>[cD£2 deUGdeK>dgSDE)R#FNGU-EGF<HVK<HVUBYGD _ 19 D€XY˜½§R#EGF<D]4‚:ð†nC½§R#UGP`bgDÔSLUGdeK>l)R#bzD]>KYGD7„$ð ◦ DLKbgD7THzDLNGF‘YGD7bqR
PQDLF>[DuD]>Ko]>deK>N SO» _ Ç h×YNŠYGSLK<DTLK<DL NGFn*CD]oYd“—cSLF<DLU*K<D]vTHVUK>F>de[GNGK>dgHVU ]R#N']>deP`U)R#b]<HVUKoF<DLEGF<S]<DLUK<SD]A”
beNGhdgiLF<z
D (‘T/l DLF<DLU Ç  HwmYdeF<DTLK<D"HVNmYd“—N ]/SDxDLK• N HVF<D]<TDLU TD`n
C D]BE)R#F>K>dgTLNGbgD]BT/l)R#F>PQSD]@YGDfbqR¬PQDLF>[cDfdeU YNGdg]/[email protected] R#U ]"b½§R#deFBNGU DkSLhdg]<]>dgHVUÆYGD• N HVF<D]<TDLU TD'dg]<H#
K>F<HVEDfhkR#dg]BSLPR#bgDLhfDLU*KBNGU2F/R¼QHVUGU DLhfDLUKs(T<l DLF<DLU Ç HwÆTHVbebedehfSkY R#U ]"bqRˆYdeF<DTLK>dgHVUÅYGDkEGF<HVE)R#PR#K>dgHVUn
‡rD–F/R¼QHVUGU DLhfDLUK–THVU ]>K>deK>N D–EHVNGFvbqRBhfD]>NGF<DxYGD–bqR"• N HVF<D]<TDLU TDANGUp[GF>NGdeK}YGDOŒ’HVU YpYGHVUKdebŒR#NGKv]½§RŒò
ŒF/R#U T/lGdeF"‚ _†nGC‘RB­ P`NGF<D ~ n§¦\F<DLEGF<S]/DLUK<D-bgD]Yd“—˜SLF<DLUK<D]OTHVUK>F>de[GNGK>dgHVU ]AR#Nˆ]>deP`U)R#bDLUˆ]<HVF>K>dgDxY˜½ NGUmÉ4Ï
Ø
DLU¬ŒŽHVU TLK>dgHVU%YGDxbqRfEGF<H#Œ’HVU YGDLNGFAR#K>hfH`]>EGl SLF>dgjN D`nC‘[email protected]<}
D (‘T/l DLF<DLU Ç HwmEcDLNGKOyLK>F<D"Yd“—N ]<SD"E)R#F}bgD]
hfyLhfD]7EGF<HzTD]<]>N ]7j*NGd)R—cDTLK<DLU*KbqRO• N HVF<D]<TDLU TDBœ R¼zbgDLdeP`lfDLKÏmdgD;DLKYGDLwzdgDLUK4NGU DE)R#F>K7dehEHVF>K/R#U*K<D
a
YNˆ]>deP`U)R#bEGF<i]}YN%]<HVb4œEHVNGFubgD]P`F/R#U YGD]}EGF<H#ŒŽHVU YGDLNGF<X
] 7-tn
É HVNGFvYGD]rPQDLF>[cD]HV[ ]<DLF>wQSD]v]<HVN ]rNGUŠŒŽR#de[GbgDAR#UGP`bgDADLUK>F<DAb½§RsGDODLKrbgDAYGSLK<DTLK<DLNGFxœdeUŒŽSLF>dgDLNGFu»\Ñ`_ ◦ ta
;
bqR'beNGhdgiLF<Ds(‘T/l DLF<DLU Ç Hw2j*NGd7R#K>K<DLdeU*K"YdeF<DTLK<DLhfDLUK"[email protected]<SLbgD]<[email protected]#F>PQDLhfDLUK"YGHVhdeU DLFBDLUªdeUK<DLU
]>deK<SxbqR\• N HVF<D]/TDLU TD`nc‡rDxKÞ¼*EcD-Y˜½¾SLwQSLU DLhfDLU*K<]–THVU*K<DLU)R#UKANGU D-K>F<HVEˆŒ’HVF>K<DBTHVUK/R#hdeU)R#K>dgHVU¤DLUmbeNGhdgiLF<D
(‘T/lGF<DLU Ç HwmD]>KF<D¯I>DLK<SxYGDxb½§R#U)R#be¼]<D`n
™ U D-ŒŽHVdg]ubgD]–E)R#F/R#hfiLK>F<D]
O
a
DLK
YGSLK<DLF>hdeU S]ab½¾SLU DLF>P`[email protected]"bqRkE)R#F>K>[email protected]>dehkR#deF<D
Xmax Nmax
Xint
D]>K}D]>K>dehfSDxE)R#Fb½ deU*K<DLF>hfSYdqR#deF<[email protected]½ deUK<SLP`F/R#bgD\YN¬EGF<H#­ bbgHVUGP`deK>N YdeU)R#bo”
¿˜Àq ï ÃQ>f‘V&Ã`fÛ;%õÞeÁf#(
Eem
Wv
Ec
=
×
X0
Z
Ne (X) dX
œ ~ n¤÷`
VH N
D]>KAb½¾SLU DLF>P`dgDK<HVK/R#bgDfYdg]<]>deEcSD\E)R#FxbqRŠTHVhEH`]/R#U*K<DfSLbgDTLK>F<HVhkR#P`U SLK>dgj*N [email protected]'PQDLF>[D`a
D]>K
Eem
Ec
b½¾SLU DLF>P`dgD"TLF>deK>dgj*N D-YGD]uSLbgDTLK>F<HVU ]–DLK
[email protected]`N DLNGF–YGDxF/RVYdqR#K>dgHVU%Y R#U ]ub½§R#deFBœ’TtŒo‡ol)R#EGdeK>F<D\ADLF>[cD]Νtn
X0
C½¾SLU DLF>P`dgDªYGD!bqR2E)R#F>K>dgTLNGbgDª]½¾HV[GK>dgDLU*KˆYGHVU T¤DLU«THVhEGK/R#U*KˆbgD%U HVh"[GF<DªK<HVK/R#bxY˜½¾SLbgDTLK>F<HVU ]¬EGF<HzYNGdeK<]a
h"NGbeK>deEGbedgSxE)R#Fvb½¾SLU DLF>P`dgD"»BbqRVj*N DLbebgDAdebg]r]/HVUKEGF<HzYNGdeK<]
n|£sESLF>dehfDLU*K/R#bgDLhfDLUKabqRBK/R#debebgDxYGDObqR\PQDLF>[cD
Ec
hfD]>NGF<SDuTHVF>F<D]<EHVU YŠR#NkU HVh"[GF<D–YGD}E)R#F>K>dgTLNGbgD]rT<l)R#F>PQSD]»-bqRAEGF<H#Œ’HVU YGDLNGF a*TDLEDLU Y R#U*KobgD]
Ne (X)
X
SLbgDTLK>F<HVU ]–SLK/R#U*KA[DR#N THVNGE!EGbeN ]uU HVhB[GF<DLNscab½¾DLF>F<DLNGFxTHVhhdg]<D\DLU¤RV]/]>dehdebqR#UK–bgD
hfD]>NGF<[email protected]#N
Ne (X)
w*F/R#dD]>KK<HVK/R#bgDLhfDLU*KOU SLP`bedePQDR#[GbgD`n
™uU D–THVF>F<DTLK>dgHVUmYG”
D ð\» ð Y D]>K}RI>HVNGK<SDx»"b½¾SLU DLF>P`dgD`a R­ U¬YGD–EGF<DLU YF<DxDLU¬THVhEGK<D–bqR-Œ’F/RVTLK>dgHVUˆYGD

b½¾SLU DLF>P`dgD}DLhEHVF>K<SD}E)R#F7bgD]4R#NGK>F<D]THVhEcH`]/R#UK<D]oYGDbqR–PQDLF>[D]u”Vh"N HVUGdgj*N D`a`U DLNGK>F>deU H-DLKl)RVYF<HVUGdgj*N D`n
‡rDLK>K<DxTHVF>F<DTLK>dgHVUmD]>KvŒR#de[GbgDLhfDLUKuYGSLEcDLU Y R#UK<D-YN¬hfHzYGiLbgDxYGDx]>dehBNGbqR#K>dgHVU%YGD]vPQDLF>[cD]NGK>debedg]<SxDLKvK<HVNGK
E)R#F>K>dgTLNGbedgiLF<DLhfDLUK-YNmhfHzYGiLbgD-l)RVYF<HVUGdgj*N D`ab½¾DLF>F<DLNGF–]>¼]>K<SLhkR#K>dgj*N D"]>NGFObqRfTHVF>F<DTLK>dgHVU¤DLU%SLU DLF>P`dgD\D]>K
D]>K>dehfSD"»@DLUwzdeF<HVU ~eY ‚ ÑV†n
Ø
[
!"#%$'&()*
Ì
Ä ¾fÀ
ÏÀ¾
ÏÇÂÀ†¾
Ë
ÍÏ΂À Á
ÉÊÀ Á5ÁÈÀ
Ì
ÏÀ
u¼
½¿¾fÀÂÁfÃ Ä ÆÈÇÉÊÁÈÀ
!
Ì
"
#$%&
()(
À
*
'
Æ5Ä+( À
Í
Á,†¾fÀ†ÄÍ
ÏÀ
¾ ÍÏÁ,-—Ä( À
Ð ÑÒnÓ]ÔÕÑÖ!ÓY×nØ6ÙÚ#ÛÝÜÝÞ߶à
õâá Ù×>×ߏÚuÜ Ù!ð#×ßtÖ*ÓÕÞ>ÑÒ/ßÙ*Ó%Ö1ßÒnÓ]ޕñÙ1×ßùÖ1ߊê>á ã—äÒAßnÞåCÙ*Ü æ!ەÞßçÛ=ßnÞnÞ>ßTèŠÓ!é1ßޗÞ>ߏÚ#Þ>ÑÒ/ßnð#܏ß{êúÙâÒnÓ]ޕñÙ1×ßtߕõŠß×/ÜÝەåCß
Ö1ßtÖ1Ñ܏ß>×/ÜÝÛÝæ*ð|ەð‡ÜÑ=é!Þ>Ñ>ßvÒÓIފê>á Ù!ð]é!êxߊÒæ*ê;Û=Ö1ßvë/Óá Ù¹Ù×>×/Ó]Ô¥ÓIêFÑâêxßtÖ1Ñ܏ß×A܏ßnÓ]ÞTڑßnð#ÖÙ*ð‡ÜÊÓ]ðuß Ú‘ÑnÞnÛ=æAÖ1ßvÖ]ß Ú#ÞnÛhÒAß
Ö1ßNÖæ!ð~ð_ÑßÒû¹ßnêGêxßaÒ/ßYÔÕßÒnÓ]Þ>ß ßnð Ô 2 úÒ úÒnÞ õ3Ùþ×>æ*ð~ð#Ù!ÛGÒÒ/Ù!ð#×ßfÖ1ßYê>á Ù×>×ߏÚuÜ Ù!ð#×ß5ßÒnÜð_Ñ×ßÒÒ/Ù!ÛGÞ>ßÕÚ6æ*Ó]Þ
ð#æ!ÞnÔ¶Ù*ê;ÛGÒ/ßnފêFß{Ò Ú‘ß×AÜÝÞßßnðNÑnðußnÞÝé*Û=ßâÖ1ßÒtÑåCÑnð_ßnÔÕßnð#ܕÒû‹Ù .,ð|Ö á æ1ä܏ßð~ÛGފÓ]"ð Ó*ï¹Ö1ß{Þ0Ù /æ!ð‘Òt×>æCÒnÔÛ=ëÓ~ßÒâßnð
EeV.m2 .s.sr
−1 è .,ðfÖ á æ1ä܏ßnð~ەÞtÓIðYêFæ*Ü ÖWá Ñnå!Ñnð_ßnÔÕßnð‡Ü•ÒŸÙ2/*Ù*ð‡ÜÓIð_ߥäæ!ð~ð_ߟëӑÙ!ê„ە܏ÑÖ1ßâÞ>ß×>æ!ð‘ÒÜÝÞӑ×/ÜÝÛ=æ!ð‘û‹Û•êWßÒÜ,ð_Ñ×ßÒ43
1
ÒÙ!ەÞßaÖ á ەԊÚ6æCÒAßnÞÕÓIðl×Þnە܏ÿÞß|Ö1ßdÒ/ÑêFß×AÜÝÛÝæ!ð Ö1ßÒaÑnåCÑnð_ßÔ¹ßð‡Ü•ÒdÒnÓAÚ1Ú#êFÑÔ¹ßð‡Ü Ù!ەÞßYàêxßwö15Cûùë/Ó]ÛtÖæ!ÛÝÜ6ÜÝÞ>ß
ÛGԊÚ#êxÑnÔÕßnð‡ÜÑŸÖÙ!ð‘Òtêxß×>Ù!êF×Ó]êÖ1ßvêá Ù×>×ߏÚ_Ü Ù!ð#×ß
7âÙ*ð‘Ò›êFÙ|é1Ù!Ô¥Ô¹ßYÖ á Ñnð_ßnÞ=é!Û ßN0
æ 8îêúÙ:Ú#Þæ1äÙ1äÛGê„ە܏ÑNÖ1ßYÖ1Ñ܏ß>×/ÜÝÛÝæ*ð ÖWá ÓIð Ñnå!Ñnð_ßnÔÕßnð‡Ü¶ßÒÜvەðAñ/ÑÞÛ ßnÓIÞßYì
òûׇá ßÒnܶìîÖ!ەÞ>ß¹Ú6æ!ÓIÞ E . 3 9 ߑíuûêúÙÈÒÓIޕñÙ×ßwߕõŠß×/ÜÝەåCß5Ö1ßfÖ1Ñ܏ß>×/ÜÝÛÝæ*ð ڑßnÓ6ÜÒ/ßf×>Ù!êF×ÓIêFßÞ ì|ڑÙ!ÞnÜÝەÞ
Ö1ßÒñæ!ð_×/ÜÝÛÝæ*ð‘Ò õö§çûTëÓIۗÖæ!ð‡ð_ßnð‡ÜTêFÙÚ_Þ'æ1ä>Ù1änەê;ÛÝ܏ћÖ1ßÕÖ]Ñ×êxßnð#×nØ6ßÔ¹ßð‡Ü{ÛGð#Ö*ÛGåAÛ=Ö!Ó~ßnê•êFßdÖ á Ó]ð_ߥÒnÜ Ù*ÜÝÛ=æ!ðîßnð
ñæ!ð_×/ÜÝÛÝæ*ð%Ö1ßÒ/ÙÖ*ÛhÒÜ Ù!ð#×ßùìŸê>á ÙAï]ß ~çTæ!ÓIޗêxßÒ{ÑnðußnÞÝé*Û=ßҋÚ#ê„Ó1Ò é!ÞÙ!ð#Ö]ßÒ{ëÓ~ß ∼ 3 9 ߑíuûWêúÙâÒÓIޕñÙ×ßtߕõŠß×/ÜÝەåCß
Ö1ßÖ1Ñ܏ß×/ÜÝÛ=æ!ðÈßÒܧÚ#ÓIÞ>ßnÔÕßnð‡ÜÊé1Ñæ!ÔÕÑÜÝÞnÛÝë/ӑß
õ3Ù:ÒÓIÞGñ/Ù×ߛߕõŠß×/ÜÝەåCß%Ö1ßÕÖ1Ñ܏ß×AÜÝÛÝæ!ðîÖ á Ó]ðußÕ×>æ*:
ð .Wé!ÓIÞ'ÙÜÝÛÝæ!ðîÖæ!ð‡ð_Ñ>ßdÖ!ÓfÞÑÒAßÙ!Ó%Ú~ßnÓIÜÒ/ßd×>Ù!êF×ÓIêFßnÞ%ßnð
×>æ!ԊÚ_Ü Ù!ð‡Ü_êxß ð#æ!ÔÕänÞ>ßÖ1ßT×ßêGê„Ó]êxßÒ Ø6ßïÙAé1æ!ð#Ù!êxßÒWÚ#ÞÑnÒ/ßnð‡ÜßnÒÖÙ!ð~Ò,êFß Þ>ÑÒ/ß>Ù!Ó1û ßÜ3ßnð¶Ô¥ÓIê„ÜÝÛ Ú#ê„ÛÝÙ!ð#Ü_×ßð#æ!ÔÕänÞ>ß
Ú6Ù!ޗêFٟÒÓIޕñÙ×ßvߕõŠß×/ÜÝەåCßâÖ á ÓIð_ßù×ßêGê„Ó]êxß yõâá Ù×>×ߏÚ_Ü Ù!ð#×ßv×ÓIÔ¥Ó]êxÑ>ß{ÒnÓ]ފÓ]ðuߋڑÑnÞnÛÝæEÖ1ßtÖ1ß,Ú#ÞnÛGÒ/ßùÖ1ßùÖæ*ð~ð_Ñ>ßÒ
Ò/ߥ×>Ù!êF×ÓIêFßdßnð ەð‡ÜÑ=é!ÞÙ!ð‡Ü ÒÓIÞâêxß܏ßnԊڇÒâê>á Ù×>×ߏÚuÜ Ù!ð#×ß¹ÛGð~ÒnÜ Ù!ð‡Ü Ù*ð_Ñ>ߥÖ!ÓYÞÑÒAßÙ!Ó]ûßð›Ú#Þßð#Ù!ð‡ÜßnðÈ×>æ*ԗÚu܏ß
êá Ñnåæ!ê„Ó6ÜÝÛÝæ!ðwÖ1ßâêFÙd×æ!:ð .Wé!ÓIÞÙ*ÜÝÛÝæ*ð|é1Ñæ!ÔÕÑÜÝÞnÛÝë/ӑ߶Ö*<
Ó ;=7 Ù!ÓY×>æ*Ó]ÞÒÖ!Ó|܏ßԗÚ#Ò [zú
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
Š",/.
ƒ¶£=\1\:?:[email protected]}1CB
>
É HVNGFuTLbqR#F>d“­)DLFb½¾DtszEcH`]<SxDLKdeU*K>F<H*YNGdeF<DxbgD]Yd“—˜SLF<DLUK<]OTHVU TDLEGK<]F<DLbqR#K>d“ŒŽ]u»Bb½§RVTTDLEGK/R#U TD"YNpF<S]<DR#N
;
YGDx]>NGFJŒRVTD-YGDAb½ š [ ]<DLF>w`R#K<HVdeF<D"É7dgDLF>F<D-MONGPQDLFaGU HVN ]NGK>debedg]/DLF<HVU ]}bgD]}YGSt­ UGdeK>dgHVU ]–]>NGdew`R#UK<D]"”
?
’œ DLUph 2 u”*bqR\]>NGFJŒRVTD–Dt—cDTLK>dewQDxYGD–YGSLK<DTLK>dgHVU
YNˆF<S]<DR#N%¡zÕ YGS]>deP`U D-b½ deUK<SLP`F/R#bgD"YGD"]/R\]>NGFJŒŽRVTD-EHVU YGSLF<SDxE)R#FbqR\EGF<HV[)R#[GdebedeK<SBYGDxYGSLK<DTLK>dgHVUŔ
Seff (E, θ, ϕ)
D1EGFIHKJMLONPNAQRNOLTSIUWV=NPXMNYX[ZTS\N]L^SIU`_ba
Seff (E, θ, ϕ) =
HVÈ
Z Z
dxdy PE,θ,ϕ (x, y)
œ„ n 

D ]>KobqR"EGF<HV[)R#[GdebedeK<SxYGDAYGSLK<DTLK<DLFvNGUŠSLwQSLU DLhfDLU*KuY˜½¾SLU DLF>P`dgD
Y R#U ]rbqRBYdeF<DTLK>dgHVU
E
DLUŠNGUkEcHVdeUK}R#N']<HVb
YNŠF<S]<DR#NnT–HVN ]oEF<STLdg]/DLF<HVU ]ob½¾STLF>deK>NGF<DAYGDubqR-EGF<HV[)R#[GdebedeK<SAYGD
(θ, ϕ)
(x, y)
YGSLK<DTLK>dgHVUmE)R#[email protected]]>NGdeK<DxbgHVF<]}[email protected]­ UGdeK>dgHVU!YGD]UGdewQDR#NsˆYGD-YGSTLbgDLU T<l DLhfDLU*K3(v„'DLKX(Xðn
œ’DLU'h 2 n¾]>FΝu”YGS]>deP`U DOb½ deUK<SLP`F/R#bgDxYGDObqRB]>NGFJŒŽRVTDODt—cDTLK>dewQDAYGD
? cdLOL]N:e1S\J[a1L]N<UWa1f]S\Jba1S\J[agZON a(E)
YGSLK<DTLK>dgHVU
]>NGFb½§R#UGP`bgD-]/HVbedgYGD-THVNGwQDLF>K}E)R#F}bgDxYGSLK<DTLK<DLNGF\”
PE,θ,ϕ (x, y)
Seff (E, θ, ϕ)
a(E) =
Z
θmax
sin θ cos θ dθ
Z
2π
dϕ Seff (E, θ, ϕ)
œ„ n§ÑQ
0
0
HVÈ
D]>Kvb½§R#UGP`bgDAÔSLUGdeK>l)R#bhkRszdehkR#b‘R#N j*N DLb]½¾Dt—˜DTLK>N DAbqRBF<DTHVU ]>K>F>N TLK>dgHVU%YGD]SLwQSLU DLhfDLU*K<]n
θmax
CDBŒRVTLK<DLNGF
EGF<HwzdgDLUK-YN%ŒŽR#deKAj*N D"bgD\YGSLK<DTLK<DLNGFxD]>KOEGbqR#UaDLKAjN½ deb;EGF<S]<DLUK<[email protected] D\]>NGFJŒRVTD
cos θ
Y R#U ]}[email protected]>K>dgHVU%Y˜½§R#UGP`bgD-]<HVbedgYGD
TDLU*K>F<SDBR#NGK<HVNGFuYGD n
S × cos θ
dΩ
θ
œ’DLU'h 2 n¾]>Fn¾]t”D]<K4b½ deUK<SLP`F/R#bgDAYGDub½§RVTTDLEGK/R#U TDOdeU ]<K/R#UK/R#U SDA]>NGF4bgDuK<DLhE ]n
? cdLOL]N:e1S\J[a1L]N A(E)
É£R#F£DtsGDLhEGbgD`a#bgHVF<]YGDobqROTHVU ]>K>F>N TLK>dgHVUYNB]>deK<Dr]>N Y"YGDob½ š ÉMBa#bgDoU HVh"[GF<DvYGDo]<K/R#K>dgHVU ]YGSLEGbgH¼QSD]
R#NGP`hfDLUK<D}RwQDT4bgD4K<DLhE ]a#DLK;YGHVU T4b½§RVTTDLEGK/R#U TDYGHVdeKyLK>F<DrTR#bgTLNGbgSDrDLU\]<HVhhkR#UK£bgD];Yd“—cSLF<DLU*K<D]
RVTTDLEGK/R#U TD]OdeU ]>K/R#UK/R#U SD]–TR#bgTLNGbgSD]}EcHVNGF}T<l)RVj*N D-EcSLF>dgH*YGDxHVÈmbgDAF<S]<DR#NmT<l)R#UGPQD`n
œ’DLUh 2 n¾]Νo”`YGS]>deP`U Dvb½ deU*K<SLP`F/R#bgD}]>NGF£bgDvK<DLhE ]
? hjikNKlKem_[fOUnSIUn_[aoXpikEGagNjF:ZrqbUn_[asX[EtLTUnNTu EE (α, δ)
[email protected]Š]>NGFJŒRVTD\Dt—˜DTLK>[email protected]@YGSLK<DTLK>dgHVU2EDLF>EcDLU YdgTLNGbqR#deF<Df»'NGU DYdeF<DTLK>dgHVU2YGHVUGU SD
]>NGFAbqR
(α, δ)
]>EGl iLF<D-TSLbgD]<K<Dkœ’TtŒJnGDLU TRVYF<SŠ÷‡rH*HVF<YGHVUGU SD]OSjN)R#K<HVF>dqR#bgD]÷-YNˆT<l)R#EGdeK>F<u
D ÷`A”
EE (α, δ) =
Z
cos θ Seff [E, θ(α, δ, t), ϕ(α, δ, t)] dt
œ„ n ~ 
t
HVÈfbqRATHVU*wQDLF<]>dgHVU'DLU*K>F<D}bgD]THzHVF<YGHVUGU SD]bgHzTR#bgD]Oœ a £DLKbgD]THzHVF<YGHVUGU SD]4Sj*N)R#K<HVF>dqR#bgD]Oœ a 
θ ϕ
αδ
YGSLEcDLU YŠYGDObqRBbqR#K>deK>N YGDAYGDub½¾HV[ ]<DLF>w`R#K<HVdeF<DADLKYN'K<DLhE ] nzC‘R\THVUGU)R#dg]<]/R#U TDAYGDOb½¾DtszEcH`]>deK>dgHVU¬YGD
t
b½ š ÉM ]>NGF–K<HVNGK<D\bqRŠ]>EGl iLF<[email protected]]>KOU STD]/]/R#deF<D\EHVNGFATR#F/RVTLK<SLF>dg]<DLFxbqRŠYdg]>K>F>de[GNGK>dgHVU2YGD]AYdeF<DTLK>dgHVU ]
úÀwv¤óô òGò>f‘
[
˜Y ½§R#F>F>dewQSD]¬YGD]ŠF/R¼QHVU ¬
] TH`]>hdgj*N D]arDLU«SLK>N YdqR#U*KpU HVK/R#hhfDLUKˆbqR‰EGF<S]<DLU TD¤Y˜½§R#UGdg]<HVK>F<HVEGdgD]ˆ»
P`F/R#U YGD"HVN¬EcDLK>deK<D-ST<l LD bebgD\R#UGP`NGbqR#deF<D`n
Š",¯®
: oº : }1
[email protected]? I{ z? \@ B
C DuTR#bgTLNGbYGDb½§RVTTDLEGK/R#U TD–YNkF<S]<DR#NŠYGD}]>NGFJŒRVTD}YGD}b½ š ÉMžTHVU ]>K>deK>N D}NGU DuEGl)RV]<D}dehEcHVF>K/R#UK<D–YN
YGSLwQDLbgHVEGEDLhfDLU*K\YN2EGF<H#I>DLKBDtsEcSLF>dehfDLUK/R#ba;DLK"D]>K"Y˜½ NGU DP`F/R#U YGD'dehEcHVF>K/R#U TDfEcHVNGF-b½¾SLK/R#[Gbedg]<]<DLhfDLU*K
Y˜½ NGUm]>EcDTLK>F<DxDLUˆSLU DLF>P`dgD`n
|7UpDt—cDLKazR­ UŠY˜½¾D]>K>dehfDLFvbqR-U HVF>hkR#bedg]/R#K>dgHVU¬YNŠ]>EcDTLK>F<DuY˜½¾SLU DLF>P`dgDAYGD]oF/R¼QHVU ]TH`]>hdgj*N D]ahfD]>NGF<S
DLU
a deb˜ŒR#NGKuYdewzdg]<DLFubgDxU HVhB[GF<DBY˜½¾SLwQSLU DLhfDLU*K<]AhfD]>NGF<S]}E)R#FOdeUK<DLF>w`R#bebgDBY˜½¾SLU DLF>P`dgDBE)R#F
[eV.s.m2 .sr]−1
bqR-]>NGFJŒRVTDuYGDOYGSLK<DTLK>dgHVUŠj*N½§RxEGF<S]<DLU*K<SubgDOYGSLK<DTLK<DLNGFvR#NŠTHVNGF<]oYNŠK<DLhE ]rYGD}EGF>dg]<DOYGDuYGHVUGU SD]azT`½¾D]>KJ
»†YdeF<DuE)R#Frb½§RVTTDLEGK/R#U TD
n –HVK<HVU ]rSLPR#bgDLhfDLU*KjN DObqR-w`R#F>dqR#K>dgHVUpYGDub½§RVTTDLEGK/R#U TD-RwQDTOb½¾SLU DLF>P`dgD
%
A(E)
deU• N DLU TD\YdeF<DTLK<DLhfDLU*KAbqRŒŽHVF>hfDBYN¤]>EcDTLK>F<D`n (HVNGK<DtŒ’HVdg]aTHVhhfD\U HVN ]ObgD"wQDLF>F<HVU ]–E)R#F–bqRk]>NGdeK<D`acR#N
YGD]<]>N ]"Y˜½ NGU D'SLU DLF>P`dgD'YGDkb½¾HVF<YF<D'YGD
|D†€\a‘b½§RVTTDLEGK/R#U TD'U DkwVR#F>dgDkEGbeN ]\RwQDT
œ’DLbebgDkD]<K
Esat ' 3
E
ÒÙ*ÜÝÓIÞÑߝvTR#FK<HVNGK<D-PQDLF>[DxY˜½¾SLU DLF>P`dgD E > E deU TLdgYGDLU*K<D-]>NGFbgD–F<S]<DR#NˆD]>K}YGSLK<DTLK<SDB»@THVNGEˆI] |GFnCD
sat
]>EDTLK>F<[email protected]>F<DkR#bgHVF<]-TR#[email protected]ŒRVWHVUª[cDR#N THVNGEªEGbeN ]A­R#[GbgDEcHVNGF-YGD]xSLU DLF>P`dgD]"]>NGEcSLF>dgDLNGF<D]"»
b½¾SLU DLF>P`dgD
YG¥
D Ò/Ù*ÜÝÓIÞ'Ù*ÜÝÛ=æ!ð¤YGDxb½§RVTTDLEGK/R#U TD`n
Esat
É;R#FAR#debebgDLNGF<]a bgDBTR#bgTLNGbYGD"b½§RVTTDLEGK/R#U TDBYGHVdeKOEGF<DLU YF<D"DLU%THVhEGK<D"K<HVN ]}bgD]–TLF>deK<iLF<D]OYGD"]<SLbgDTLK>dgHVU
R#EGEGbedgjN S]pR#NsËSLwQSLU DLhfDLUK<]avR­ UY˜½¾D]>K>dehfDLFŠTHVF>F<DTLK<DLhfDLUKpbqR2]<NGFJŒŽRVTDˆDt—˜DTLK>dewQD!YGDˆYGSLK<DTLK>dgHVU‹j*NGd
THVF>F<D]>EHVU Y'»–bqRxT<l)R#deU D}YGDuYGSTLbgDLU T<l DLhfDLUKnC½¾D]<K>dehkR#K>dgHVU'YGDvb½§RVTTDLEGK/R#U TD–D]>KYGHVU TK<HVK/R#bgDLhfDLUKrbedgSD
R#NÅT<l HVd“s‰YGD]@R#bePQHVF>deK>lGhfD]@YGDk]<SLbgDTLK>dgHVUZYGD]\SLwQSLU DLhfDLUK<]^
n ArdgDLUÅj*N Dfb½ š [ ]<DLF>w`R#K<HVdeF<DŠÉ7dgDLF>F<D'MONGPQDLF
]<[email protected]>dqR#bgDLhfDLUKkTHVU WLNXEcHVNGFYGSLK<DTLK<DLFfbgD]@F/R¼QHVU ]fTH`]<hdgjN D]YGDŠhkR#UGdgiLF<D¬HVEGK>dehkR#bgDm»%E)R#F>K>deFfYGD
D†€\a)U HVK>F<D"P`F<HVNGEcDBYGDBbì½ D¯}
É BaRV]<]<HzTLdgSB»'TDLbeNGd;YGDBb½¾Sj*NGdeEDBYG}
D (‘NGF>deUacRfEGF<HVEH`]/S"NGU DBYGSt­ UGdeK>dgHVU
1019
YGD]uR#bePQHVF>deK>lGhfD]uYGDxYGSTLbgDLU T/l DLhfDLUKxYGDAl)R#NGK}UGdewQDR#NÅ4œ (v„fDL
K (XðQrj*NGd˜EcDLF>hfDLKOR#N¬F<S]<DR#NmYGDx]>NGFJŒRVTD
Y˜½¾yLK>F<DxK<HVK/R#bgDLhfDLUK–Dt¥kTRVTD"YGi]
|D†€\n
'3
™uU DmTHVUGU)R#dg]<]/R#U TD%EGF<STLdg]<DmYGDmb½¾DtszEcH`]>deK>dgHVU«YN TLdgDLb
D]>KŠSLPR#bgDLhfDLUKpU STD]<]/R#deF<DmEHVNGF
EE (α, δ)
hfD]>NGF<DLFfNGU DmR#UGdg]<HVK>F<HVEGdgD%YGDˆb½¾HVF<YF<DˆYN EHVNGFkTDLU*K'Y R#U ]kbqRªYdg]>K>F>de[GNGK>dgHVU‹R#UGP`NGbqR#deF<D`nrMu­ U YGD¬U D
E)RV]TLF<SDLFuR#F>K>d“­)TLdgDLbebgDLhfDLUKOY˜½¾DtsGTi]}Y R#U ]vNGU DxYdeF<DTLK>dgHVUmYN¬TLdgDLbadeb˜THVUwzdgDLUK–YGD–hfH*YGSLbedg]<DLFuR#N¬hdgDLNs
bgD]BEcDLFJŒŽHVF>hkR#U TD]YNÆF<S]/DR#NZ]<NGF\K<HVNGK<DŠbqR!YNGF<SD¬YGDŠbqRˆEGF>dg]/DŠYGDpYGHVUGU SD]n7É;HVNGFbgD]@SLU DLF>P`dgD]fDLU
YGD]<]<HVN ]"YGDfbqRˆ]/R#K>NGF/R#K>dgHVUÆYGDfb½§RVTTDLEGK/R#U TD`abqR¬w`R#F>dqR#K>dgHVUÅYGDfb½¾Dt¥kTRVTLdeK<SŠYGDkYGSLK<DTLK>dgHVUZRwQDT'b½§R#UGP`bgD
ÔSLUGdeK>l)R#b–YGH gKŠDLU EGbeN ]kyLK>F<D%EGF>dg]<DˆDLU‹THVhEGK<D`aoTHVhEGbedgj*N)R#UK¬DLU THVF<D%b½§R#U)R#be¼z]/D`n4CD]'hfSLK>l H*YGD]ŠYGD
YGSLK<DLF>hdeU)R#K>dgHVU2YGD"bqRŠTR#F>K<[email protected]>K>NGF<DYGDBb½ š [ ]<DLF>w`R#K<HVdeF<DÉ7dgDLF>F<[email protected])R#U*Kxb½¾DtszEcH`]>deK>dgHVU
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
[W
F<DLbqR#K>dewQD-YGD]}Yd“—˜SLF<DLUK<D]OF<SLP`dgHVU ]uYNˆTLdgDLba ]/DLF<HVUK}EGF<S]<DLU*K<SD]OY R#U ]}bgD-T<l)R#EGdeK>F<D\¦n
ÕADpEGbeN ]a£b½¾Dt¥kTRVTLdeK<SmYGDmYGSLK<DTLK>dgHVU YNËF<S]<DR#NËYGDˆ]<NGFJŒŽRVTD¬YGD¬b½ š ÉM¹YGSLEDLU Y»![)RV]<]<D¬SLU DLF>P`dgD
œ
YGD"bqRU)R#K>NGF<D\YGD]uE)R#F>K>dgTLNGbgD]OEGF>dehkR#deF<D]nMudeU ]>da»fSLU DLF>P`dgD\SLPR#bgD`a NGU%U H¼R#NªYGDxŒŽDLFOR#NGF/R
E < Esat
EGbeN ]kYGD%T<l)R#U TD!Y˜½¾yLK>F<DmYGSLK<DTLK<Smj*N½ NGU EGF<HVK<HVU6
n DÞbuD]>KfEcH`]<]>de[GbgDˆY˜½¾DtsEGbgHVdeK<DLFpTDLK>K<DˆEGF<HVEGF>dgSLK<S%EHVNGF
SLK>N YdgDLFfbqR¤THVhEcH`]>deK>dgHVU YGD] ‡r]fY R#U ]bqR!PR#hhfDˆY˜½¾SLU DLF>P`dgD%THVhEGF>dg]<D¬DLUK>F<D
DLK ~ |D€@a7bq»¤HVÈ
a

EGF<STLdg]<SLhfDLUKA]/RkTHVUGU)R#dg]<]/R#U [email protected]#EGEHVF>K<DLF/R#deK–NGU D-deUŒŽHVF>hkR#K>dgHVU%EGF<STLdgDLN ]/D"EcHVNGFOTHVhEGF<DLU YF<D\b½¾HVF>deP`deU D
YGD] ‡r]n
a
|7U­ UaU HVN ]ORwQHVU ]OdeU*K>F<H*YNGdeK–bgD"THVU TDLEGKAYGDxŒŽHVU TLK>dgHV’
U õö§çٜ’C‘R#K<DLF/R#
b (F>deP`PQDLFAÉ7F<HV[)R#[GdebedeK†¼ tacjNGd
YGHVUGU D\bqRkEGF<HV[)R#[GdebedeK<SkYGDBYGSTLbgDLU T/l DLhfDLUKBY˜½ NGU D]<K/R#K>dgHVU!bgHVF<]xYN¤E)RV]<]ÎR#PQD\Y˜½ NGU D\PQDLF>[[email protected]%ŒŽHVU TLK>dgHVU
YGD]/R¬Ydg]>K/R#U TD'»pb½§RsDYGDkTDLbebgDt†TLdn;‡rD]AŒŽHVU TLK>dgHVU ]'œNGU DE)R#FBSLU DLF>P`dgD`aE)R#FBR#UGP`bgDfDLK"E)R#F-U)R#K>NGF<DkYGD
bqRªE)R#F>K>dgTLNGbgD%EGF>dehkR#deF<DkEDLF>hfDLK>K<DLU*KŠYGD!TR#bgTLNGbgDLF'bqR‰]>NGFJŒRVTDmDt—˜DTLK>dewQD%YGD%YGSLK<DTLK>dgHVU‹YN¨¡zÕ@ahkR#dg]
SLPR#bgDLhfDLUKxDLbebgD]–THVU ]>K>deK>N DLUKAYGD]OHV[ ]/DLF>wVR#[GbgD]OEGl¼]>dgj*N D]Oj*NGd‘EcDLF>hfDLK>K<DLUKAYGD-K<D]>K<DLFADLKOTHVU*K>F/R#deU YF<D
bgD]hfHzYGiLbgD]}YGD-]>dehBNGbqR#K>dgHVU ]uYGD]}PQDLF>[cD]}DLK}YNmYGSLK<DTLK<DLNGFn
Š",†+
·c:?²@¸¹5gB¨³B5±³\¶79<ó/571B9c²\5¸¹571€?ô³B5±²\º}6 ¨
? 1\:I{}57º}6SŠ
5‚?tñ
~„ó°ö@
C D]BSLwQSLU DLhfDLUK<]j*NGd4]/R#K>dg]JŒŽ[email protected]#NsªTLF>deK<iLF<D]\YN‰UGdewQDR#NÆYGDkYGSTLbgDLU T/l DLhfDLUKs( ~ a‘j*NGd4DtsdePQDNGU D
THVl SLF<DLU TDvK<DLhEHVF<DLbebgDDLK£NGU DTHVhE)RVTLdeK<SvhfHzYGSLF<SD-œ’TtŒT<l)R#EGdeK>F<D ~ ta#THVhEcHVF>K<DLUKoDLU THVF<DvNGU D4Œ’F/RVTLK>dgHVU
dehEHVF>K/R#U*K<DBY˜½¾SLwQSLU DLhfDLU*K<]–Œ’HVF>K>NGdeK<]OHVN%Yd“¥kTLdebgDLhfDLU*K–F<DTHVU ]<K>F>N TLK>de[GbgD]nC‘D"TLF>deK<iLF<DH( ~ D]>K}wQHVbgHVU*K/R#d“
F<DLhfDLUKBEDLF>hdg]<]<d“ŒJacdebEcDLF>hfDLKBYGDkTHVU ]>K>deK>N DLF"NGU2bgHVKBYGDfYGHVUGU SD]BY R#U ]"bgD]<j*N DLbg]xU HVN ]"]<HVhhfD]B]I|GF<]
YGDxU DxE)RV]OhkR#U jN DLFObgD]uSLwQSLU DLhfDLU*K<]OEGl*¼z]<dgjN D]a)YGHVUKubgD]uEGF<HVEGF>dgSLK<S]uEcDLNGwQDLUKAyLK>F<D-K>F<i]OYd“—cSLF<DLU*K<D]
]>NGdew`R#UKrb½§R#UGP`bgDOÔSLUGdeK>l)R#b7”zDLUŠDt—˜DLKa*bgD]4PQDLF>[cD]R¼R#UK
U DOEDLNGwQDLU*KvE)RV]ryLK>F<D–]<SLbgDTLK>dgHVUGU SD]vUGd
θ & 60◦
F<DTHVU ]>K>F>NGdeK<D]vYGD}bqR-hfyLhfD}ŒRVWHVU'j*N D}bgD]4PQDLF>[D]ohfHVdeU ]rdeU TLbedeU SD]n*C‘HVF<]oYGDub½§R#U)R#be¼z]<DOYGD]oYGHVUGU SD]adeb
ŒŽR#NGKOYGHVU T-DtszdePQDLF–YGD]}THVU YdeK>dgHVU ]–EGbeN ]u]>K>F>dgTLK<D]uEHVNGFO]<SLbgDTLK>dgHVUGU DLFubgD]OSLwQSLU DLhfDLUK<]–EGl¼]>dgj*N D]a T`½¾D]>K
bgDŠF<ÚVbgDŠYNZUGdewQDR#NËYGD¬YGSTLbgDLU T/l DLhfDLUK Û (v„ [email protected]ŒŽR#NGK]½§RV]/]>NGF<DLFYGD'bqR%[cHVUGU DŠF<DTHVU ]>K>F>N TLK>dgHVU YGD]
E)R#F/R#hfiLK>F<D]uYGDxbqR\PQDLF>[cD`a TDxj*NGd‘D]>KbqRBŒŽHVU TLK>dgHVUmYN
N (Xðn
܄ƒ#ê§ßJàvèéßcä§ßGm†£ß¯á)ä§ßGG‡à#ß}ˆûO9JçJä§ßJàçýßJàáAˆñ#ãÎë é#àߘãÎç>èéê›ëŽê¾áê§ìtàðLàìté#ëGç¯ìtàáê§àé߯âìtàë%û$ü éáê§ä§ê§ëŽß¯â)ä›ßcáßÞâïßVûߘàꊉtß>ãÎé
ûßûO9Jç¯ä›ß¯à#çý#ßJïßJàá˜ìÎ鋍ŒWŽŠI‘nŒn
úÀg¿2$Šõò()Vò@h<i‰qÛ;NÛ;ÁÃõfÃ_h‘gi2fbÛNh &‘efò* &tm"ú‰emB
[[
xÛÁ*ÃõfÃhi‰fHÛR’&‘V0{i‰Zf;ò &Ùknm"úVo D]<K4NGUpTLF>deK<iLF<DuEcHVNGFo]<SLbgDTLK>dgHVUGU DLFvbgD]rSLwQSLU Dt
hfDLUK<]OEGl¼]>dgjN D]n ÉHVNGFObgD]}PQDLF>[cD]OR¼R#UK–NGU%R#UGP`bgD-ÔSLUGdeK>l)R#b;deUŒ’SLF>dgDLNGFA» _YGDLP`F<S]a)bgDz(v„fF<Dj*NGdgDLF>K
Ø
]<HVdeK ~ ]>K/R#K>dgHVU z
] ( š (ƒDLUÅTHVU­ P`NGF/R#K>dgHVUÅTHVhE)RVTLK<D ~ ‡ œ’Y R#U ]"bqRŠEGF<DLhdgiLF<DkTHVNGF<HVUGU Dta]<HVdeKx„ˆ]>K/R

K>dgHVU )
] (}ÑOY R#U ];bqROEGF<DLhdgiLF<DrTHVNGF<HVUGU Dxœ„*‡ ta#bgD]£]>K/R#K>dgHVU ]£YGDLwVR#U*K£wQSLF>d“­)DLF;NGUTLF>deK<iLF<DvYGDrTHVhE)R#K>de[GbedeK<S

K<DLhEHVF<DLbebgD`nCz
D (v„ ~ ( š (¨EcDLF>hfDLKuYGD-THVU ]/DLF>wQDLFOJ^ ð Y YGD]}SLwQSLU DLhfDLU*K<]–EGl¼]>dgjN D]aGbgD
] ð Y F<D]>K/R#U*K
]<HVUKF<STLNGEcSLF<S]E)R#Fvbg”
D (v„„*‡ nGCDf÷t[GF>NGdeK}YGD–ŒŽHVU Y÷Vaz=Y |mR#NspT_H pŽU TLdgYGDLU TD]ŒŽHVF>K>NGdeK<D]}DLUK>F<D"]>K/R#K>dgHVU ]

D]>KuR#deU ]>dD]<]<DLU*K>dgDLbebgDLhfDLUK–UNGbn
,6-0/.1„ n
DLEGF<S]<DLU*K/R#K>dgHVUÆYGD]BEH`]<deK>dgHVU ]BYGD'T‘“ANGF<]BYGDkPQDLF>[D]@R#NGK<HVF>dg]<SD]\E)R#F"bgD'TLF>deK<iLF<Db(Xð‰”bqR

u
å
2
a
TLNGwQD'TDLU*K>F/R#bgDŠD]<KBTDLbebgDŠjNGdrEH`]/]<iYGDfbgD'EGbeN ]-ŒŽHVF>[email protected]]>deP`U)R#bn£‡rDLbebgDt†TLdvD]>[email protected]<HVNGF<SD¬»mPR#N T<l DŠE)R#F\]<d“s
wQHVdg]>deU D]vDLUpSLK/R#KYGD–hkR#F<T<l D`aGDLK»\YF<HVdeK<D–E)R#FvTLdeU jk]<DLNGbgDLhfDLUKn CR"]>NGFJŒRVTDADt—cDTLK>dewQDAYGDAYGSLK<DTLK>dgHVUˆYGD
[email protected] DtsGR#PQHVU)R#bgDBYGDxYF<HVdeK<D"D]>K}YGHVU T-EGbeN ]vŒŽR#de[GbgD"Y˜½ NGU¬ŒŽRVTLK<DLNGF
j*N DxTDLbebgD-YGDxPR#N T/l D`n
2/3
eÛ;ÁÃ*õògf‘Ã_hi2gfzÛ'Ã4f’&;0{i‰sfò* &tknm\>o D]<KNGUkTLF>deK<iLF<D}YGD}j*N)R#bedeK<S}YGDF<DTHVU ]<K>F>N TLK>dgHVU
YGD7b½¾SLwQSLU DLhfDLUKn#CD]PQDLF>[cD]˜K<HVhB[)R#UK£]>NGFbgD7[cHVF<Y-YN-F<S]<DR#NBHVN"Y R#U ]NGU D7F<SLP`dgHVUBTHVhEGF<DLU)R#UK£NGU DHVN
EGbeN ]>dgDLNGF<]A]>K/R#K>dgHVU ]–deU)RVTLK>dewQD]AF>dg]<jN DLU*KxY˜½¾yLK>F<DBdeU THVF>F<DTLK<DLhfDLU*KxF<DTHVU ]>K>F>NGdeK<D]x[GdgDLU¤jN½¾DLbebgD]–wQSLF>d“­)DLU*K
bgD]oTHVU YdeK>dgHVU ]Yx
N (v„ a*DLKvYGHVdewQDLUKYGHVU T–yLK>F<DOSLbedehdeU SD]YGD]vYGHVUGU SD]nz™OU DuE)R#F>K>dgD–YN'EGF<S]/DLUKvK>F/Rw`R#deb
YGDuK>l i]<DARBSLK<SOTHVU ]/RVTLF<SDx»"bqR"YGSLK<DLF>hdeU)R#K>dgHVUˆY˜½ NGUpTLF>deK<iLF<DAYGB
D (Xð\Dt¥kTRVTD–DLKr­R#[GbgD`a*j*NGdU D–]<HVdeKrE)RV]
K>F<HVEm]>K>F>dgTLKU HVUˆEGbeN ]EcHVNGF}EcDLF<YF<DAbgDxhfHVdeU ]}EcH`]<]>de[GbgDxYGDxYGHVUGU SD]n
C‘R'THVU YdeK>dgHVU¤YGu
D (Xð'jN D"U HVN ]xRwQHVU ]AEGF<HVEcH`]<SD`aH#¥kTLdgDLbebgDLhfDLU*KBRVTTDLEGK<SD\E)R#FObqR'THVbebqR#[cHVF/R#K>dgHVUa
YGDLhkR#U YGDOj*N D-” 7bqRx]>K/R#K>dgHVUpR¼R#UKvbgDuEGbeN ]7ŒŽHVF>Ko]>deP`U)R#b]<HVdeK4DLUK<HVNGF<SDAY˜½§R#N'hfHVdeU ]oTLdeU [email protected] ]4EGF<H*T/l D]

wQHVdg]>deU D]¬DLU‹SLK/R#KˆYGD%ŒŽHVU TLK>dgHVUGU DLhfDLUKa}DLK¬j*N DªÑQbgD¤‘T “ANGFˆYGD!bqR2PQDLF>[cD!F<DTHVU ]>K>F>NGdeKˆ]<HVdeK¬]>deK>N Sª»
b½ deUK<SLF>dgDLNGF\NGU2K>F>dqR#UGP`bgDkSj*NGdebqR#K<SLF/R#boYGDf]<K/R#K>dgHVU ]BRVTLK>dewQD]n‡rDLK>K<D'THVU YdeK>dgHVUÅD]>KBYGHVU TfDtszEGF>dehfSDkK>F<i]
]>dehEGbgDLhfDLUK–]<HVN ]bqRBŒŽHVF>hfD-Y˜½ NGU D-THVUK>F/R#deU*K<D"PQSHVhfSLK>F>dgj*N D`n
CD]AEcH`]>deK>dgHVU ]xYGD‘T “ANGF<]-YGD\PQDLF>[DfR#NGK<HVF>dg]/SD]xE)R#FxTDTLF>deK<iLF<DfR#NªwQHVdg]>deU)R#PQDY˜½ NGU DTLNGwQDYGHVUGU SD
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
[c
,.-F/.1z„ n§Ñ
DLEGF<S]<DLU*K/R#K>dgHVUˆYGD]vEH`]>deK>dgHVU ]vYGDAT‘“ANGF<]vYGD–PQDLF>[D]R#NGK<HVF>dg]<SD]-œ’DLU¬HVh"[GF<Sx]<NGFobqR"­ P`NGF<D
E)R#FubgDBTLF>de2s
K<iLa F<D}(Xð!÷tl DtsGR#PQHVU DQ÷VnM” PR#N T<l Dp”Y R#U ]ubgDBTRV]OY˜½ NGU D\THVU­ P`NGF/R#K>dgHVU2THVhE)RVTLK<DBYN%F<S]<DR#N
œK<HVNGK<D]bgD]u]<K/R#K>dgHVU ]}]<HVUK–RVTLK>dewQD]Νta)»@YF<HVdeK<Dk”bgHVF<]/jN½ NGU D-]<K/R#K>dgHVUmD]>KdeU)RVTLK>dewQD`n
]<HVUK4F<DLEGF<S]<DLUK<SD]vDLUkHVhB[GF<Su]>NGFbqRA­ P`NGF<D}„ n aQEcHVNGFbgD]oYGDLNs]/DLNGbg]4TRV]7EcH`]<]>de[GbgD]O”`NGU D}]>K/R#K>dgHVUpRwQDT

]>d“sˆwQHVdg]>deU D]ADLU¤SLK/R#K–YGD"Œ’HVU TLK>dgHVUGU DLhfDLU*K'œŽ»kPR#N T<l [email protected]]>NGF–[email protected]­ P`NGF<DtacHVN!TLdeU jˆwQHVdg]>deU D]A]<DLNGbgDLhfDLUKkœŽ»
YF<HVdeK<Dtn ÕxR#U ]}bgD-]/DTHVU YˆTRV]abqR]>NGFJŒRVTD"R#Nm]/HVbTHVF>F<D]>EHVU Y R#U*K–R#Ns¬EcH`]>deK>dgHVU ]uYGDxT‘“ANGF<]–R#NGK<HVF>dg]<SD]
F<DLEGF<S]<DLUK<D"NGU D–ŒF/RVTLK>dgHVU
YGDxTDLbebgD-YGDxPR#N T<l D`n
2/3
Õ ½§R#NGK>F<D]vTHVU YdeK>dgHVU ]rHVU*KSLK<SuSLK>N YdgSD]aTHVhhfDOE)R#FrDtsGDLhEGbgDubgD3(Xð-YdeKx÷tK>F>dqR#UGP`bgDQ÷–jNGdcYGDLhkR#U YGD
@
]>dehEGbgDLhfDLUKvj*N D}bgD–TH*DLNGFoF<DTHVU ]>K>F>NGdeKr]/HVdeK4deU TLbeNGKvY R#U ]4NGU'K>F>dqR#UGP`bgD–Sj*NGdebqR#K<SLF/R#bcYGDu]>K/R#K>dgHVU ]vRVTLK>dewQD]a
HVNˆDLU THVF<Dxbg”
D (Xðˆ÷tl Dts R#PQHVU DQ÷xj*NGdYGDLhkR#U YGD-jN DAbqRf]>K/R#K>dgHVUmR¼R#U*KubgDAEGbeN ]ŒŽHVF>K}]>deP`U)R#b]<HVdeK}DLU*K<HVNGF<SD
YGDx]>d“spwQHVdg]>deU D]}DLU¬ŒŽHVU TLK>dgHVUGU DLhfDLUKn
,.-F/.1z„ n ~
DLEGF<S]<DLU*K/R#K>dgHVUˆYGD]vEH`]>deK>dgHVU ]vYGDAT‘“ANGF<]vYGD–PQDLF>[D]R#NGK<HVF>dg]<SD]-œ’DLU¬HVh"[GF<Sx]<NGFobqR"­ P`NGF<D
2
s
a
E)R#F-bgDkTLF>de<K iLF<Db(Xð‰÷tK>F>dqR#UGP`bgDQ÷Vn‚”ÒPR#N T<l Dm”Y R#U ]-bgDfTRV]\Y˜½ NGU DkTHVU­ P`NGF/R#K>dgHVUZTHVhE)RVTLK<DŠYN2F<S]<DR#N
œK<HVNGK<D]bgD]u]<K/R#K>dgHVU ]}]<HVUK–RVTLK>dewQD]Νta)»@YF<HVdeK<Dk”bgHVF<]/jN½ NGU D-]<K/R#K>dgHVUmD]>KdeU)RVTLK>dewQD`n
úÀg¿2$Šõò()Vò@h<i‰qÛ;NÛ;ÁÃõfÃ_h‘gi2fbÛNh &‘efò* &tm"ú‰emB
[v
C D|(Xðf÷tK>F>dqR#UGP`bgDQ÷vEcDLF>hfDLK4YGDvF<STLNGEcDLF<DLFNGUfhkRsdeh"NGh×Y˜½¾SLwQSLU DLhfDLU*K<]azTR#F7deb THVU ]<DLF>wQDbgD]7PQDLF>[cD]
K<HVh"[cSD]r]<NGF4bgDO[HVF<YŠYNkF<S]<DR#Nªœ’TtŒ­ P`NGF<DO„ n ~ ta*DLKrU DOEGF<HwQHzjN DOE)RV]4NGUŠP`F<H`]x÷tK>F<HVN‘÷uY R#U ]rbgDOF<S]<DR#N
]>doNGU D¬]<K/R#K>dgHVUZD]<K\deU)RVTLK>dewQD`n4‡rDLEcDLU Y R#UKabgD]SLwQSLU DLhfDLUK<]f]>deK>N S]]<NGF\bgDŠ[cHVF<YG]]<HVUK]<HVNGwQDLU*KhkR#b
F<DTHVU ]>K>F>NGdeK<]a7DLKb½¾SLw`R#beN)R#K>dgHVUËEGF<STLdg]<D¬YGDpbqR%Œ’F/RVTLK>dgHVU YGDŠPQDLF>[D]hkR#bF<DTHVU ]<K>F>NGdeK<D]kj*N DpbqR#dg]<]<DLF/R#deK
E)RV]<]<DLFNGU¬K<DLb‘TLF>deK<iLF<D-D]<K}YGSLbedgTR#K<D`n
C D“(Xð%÷tl DtsGR#PQHVU DQ÷VacD]>KONGU%TLF>deK<iLF<DBdeU YGSLEDLU Y R#U*K–YGDBbqRF<DTHVU ]<K>F>N TLK>dgHVUaj*NGd‘DtsGTLbeNGKONGU D"F/R#UGPQSD
YGD']>K/R#K>dgHVU ]@DLUZEGbeN ]@jN DŠTDLbebgD]YGSt­ UGdg]<]/R#U*[email protected][cHVF<YZYNÅF<S]<DR#N¨œ’TtŒ}­ P`NGF<DŠ„ n§ÑQ'”debrD]>[email protected] TŠEGbeN ]
THVU ]<DLF>wVR#K>d“Œ£]>NGFTDAEHVdeU*KjN DAbgD"(XðBK>F>dqR#UGP`bgD`n É£R#F}THVUK>F<D`aGdebcEGF<HwQHzj*N DANGU DAP`F<H`]<]<D–EDLF>K<DAYGDx]>NGFJŒRVTD
Dt—cDTLK>dewQD-]<d˜NGU Dx]>K/R#K>dgHVU%»@b½ deU*K<SLF>dgDLNGFOYNˆF<S]<DR#NˆD]<KdeU)RVTLK>dewQD'œEcDLF>K<D-YGD“÷\l Dts R#PQHVU D]Νtn
C D!TLF>deK<iLF<D!YGD!(XðÅjN D%U HVN ]¬RwQHVU ]phdg]¬R#NEcHVdeUKa|ð‰wQHVdg]>deU D]*? T‘“ANGFŠF<DTHVU ]>K>F>NGdeKˆY R#U ]ŠNGU
K>F>dqR#UGP`bgDxD]>KoNGU¬THVhEGF<HVhdg]}DLU*K>F<DAbgD"(Xð"K>F>dqR#UGP`bgDxDLKvbgD3(XðBl DtsGR#PQHVU D”zdebcK>F/R#deK<D–bgD]v[HVF<YG]vTHVhhfDAbgD
(Xðul Dts R#PQHVU D`aQDLK£bgD]}÷tK>F<HVN ]÷vY R#U ]£bgDvF<S]<DR#NfTHVhhfDvbg.
D (Xð–K>F>dqR#UGP`bgD`n¡*dGb½¾HVUEGF<DLU YTHVhhfDvF<StŒŽSLF<DLU TD
bgDOU HVhB[GF<D"Y˜½¾SLwQSLU DLhfDLU*K<]O]<SLbgDTLK>dgHVUGU S]}E)R#FbgDxTLF>deK<iLF<z
D (Xð\K>F>dqR#UGP`bgD`aGbg”
D (Xð\l Dts R#PQHVU DxDLUˆTHVU ]<DLF>wQD}÷_ð
a R#bgHVF<]}jN DAU HVK>F<D"TLF>deK<iLF<D-EcDLF>hfDLKuY˜½¾DLU¬PR#F<YGDLF–^`_ n
%
%
,6-0/.1„ n „
SLE)R#F>K>deK>dgHVU2]>NGFAbgDBF<S]<DR#N2YGD"bqRkŒ’F/RVTLK>dgHVU2Y˜½¾SLwQSLU DLhfDLUK<]"F<D¯IJDLK<SD\E)R#FAbgDu(Xð aE)R#FAEGd“sGDLbg]
N
2
a
YGD Ç h 2 R#N¤]<HVbn)C4½¾ST<l DLbebgDYGD\THVNGbgDLNGF–deU Ydgj*N D\DLU%F<HVNGPQDBbgD]OŒF/RVTLK>dgHVU ]–dehEcHVF>K/R#UK<D]xY˜½¾SLwQSLU DLhfDLUK<]
F<D¯IJDL K<S]aDLUm[GbgDLN%bgD]uEGbeN ]uŒŽR#de[GbgD]n)CD\TLF>deK<iLF<D“(Xð'DtsTLbeNGKObgD]uPQDLF>[cD]OK<HVh"[cSD]O]>NGFObgD-[cHVF<YmYN!F<S]<DR#N
HVN¬EGF<i]uY˜½ NGU D-]>K/R#K>dgHVUmhkR#U jN)R#U*K<D"HVN¬deU)RVTLK>dewQD`n
ÉHVNGF-]/DBTHVU*wVR#deU TLF<DfYGD\bqR'U STD]<]>deK<[email protected]˜½ dehEcH`]<DLFANGUå(XðazIL½§R#d£SLK>N [email protected]'F<SLE)R#F>K>deK>dgHVU‰]>E)R#K>dqR#[email protected]]
SLwQSLU DLhfDLUK<]}j*NGdU DA]/R#K>dg]>Œ’HVU*KrE)RV]R#N¬TLF>deK<iLF<D"(Xð\H#¥kTLdgDLbnGC‘R"­ P`NGF<DO„ n „@hfHVU*K>F<DAbqR-Œ’F/RVTLK>dgHVU¬Y˜½¾SLwQSLU Dt
c)
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
hfDLUK<]vF<D¯I>DLK<SD–E)R#FrbgD3(Xð\DLUŠŒ’HVU TLK>dgHVU¬YGD–bqR-EcH`]>deK>dgHVU¬]>NGFrbgD–F<S]<DR#N2œbgDOYGSLEGbgHVdgDLhfDLU*KuSLK/R#UK}DLU¬THVNGF<]a
bgDŠF<S]<DR#NËD]>[email protected]#deUK<DLU)R#U*KkEGbeN ]SLK<DLU YNËj*N D¬TDŠj*N DŠb½¾[email protected]]>[email protected]%­ P`NGF<Dg•›tn7C½¾ST/l DLbebgDmYGD
THVNGbgDLNGFoTHVF>F<D]>EcHVU Yk»AbqRAŒF/RVTLK>dgHVUŠY˜½¾SLwQSLU DLhfDLUK<]rF<D¯IJDLK<S]oY R#U ]NGU D}]<NGFJŒŽRVTD}YGD Ç h 2 R#N']<HVbn*‡rHVhhfD

HVU2EcDLNGKxbgDfTHVU ]>K/R#K<DLFB]<NGF-bqRŠ­ P`NGF<D„ n „ a˜bge
D (Xð¬DtsTLbeNGK"[GdgDLU‰bgD]xPQDLF>[cD]-K<HVhB[)R#[email protected]]>NGF-bgD]x[cHVF<YG]"YN
F<S]<DR#NˆDLK}EGF<i]}Y˜½ NGU D"]>K/R#K>dgHVUˆhkR#U j*N)R#UK<D"HVNmdeU)RVTLK>dewQD`n
‡ D]7SLwQSLU DLhfDLUK<]oDLU[HVF<YNGF<DYNF<S]/DR#Nf]<HVUK7PQSLU SLF/R#bgDLhfDLU*K4hkR#bF<DTHVU ]>K>F>NGdeK<]aQTHVU YNGdg]ÎR#UKo»ONGU D
r
]>NGF<D]>K>dehkR#K>[email protected]½¾SLU DLF>P`dgDfYGD\bqRŠPQDLF>[cD`n š Fa˜THVhEGK<[email protected]<DLUN‰YGD\bqR'ŒŽHVF>K<[email protected]<HVdg]<]/R#U TDfYN!• Ns2RwQDT
b½¾SLU DLF>P`dgD`aTD]ojN DLbgj*N D]4SLwQSLU DLhfDLU*K<]rYGHVU*K4b½¾SLU DLF>P`dgDOD]>K4]>NGF<D]>K>dehfSDuEDLNGwQDLU*KrRwQHVdeF4NGU D}THVU*K>F>de[GNGK>dgHVU
U HVU¬U SLP`bedePQDR#[GbgD"Y R#U ]ub½¾D]>K>dehkR#K>dgHVU%YNˆ]>EcDTLK>F<D"»\l)R#NGK<D-SLU DLF>P`dgD`n
,.-F/.1„ :n ð
”
PR#N T<l D!”RQ SLE)R#F>K>deK>dgHVU‰YGDfbqRŠŒ’F/RVTLK>dgHVU‰Y˜½¾SLwQSLU DLhfDLU*K<]\YGHVU*K"b½¾SLU DLF>P`dgDkF<DTHVU ]>K>F>NGdeK<D
D]>KO]<NGESLF>dg2 DLNGF<[email protected]» |D€@ac]/R#aU ]–R#EGEGbedgj*N DLFAYGD}(XðnC‘R'YdeF<DTLK>dgHVUªR#ÔdehBNGK>l)R#[email protected]¼QDLUGU [email protected]˜½§R#F>F>[email protected]]
Ç h×D]>K7F<DLEGF<S]<DLU*K<SDuE)R#FNGUK>F/R#deK4F<HVNGPQDu]<DvK<DLF>hdeU)R#U*K
SLwQSLU DLhfDLUK<]rY R#U ]45T/l)RVjN D"÷tEGd“sGDLb†÷YGD Ç h

×
]>NGFbgDAEGd“sGDLb–n ” YF<HVdeK<Dk”[R#EGF<i]uR#EGEGbedgTR#K>dgHVU%YN€(Xð
C R­ P`NGF<D4„ n:ðGœŽRQ˜deU YdgjN DobqRF<SLE)R#F>K>deK>[email protected]bqRŒF/RVTLK>[email protected]˜½¾SLwQSLU DLhfDLUK<]7YGHVU*K‘b½¾SLU DLF>P`dgDoF<DTHVU ]>K>F>NGdeK<D
‘
D]>Ko]<NGESLF>dgDLNGF<Dx» |D€@azbgHVF<]<j*N DubgD3(Xð-U½¾D]<K4E)RV]R#EGEGbedgjN S`nCDuŒR#deKrj*N D}bgD]vSLwQSLU DLhfDLUK<]vU DOE)RV]<]/R#U*K
5
E)RV]7bg|
D (XðA]<HVdgDLU*K4]>¼]>K<SLhkR#K>dgjN DLhfDLU*KF<DTHVU ]>K>F>NGdeK<]rRwQDTNGU D}SLU DLF>P`dgD}EGbeN ]SLbgDLwQSD"œ’TR#F>F<S]7F<HVNGPQD]rRwQDT
bgD]A•)iT<l D]tAdeU Ydgj*N DfjN DfbgDLNGF-F<DTHVU ]>K>F>N TLK>dgHVUZDLU‰SLU DLF>P`dgDkD]>KxhkR#NGw`R#dg]<D`nCRˆYdeF<DTLK>dgHVUÆR#ÔdehBNGK>l)R#bgD
Ç hÌD]>KSLPR#bgDLhfDLU*K‘F<DLEGF<S]<DLU*K<SD
hfH¼QDLUGU DoY˜½§R#F>F>dewQSDYGD]‘SLwQSLU DLhfDLU*K<]Y R#U ]T/l)RVjN D–÷tEGd“sDLb†÷£YGD Ç h

×
E)R#FONGU!K>F/R#deKAF<[email protected]]<D"K<DLF>hdeU)R#U*K-]>NGF–bgD"EGd“sGDLbnc‡rDLK>K<[email protected]<DTLK>dgHVUªhfH¼QDLUGU [email protected]<DTHVU ]>K>F>NGdeK<[email protected]]OPQDLF>[cD]
U DfE)RV]<]ÎR#UK\E)RV]"bgZ
D (Xðm]/DLh"[GbgDŠyLK>F<D']>¼]>K<SLhkR#K>dgj*N DLhfDLUKYdeF>dePQSD'bgDfbgHVUGP¤YGD]B[cHVF<YG]\YN‰F<S]<DR#Nn;|7U
R#EGEGbedgjN)R#U*Kbg6
D (XðaTD];SLwQSLU DLhfDLUK<]7]/HVN—F/R#U*K;Y˜½ NGU DohkR#NGwVR#dg]/DF<DTHVU ]>K>F>N TLK>dgHVUDLU\R#UGP`bgDrDLK£DLUBSLU DLF>P`dgD
úÀeú‰%ÃQfÃQܑeÛ;ˆÂÞfÃ`f >m-
cÄ
]<HVUKuSLbedehdeU S]a THVhhfDxbgDAhfHVUK>F<D"bqR\­ P`NGF<Dx„ n:ðGœ[tn
Š", Š
°p5±9‚@}1B9;5£µC?̳B5±Ó‘@}1\9G?:\@}1
ü‚ü†ý
°—#˜
þåQ .
™ J
M ¼R#UK}YGSt­ UGdcbgD]R#bePQHVF>deK>lGhfD]}YGDA]<SLbgDTLK>dgHVU¬YGD]SLwQSLU DLhfDLUK<]}j*NGdc]<HVUKNGK>debedg]<S]vY R#U ]vbgDOEGF<H*TD]/]>N ]
£
Y˜½§R#U)R#be¼z]<DpYGD]YGHVUGU SD]YNZF<S]<DR#NXYGDp]>NGFJŒRVTD`adebrŒR#NGKf»mEGF<S]/DLUKfD]>K>[email protected]!]>NGFJŒRVTDŠDt—˜DTLK>dewQD¬YGD
YGSLK<DTLK>dgHVU
THVF>F<D]>EcHVU Y R#UKx»TD]uTLF>deK<iLF<D]nArdgDLU%j*N D-TDLbqRU D-]/HVdeK}E)RV]uSLwzdgYGDLUKx»@EGF>dgHVF>da
Seff (E, θ, ϕ)
debD]>KrEH`]/]>de[GbgDuYGD–YGSLK<DLF>hdeU DLFvbqR"]>NGFJŒRVTD–Dt—cDTLK>dewQD–YGDOYGSLK<DTLK>dgHVU¬YNˆ¡zÕÌ»"E)R#F>K>deFrYGDObqR"THVUGU)R#dg]/]/R#U TD
YGD]EGF<HV[)R#[GdebedeK<S]OYGDxYGSTLbgDLU T/l DLhfDLUKAdeU YdewzdgYN DLbebgD]uYGD]}]>K/R#K>dgHVU ]n
OHVN ]RwQHVU ]deUK>F<HzYNGdeKbgDATHVU TDLEGKvYGDOŒŽHVU TLK>dgHVA
U õö§ç׌œ õ Ù*܏ßnÞÙ!ê ö~ÞÛFéAé1ßnފçÞ'æ1ä>Ù1änەê;ÛÝܚ/rY R#U ]TD–[GNGK
EGF<STLdg]B”)»fNGU D"SLU DLF>P`dgD`aNGU!R#UGP`bgDBDLKuNGUmKÞ¼*EcD"YGD-E)R#F>K>dgTLNGbgD"EGF>dehkR#deF<D"­GsS]a [email protected]ŒŽHVU TLK>dgHVU%gC (É«YGHVUGU D
bqRpEGF<HV[)R#[GdebedeK<S'YGDkYGSTLbgDLU T<l DLhfDLUKY˜½ NGU D']>K/R#K>dgHVU‹4œ (}Ѫ”‘]<HVdeK"YGDfKÞ¼*Ecs
D (JH (xa‘]<HVdeK"YGDfK†¼zEDf]/DLNGdebòxDLU
Œ’HVU TLK>dgHVUpYGDu]/R"Ydg]>K/R#U TDA»-b½§RsGDuYGDObqRxPQDLF>[cD`n*C‘D]4ŒŽHVU TLK>dgHVU ]rC (ɏ]vF/RV]/]<DLh"[GbgDLU*KrK<HVNGK<DOb½ deUŒŽHVF>hkR#K>dgHVU
NGK>debgDkR#NÅTR#bgTLNGbrYGDbqR¬]>NGFJŒRVTDkDt—˜DTLK>dewQDfYGDkYGSLK<DTLK>dgHVUÆDLKBYGDb½§RVTTDLEGK/R#U TD`aF<S]>NGhkR#U*K\DLU2NGU D']<DLNGbgD
]<SLF>dgDxYGDAŒŽHVU TLK>dgHVU ]bgD]EGF<HVEGF>dgSLK<S]–YGDxYGSLwQDLbgHVEGEDLhfDLU*K–YGD]PQDLF>[cD]uDLKbqR\F<SLEcHVU ]<D-YNˆYGSLK<DTLK<DLNGFn
CDxTR#bgTLNGbYGDxTD]vŒŽHVU TLK>dgHVU ]}]½¾Dt—˜DTLK>N DxDLUˆ]>deh"NGbqR#U*K}bgDxYGSLwQDLbgHVEGEcDLhfDLUKOYGD]}PQDLF>[D]DLKbqR\F<SLEcHVU ]<D
YN!YGSLK<DTLK<DLNGFxTHVF>F<D]>EcHVU Y R#UK<D`Gn –HVN ]OwQDLF>F<HVU ]AE)R#F–bqR']>NGdeK<D\jN DBbgD]AC (É4]AEDLNGwQDLU*KAyLK>F<[email protected]#bgDLhfDLU*K
Ô¹ßnÒÓIÞÑ>ßnÒtaw`R#bedgY R#[email protected]#deU ]>d7bqRphfSLK>l HzYGDfDLKxEcDLF>hfDLK>K/R#UKBYGDfK<D]>K<DLFxbqR¬j*N)R#bedeK<SfYGD]-]>dehBNGbqR#K>dgHVU ]n‘Õ@½ NGU
EHVdeU*KYGDowzN DrEGF/R#K>dgjN D`atI½§R#dG]>deh"NGbgSvbqROF<SLEcHVU ]<DvY˜½ NGUF<S]<DR#NfYGDvYGSLK<DTLK<DLNGF<]7dgYGDLUK>dgj*N D]o»–TDLNs\NGK>debedg]<S]
[email protected]¡zÕ YGD"b½ š ÉMBaYGHVUKAbgD]–]>K/R#K>dgHVU ]A]<HVUKAYdg]>EcH`]<SD]aY R#U ]ObgDBF<StŒŽSLF<DLUK>dgDLb7YGD"bqRkPQDLF>[cD`a]>NGF–YGD]
TDLF<TLbgD]}THVU TDLUK>F>dgj*N D]uYGHVU*K}bgDAF/R¼QHVU%R#NGP`hfDLU*K<D-E)R#FE)RV]}YGz
D ð`_\hfiLK>F<D]nGCDxF<S]<DR#NmRR#deU ]>d˜bqRBŒŽHVF>hfD
Y˜½ NGU D'SLK<HVdebgD¬»%YGHVNGÔDk[GF/R#U T/l D]a£EDLF>hfDLK>K/R#U*KY˜½¾ST<l)R#UK>debebgHVUGU DLF'[DR#N THVNGEZEGbeN ]B­ U [email protected]'EGF<H#­ b
bqR#K<SLF/R#b£YGDBbqRkPQDLF>[DŠ”)K<HVN ]–bgD"
] ð`_'hfiLK>F<D]a)bqRkF<SLEcHVU ]<DBYGD Ñ']>K/R#K>dgHVU ]fœ’Ydg]>EcH`]<SD]uF<SLP`NGbedgiLF<DLhfDLU*K"DLU

R#Ôdeh"NGK>lD]<K}]>deh"NGbgSD`n
ÉHVNGFNGU"DLU ]<DLhB[GbgD4YGD7w`R#bgDLNGF<]‘YGD7b½¾SLU DLF>P`dgD aYGD7b½§R#UGP`bgD4ÔSLUGdeK>l)R#b aDLK‘EHVNGFNGU"K†¼zEDYGDEGF>dehkR#deF<D
E
θ
Yd“—cSLF<DLU*K œEGF<HVK<HVU¬HVNfŒŽDLFΝtaI½§R#dSLK>N YdgSObqRxF<SLEcHVU ]<DuYGD–TD}F<S]/DR#N!÷LDLUpSLK<HVdebgDQ÷VaEcHVNGFoYd“sfF<SR#bedg]/R#K>dgHVU ]
Z
Yd“—cSLF<DLU*K<D]ŠYGDˆPQDLF>[D]Š]>deh"NGbgSD]¬RwQDT!‡rHVF<]>d Ç RaNGK>debedg]/R#U*K'bgDmhfH*YGiLbgDml)RVYF<HVUGdgjN å
D x-¡zùQ
| ( n4C‘R

]>deh"NGbqR#K>dgHVUZYNÆYGSLK<DTLK<[email protected]*N DxI½§R#d4NGK>debedg]<SDŠD]>KBTDLbebgDŠj*NGdrRˆSLK<S'YGSLwQDLbgHVEGESD¬»ˆbì½ DÞ}
É \a‘YGHVU*KBbgDkU HVh
YGD"[)R#EGK<yLhfDD]>Kf÷ Ù!Ò4/];‡Û•Ô 5 ÷Vnc™OU [email protected]]<TLF>deEGK>dgHVU2YGSLK/R#debebgSDYN%EGF>deU TLdeEcDDLK–NGU DTR#F/RVTLK<SLF>dg]/R#K>dgHVU2YGD]
9
8n›eã/ê›ëã/àá;çJßÞâŽá†ãÎê§à#ßJï߯àáâ 9`›É9¯âß¯à#çJßÿrëãrëŽê§ïñä›ê§çJê¾á 9 û ü éáê§ä›ê§ëã<áê›ìÎàœ
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
c[v
,.-F/.1„ n
”PR#N T<l D\”QDtsGDLhEGbgD]oYGDŒ’HVU TLK>dgHVU ]rC(ÉÆTHVF>F<D]>EcHVU Y R#UK»-YGD]4PQDLF>[D]4EGF<H*YNGdeK<D]4E)R#FoYGD]
EGF<HVK<HVU ]uØBY˜2 ½§R#UGP`bgD"ÔSLUGdeK>l)R#b
aEHVNGFuYd“—˜SLF<DLUK<D]–SLU DLF>P`dgD]aGdeU YdgTSD]OE)R#F
a)HVÈ
D]<KuDLU
θ = 25◦
log10 (E)
E
|D€@n ”ÌYF<HVdeK<Dk”zSLwQHVbeNGK>dgHVU%YGDAbqRBŒ’HVU TLK>dgHVUˆCg(É Y˜½ NGU¬EGF<HVK<HVU%»
18 D€ôEcHVNGF}Yd“—cSLF<DLU*K<]OR#UGP`bgD]
ÔSLUGdeK>l)R#Nscn)™uUˆRIJN ]>K<DLhfDLUKuYGD"T<l)RVj*N D–Œ’HVU TLK>dgHVUmD]<KK>F/RVWSxDLU¬EE
HVdeU*=K>debe10
bgS]n C‘R]<deh"NGbqR#K>dgHVUmYGD]PQDLF>[cD]
[email protected]<S-Dt—˜DTLK>N SD"RwQDTB‡rHVF<]<d Ç Ra)RwQD}
T x-¡zùQ
| (¨EcHVNGFhfH*YGiLbgDxl)RVYF<HVUGdgj*N D`n
F<S]>NGbeK/R#K<]}Y˜½¾|4RV]<¼G¡*deh©D]<K}EGF<S]<DLUK<SD"Y R#U ]"‚ †n

C‘RˆEGF<HV[)R#[GdebedeK<SŠYGD'YGSTLbgDLU T<l DLhfDLU*Kk»¬NGU DkYdg]<K/R#U TD YGDfb½§RsGDkYGDkbqR¬PQDLF>[cDk]½¾HV[GK>dgDLU*K\YGHVU TŠDLU
r
TR#bgTLNGbqR#UKbgD}F/R#EGEHVF>KYNU HVhB[GF<DuYGD]>K/R#K>dgHVU ]4R¼R#UKrYGSTLbgDLU T<l S
R#NU HVhB[GF<D}K<HVK/R#b)YGD]<K/R#K>dgHVU ]
NT2 (r)
EGF<S]<DLUK<D]@»ˆTDLK>K<D'Ydg]>K/R#U TD
ajNGdrY R#U ]-U HVK>F<DŠTHVU­ P`NGF/R#K>dgHVUXYGDfF<S]<DR#NÅwVR#NGKBK<HVNVIJHVNGF<] Ñ`_
Ntot (r)

œ _\PQDLF>[D]
Ñ]>K/R#K>dgHVU ]E)R#FuTDLF<TLbgDx”

× 
PE,θ,Z (r) =
ü‚ü0E
žuIL
NT2 (r)
Ntot (r)
œ„ n „*
IV QJ$I> ÿ%PŸ< b
ž
C RO­ P`NGF<D„ n debebeN ]>K>F<Dvb½¾SLwQHVbeNGK>dgHVU'YGD]Cg(ɏ]rDtszdePQDR#U*KNGUfYGSTLbgDLU T<l DLhfDLU*K|(}Ñ"(‘HJ( RwQDT}b½¾SLU DLF>P`dgD
‘
Ø
DLK7b½§R#UGP`bgDÔSLUGdeK>l)R#b EcHVNGFYGD]7EGF<HVK<HVU ]n*‡rHVhhfDOR#K>K<DLU YNaQbqR–EGF<HV[)R#[GdebedeK<SuYGDYGSTLbgDLU T/l DLhfDLUKvD]>KSLPR#bgD
» EGF<HzT<l D\YGDxb½§RsGD"YGD"bqRPQDLF>[cD"DLKuK<DLU YmwQDLF<]–_k»P`F/R#U YGD\Ydg]>K/R#U TD`n)CD"F/R¼QHVU¤YGD-K>F/R#U ]>deK>dgHVUa
a
r1/2

K<DLbj*N D
R#NGP`hfDLU*K<D-RwQDT–b½¾SLU DLF>P`dgD`aEGNGdg]<j*N DubqR"PQDLF>[cDOD]>KrEGbeN ]SLK<DLU YN Dx»-P`F/R#U YGD
PE,θ,Z (r1/2 ) = 1/2
SLU DLF>P`dgD`a*DLKrYdehdeUN DARwQDTOb½§R#UGP`bgDuÔSLUGdeK>l)R#bcTR#F4bqR-PQDLF>[D–RAE)R#F<THVNGF>NpNGU D}EGbeN ]oP`F/R#U YGDuYdg]>K/R#U TD–Y R#U ]
b½§R#K>hfH`]>EGl iLF<D-DLKuD]<KE)R#FuTHVU ]<Sj*N DLUK}EGbeN ]–R#K>K<SLUN SD`n
DÞb4D]>K"EcH`]<]>de[GbgDkYGDfE)R#F/R#hfSLK>F>dg]<DLFTD]xŒŽHVU TLK>dgHVU ]@YGDfhkR#UGdgiLF<D'K>F<i]-EGF<STLdg]<D`a;THVhhfD'bgDfhfHVUK>F<DLU*K
úÀq –¢¡ ÃQGÃòÁ!ۀÛÁÃ`>f
c¿
gb D]–RIJN ]>K<DLhfDLUK<]AK>F/RVTS]A]>NGFObqR­ P`NGF<D\„ n nC‘RkEGF<HV[)R#[GdebedeK<SYGD"YGSTLbgDLU T<l DLhfDLU*K-EcDLNGK–]½¾STLF>deF<[email protected]]<HVN ]ObqR
Ø
Œ’HVF>hfD”
PE,θ,Z (d) =
"
= 1
(r − r0 )2
1 + 2 exp −
λ2
−9 #−1/9
œ„ n:ðQ
si (r > r0 )
œ„ n 
Ø
œ„ ¤n ÷`
si (r < r0 )
VH ȉbgD]\YGDLNs2E)R#F/R#hfiLK>F<D]Bbede[GF<D]\]<HVU*KBbgD]\Ydg]>K/R#U TD]
DLK n‘C4½¾SLwQHVbeNGK>dgHVUZYGD'TD]"E)R#F/R#hfiLK>F<D]RwQDT
r0
λ
b½¾SLU DLF>P`dgD'DLK\b½§R#UGP`bgD'D]<K"F<DLEGF<S]<DLU*K<SD¬]>NGF"bqR¬­ P`NGF<D'„ n¤÷znC‘RmwVR#[email protected] ajNGdrdeU Ydgj*N DkbqRmYdg]>K/R#U TD
r0
,6-0/.1c„ ¤n ÷
wQHVbeNGK>dgHVUÆYNªE)R#F/R#hfiLK>F<D a˜»ŠPR#N T<l D`aDLK-YGD a˜»pYF<HVdeK<D`a‘RwQDTbgDTH`]>deUN ]"[email protected]½§R#UGP`bgD
ÔSLUGdeK>l)R#baG2¤
EHV£ NGFOYGD]PQDLF>[D]uYGDxEGF<HVK<HVUm]>rde0hBNGbgSD]}EcHVNGFuYGD]}SLU λDLF>P`dgD]OR#bebqR#U*KuYGD 17.8 » 18.6 D€@n
10
10
THVF>F<D]>EHVU Y R#U*KrR#NfYGSL[GNGK4YGDvbqR-T<lNGK<DOYGDvbqRAEGF<HV[)R#[GdebedeK<S}DLU†YGD]<]/HVN ]4YGD aR#NGP`hfDLU*K<DuTHVhhfDb½¾HVUk]½ ¼

R#K>K<DLU YpRwQDT–b½¾SLU DLF>P`dgD–DLKvYdehdeUN D-RwQDT–b½§R#UGP`bgDOÔSLUGdeK>l)R#bnGCR-w`R#bgDLNGFYGD a*j*NGdF<DLEGF<S]/DLUK<DAbqR"bqR#F>PQDLNGF
λ
YGD}bqRxÔHVU DOYGDuYGSTLF<HVdg]<]ÎR#U TDOYGDubqRxEGF<HV[)R#[GdebedeK<S`aGR#NGP`hfDLUK<DxRwQDTOb½¾SLU DLF>P`dgD–DLK4b½§R#UGP`bgD`azTHVU ]<SjN DLU TD–YGD
b½§R#NGP`hfDLUK/R#K>dgHVU‰YGDBbqRfK/R#[email protected]'PQDLF>[[email protected] aDLKxYGD"bqRkhfHVdeU ]–Œ’HVF>K<D\YGSTLF<HVdg]<]/R#U [email protected]¤]>deP`U)R#b7DLU
E
Œ’HVU TLK>dgHVU"YGD4bqRYdg]>K/R#U TD4»b½§RsD4EHVNGFbgD]‘PQDLF>[D]R¼R#UK;K>F/RwQDLF<]/SNGU DP`F/R#U YGDoSLE)R#dg]<]<DLNGF‘Y˜½§R#K>hfH`]<EGl iLF<D`n
Š",†ñ
0¦¥ 9º}9;:?¶ ³\5Ò³B¶‚?579?:\@u1
M E)R#F>K>deFrYGD]7ŒŽHVU TLK>dgHVU ]oCg(ɏ]aU HVN ]4Ydg]>EcH`]<HVU ]YGDuK<HVNGK<Db½ deUŒŽHVF>hkR#K>dgHVU'U STD]<]ÎR#deF<D}EcHVNGF4TR#bgTLNGbgDLF
b½§RVTTDLEGK/R#U TD"YN¬F<S]<DR#NmYGD"]>NGFJŒŽRVTD`[email protected]<H*TSYNGF<D"D]>K}YGHVU TxbqR]>NGdew`R#UK<Dk” R­ UmY˜½ NGK>debedg]<DLFOR#NˆhdgDLNs
c ú
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
bqR-]>¼zhfSLK>F>dgD}l Dts R#PQHVU)R#bgD–YNp¡zÕ¨YGDub½ š ÉM\aU HVN ]rTR#bgTLNGbgHVU ]ob½¾Dt¥kTRVTLdeK<S–YGDOYGSLK<DTLK>dgHVUŠDLKob½§RVTTDLEGK/R#U TD
deU ]>K/R#UK/R#U SDpY˜½ NGU DŠTDLbebeNGbgDkl Dts R#PQHVU)R#bgDpSLbgSLhfDLUK/R#deF<Dªœ ]>K/R#K>dgHVUÆTDLU*K>F/R#bgDpDLUK<HVNGF<SDpYGD wQHVdg]>deU D]Νta

Ø
EGNGdg]uU HVN ]–hBNGbeK>deEGbedgHVU ]xbgD"F<S]<NGbeK/R#KOE)R#F–bgD"U HVhB[GF<DYGDBTDLbebeNGbgD]–EGF<S]<DLU*K<D]xY R#U ]–bqRkTHVU­ P`NGF/R#K>dgHVU2YN
F<S]<DR#NTHVU ]>dgYGSLF<SD`n`C½§RVTTDLEGK/R#U TD]>NGF£NGU DrEcSLF>dgH*YGDvYGDrEGF>dg]<DrYGDrYGHVUGU SD]7]/DrTR#bgTLNGbgDR#bgHVF<]7]>dehEGbgDLhfDLU*K
DLUˆTHVhEGK/R#UKObgDxU HVh"[GF<DBYGDxTDLbebeNGbgD]J†]<DTHVU YGD]ARVTTLNGh"NGbgSD]OEDLU Y R#U*K}bqRYNGF<SD-YGDAŒŽHVU TLK>dgHVUGU DLhfDLUKn
ü¨§£üÞý
ÿSK·gÿ–ª IV`JQÿS Q`ÿ%g IVN+ÿSÿ%KKK†ÿ~#ÿ–«^ILM'< IVKÿ
©bIVKK3
C ½¾Dt¥kTRVTLdeK<S–YGD}YGSLK<DTLK>dgHVUpY˜½ NGU D}TDLbebeNGbgD}l Dts R#PQHVU)R#bgDOYGSLEcDLU YfYGD]oTHVU YdeK>dgHVU ]oYGDOYGSTLbgDLU T<l DLhfDLUKx”

EHVNGFE)RV]/]<DLFfbgDˆTLF>deK<iLF<D€(v„ a7NGU SLwQSLU DLhfDLUKpYGHVdeK'RwQHVdeFŠR#NËhfHVdeU ] ~ ]>K/R#K>dgHVU ]fU HVU‹R#bedeP`U SD]'R¼R#U*K
YGSTLbgDLU T<l SBNG€
U (}e
Ñ (‘JH ( œbgD"EGF>deU TLdeEDBYN%TR#bgTLNGb;D]>KubgDxhfyLhfD"EHVNGFObqRfTHVU­ P`NGF/R#K>dgHVU!„
(}ÑQtnb–HVN ]
C1
THVUGU)R#dg]<]<HVU ]YGS¯I<»!bqR%EGF<HV[)R#[GdebedeK<SˆjN½ NGU D¬TLNGwQD¬YGSTLbgDLU T/l D¬NG
U (‘JH ( DLUZŒŽHVU TLK>dgHVUËYGD¬]/R!Ydg]>K/R#U TDˆ»
b½§RsDŠYGD'bqRˆPQDLF>[cD`a‘P`F/ÖVTDpR#Ns2ŒŽHVU TLK>dgHVU ]@C (É;n7ÉHVNGFfT/l)RVjN D'EcH`]>deK>dgHVUZEH`]<]<de[GbgDkYNZ‘T “ANGFadebv]>N¥fK
R#bgHVF<]-YGDfTR#bgTLNGbgDLF-bqRŠEGF<HV[)R#[GdebedeK<S'YGDfYGSTLbgDLU T<l DLhfDLU*K\THVUVI>HVdeUK<D'Y˜½§R#N2hfHVdeU ] ~ ]<K/R#K>dgHVU ]-DLU2K>F>dqR#UGP`bgD
SjNGdebqR#K<SLF/R#ba)THVhEGK<D-K<DLUN%YGDxbgDLNGF}Ydg]>K/R#U TD"»@b½§RsGD`n
É;HVNGF4NGU D}SLU DLF>P`dgDODLKNGU'R#UGP`bgDuY˜½ deU TLdgYGDLU TD–YGHVUGU S]a`deb ŒŽR#NGK4THVhhfDLU TDLFrE)R#F4TR#bgTLNGbgDLFobqRxYdg]>K/R#U TD
YGDvT<l)RVj*N Dr]<K/R#K>dgHVUk»ub½§RsGDrYGDobqROPQDLF>[cD`a`[email protected]<Dv]/ROEGF<HV[)R#[GdebedeK<SvdeU YdewzdgYN DLbebgD}YGDvYGSTLbgDLU T<l DLhfDLU*K
DLK
bgD]OEGF<HV[)R#[GdebedeK<S]BjN DBbqRŠ]>K/R#K>dgHVU
»fE)R#F>K>deF-YGD\bqRŠgC (ÉÌTHVF>F<D]>EcHVU Y R#UK<D`n¡zHVdgDLU*K
Pi
Qi = 1 − P i
i
œ EGF<DLU)R#UKObgD]wVR#bgDLNGF<]uYGD e
» ÷`rYGSTLbgDLU T<l DBDLKaGF<D]>EcDTLK>dewQDLhfDLUKa)U D-YGSTLbgDLU T<l D"E)RV]}YG”
D (}q
Ñ (‘JH (xn)C‘R

i
Qi=7 n
EGF<HV[)R#[GdebedeK<S"j*N½§R#N TLNGU D-]>K/R#K>dgHVUˆU½§R#deKOYGSTLbgDLU T<l S"]½¾STLF>deK
P(0) =
i=1 Qi
,.-F/.1Q„ n§¦ ‡rHVUwQDLUK>dgHVU ]rPQSHVhfSLK>F>dgj*N D]NGK>debedg]<SD]4EHVNGF7bgDTR#bgTLNGbYGDvbqRAEGF<HV[)R#[GdebedeK<S}YGDX(v„ aVbqRAEcH`]>deK>dgHVU
YNˆT‘“ANGFu2 YGDxbqR\PQDLF>[cD-D]>KdeU Ydgj*N SDxE)R#F}[email protected]<HVd“s˜n
ÕAD!hfyLhfD`avbqRÆEGF<HV[)R#[GdebedeK<S2jN½ NGU DªDLK¬]<DLNGbgDLhfDLU*KˆNGU ª
D ]>K/R#K>dgHVUÌR#deKˆYGSTLbgDLU T/l S2]½¾STLF>deK
P(1) =
P
arDLK'bqR‰EGF<HV[)R#[GdebedeK<S¤Y˜½§RwQHVdeFmÑÅ]>K/R#K>dgHVU ]x‘
( HJ(¹D]>K'YGHVUGU SD!E)R#F
P(0) ×
i
Pi /Qi
P(2) = P(0) ×
úÀq –¢¡ ÃQGÃòÁ!ۀÛÁÃ`>f
P
c
Gn {;deU)R#[email protected]<HV[)R#[GdebedeK<SBEHVNGFuj*N½ NGU DxPQDLF>[cDxK<HVh"[)R#U*K–DLUmNGUˆEHVdeU*KOYGD-THzHVFJ
;E)RV]/]<DvbgDUGdewQDR#N'YGDYGSTLbgDLU T/l DLhfDLUK|(v„'œ ~ (HJ( DLUfK>F>dqR#UGP`bgDOSjNGdebqR#K<SLF/R#bòD]>KYGHVUGU SD
j>i Pi Pj /Qi Qj
GY HVUGU SD]Oœ
xc , y c
E)R#F-”

X Pi X Pi Pj
+
+
PT4 (xc , yc ) = 1 − P(0) 1 +
Qi
Qi Qj
i
j>i
X
i,j,k align

Pi Pj Pk 
Qi Qj Qk
œ„ n§¦Q
HVȬbgD-YGDLF>UGdgDLF}K<DLF>hfD-THVhEGK/R#[Gdebedg]<D"bgD]}THVU­ P`NGF/R#K>dgHVU ]–HVÈ ~ ]>K/R#K>dgHVU ]u]/HVUKuR#bedeP`U SD]Bœ’TtŒ£­ P`NGF<Dx„ n§¦Qtn
‡rDLK>K<D"EGF<HzTSYNGF<DBEDLF>hfDLKAYGDBTR#bgTLNGbgDLF–bqRkEGF<HV[)R#[GdebedeK<S\YGD\YGSTLbgDLU T<l DLhfDLU*KxEHVNGF–U½ dehEcHVF>K<DBj*N DLbg]
E)R#F/R#hfiLK>F<D]7YGDoPQDLF>[cDAœ’SLU DLF>P`dgD`aQR#UGP`bgDrDLK£KÞ¼*EcDoYGDrEGF>dehkR#deF<Dtn`MXE)R#F>K>deF7YGDrTDLK>K<DrEGF<HV[)R#[GdebedeK<S}Y˜½¾HV[GK<DLUGdeF
NGx
U (v„ [email protected]]<NGFJŒŽRVTDxDt—˜DTLK>dewQDxYGDxYGSLK<DTLK>dgHVUmY˜½ NGU DxTDLbebeNGbgD-SLbgSLhfDLU*K/R#deF<Dxl DtsGR#PQHVU)R#bgD"]<DATR#bgTLNGbgD-THVhhfD
b½ deUK<SLP`F/R#bgD"YGD"]/[email protected]]>NGFJŒRVTDx]>[email protected]<SLP`dgHVU%R#NGK<HVF>dg]<SD-E)R#F}bg”
D (XðaGEcHVU YGSLF<SDxE)R#F}bqR\EGF<HV[)R#[GdebedeK<S"YG“
D (v„ˆ”
Seff =
Z
cellule
PT 4 (xc , yc ) HT 5 (xc , yc ) dS
œ„ n§^Q
D]>K}NGU D"Œ’HVU TLK>dgHVU¤j*NGdwVR#NGK bgHVF<]/jN D"bgD"T‘“ANGFOYGDBbqRPQDLF>[cD"]/D"]>deK>N DBY R#U ]uNGU D"ÔHVU D
HVÈ

HT 5 (xc , yc )
R#NGK<HVF>dg]<SD}E)R#FbgDuTLF>deK<iLF<DB(XðaQDLKr_-Y R#U ]bgDuTRV]4THVUK>F/R#deF<[email protected]œ’TtŒ>nV­ P`NGF<D}„ n EHVNGFbgDu]<T<l SLhkR"YGD}bqRx]>NGFJŒRVTD

R#NGK<HVF>dg]<SDxE)R#F}bg”
D (XðQtn
,6-0/.1˜„ n§^
”×PR#N T/l Dm”SLwQHVbeNGK>dgHVUÆYGDbqR¬]>NGFJŒRVTDDt—˜DTLK>dewQDkYGDfYGSLK<DTLK>dgHVUÆY˜½ NGU DfTDLbebeNGbgDfl Dts R#PQHVU)R#bgD
œ U HVF>hkR#bedg]<2 SDˆ» BDLUZŒ’HVU TLK>dgHVUËYGDpb½¾SLU DLF>P`dgDpEcHVNGFfYd“—cSLF<DLU*K<]kR#UGP`bgDpÔSLUGdeK>l)R#Ns˜aY R#U ]@bgD¬TRV]Y˜½ NGU D
PQDLF>[DdeUGdeK>dgSDOE)R# F4NGUkEGF<HVK<HVUnT”«YF<HVdeK<D"”*THVhE)R#F/R#dg]<HVUpYGDbqR-]>NGFJŒRVTD}Dt—˜DTLK>dewQDOYGDuYGSLK<DTLK>dgHVUŠHV[GK<DLU*N D
EHVNGF}YGD]U H¼R#NsmYGD–Œ’DLF}HVNmEHVNGF}YGD]EGF<HVK<HVU ]n
C‘R'­ P`NGF<D„ n§^¬RQADLKx[tacEGF<S]/DLUK<DbgD]xF<S]<NGbeK/R#K<]-YN2TR#[email protected]¬]>NGFJŒRVTDDt—˜DTLK>dewQDYGDfYGSLK<DTLK>dgHVU
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
c%W
˜Y ½ NGU D"TDLbebeNGbgD"l DtsGR#PQHVU)R#bgDBDLUˆŒŽHVU TLK>dgHVU%YGDxb½¾SLU DLF>P`dgD`aEHVNGFOYd“—cSLF<DLU*K<]–R#UGP`bgD]OÔSLUGdeK>l)R#NsmDLKuKÞ¼*EcD]uYGD
EGF>dehkR#deF<D]nGC‘[email protected]]>NGFJŒRVTDADt—cDTLK>dewQDxYGDxYGSLK<DTLK>dgHVU%]/R#K>NGF<D–wQDLF<] ~ |D€ôEHVNGFYGD]EGF<HVK<HVU ]}EDLNpdeU TLbedeU S]}DLK
wQDLF<]x„m|D†€±EHVNGF-bgD]-PQDLF>[D]BY˜½§R#UGP`bgDÔSLUGdeK>l)R#b
HVN
n‡rDLK>K<DfSLwQHVbeNGK>dgHVUÆYGD
RwQDT
θ = 0◦
θ = 60◦
Seff
b½§R#UGP`bgD-EGF<Hw*dgDLU*K–YGD"bqRfTHVhEcSLK>deK>dgHVU%DLUK>F<DBb½§R#NGP`hfDLUK/R#K>dgHVUªYGD"bqRK/R#debebgD"YGD"b½¾DLhEGF<DLdeUK<D\YGD-bqRfPQDLF>[D
R#Nk]<HVbRwQDT a`j*NGdRAK<DLU Y R#U TD–»–ŒŽRwQHVF>dg]<DLF4bgDuYGSTLbgDLU T<l DLhfDLUK}Yb
N (v„ a*RwQDTub½§R#K>K<SLUN)R#K>dgHVUpYGDbqRxPQDLF>[cD
θ
]>NGdeK<Dk»p]<HVU‰EGbeN ]-P`F/R#U YªE)R#F<THVNGF<]\Y R#U ]-b½§R#K>hfH`]>EGl iLF<D`aTDfj*NGd4YdehdeUN DkbgDf]>deP`U)R#bhfD]>NGF<SkY R#U ]-bgD]
]>K/R#K>dgHVU ]n
‡ [email protected]%EcDLNGKubgD\THVU ]>K/R#K<DLFx]>NGF–bqR­ P`NGF<DB„ n§^k[ta)bqR']>NGFJŒRVTDBDt—˜DTLK>[email protected]<DTLK>dgHVU¤U DBwVR#F>dgD
r
E)RV]@YGD'bqR%hfyLhfDphkR#UGdgiLF<D¬]>NGdew`R#[email protected]ŠK†¼zEcDŠYGDŠE)R#F>K>dgTLNGbgDpEGF>dehkR#deF<D`a£bgD]@U H¼R#NsÆYGD'Œ’DLFfR¼R#UKNGU D
EGF<HV[)R#[GdebedeK<SˆEGbeN ]kP`F/R#U YGDˆYGDˆYGSTLbgDLU T<l DLFŠNGU (v„‰j*N D¬bgD]EGF<HVK<HVU ]n4CDpŒR#deKfj*N D¬bqRª]>NGFJŒRVTDˆYGDˆYGSt
K<DTLK>dgHVU2YGSLEDLU YGDkY˜½ NGU [email protected]¼zEHVK>l i]<Dk]>NGFAbqRpTHVhEH`]>deK>dgHVUÅYNªF/R¼QHVUGU [email protected]`]>hdgj*N [email protected]]/DLh"[GbgDLF
EGF<HV[GbgSLhkR#K>dgjN DB»"EGF<DLhdgiLF<D-w*N D`azhkR#dg]U HVN ]vwQDLF>F<HVU ]}j*N DxTDLK>K<DxTR#F/RVTLK<SLF>dg]>K>dgj*N DxEcDLNGKvyLK>F<DxDtsEGbgHVdeK<SD
U½¾D]>KoE)RV]v]/R#K>NGF<SD`nCD–YGHVhkR#deU DxY˜½¾SLU DLF>P`dgD
EHVNGFvTHVUK>F/R#deU YF<DAbqRBTHVhEcH`]>deK>dgHVU¬Y R#U ]obqR"F<SLP`dgHVU¬HVÈ
Seff
DLUXjN D]>K>dgHVUX]<Dp]>deK>N D¬DLU*K>F<D DLK„¤|4D€@a£EGF<STLdg]<SLhfDLUKfbq»!HVÈXbqR!THVUGU)R#dg]<]/R#U TD¬YGDpbqR!THVhEcH`]>deK>dgHVU

R#EGEHVF>K<DLF/R#deKBNGU DdeUŒŽHVF>hkR#K>dgHVUÅ]>NGFxbqRpEcH`]>deK>dgHVU‰YGDfbqRpK>F/R#U ]>deK>dgHVU‰PR#bqRVTLK>dgj*N DwQDLF<]BDtszK>F/RPR#bqRVTLK>dgj*N D`n
OHVN ]vEGF<S]/DLUK<DLF<HVU ]uY R#U ]}bgD–T/l)R#EGdeK>F<d
D ¬=¬@bgD]vhfSLK>l H*YGD]}j*N DAU HVN ]uRwQHVU ]}YGSLwQDLbgHVEGESAEcHVNGF}SLK>N YdgDLF}bqR
U)R#K>NGF<D-YGD]F/R¼QHVU ]uTH`]>hdgj*N D]uDLU*K>F<D DLK„Q|D†€\n

Mu­ UÅYGDŠ]½§R—F/R#U T/lGdeF\YNÆEGF<HV[GbgiLhfDpYGDfbqRmwVR#F>dqR#K>dgHVUZDLUÅSLU DLF>P`dgDpYGDkbqRm]>NGFJŒRVTDkDt—˜DTLK>dewQDŠYGD'YGSt
K<DTLK>dgHVUaR#deU ]>d–j*N D!YGDmb½ l*¼*EcHVK>l i]<D¤]>NGFŠbqR‰THVhEcH`]>deK>dgHVUardebx]>N¥fKŠYGD!THVU ]>dgYGSLF<DLFpNGUGdgj*N DLhfDLUK¬bgD]
SLwQSLU DLhfDLUK<]AR#N†YGD]<]>N ]}YGD-b½¾SLU DLF>P`dgD-YGD-]/R#K>NGF/R#K>dgHVUa ]/HVdeK
|D†€\n
E>3
$ ÃQÃQܑfÃQB4fffÁ“Ûog&<fxÃQzõõ’&;õzh‘$’G()f õ®­ ”ÅE)R#F>K>deF‘YGD7b½¾DtsEGF<D]<]>dgHVU"YGD£bqR]<NGFJŒŽRVTD
@
Dt—cDTLK>dewQD"YGDxYGSLK<DTLK>dgHVU
a b½§RVTTDLEGK/R#U TD"deU ]>K/R#UK/R#U SDBY˜½ NGU D-TDLbebeNGbgDxl Dts R#PQHVU)R#bgD`aGhfD]>NGF<SD
Seff (E, θ, ϕ)
DLU¬h 2 n¾]>FneaG]½¾HV[GK>dgDLU*KO]>dehEGbgDLhfDLU*KODLUˆdeU*K<SLP`F/R#UK
]>NGFb§½ R#UGP`bgD-]/HVbedgYGDf”
Seff (E, θ, ϕ)
a(E) =
Z
Seff (E, θ, ϕ) cos θ dΩ
θ≤60◦
œ„ n Q_ 

VH ÈmbgD-ŒŽRVTLK<DLNGF
EGF<DLU Y!DLU%THVhEGK<DBbgDxŒR#deKOj*N D-bgD"YGSLK<DTLK<DLNGFAEGF<S]<DLU*K<D"NGU DB]>NGFJŒRVTD-EGF<H#I>DLK<SD\R#Ns
cos θ
PQDLF>[D]}deU TLdgYGDLUK<D]–]<HVN ]b½§R#UGP`bgD n
θ
úÀ‚[email protected]ÃQÃQܑfÃQª4f‘ff‘Á*%kGÃQÃQܑfÃQª4fÁ()#Á€Û;& Á* &tÛ;m&ÂÞGÃQ
Š",W¯
cV[
ž9;9;5£µ?ºu1B9;5 :/1CB–?º}1¢?º}1\¶75Ì5‚?Zº}99;5£µC?º}1\95 :<1¢?¶ö·c¶75̳\6 ·c¶gB)57ºu6¢³\5
~
6"·cÓº}9;5
B
‡ HVhhfD\U HVN ]Ab½§RwQHVU ]AhfDLUK>dgHVUGU SfEGF<STSYGDLhhfDLUKa˜b½§RVTTDLEGK/R#U [email protected] ]>K/R#U*K/R#U SDY˜½ NGU [email protected]­ P`NGF/R
r
K>dgHVU2YGHVUGU SDYNªF<S]<DR#N¤EDLNGK-yLK>F<DHV[GK<DLU*N Df]>dehEGbgDLhfDLUKBDLUªh"NGbeK>deEGbedqR#UKBb½§RVTTDLEGK/R#U TDdeU ]>K/R#U*K/R#U SD
Y˜½ NGU D-TDLbebeNGbgDxl Dts R#PQHVU)R#bgD-SLbgSLhfDLUK/R#deF<DBE)R#FbgDxU HVh"[GF<DBYGDxTDLbebeNGbgD]}EGF<S]<DLU*K<D]uY R#U ]bqRfTHVU­ P`NGF/R#K>dgHVU
DLU2jN D]>K>dgHVUn‘‡ol)RVj*N D]>K/R#K>dgHVUÅR¼R#UK"]<d“s%wQHVdg]>deU D]"RVTLK>dewQD]aTHVhhfDf]>NGFAbqR'­ P`NGF<D„ n RQta˜THVU*K>F>de[GN D

»fb½§RVTTDLEGK/R#U [email protected] ]>K/R#U*K/R#U [email protected]–NGU [email protected]@SLbgSLhfDLUK/R#deF<D`aR#bgHVF<]AjN½ NGU [email protected]]>K/R#K>dgHVU2R¼R#U“
K ðkwQHVdg]>deU D]
THVhEGK<DAEHVNGF
YGDxTDLK>K<Dxw`R#bgDLNGFaGTHVhhfDxdebebeN ]>K>F<Sx]>NGFbqRB­ P`NGF<DA„ n [tnGC‘D]}]>K/R#K>dgHVU ]uR¼R#UKuhfHVdeU ]
2/3

YGz
D ð@wQHVdg]>deU D]uRVTLK>dewQD]uU DxTHVUK>F>de[GN DLUKAE)RV]}YN¬K<HVNGK–»@b½§RVTTDLEGK/R#U TD`n
,6-0/.1;„ n _
‡rHVhE)R#F/R#dg]<HVUËYGDŠb½§RVTTDLEGK/R#U TD¬deU ]>K/R#UK/R#U SD¬HV[GK<DLU*N DpE)R#[email protected]>l HzYGD¬YGDŠTHVhEGK/R#PQD

2
YGD]ul Dts R#PQHVU D]œ’THVNGF>[D\bedg]<]<DtaRwQDT\NGU¤TR#bgTLNGb£Ï!HVUK<D‡vR#F>bgHŠ[)R#bqR¼R#U*KxK<HVNGK<D"bqR']>NGFJŒRVTDBYN!F<S]<DR#N
EHVNGF4YGD]4PQDLF>[D]o]>deh"NGbgSD]v»xEGbeN ]<dgDLNGF<]4SLU DLF>P`dgD]a*EHVNGFoYGD]EGF<HVK<HVU ]AœEHVdeU*K<]4F<HVNGPQD]Νtn*CD}F<S]/DR#NkSLK/R#deK
THVU ]>K>deK>N SuYG"
D ÷ l Dts R#PQHVU D]rDLUpRVH0|GKrÑ`_`_#„ abqR-THVNGF>[cDu[GbgDLN DOTHVF>F<D]>EHVU Yp»Ab½§RVTTDLEGK/R#U TD–deU ]>K/R#U*K/R#U SD
Y˜½ NGU D-TDLbebeNGbgDxl Ø Dts R#PQHVU)R#bgD-h"NGbeK>deEGbedgSD"E)R#3
F ÷ n
Ø
| U-EGF>deU TLdeEcD`abqRvhfSLK>l HzYGD4YGD4THVhEGK/R#PQDoYGD]˜l Dts R#PQHVU D]EHVNGF>F/R#deKU D7E)RV]‘yLK>F<D]<N¥k]/R#hfDLUKEGF<STLdg]<D`a
7
EGNGdg]<jN [email protected]<S]<DR#NÅYGDf]<NGFJŒŽRVTDF<SDLboYGDb½ š [ ]<DLF>w`R#K<HVdeF<DkÉ7dgDLF>F<D'MuNGPQDLF-U DfEH`]<]/[email protected])RV]-NGU DE)R#FJŒR#deK<D
]>¼*hfSLK>F>dgD"l DtsGR#PQHVU)R#bgDŠ”)TDLF>K/R#deU D]A]>K/R#K>dgHVU ]OU½§R¼R#U*KxE)RV]}EGN¤yLK>F<D"YGSLEGbgH¼QSD]xDtsGRVTLK<DLhfDLU*Kx»fb½¾DLU YF<HVdeK
EGF<SLw*Nn‡rDLEcDLU Y R#UKa;U HVN ]\RwQHVU ]BwQSLF>d“­)Skj*N Dfb½¾DLF>F<DLNGF"F<S]<NGbeK/R#UK<DŠ]>NGF-b½§RVTTDLEGK/R#U TDŠdeU ]>K/R#UK/R#U SDŠYN
F<S]<DR#NŠD]<K4K<HVK/R#bgDLhfDLUKU SLP`bedePQDR#[GbgD`nGCRx­ P`NGF<DO„ n _-debebeN ]<K>F<DOb½¾DtsGTDLbebgDLUK}RVTTHVF<YpDLUK>F<DObqR"hfSLK>l H*YGDAYGD

[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
c%c
T HVhEGK/R#PQDkYGD]Bl DtsGR#PQHVU D]BDLK"NGU‰TR#bgTLNGbvYGD
Dt—˜DTLK>N SkDLU2[)R#bqR¼R#[email protected]<HVNGK<DfbqRm]>NGFJŒŽRVTDkYN2F<S]<DR#N
Seff
E)R#FE)RV]YGDzð`_#hˆnGC‘[email protected]>l HzYGDxYGDxTHVhEGK/R#PQD-D]>K}YGHVU T–­R#[GbgDxDLKEcDLF>hfDLK}Y˜½¾STHVU HVhdg]<DLFuP`F/R#U YGDLhfDLU*K
YN¬K<DLhE ]uYGDxTR#bgTLNGbEHVNGFb½¾D]>K>dehkR#K>dgHVU!YGDAb½§RVTTDLEGK/R#U TD`n
C ½§RVTTDLEGK/R#U TD"deU ]>K/R#UK/R#U SD-Y˜½ NGU D"TDLbebeNGbgDAl DtsGR#PQHVU)R#bgD-D]<K}YGHVU TAb½ NGUGdeK<SxYGDA[)RV]<DxYGDAb½§RVTTDLEGK/R#U TD

YNmF<S]<DR#N¤YGD"]<NGFJŒŽRVTD`n>D†b;D]>KuEcH`]<]>de[GbgDBYGD-bqR'TR#bgTLNGbgDLFONGU D"ŒŽHVdg]}EcHVNGFOK<HVNGK<D\DLKuhfyLhfD\YGD"bqRfhfD]>NGF<DLF
P`F/ÖVTD¬R#NsÆYGHVUGU SD]@l¼z[GF>dgYGD]HV[GK<DLUN D]'RwQDTpbqRm• N HVF<D]<TDLU TD`n7MON†YGD]<]>N ]@YGDŠbqR%]ÎR#K>NGF/R#K>dgHVUa£EHVNGF
|D†€\abqRˆ]>NGFJŒRVTDkYGD'YGSLK<DTLK>dgHVUZE)R#FBTLNGwQDpR¼R#UK]>d“sªwQHVdg]>deU D]@RVTLK>dewQD]@D]>K"YGHVUGU SDŠE)R#F 2
E > 3
d ×
√
Ç h 2 aGHVÈ
Ç h©D]<[email protected]]>K/R#U TD-DLUK>F<DBYGDLNsp]<K/R#K>dgHVU ]n C½§RVTTDLEGK/R#U TD"deU ]>K/R#UK/R#U SD
3/2 ≈ 1.95
d = 1.5
Ç h 2 ]>Fn
THVF>F<D]>EcHVU Y R#UK<D`a deU*K<SLP`F<SD"]>NGFbgD]OR#UGP`bgD]
◦ aGD]<K}YGHVU T
θ < 60
acell ≈ 4.59
,.-F/.1‘„ n
¡*l SLhkRˆF<S]>NGhkR#UK\bqRpEGF<HzTSYNGF<DkYGD'TR#bgTLNGboYGDfb½§RVTTDLEGK/R#U TDŠYN2F<S]<DR#NÆYGDf]>NGFJŒRVTDkYGD
b½ š ÉM\n `u2
{ deU)R#bgDLhfDLUKa`EcHVNGFTR#bgTLNGbgDLF7b½§RVTTDLEGK/R#U [email protected]<S]<DR#NkR#N†YGD]/]>N ]7YGDvbqR–]/R#K>NGF/R#K>dgHVUk]>NGF£NGU DvEcSLF>dgH*YGD
;
YGDAK<DLhE ]}YGHVUGU SD`aGdeb]>N¥fKOYGDxTHVhEGK<DLF}bgDAU HVh"[GF<DBYGDxTDLbebeNGbgD]}l Dts R#PQHVU)R#bgD]uTLNGh"NGbgSD]O]>[email protected]<SD
Ç h 2 ]<FnC‘RÆTHVU­ P`NGF/R#K>dgHVU YN«F<S]<DR#N
YGD!hfD]>NGF<DªDLKˆYGD!h"NGbeK>deEGbedgDLFmbgD!F<S]>NGbeK/R#K¬E)R#F
acell ≈ 4.59
[email protected]<l)R#UGPQDLFfRwQDT'bgD'K<DLhE ]\[email protected]'h"NGbeK>deEGbgD]@F/R#dg]<HVU ]p”‘YGD'U HVNGwQDLbebgD]]>K/R#K>dgHVU ]@]<HVU*K\YGSLEGbgH¼QSD]
œŽRVT<l iLwQDLhfDLU*K–YN%¡zÕôwQDLF<]­ UmÑ`_`_ ÷znenen›K<HVNGKuR#NmTHVNGF<]}YGDxb½§R#UGU SD`a)TDLbebgD]J†TLdEcDLNGwQDLUKuSLPR#bgDLhfDLU*KOyLK>F<D
YGSt­)TLdgDLUK<D]–YGD–ŒŽRVWHVUmE)RV]<]/R#PQiLF<DxDLKuYGHVdewQDLU*KOyLK>F<D-THVU ]>dgYGSLF<SD]OTHVhhfD"R#[ ]<DLU*K<D]uYNˆF<S]<DR#Nn
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
cv
C R‰­ P`NGF<D¤„ n F<S]>NGhfD¤bgD!EGF>deU TLdeED¤YNžTR#bgTLNGb-YGD!b½§RVTTDLEGK/R#U TD2YNÌ¡zÕ ]>NGFŠNGU D¤EcSLF>dgH*YGD¤YGD
‘
`
EGF>dg]<[email protected] SD]n˜[email protected]]>NGdewzd£YGD]xT/l)R#[email protected]—cDTLK/R#U*K-bgDf¡zÕ±R#NªTHVNGF<]-YN¤K<DLhE ]-D]<KADt—˜DTLK>N SDLU
]>NGF>wQDLdebebqR#UK}bgD]}K/R#NspYGDxYGSTLbgDLU T/l DLhfDLUK–bgH*TR#NsˆRwQDTANGU DxF<S]/HVbeNGK>dgHVUˆK<DLhEHVF<DLbebgD"Y˜½ NGU Dx]<DTHVU YGD`a TD
jNGd˜D]>KbqR#F>PQDLhfDLU*KO]>N¥k]/R#U*KEHVNGFbgDxTR#bgTLNGbYGDAb½§RVTTDLEGK/R#U TD`n)M¨E)R#F>K>deF}YGD]}deUŒ’HVF>hkR#K>dgHVU ]OTHVUK<DLUN D]
Y R#U ]ubgD]u­)T<lGdgDLFxYGD"]<NGF>wQDLdebebqR#U TD\Y!
N (}Ñ%4œ (}Ñ­ bgD]A]>NGFubqR­ P`NGF<Dta)bqR'THVU­ P`NGF/R#K>dgHVU¤PQSHVhfSLK>F>dgjN [email protected]
F<S]<DR#N%EcDLF>hfDLKAYGDBTR#bgTLNGbgDLFAbgDBU HVh"[GF<D
YGD\TDLbebeNGbgD]–l DtsGR#PQHVU)R#bgD]-R#N!K<DLhE ] aDLKOb½§RVTTDLEGK/R#U TD
Ncell
t
D]>KudeU TLF<SLhfDLU*K<SDYGD-bqRkwVR#bgDLNGF
aHVÈ
D]>KOb½ deUK<DLF>w`R#bebgDYGD-K<DLhE ]–EcDLU Y R#UKAbgDjN DLb
Ncell × acell × ∆t
∆t
[email protected]­ P`NGF/R#K>dgHVU!YN¬F<S]<DR#NmU½§[email protected])RV]}T<l)R#UGPQS`n
”«K>deK>F<DOY˜½¾DtsGDLhEGbgD`aR#NŠTHVNGF<]rYGDubqR-ESLF>dgHzYGDAR#bebqR#UKvYN
I<R#UwzdgDLFvÑ`_`_#„R#N ~ IJNGdebebgDLKvÑ`_`_ deU TLbeN ]a
 er
Ø
bgD4F<S]<DR#[email protected]]>NGFJŒŽRVTDrYGDob½ š [ ]<DLF>wVR#K<HVdeF<DvÉ7dgDLF>F<DvMONGPQDLF£R–RVTTLNGh"NGbgS
TDLbebeNGbgD
] (Xð†]/DTHVU YGD]a
10
≈ 2.94 10
Ç
Ç
]<HVdeK4NGU D–RVTTDLEGK/R#U TDOK<HVK/R#bgD–YGD
h
>
]
4
F
]
*
a
V
H
È
h
>
]
2
2 FrR#UaEHVNGFrYGD]oSLwQSLU DLhfDLU*K<]}R¼R#U*K
1.35 1011
4280
YGD]vR#UGP`bgD]
DLKvNGU DASLU DLF>P`dgDA]>NGESLF>dgDLNGF<Dx» ~ |4D€@nzÉ;HVNGF}THVhE)R#F/R#dg]<HVUab½§RVTTDLEGK/R#U TDxHV[GK<DLU*N D
θ < 60◦
E)R#Fob½¾DtsESLF>dgDLU TDAMuAMx¡zM¨DLUpYd“sŠR#U ]vYGDOYGHVUGU SD]vD]>KrNGUŠEDLNŠEGbeN ]vYGD–YGDLNskŒ’HVdg]rDLKrYGDLhdcdeUŒŽSLF>dgDLNGF<D
»@TDLK>K<DxwVR#bgDLNGFn
Š",³
}6 }5ó·ó<[email protected]¶B5£·[{rºz¶uó<5gBÙó<:<¶75gB·wóOõº}9;95£µC?º}1\9;5
´µ@ C{
CHVF<]-YGDb½¾SLbqR#[cHVF/R#K>dgHVUÅ[email protected]¬hfSLK>l H*YGDfYGDTR#bgTLNGboYGDb½§RVTTDLEGK/R#U TD`aEGbeN ]>dgDLNGF<]BHV[ ]<DLF>w`R#[GbgD]xEGl¼*
]>dgjN D]OK<DLbebgD]–jN DBbgD]}ŒŽHVU TLK>dgHVU ]AC (É4]x]<D"]<HVU*KARwQSLF<SD]xyLK>F<DBYGD]–P`F/R#U YGDLNGF<]–deUK<SLF<D]/]/R#UK<D]-»'SLK>N YdgDLF
DLK4hfD]>NGF<DLFrDtszEcSLF>dehfDLUK/R#bgDLhfDLU*KnGC‘R-EGbeNGE)R#F>KrYGDOTD]4P`F/R#U YGDLNGF<]vYGSLEDLU YGDLU*KrYGDObqRxU)R#K>NGF<DOYGD]4F/R¼QHVU ]
TH`]>hdgjN D]a deb£D]>K–R#deU ]>d;deUK<SLF<D]/]/R#UKxY˜½¾SLK>N YdgDLFAbgDLNGFOEcHVNGwQHVdeFxYdg]<TLF>dehdeU)R#UKncMOEGF<i]xRwQHVdeFxDtszEGbgHVF<SBNGU
TDLF>K/R#deU‰U HVh"[GF<D'YGDfEGdg]>K<D]BYd“—cSLF<DLU*K<D]aEcDLF>hfDLK>K/R#[email protected]`R#bedgYGDLFBDLK-K<D]<K<DLF-bgD]-F<S]>NGbeK/R#K<]BHV[GK<DLUN ]@DLU
]>deh"NGbqR#K>dgHVUaI>DuEGF<S]<DLU*K<DA[GF>dgiLwQDLhfDLUK}Y R#U ]vTDLK>K<DA]<DTLK>dgHVUŠbgD]rHV[ ]/DLF>wVR#[GbgD]j*NGd]/DLh"[GbgDLUKyLK>F<D–bgD]rEGbeN ]
EGF<HVhfDLK>K<DLN ]<D]n
üš¸ü†ý
¹
ÿ%_Qÿ¤ÿ « |Q$ÿJIVKÿÿ*K†ÿ%jTUg ºŸ) b
ž
C D]OŒ’HVU TLK>dgHVU ]xCg(ÉÌ]/HVUK-»fbqR'[)RV]<[email protected]¤TR#bgTLNGbYGD\b½§RVTTDLEGK/R#U TD¬”DLbebgD]–EcDLF>hfDLK>K<DLUKa‘»kE)R#F>K>deF-YGD
bqR¬THVUGU)R#dg]<]/R#U TD'YGDbqR¬EGF<HV[)R#[GdebedeK<S'deU Ydew*dgYN DLbebgD'YGDfYGSTLbgDLU T<l DLhfDLU*KfY˜½ NGU Dk]>K/R#K>dgHVUÅDLU2ŒŽHVU TLK>dgHVUÅYGD
]/RŠYdg]>K/R#U TDk»Šb½§RsGDY˜½ NGU DPQDLF>[DY˜½¾SLU DLF>P`dgDkDLK"Y˜½§R#UGP`bgDfYGHVUGU S`aYGDTR#bgTLNGbgDLF-bqRp]<NGFJŒŽRVTDDt—˜DTLK>dewQDfYGD
YGSLK<DTLK>dgHVU\YN-F<S]/DR#NBYGD4]<NGFJŒŽRVTD`nC‘D]ŒŽHVU TLK>dgHVU ]Cg(ɤ]<HVUK;HV[GK<DLUN D]£DLU"]>dehBNGbqR#K>dgHVUaVY R#U ]b½ l¼zEcHVK>l i]<D
Y˜½ NGU D\THVhEcH`]>deK>dgHVUªbgSLPQiLF<D¬œEGF<HVK<HVU ]Ν–HVN¤EGbeN ]AbgHVNGF<YGD¬œU H¼R#NsªYGDBŒ’DLFΝtncC½¾SLwQHVbeNGK>dgHVUÅ[email protected]½¾SLU DLF>P`dgD
v)
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
YGDubqR\]>NGFJŒRVTD–Dt—cDTLK>dewQDAYGD–YGSLK<DTLK>dgHVUˆYGSLEDLU YpYGD–TDLK>K<DOl*¼*EcHVK>l i]<DAYGDATHVhEH`]<deK>dgHVUpDLKvYNphfH*YGiLbgDAYGD
]>deh"NGbqR#K>dgHVU¬YGD]oPQDLF>[D]rDLKvYNŠYGSLK<DTLK<DLNGFnMu­ U'YGDOwVR#bedgYGDLFrTD]oF<S]<NGbeK/R#K<]4K>l SHVF>dgj*N D]azdebD]>KodehEHVF>K/R#U*K
YGDxTHVhE)R#F<DLFubgD]F<S]<NGbeK/R#K<]uYGDx]>dehBNGbqR#K>dgHVU ]–RwQDTxbgD]}YGHVUGU SD]n
É;HVNGFuhfD]>NGF<DLFObgD]ŒŽHVU TLK>dgHVU ]OCg(ɍ]<NGF}bgD]uYGHVUGU SD]–YN¤¡zÕ@aGdeb˜ŒR#NGKuTR#bgTLNGbgDLFaEHVNGFuNGUˆdeUK<DLF>w`R#bebgD
Y˜½¾SLU DLF>P`dgD-DLK}Y˜½§R#UGP`bgD-ÔSLUGdeK>l)R#bYGHVUGU S`aGbqR\EGF<HV[)R#[GdebedeK<S"YGD-YGSTLbgDLU T/l DLhfDLUKAY˜½ NGU D-]>K/R#K>dgHVUmDLUpŒŽHVU TLK>dgHVU
YGD¬]/R¤Ydg]>K/R#U TDm»!b½§RsGD¬YGD¬bqR!PQDLF>[cDª”£EHVNGFkNGUXdeUK<DLF>w`R#bebgDmYGD¬Ydg]>K/R#U TDˆTDLU*K>F<SmR#NGK<HVNGFkYGD a;NGU D
r
]>K/R#K>dgHVU¬]>debgDLU TLdgDLN ]<DATHVUK>F>de[GN D-EHVNGF_\Y R#U ]vbqR-w`R#bgDLNGFvhfH¼QDLUGU D-YGD–bqRBC(Éuœ taGDLKvNGU DA]>K/R#K>dgHVU¬jNGdR
r
YGSTLbgDLU T<l SxNG*
U (}Ñ\EcHVNGF nGÉ;R#FOTDLK>K<DAhfSLK>l H*YGDxHVUˆTHVU ]>K>F>NGdeKuNGU DxC (ɍDtszEcSLF>dehfDLUK/R#bgD"RwQDTxK<HVN ]vbgD]

SLwQSLU DLhfDLUK<]–YN%¡zÕ@n
‡rDLEcDLU Y R#UKadebŒŽR#NGKyLK>F<D–EGF>N YGDLUKOY R#U ]vbqRBTHVhE)R#F/R#dg]/HVUmRwQDTAbgD]]>dehBNGbqR#K>dgHVU ]}TR#FvEHVNGFTR#bgTLNGbgDLF
bqRxCg(ÉÆ]<NGFbgD]4YGHVUGU SD]adebGŒR#NGKoj*N D}TD]4SLwQSLU DLhfDLU*K<]vR#dgDLU*KowQSLF>d“­)SubgD]4THVU YdeK>dgHVU ]rYNb( ~ •*‡rD]rCg(ɏ]
DtszEcSLF>dehfDLUK/R#bgD]O]<HVUKOYGHVU Tk÷LTHVU YdeK>dgHVUGU DLbebgD]÷VaT`½¾D]>KJÞ»†YdeF<D"j*N½¾DLbebgD]F<DLEGF<S]<DLU*K<DLUK–[email protected]<HV[)R#[GdebedeK<S"YGD
YGSTLbgDLU T<l DLhfDLU*KODLU'Œ’HVU TLK>dgHVUˆYGD–bqR\Ydg]>K/R#U TDx»Bb½§RsD`a#ÒÙ1×nØIÙ!ð‡Ü ëӑßvê>á Ñnå!Ñnð_ßnÔÕßnð‡ÜÙ{Ú6Ù!ÒÒ/ÑtêFߛö±øQngD†bcD]>K
YGHVU TBSLwzdgYGDLUKxj*N D"K/R#U*KxjN DBbqRfEGF<HV[)R#[GdebedeK<[email protected]\YGSLK<DTLK>dgHVUa˜T`½¾D]>KJÞ»†YdeF<D\bqRfEGF<HV[)R#[GdebedeK<[email protected]˜½¾HV[GK<DLUGdeFxNGU
( ~ azU½¾D]>KvE)RV]odeU YGSLEcDLU Y R#UK<D"YGDOb½¾SLU DLF>P`dgD`abgD]gC (ɏ]hfD]<NGF<SD]]>NGFvbgD]rYGHVUGU SD]]/HVUK}HV[GK<DLUN D]ORwQDT
NGUŠST/l)R#UK>debebgHVU!YGDuPQDLF>[cD]Y R#U ]vbgDjN DLbcbgD]o• N TLK>N)R#K>dgHVU ]EcH`]>deK>dewQD]R#N†YGD]<]>N ]YN¬]<DLNGdebdehEH`]/SOE)R#FvbgD
( ~ ]<HVU*KuŒRwQHVF>dg]/SD]n š U¤]½§R#K>K<DLU Y!YGHVU T\»kTDBj*N D-bgD]–gC (ɏ]AhfD]>NGF<SD]ARwQDT\bgD\¡zÕ DLU†YGD]<]<HVN ]–YGD"bqR
]/R#K>NGF/R#K>dgHVUm]½¾SLK<DLU YGDLU*KuEGbeN ]bgHVdeU%DLUmYdg]>K/R#U TD-j*N DAbgD]}gC (ɏ]O]<deh"NGbgSD]n
C Rk]<HVbeNGK>dgHVU%THVU ]>dg]<K<D"»dehEcH`]<DLF}bgDBTLF>deK<iLF<D“( ~ ]<NGF}bgD]uCg(ɏ]A]>deh"NGbgSD]a EcHVNGFOHV[GK<DLUGdeFOYGD]–Cg(ɏ]
‘
THVU YdeK>dgHVUGU DLbebgD]a}THVhE)R#F/R#[GbgD]m»ÆTDLbebgD]¬hfD]<NGF<SD]n}‡rDLEDLU Y R#U*KadebxD]>KpdehEH`]<]<de[GbgD!Y˜½§R#EGEGbedgj*N DLFˆbgD
TLF>deK<iLF<q
D ( ~ Y R#U ]xbqRŠTHVU­ P`NGF/R#K>dgHVUÅYGD\F<S]<DR#N2DLU2SLK<HVdebgDj*NGd7RŠSLK<[email protected]>debedg]<[email protected]#bgTLNGbgDLFxbgD]-Cg(ɏ]n
ÉHVNGF"dehEGbgSLhfDLUK<DLF\bgq
D ( ~ a˜deb£ŒŽR#NGKB]>deh"NGbgDLF-bgD]-PQDLF>[D]-]<NGFxNGU2F<S]<DR#NªdgYGDLUK>dgj*N D'»pTDLbeNGdoYGDb½ š ÉM\n
CDfEGF<HV[GbgiLhfDŠD]>KBR#bgHVF<]BEGNGF<DLhfDLUK\K<DT<lGUGdgj*N D!”deb7ŒR#NGKB]>dehBNGbgDLF"NGUÅP`F/R#U Y‰U HVhB[GF<D'YGDPQDLF>[cD]@RwQDT
Yd“—cSLF<DLU*K<D]pEH`]<deK>dgHVU ]ŠYGD!‘T “ANGFavDLKŠTD!EcHVNGFŠT/l)RVjN DªR#UGP`bgD¤DLKpSLU DLF>P`dgD`a}R­ U‹Y˜½¾HV[GK<DLUGdeFˆYGD]pgC (ɏ]
TR#bgTLNGbgSD]'RwQDTm]>N¥k]/R#hhfDLU*KŠY˜½¾SLwQSLU DLhfDLU*K<]ŠE)R#FkdeUK<DLF>w`R#bebgDmYGDmYdg]>K/R#U TD`n4ù ½§R#dAR#bgHVF<]kHVF>dgDLU*K<SˆhfHVU
SLK>N YGDxwQDLF<]NGU D"R#NGK>F<D-hfSLK>l H*YGD`aGNGK>debedg]ÎR#UKubgD]uSLwQSLU DLhfDLUK<]Ol¼z[GF>dgYGD]n
É HVNGF4hfD]>NGF<DLFbgD]oC(É4]oDLU†YGD]<]/HVN ]4YGDbqRx]ÎR#K>NGF/R#K>dgHVUaT`½¾D]>KJÞ»†YdeF<DI¯N ]>K<DLhfDLU*KoY R#U ]4bqRAF<SLP`dgHVU'HVÈ
;
bgDOTR#bgTLNGbY˜½§RVTTDLEGK/R#U TD"D]>KoU STD]/]/R#deF<D`azdebcD]<KoEcH`]<]>de[GbgDAY˜½ NGK>debedg]<DLFvbgD]SLwQSLU DLhfDLUK<]}l¼z[GF>dgYGD]ahfD]>NGF<S]
E)R#FvbgDxYGSLK<DTLK<DLNGF}YGD–• N HVF<D]<TDLU TDxDLKbgD-¡zÕ\nGC‘D–{£Õ D]>KTR#E)R#[GbgDxY˜½¾HV[ ]<DLF>wQDLFbgD]vPQDLF>[cD]u»BEGbeN ]v[)RV]<]<D
SLU DLF>P`dgDˆj*N D¬bgD%¡zÕ@aDLKNGU Dm]/DLNGbgD¬]>K/R#K>dgHVU‹R#N ]/HVbK<HVN T<l SDm]<N¥fKfEHVNGF'THVU ]>K>deK>N DLFkNGU SLwQSLU DLhfDLU*K
l¼z[GF>dgYGD`no™OU«DLU ]<DLh"[GbgD!YGD%TLF>deK<iLF<D]pYGD!jN)R#bedeK<S!D]>KpR#EGEGbedgjN S¤»ªb½¾SLwQSLU DLhfDLU*Ka}R­ U‹Y˜½¾SLbedehdeU DLFpbgD]
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
vÄ
,6-0/.1„ n Ñ
‡rHVhE)R#F/R#dg]<HVU\DLUK>F<DobqRgC (É%hfD]<NGF<SDoRwQDT4bgD]‘YGHVUGU SD]l¼z[GF>dgYGD];RwQDTbgD]C(ɏ];]>dehBNGbgSD]

2
EHVNGFvYGD]rEGF<HVK<HVU ]DLKrYGD]rU H¼R#NspYGDOŒ’DLFa*EHVNGFrNGU D–SLU DLF>P`dgDxDLU†YGD]<]<HVN ]vYGDubqR\]/R#K>NGF/R#K>dgHVU2œ
18
D€-tnM»×PR#N T<l D¬”EcHVNGF–NGU2R#UGP`[email protected]ÔSLUGdeK>l)R#bYGD ◦ aDLK"»ŠYF<HVdeK<DBEcHVNGF
n◦ C‘D]A[)R#F>F<D]-EY˜∼
½¾DLF>10
F<DLNGF
25
' 53
l HVF>deÔHVUK/R#bgD]–THVF>F<D]>EHVU YGDLU*K–»\NGU DxdeU TDLF>K>deK>N YGDB
YGD“ð`_\h©]>NGFbqR\EcH`]>deK>θdgHVUm
YNˆT‘“ANGFn
YGSTLbgDLU T<l DLhfDLUK<]fŒ’HVF>K>NGdeK<]@[email protected]]½§RV]<]>NGF<[email protected]'bqRˆ[cHVUGU DŠF<DTHVU ]>K>F>N TLK>dgHVUËYGDkb½¾SLwQSLU DLhfDLU*Kn4‡rD]@TLF>deK<iLF<D]
]<HVUKuYGSLK/R#debebgS]OY R#U ]bgDxT<l)R#EGdeK>F<Duða R#deU ]>dj*N DxbgD]}TR#F/RVTLK<SLF>dg]>K>dgj*N D]uYNmbgHVK}YGD-YGHVUGU SD]}l*¼*[GF>dgYGD]n
CD]}SLwQSLU DLhfDLU*K<]ul*¼*[GF>dgYGD]uU Dx]<HVUK}YGHVU TxE)RV]u]<HVNGhdg]uR#NˆTLF>deK<iLF<Dz( ~ a TDxj*NGd˜EcDLF>hfDLK}YGD-TR#bgTLNGbgDLF
YGD]-C (É4Õ
] ð#æ*ð ×>æ!ð_Ö!ەÜÝÛ=æ!ð~ðußnêGêxßÒLa‘RwQDTkbqRŠhfyLhfDfhfSLK>l HzYGDfjN DfU HVN ]"RwQHVU ]\DtszEGbedgj*N SDkR#NGE)R#F/RwVR#U*Kn
C‘R%­ P`NGF<D¬„ n Ñ!debebeN ]>K>F<DpbqR¤THVhE)R#F/R#dg]<HVU‹YGD]fgC (ɏ]fl*¼*[GF>dgYGD]kRwQDT¬bgD]fgC (ɏ]k]<deh"NGbgSD][email protected] D

SLU DLF>P`dgD-wQHVdg]>deU D-YGD |D€@n)CREGF<DLhdgiLF<D"THVU ]<K/R#K/R#K>dgHVU%j*NGdEDLNGKOyLK>F<DxŒR#deK<D-D]>KbgD-F<DLbqR#K>dewQDLhfDLU*KO[cHVU

RVTTHVF<Y\DLU*K>F<DvbgD]£EGF<SYdgTLK>dgHVU ]7K>l SHVF>dgj*N D]7DLK;bgD]£YGHVUGU SD]n`CROgC (ɪl*¼*[GF>dgYGD]<DLhB[GbgDTDLEDLU Y R#U*KT<l*NGK<DLF
bgSLPQiLF<DLhfDLUK"EGbeN ]AK<ÚVKBjN D\bgD]-gC (ɏ]"]<deh"NGbgSD]aR#EGE)R#F<DLhhfDLU*K\YGD\ŒRVWHVUªEGF>dew*debgSLP`dgSDfEHVNGFxbgD]AEcDLK>deK<]
R#UGP`bgD]vÔSLUGdeK>l)R#Nscn)‡rHVhEGK<DxK<DLU*NˆYGD–bqR\EGF<STLdg]>dgHVUˆjNGdcEcDLNGKyLK>F<D–HV[GK<DLU*N D"[email protected]]<K/R#K>dg]>K>dgjN DAYdg]>EcH#
UGde[GbgD}RVTLK>N DLbebgDLhfDLUKa*deb D]>K7dehEH`]/]>de[GbgDY˜½¾SLK>N YdgDLFoYdeF<DTLK<DLhfDLU*K4bqRATHVhEcH`]>deK>dgHVUkYGD]7F/R¼QHVU ]4TH`]>hdgj*N D]
RwQDT\TDLK>K<D-hfSLK>l HzYGD`a)bgD]–Yd“—cSLF<DLU TD]ADLU*K>F<DBgC (ɏ]OŒ’DLF–DLKuEGF<HVK<HVU¤SLK/R#UK–K>F<HVEmŒR#de[GbgD]–THVhE)R#F<SD]AR#Ns
deU TDLF>K>deK>N YGD]n
Õ D}ŒRVWHVUpF>dePQHVNGF<DLN ]<D`ab½¾Dt¥kTRVTLdeK<SxYGDAYGSLK<DTLK>dgHVUˆYGD]vPQDLF>[D]RwQDTAbgDA{£Õ¨U½¾D]<KrE)RV]v]>K>F>dgTLK<DLhfDLU*K}bqR
A
hfyLhfD-»"K<HVNGK<D]}bgD]SLU DLF>P`dgD]}DLKvbgD]}R#UGP`bgD]ÔSLUGdeK>l)R#NscnGC‘D]}TLF>deK<iLF<D]}YGDA]<SLbgDTLK>dgHVUˆj*NGd]<HVUK}R#EGEGbedgj*N S]
R#Ns'SLwQSLU DLhfDLUK<]l*¼*[GF>dgYGD]}EDLNGwQDLU*KvSLPR#bgDLhfDLU*K}deUK>F<HzYNGdeF<DxYGD]r[GdqR#dg]]>NGFrbgD–TR#bgTLNGb˜YGD–bqR"EGF<HV[)R#[GdebedeK<S
YGDYGSTLbgDLU T<l DLhfDLUKY˜½ NGU D}]<K/R#K>dgHVUkYNŠ¡zÕ@n DÞb THVUwzdgDLUKoYGHVU TuY˜½¾yLK>F<DEGF>N YGDLUKrjN)R#U*Ko»Ab½ deUK<DLF>EGF<SLK/R#K>dgHVU
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
v[v
GY D]–F<S]>NGbeK/R#K<]@”U D\]/RVT/l)R#UKxE)RV]AjN)R#U*K>d“­)DLF-DtsGRVTLK<DLhfDLU*KxbgD]ASLwQDLUK>N DLbg]x[GdqR#dg]A]>N ]<TDLEGK>de[GbgD]xY˜½§R—˜DTLK<DLF
[email protected]"YGDfYGHVUGU SD]"l¼z[GF>dgYGD]Šœ’j*NGd]<HVUKBDLU THVF<D'»pb½¾SLK>N YGDkR#NÅ]<DLdeU‰YGD]xP`F<HVNGEcD]"]>EcSTLdqR#bedg]>K<D]-YNÅ{;Õ
Y R#U ]MuNGPQDLFΝta;U HVN ]THVU ]>dgYGSLF<HVU ]YGHVU T¬TDLK>K<DpSLK>N YGD¬THVhhfDpNGUZK<D]>K\EGF<SLbedehdeU)R#deF<D¬EcDLF>hfDLK>K/R#UKkYGD
wQSLF>d“­)DLFkbgD]'F<S]>NGbeK/R#K<]ŠYGD]']>dehBNGbqR#K>dgHVU ]¬»ªNGUYGDLP`F<S%YGDˆEGF<STLdg]>dgHVU«j*N½ debO]<DLF/R2EH`]<]<de[GbgDˆY˜½§R#hfSLbedgHVF<DLF
NGbeK<SLF>dgDLNGF<DLhfDLUKn
‡rHVhhfDU HVN ]7bgDvwQDLF>F<HVU ]rR#NfT<l)R#EGdeK>F<
D ðaQbgHVF<]<jN Dvb½¾HVUfD]<K>dehfDvb½¾SLU DLF>P`dgDuYGD]£PQDLF>[cD]4»–E)R#F>K>deF4Y˜½ NGU D
F<DLbqR#K>dgHVUŠYGSYNGdeK<D–YGD]4]<deh"NGbqR#K>dgHVU ]aHVU'HV[GK>dgDLU*KoNGU DuwVR#bgDLNGFoEGbeN ]oSLbgDLwQSD`azYGD}b½¾HVF<YF<D–YGD
azjN DuTDLbebgD
20%
YGHVUGU SD\E)R#F–bqRf• N HVF<D]<TDLU TD`n‘¡*d£b½¾HVU¤EGF<DLU Y¤THVhhfDBF<StŒŽSLF<DLU TD\b½¾ST<l DLbebgDY˜½¾SLU DLF>P`dgDYGD]A]>dehBNGbqR#K>dgHVU ]a
bqRˆgC (É l¼z[GF>dgYGDpD]>KBYGHVU TpTR#bgTLNGbgSDpRwQDTpYGD]\PQDLF>[D]\Y˜½¾SLU DLF>P`dgD¬DLUÆhfH¼QDLUGU D
EGbeN ]@SLbgDLwQSD`a£DLK
20%
YGDLw*F/R#deKOYGHVU TxyLK>F<DxbgSLPQiLF<DLhfDLUK ø EGbeN ]uSLK<DLU YN D]uj*N DAbgD]}gC (ɏ]–]>deh"NGbgSD]n
,.-F/.1„ n ~ ‡rHVhE)R#F/R#dg]/HVUBDLUK>F<DobqRCg(É!hfD]>NGF<SDoRwQDTbgD]‘YGHVUGU SD]l¼z[GF>dgYGD]£RwQDTbgD]C(É4]]>dehBNGbgSD]
EHVNGFAYGD ]AEG2 F<HVK<HVU ]xDLKxYGD]–U H¼R#NsªYGD-ŒŽDLFaEHVNGF–NGU [email protected] DLF>P`[email protected]]>deU [email protected]']/R#K>NGF/R#K>dgHVU œ
E ∼ 1018.6
D€-tn ”ÌPR#N T/l Dk”EcHVNGF}NGUmR#UGP`bgDxÔSLUGdeK>l)R#b;YGD ◦ aGDLKO»@YF<HVdeK<DxEcHVNGF
◦n
25
θ ' 53
C DpŒR#deKkj*N Dˆ]>NGFfbqR¤­ P`NGF<Dˆ„ n ÑabqR2C(ÉÒl¼z[GF>dgYGDmT/lNGK<D%EGbeN ]fK<ÚVKkj*N D¬bgD]']>deh"NGbqR#K>dgHVU ]arDLKkTD

Yd“—cSLF<DLhfDLU*[email protected]½§R#UGP`[email protected]ÔSLUGdeK>l)R#baEHVNGF>F/R#deK-YGHVU TyLK>F<D\bqRpTHVU ]<Sj*N DLU TDY˜½ NGUª[GdqR#dg]AEGF<HwQDLU)R#U*KBYGD]
]<SLbgDTLK>dgHVU ]]>[email protected]]SLwQSLU DLhfDLUK<]kl¼z[GF>dgYGD]aj*NGdrŒŽRwQHVF>dg]<DLF/R#deKbgD]@PQDLF>[D]@F<DTHVU ]<K>F>NGdeK<D]'»%EGbeN ]@[)RV]<]<D
SLU DLF>P`dgD`a7ŒRwQHVF>dg]/R#U*KˆR#deU ]>dubgD]kEDLK>deK<]Š]>deP`U)R#Ns » _`_`_2hfiLK>F<D]+
n DÞbuD]>K'SLPR#bgDLhfDLUKpEH`]/]>de[GbgDmj*N D¬bgD]

]>deh"NGbqR#K>dgHVU ]U DoF<DLEGF<HzYNGdg]<DLU*KE)RV];E)R#FJŒŽR#deK<DLhfDLU*KbgD]>deP`U)R#bhfD]>NGF<SvY R#U ]7bgD]£]>K/R#K>dgHVU ]7DLK]<[email protected]>dgHVU
ùJä›ß¯ë£ñVã<â†ãÎz
ï ¼¯áŽâßJë)ûßoä 〽¾V;<‰tã<âê›ß¯àá;ä§ßJàáßJïßJàá£ãI‰L߯çräçü 9Jà#߯â úÎê§ß—¿eç`›À2Áútéâß†¾ Ã4Ä)ûßoëŽìÎâŽáßrèé$ü é#àzûO9¯ç>ãÎä›ãœúÎß|ûß
ßJàB9Jà#ßÞâ útê§ß7à#ß û߉â†ãÎê¾áñVã/ë‘çýVã/àúÎ߯âúÎâ†ãÎàûßJï߯àáä›ã› ìÎâïßûßJë½OV; ë
20%
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
v¿
RwQDTxb½§R#UGP`bgDxÔSLUGdeK>l)R#bn
C‘R ­ P`NGF<DƄ n ~ debebeN ]<K>F<DÅbgD]¤F<S]<NGbeK/R#K<]!HV[GK<DLU*N ]ªEcHVNGF!NGU DZSLU DLF>P`dgDÆwQHVdg]>deU DZYGDÆbqR‹]/R#K>NGF/R#K>dgHVU


D
@
€
n
‘
C
D"U HVhB[GF<DY˜½¾SLwQSLU DLhfDLUK<]xl*¼*[GF>dgYGD]ASLK/R#UKx[cDR#N THVNGE!EGbeN ]uŒR#de[GbgD`abgD]–deU TDLF>K>deK>N YGD]
E ∼ 1018.6
]>NGFbqRªgC (É×l*¼*[GF>dgYGD%]<HVUKfF<DLbqR#K>dewQDLhfDLU*K'P`F/R#U YGD]no‡rDLEDLU Y R#U*Ka4[GdgDLUjN½ debu]/HVdeKdehEH`]<]<de[GbgDˆY˜½¾yLK>F<D
TR#K<SLPQHVF>dgjN D`a)bgD]}hfyLhfD]OF<DLhkR#F<jN D]–jN D-EGF<STSYGDLhhfDLUKx]<DLh"[GbgDLU*K–]½§R#EGEGbedgjN DLFOEHVNGFuTDLK>K<D"PR#hhfD
Y˜½¾SLU DLF>P`dgD`n`‡rD]F<S]>NGbeK/R#K<];hfHVUK>F<DLU*KjN½ debzD]>K;U STD]<]/R#deF<DrY˜½¾SLwVR#beN DLF;EGF<STLdg]<SLhfDLUK£b½¾Dt—˜DLK;YGD]£TLF>deK<iLF<D]£YGD
]<SLbgDTLK>dgHVUmR#EGEGbedgjN S]OR#NspSLwQSLU DLhfDLU*K<]}l¼z[GF>dgYGD]u]<NGFrbgD]uYGHVUGU SD]}YN%¡zÕ@nGÉHVNGFOR#NGK/R#U*Kuj*N½¾HVU¬EGNGdg]<]<D
DLUmIJNGPQDLFaTDLK>K<DpSLK>N YGDp]>NGP`PQiLF<DŠYGHVU TpjN D'[email protected] SD]\l¼z[GF>dgYGD]D]>KB[GdqR#dg]/S`a;TDŠj*NGdrD]>KBNGU D
THVU TLbeN ]>dgHVU¬dehEHVF>K/R#U*K<DxY R#U ]bqRBhfD]>NGF<DAHVȬTD]SLwQSLU DLhfDLUK<]}l*¼*[GF>dgYGD]u]<HVUKvNGK>debedg]<S]vEcHVNGFSLK/R#bgHVUGU DLF
bqR\F<DLbqR#K>dgHVU
 aR­ UˆYGDxF<DTHVU ]>K>F>NGdeF<D-bgD]uSLwQSLU DLhfDLUK<]OhfD]>NGF<S]uRwQDT"bgD"¡zÕٜ’TtŒ£T/l)R#EGdeK>F<}
D ðQtn
S1000 E
üš¸üFE
4ÿ ÿ%KK†ÿ~GÆ6ÿ%J>ImÆ6KÿÈÇ6KIɟXIV+„Æ|V
ŕ
C D–K<DLF>hfDxYGD÷LC‘RVYde[HVK÷-YGS]>deP`U D–bqR§õ Ù*܏ßnÞÙ!êM7vÛhÒÜÝÞÛ änÓ6ÜÝÛÝæ*ðYæ'ñ€ÊTەð‘Ò¥ãåCßnÞdö Ø1Þ>ßÒØ6æ!êúÖQaGjNGd˜D]>KrNGU D
Œ’HVU TLK>dgHVUf]<DLhB[GbqR#bgDu»ObqRAC(ɍ”VDLbebgDvF<DLEGF<S]<DLUK<D`a*DLUBŒŽHVU TLK>dgHVUkYGDrbqRAYdg]>K/R#U TD}»–b½§RsGD`aVbgDrU HVh"[GF<D}hfH¼QDLU
U ]jNGdcHVUKNGUˆ]>deP`U)R#b˜]>NGEcSLF>dgDLNGF}R#Nˆ]<DLNGdeb˜YGD
Y˜½ deUK<DLF>w`R#bebgD]"œU HVN ]vNGK>debedg]<DLF<HVU ]bgD–K<DLF>hfDxYG¹
D änەð˜oYGD
25
€A|ÏÅn™OU D"]>K/R#K>dgHVU2R'YGSTLbgDLU T<l [email protected]
U (‘JH ( ]>dDLbebgD\EcH`]<]<iYGD"EGbeN ]–YGD Ñ'[GdeU ]xR#N†YGD]<]>N ]–YN¤]/DLNGdebn
0.2

C‘R!C‘RVYde[HVK'D]>KYGHVU T'NGU D¬HV[ ]/DLF>wVR#[GbgDpEGbeN ]\­ U D¬j*N DŠbqR%gC (É©”£DLbebgDpF/RV]<]/DLh"[GbgDpb½ deUŒŽHVF>hkR#K>dgHVU ]>NGF
bqRkEGF<HV[)R#[GdebedeK<SfdeU Ydew*dgYN DLbebgDfYGDYGSTLbgDLU T/l DLhfDLUK\Y˜½ NGU D]>K/R#K>dgHVUachkR#dg]-SLPR#bgDLhfDLUKBDLbebgDTR#F/RVTLK<SLF>dg]<DbqR
bqR#F>PQDLNGF}K<DLhEHVF<DLbebgD"YNm]>deP`U)R#b‘DLUpŒŽHVU TLK>[email protected]]>K/R#U TDB»@b½§RsD`n
CDEGF>deU TLdeEcDfYGDbqRpYGSLK<DLF>hdeU)R#K>dgHVUZYGD]-C‘RVYde[cHVK<]"D]>KB]>dehdebqR#deF<Dk»pTDLbeNGd4NGK>debedg]/[email protected]]"C (ɏ]Š”
IL½§R#d]>dehBNGbgSfNGU2P`F/R#U Y‰U HVh"[GF<D'YGDfPQDLF>[cD]-]>NGF-NGU2F<S]/DR#NÅDLU‰SLK<HVdebgD`a‘R­ UÆY˜½¾HV[GK<DLUGdeF"bqRpEGbeN ]"P`F/R#U YGD
]>K/R#K>dg]>K>dgj*N DBEcH`]<]>de[GbgDBEcHVNGF–NGU¤deU*K<DLF>wVR#[email protected]]>K/R#U [email protected] S`a»ŠSLU DLF>P`dgDDLK-R#UGP`[email protected]ÔSLUGdeK>l)R#b£­GsS`ncÕAD
bqRkhfyLhfDBŒRVWHVUªjN D\bgD]AC (É4]aR­ UªYGDBEcHVNGwQHVdeF-Dt—˜DTLK>N DLF–NGU [email protected])R#F/R#dg]<HVU‰[email protected]]xYGHVUGU SD]abgD]
C‘RVYde[HVK<]}EDLNGwQDLU*K}yLK>F<DxTR#bgTLNGbgSD]}YGDAhkR#UGdgiLF<Õ
D ×>æ!ð_Ö!ەÜÝÛ=æ!ð~ðußnêGêxߔHVUˆTR#bgTLNGbgD-R#bgHVF<]bgD–U HVhB[GF<D-YGDA[GdeU ]
R#N†YGD]<]>N ]}YNm]/DLNGdebaGDLUˆdehEH`]ÎR#UK}NGU¬hdeUGdehBNGh¢YGD Ñ@[GdeU ]a R­ UmYGDxw`R#bedgYGDLFbqRTHVU YdeK>dgHV€
U (JH (xn

C‘Rk­ P`NGF<D\„ n „ŠdebebeN ]<K>F<[email protected]½¾SLwQHVbeNGK>dgHVU2YGDBbqR'C‘RVYde[cHVK-THVU YdeK>dgHVUGU DLbebgDkRwQDT\b½¾SLU DLF>P`dgDDLKAb½§R#UGP`[email protected]ÔSt

UGdeK>l)R#bn C‘R\ŒŽHVF>[email protected]‘RVYde[cHVKuEGF<Hw*dgDLU*[email protected]>deK>dgHVU%DLU*K>F<D"YGDLNsˆDt—˜DLK<]-”bqRYdehdeUNGK>dgHVU¤YGD
b½ deUK<DLU ]>deK<SYN]>deP`U)R#b RwQDTrbqR–Ydg]>K/R#U TD»Ob½§RsDrDLK£b½§R#NGP`hfDLUK/R#K>dgHVUŠYGD4b½¾SLE)R#dg]<]/DLNGF7K<DLhEHVF<DLbebgDvYN\ŒF<HVU*K
YGDBPQDLF>[cD`a˜j*NGd7EGF<H*YNGdeKxNGUªhkRsdeh"NGh »'NGU DYdg]>K/R#U TDkYGSLEDLU Y R#U*K"YGD\b½¾SLU DLF>P`[email protected]½§R#UGP`[email protected]ÔSLUGd“
K>l)R#bnQÉHVNGFobgD]oR#UGP`bgD]4ÔSLUGdeK>l)R#Nsk]>NGESLF>dgDLNGF<]o»
a`bqRxCRVYde[HVKrYGDLw*dgDLU*KNGU DŒŽHVU TLK>dgHVUk]>K>F>dgTLK<DLhfDLU*K
∼ 45◦
YGSTLF<HVdg]<]/R#UK<DYGD\bqRŠYdg]>K/R#U TD»'b½§RsD`acTHVU ]<Sj*N DLU [email protected]\bqRkhfHVdeU ]–ŒŽHVF>K<[email protected]>dgHVU2YGD\b½¾SLE)R#dg]<]<DLNGF-YN
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
E = 1018 eV
E = 1018.6 eV
80
E = 1019 eV
E = 1019.6 eV
60
40
Nb of bins over threshold
Nb of bins over threshold
v ú
θ = 0°
40
θ = 36°
θ = 45°
θ = 60°
30
20
20
0
1000
2000
3000
4000
Distance to axis in m
10
0
500
1000
1500
2000
Distance to axis in m
,.-F/.1V„ n „ ” PR#N T<l D-”VSLwQHVbeNGK>dgHVUkYGDvbqRACRVYde[HVK4RwQDTb½¾SLU DLF>P`dgD`a`EcHVNGF£YGD]4]>deh"NGbqR#K>dgHVU ]YGDvPQDLF>[cD]YGD
EGF<HVK<HVU ]o » 2
nK”‹YF<HVdeK<D"”QSLwQHVbeNGK>dgHVUŠYGD}bqRxC‘RVYde[cHVKrRwQDT}b½§R#UGP`bgDuÔSLUGdeK>l)R#b)EcHVNGF4YGD]4]>dehBNGbqR#K>dgHVU ]
= 36◦
YGDAPQDLF>[D]uθYGDx
EGF<HVK<HVU%»
18 D€@n
E = 10
ŒF<HVU*KYGDvPQDLF>[D}EHVNGF7bgD]£PQDLF>[cD]4R#PQSD]n`CD]Œ’HVU TLK>dgHVU ]4C‘RVYde[HVK<]EDLF>hfDLK>K<DLU*KoYGDvK<D]>K<DLF7bqR–w`R#bedgYdeK<S}YGD]
]>deh"NGbqR#K>dgHVU ]–YNmYGSLK<DTLK<DLNGF–DLKuYGD]}PQDLF>[cD]OR#K>hfH`]>EGl SLF>dgj*N D]aTR#FuTD]uHV[ ]<DLF>w`R#[GbgD]u]/HVUK}K>F<i]O]<DLU ]>de[GbgD]
»\bqRBŒ’HVF>hfD"YNm]>deP`U)R#baGDLKOR#Ns¬THVU YdeK>dgHVU ]OYGDxYGSTLbgDLU T/l DLhfDLUKxj*NGd]<HVUKOR#EGEGbedgj*N SD]n
C Rf­ P`NGF<D\„ n ðkdebebeN ]<K>F<DBEcHVNGF–NGU DBPQDLF>[cD\Y˜½¾SLU DLF>P`dgDYGHVUGU SDBbgD]ATHVU*K>F>de[GNGK>dgHVU ]xF<D]>EcDTLK>dewQD]–YGD\bqR
‘

THVhEH`]ÎR#UK<D–SLbgDTLK>F<HVhkR#P`U SLK>dgjN DxDLKrYGDubqR"THVhEH`]/R#U*K<D–h"N HVUGdgj*N D–R#NŠU HVh"[GF<DAYGD}[GdeU ]vR#N†YGD]<]>N ]vYN
]<DLNGdebœŽRwQDT4NGUBhdeUGdeh"NGh dehEcH`]<S4YGD Ñ[GdeU ]ΝtnCDoU HVh"[GF<DoK<HVK/R#bzYGD4[GdeU ];R#N†YGD]<]<N ][email protected]]<DLNGdebU½¾D]<K‘E)RV]

]>dehEGbgDLhfDLUKrYGHVUGU S}E)R#FbqR-]<HVhhfD}YGD]4YGDLNsfTHVhEcH`]/R#UK<D]O”QDLUkDt—˜DLKa`EcHVNGFNGU D}]>K/R#K>dgHVUŠ]>deK>N SDu»–NGU D
Ydg]>K/R#U [email protected] SD`acbgD]x]>deP`U)R#Ns%EGF<HzYNGdeK<]AE)R#FAbgD]Ah"N HVU ]-DLKAE)R#FAbqRŠTHVhEH`]ÎR#UK<DSLbgDTLK>F<HVhkR#P`U SLK>dgj*N D
]<DxF<DTHVNGwzF<DLUKOK<DLhEHVF<DLbebgDLhfDLU*KaYGDB]<HVF>K<D-j*N DxbgD-K<DLhE ]uE)RV]<]<S"R#N†YGD]/]>N ]OYN%]<DLNGdebU½¾D]>KOE)RV]uYGHVUGU S
E)R#FbqR]<HVhhfD-YGD]}K<DLhE ]deU Ydew*dgYN DLbg]OYGD-T<l)RVj*N D-THVhEcH`]/R#UK<D`n
É HVNGF\bgD]BPQDLF>[cD]-ŒR#de[GbgDLhfDLUKdeU TLbedeU SD]abqR%THVhEH`]/R#U*K<DpSLbgDTLK>F<HVhkR#P`U SLK>dgjN D¬D]>K\YGHVhdeU)R#UK<D`a7DLK
;
YGSLK<DLF>hdeU DbqRkŒ’HVF>hfDYGD\bqRpC‘RVYde[cHVKn˜[email protected]"NGh [email protected]ŠTHVhEcH`]/R#UK<Dfh"N HVUGdgj*N DkR'bedgDLNªEHVNGFANGU D
Ydg]>K/R#U TD'deUŒ’SLF>dgDLNGF<D¬»%TDLbebgD'YGD'bqRmTHVhEcH`]/R#U*K<DŠSLbgDTLK>F<HVhkR#P`U SLK>dgj*N D2”‘bgD]BhBN HVU ]@U D']<NG[Gdg]<]/R#UK\E)RV]
bqRZYd“—N ]<dgHVU¨h"NGbeK>deEGbgDÅ]<HVUKˆEGbeN ]mTHVbebedehfS]mY R#U ]¬bqRXYdeF<DTLK>dgHVU YGDªb½§RsDªYGD2bqRÆPQDLF>[D`a}DLKmR#F>F>dewQDLU*K
YGDfŒRVWHVUZEGbeN ]@]<DLF>F<SD'DLUZK<DLhE ]n;É;HVNGFbgD]BPQDLF>[cD]R¼R#U*KNGUÆP`F/R#U YXR#UGP`bgDŠÔSLUGdeK>l)R#b-œ
ta;bqR
θ > 45◦
THVhEH`]ÎR#UK<DBh"N HVUGdgj*N D-­ UGdeKOE)R#FOYGHVhdeU DLFx»P`F/R#U YGD\Ydg]>K/R#U TD`n|U%SLK>N YdqR#UK–bgD]–C‘RVYde[HVK<]OEcHVNGFubgD]
Total Proton
80
Muons Proton
EM Proton
Total Iron
Muons Iron
60
EM Iron
40
Nb of bins over threshold
Nb of bins over threshold
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
v
Total Proton
Muons Proton
EM Proton
Total Iron
Muons Iron
30
EM Iron
20
20
0
1000
10
0
2000
3000
Distance to axis in m
1000
2000
3000
Distance to axis in m
,6-0/.1˜„ n ð
¡zSLE)R#F/R#K>dgHVUÆYGD]-Yd“—˜SLF<DLUK<D]"THVU*K>F>de[GNGK>dgHVU ]\»ŠbqRpCRVYde[HVK-EcHVNGF-YGD]xPQDLF>[cD]-]>dehBNGbgSD]B»
 D†2 €©DLK
»¬PR#N T/l DŠDLK
◦ »mYF<HVdeK<D`n;CD]\EGF<HVK<HVU ]@]<HVUK\deU YdgjN S]\DLUZTHVNGbgDLNGF<]
E = 1019
0◦
EGbgDLdeU D]aGbgD]U H¼R#θNs¬=YGDx
Œ’DLFE)R#F}YGD]O]>¼*hBθ[=HVbgD53
]}w*dgYGD]n
QP DLF>[D]BF<DLbqR#K>dewQDLhfDLUKdeU TLbedeU SD]adeboD]<K"EcH`]<]>de[GbgDkYGD'K<D]>K<[email protected]]>dbgD]@]>dehBNGbqR#K>dgHVU ]BF<DLEGF<HzYNGdg]<DLU*K\[GdgDLUÆbqR
ŒF/RVTLK>dgHVUmYGDAhBN HVU ]}EGF<S]<DLU*K<D"Y R#U ]bgD]uPQDLF>[D]OR#K>hfH`]>EGl SLF>dgjN D]n
C‘R¬THVhE)R#F/R#dg]<HVUÆYGD]-C‘RVYde[cHVK<]\RwQDTbgD]"YGHVUGU SD]"U STD]<]>deK<DbgD]-hfyLhfD]-EGF<STR#NGK>dgHVU ]\jN DEcHVNGF
bgD]–gC (ɏ]ncù ½§R#d7THVhE)R#F<[email protected]]AC‘RVYde[cHVK<]xTHVU YdeK>dgHVUGU DLbebgD]-HV[GK<DLUN D]"DLU!]>dehBNGbqR#K>dgHVU‰RwQDT\bgD]AC‘RVYde[cHVK<]
HV[GK<DLUN D]o»–E)R#F>K>deFoYNF<S]<DR#NfYGD}]>NGFJŒRVTD`aVEHVNGF7bgD]4SLU DLF>P`dgD]4R#N†YGD]<]>N ]YGDvbqR-]/R#K>NGF/R#K>dgHVUnQCDvEGF>deU TLdeEcD
YN!TR#bgTLNGb7D]>KAdgYGDLUK>dgj*N D»ŠTDLbeNGd;NGK>debedg]/S"EcHVNGFAbgD]–gC (ɏ]”EcHVNGF–T/l)RVjN DSLwQSLU DLhfDLU*z
K ( ~ ]ÎR#K>dg]JŒŽR#dg]ÎR#UK
bgD]xTLF>deK<iLF<D]-Yå
N (v„ˆDLKAY•
N (XðacbgD\U HVhB[GF<[email protected][GdeU ]-R#N†YGD]<]>N ]xYN2]<DLNGdeb7YGDT/l)RVjN D]>K/R#K>dgHV•
U (}x
Ñ (JH (
D]>K}TR#bgTLNGbgS`a)R#deU ]>d‘j*N Dx]/[email protected]]>K/R#U TDB»\b½§RsD`n
M}­ UˆY˜½¾SLK/R#[GbedeFuNGU D-THVhE)R#F/R#dg]<HVUˆEGF<STLdg]<D`a debD]>KU STD]<]/R#deF<D-Y˜½§R#EGEGbedgj*N DLF}NGUmTLF>deK<iLF<D"YGDx]<SLbgDTLK>dgHVU
]>NGEGEGbgSLhfDLUK/R#deF<DˆR#NsÅ]<K/R#K>dgHVU ]\THVU*K<DLUN D]fY R#U ]@bgD]\SLwQSLU DLhfDLUK<]fhfD]>NGF<S]@E)R#FBbgD¬¡zÕ ”;DLUXDt—cDLKabqR
bqR#F>PQDLNGF;K<DLhEcHVF<DLbebgDYN]>deP`U)R#bhfD]>NGF<SvY R#U ]£NGU Dv]>K/R#K>dgHVUYGSLEDLU [email protected]ŒŽRVWHVUfRV]<]/DLÔr]<DLU ]<de[GbgDrYGDoŒRVTLK<DLNGF<]
K<DLbg]ˆjN D!bqRZbgHVUGP`N DLNGF%Y˜½§R#[ ]<HVF>EGK>dgHVUÌYGD¤b½¾DR#Na}HVN¨YGD]¬EGF<HVEGF>dgSLK<S]%YGD¤F<St•)DTLK>dew*deK<S‰Y
N (¼zwQD Ç œ’TtŒ>n
‚ „Vta)j*NGd‘]<HVUK–Yd“—cSLF<DLU*K<D]AY˜½ NGU D"]>K/R#K>dgHVUª»@b½§R#NGK>F<D`n™uU D-]/SLbgDTLK>dgHVU!]>NGFubqRw`R#bgDLNGFOYNmF/R#EGEcHVF>KODLU*K>F<D
Ø
b½ deUK<SLP`F/R#bgD¬[email protected]"NGh´YGD]@]>deP`U)R#N=s ËQa;jNGdD]>K\bqRmTR#F/RVTLK<SLF>dg]<K>dgjN D¬YGHVU*[email protected] YÆbqR%wVR#bgDLNGFYN
U HVh"[GF<DpYGDk[GdeU ]R#N†YGD]<]<N ]BYNX]<DLNGdeba;D]>KBDt—˜DTLK>N SDp]>NGF"bgD]@]>K/R#K>dgHVU ]aYGDkŒŽRVWHVUX»[email protected]Å[email protected]
ä›ã tã/ä§ßJéâ<ûß;çJߣâ†ãÎññ`ìÎâŽá‘éáê§ä§ê§ë`9Jß7ßJàAëŽê§ïréä›ã/áê§ìtàOßJë’á<ûß
Ì p‰
3.5
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
v%W
,.-F/.1„ n
‡rHVhE)R#F/R#dg]<HVU%DLUK>F<DAbgD]C‘RVYde[cHVK<]}THVU YdeK>dgHVUGU DLbebgD]u]<deh"NGbgSD]"œ’THVNGF>[cD]}DLUpEHVdeU*K>debebgS]ΝvDLK

“
Ø
2
bgD]uC‘RVYde[cHVK<]OhfD]>NGF<SD]–RwQDT"bgD]OYGHVUGU SD]AYN¤¡zÕ@n C½¾D]>K>dehkR#K>dgHVUªYGD"b½¾SLU DLF>P`dgD"D]>K–Dt—cDTLK>N SD\»E)R#F>K>deF
YGD]o]<deh"NGbqR#K>dgHVU ]-œ’TDLF<TLbgD]rU HVdeF<]Ν4HVNp»"E)R#F>K>deFrYGDObqR"hfSLK>l H*YGDx‡6DJ‡|?–{£ÕҜbgH`]/R#UGPQD]rw*dgHVbgDLK<]Νta*EcHVNGF4NGU D
SLU DLF>P`dgDxTDLUK>F<SD"R#NGK<HVNGF}YGD
D†€\n ”žPR#N T/l DxEcHVNGFbgD]}R#UGP`bgD]ÔSLUGdeK>l)R#NspTDLU*K>F<S]uR#NGK<HVNGFuYGD
E = 1019.6
◦ DLKu»@YF<HVdeK<D"EHVNGF
◦n
θ = 25
θ = 56
YGHVUGU SD]ATHVF>F<D]>EHVU Y R#U*K"R#Ns%]>dehBNGbqR#K>dgHVU ]ncC‘R­ P`NGF<D\„ n debebeN ]>K>F<D\bgD]uF<S]<NGbeK/R#K<]–HV[GK<DLU*N ]xEcHVNGFuNGU D
Ø
SLU DLF>P`dgD-YGD 18.6 D†€\n
10
CDOEGF<HV[GbgiLhfD–EGF>deU TLdeE)R#b˜EHVNGFrbqR"THVhE)R#F/R#dg]/HVUpDLU*K>F<D–bgD]r]<deh"NGbqR#K>dgHVU ]DLKrbgD]rYGHVUGU SD]vD]>KazTHVhhfD
Y R#U ]ŠbgD!TRV]pYGD]pgC (ɏ]avb½ deU TDLF>K>deK>N YGDª]>NGFpbgD!T<l HVd“s‹YGD!bqR‰hfSLK>l H*YGDªY˜½¾D]<K>dehkR#K>dgHVU¨YGDmb½¾SLU DLF>P`dgD2»
NGK>debedg]<DLFnGC‘[email protected]<DTHVU ]<K>F>N TLK>dgHVU%YGD]}SLwQSLU DLhfDLU*K<]–RwQDTxbgDx{£Õô]/HVN ]J†D]>K>dehfD-Y˜½¾DLU*w*deF<HVU
b½¾SLU DLF>P`dgD"E)R#F
20%
F/R#EGEHVF>KpR#Ns ]>deh"NGbqR#K>dgHVU ]aAœ’TtŒ>n4T/l)R#EGdeK>F<!
D ðQtarTDmj*NGdODtsEGbedgj*N DˆEcHVNGF<jN HVdObgD]'YGHVUGU SD]'NGK>debedg]/R#UKpbqR
F<StŒ’SLF<DLU TDkYGHVUGU SDE)R#F-bqR'• N HVF<D]<TDLU TD']<HVUK-]>deP`UGd“­)TR#K>dewQDLhfDLU*[email protected]#N†YGD]/]>N ]"YGD]-THVNGF>[cD]xK>l SHVF>dgj*N D]n
É;R#F>hdbgD]4YGHVUGU SD]vRwQDTOb½¾SLU DLF>P`dgD–D]>K>dehfSDO»-E)R#F>K>deFoYGD]r]>deh"NGbqR#K>dgHVU ]a*bqRxTHVNGF>[cDuYGD]oEGF<HVK<HVU ]o]/DLh"[GbgD
bqRpEGbeN ]xEGF<HzT<l D'YGD]-EGF<SYdgTLK>dgHVU ]BK>l SHVF>dgjN D]a‘DLK-EH`]/]<iYGDNGU [email protected]ŒŽHVF>hfDP`bgHV[)R#bgDLhfDLUK\[GdgDLU‰F<DLEGF<HzYNGdeK<D
E)R#FxbgD]x]>deh"NGbqR#K>dgHVU ]Gn OHVN ]-deU ]>dg]>K<HVU ]xTDLEcDLU Y R#UKB]<NGF–bgD\ŒR#deK-j*N D\TDLK>K<DkSLK>N YGDTHVU ]>K>deK>N DNGUªK>F/Rw`R#deb
DtszEGbgHVF/R#K<HVdeF<D–EGF<SLbedehdeU)R#deF<D`a DLKvj*N DOTHVhEGK<DAK<DLU*NpYGD]rU HVhB[GF<DLN ]<D]deU TDLF>K>deK>N YGD]aDLUK>F<D-R#NGK>F<D]vbedgSD]»
b½¾D]>K>dehkR#K>dgHVUfYGDob½¾SLU DLF>P`dgD`a`b½ deUK<DLF>EGF<SLK/R#K>dgHVUkYGD]£F<S]<NGbeK/R#K<]7YGHVdeK£yLK>F<DrEGF>N YGDLU*K<D`nQC‘Ru­ P`NGF<Dr„ n deU Ydgj*N D
Ø
SLPR#bgDLhfDLUKAjN D"bqRf]>dehBNGbqR#K>dgHVU%U D"F<DLEGF<H*YNGdeKOE)RV]uE)R#FJŒR#deK<DLhfDLUKAb½¾SLwQHVbeNGK>dgHVU¤YNmU HVhB[GF<DBYGD"[GdeU ]–R#N
YGD]<]>N ]\YNÅ]<DLNGdebrEcHVNGFBbgD]"ŒŽR#de[GbgD]\Ydg]>K/R#U TD]»mb½§RsD'YGDkbqRmPQDLF>[D`n7‡[email protected]—cDLKBEHVNGF>F/R#deK\EGF<HwQDLUGdeFfYN
E)RV]<]/R#PQDxYGDxb½ NGK>debedg]ÎR#K>dgHVUmYNˆ[)RV]PR#deUˆwQDLF<]}bgDxl)R#NGK}PR#deUa j*NGd˜EHVNGF>F/R#deKuU DAE)RV]uyLK>F<DA[GdgDLUˆhfHzYGSLbedg]<S`n
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
üš¸üòû
¹
vV[
ÿ%_QÿåK·gÿ–ª IV`JQÿS Q`ÿ%g Ìÿ1͵4ÿ%_!ÿSK ItOI> $I Õ R#U ]}bgD-TRVYF<DBYGDxU HVK>F<D"TR#bgTLNGb;YGDxbqR]>NGFJŒRVTD-Dt—˜DTLK>dewQD"YGD"YGSLK<DTLK>dgHVUa)b½¾SLwQHVbeNGK>dgHVU!YGD-b½¾Dt¥kTRVTLdeK<S
x
YGDkYGSLK<DTLK>dgHVUËRwQDTŠb½¾SLU DLF>P`dgDpD]>KBTR#bgTLNGbgSD¬»ˆE)R#F>K>[email protected]]"ŒŽHVU TLK>dgHVU ]\Cg(ɏ]n£C‘Rmw`R#bgDLNGF\YGDkb½¾Dt¥kTRVTLdeK<S
YGD-YGSLK<DTLK>dgHVU!YGSLEcDLU Y R#UKAYGDxbqRfU)R#K>NGF<D"YGD]}F/R¼QHVU ]OTH`]<hdgjN D]a)NGU D-l*¼*EcHVK>l i]<D-YGDBTHVhEH`]>deK>dgHVU!D]>K
U STD]<]/R#deF<Df]>d4b½¾HVUÅ]<HVNGl)R#deK<DfTR#bgTLNGbgDLF\bgDf]>EcDTLK>F<DfY˜½¾SLU DLF>P`dgD'DLU†YGD]<]<HVN ]\YGDb½¾SLU DLF>P`dgDkYGD']/R#K>NGF/R#K>dgHVUn
OHVN ]ŠwQDLF>F<HVU ]ŠE)R#FpbqRÅ]<NGdeK<D%j*N½ deb–D]<KŠEcH`]<]>de[GbgD!Y˜½¾DtsEGbgHVdeK<DLFpTDLK>K<D!EGF<HVEGF>dgSLK<S¤EcHVNGFŠTHVU*K>F/R#deU YF<DªbqR
THVhEH`]>deK>dgHVU%YGD]F/R¼QHVU ]OTH`]>hdgjN D]n
,6-0/.1`„ n ÷
wQHVbeNGK>dgHVUŠYGDbqRAEGF<HV[)R#[GdebedeK<S–YGD}YGSLK<DTLK>dgHVU'YN'F<S]<DR#NkYGD}]>NGFJŒRVTD"œEGF<HV[)R#[GdebedeK<S–Y˜½¾HV[GK<DLUGdeF

£
2
NGU*v
( „*vDLU¬ŒŽHVU TLK>dgHVUmYGDAb½¾SLU DLF>P`dgD`a)YGSLK<DLF>hdeU SD\RwQDTxbgD]}SLwQSLU DLhfDLU*K<]ul*¼*[GF>dgYGD]n
Õ R#U ]"bgDk[GNGKBY˜½¾SLK/R#[GbedeF\NGUÅ]>EcDTLK>F<DkDLUÆSLU DLF>P`dgD'DLU†YGD]<]/HVN ]BYGDkbqRˆ]/R#K>NGF/R#K>dgHVUZjNGdr]<HVdeK-­R#[GbgDŠDLK
x
deU YGSLEDLU Y R#U*K4Y˜½ NGU Dl¼zEcHVK>l i]<DYGD}THVhEcH`]>deK>dgHVUatIL½§R#dGNGK>debedg]/SvbgD]SLwQSLU DLhfDLUK<]ol¼z[GF>dgYGD]7EcHVNGF7hfD]>NGF<DLF
b½¾Dt¥kTRVTLdeK<S"YGDxYGSLK<DTLK>dgHVU%YN%¡zÕ DLU†YGD]/]<HVN ]}YGD |D†€\n
4
CD'EGF>deU TLdeEDpYNZTR#bgTLNGbvD]<K\]<DLhB[GbqR#[GbgD¬»mTDLbeNGdYGD]@C (É4]@l¼z[GF>dgYGD]¬”‘EcHVNGF\T/l)RVjN DpSLwQSLU DLhfDLU*K
l¼z[GF>dgYGDB]/R#K>dg]>ŒŽR#dg]/R#U*K–R#NsmTLF>deK<iLF<D]–YGD"j*N)R#bedeK<S`a]<dTDLbeNGd“†TLd£]/R#K>dg]JŒR#deKOR#N%TLF>deK<iLF<H
D (v„!œ’DLKASLPR#bgDLhfDLUK-R#N
TLF>deK<iLF<N
D (Xð2R­ U YGDm]½§RV]<]>NGF<DLFkYGDm]/R¤[cHVUGU D¬F<DTHVU ]>K>F>N TLK>dgHVUtaodebuTHVhEGK<DˆEcHVNGF Y R#U ]kb½ deUK<DLF>w`R#bebgD

Y˜½¾SLU DLF>P`dgDƜhfD]>NGF<S%E)R#FkbqR2• N HVF<D]<TDLU TDkTHVU ]>dgYGSLF<S`avDLKkEcHVNGFŠ_‰]>deU HVUnoCRª­ P`NGF<Dm„ n ÷ªdebebeN ]>K>F<D%bgD

F<S]>NGbeK/R#KoYNkTR#bgTLNGbcHV[GK<DLUN¬RwQDTubgD}hfyLhfDubgHVKoYGD}YGHVUGU SD]ol¼z[GF>dgYGD]vjN D}TDLbeNGdNGK>debedg]<S}EcHVNGFbgD]rC (É4]n
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
v%c
qb RwzN D"YGD]OF<S]>NGbeK/R#K<][email protected]­ P`NGF<DB„ n ÷za)deb£THVUwzdgDLUKxY˜½§RVYGHVEGK<DLF–bqRfhfyLhfDBEGF>N YGDLU [email protected] D"EHVNGF

b½ deUK<DLF>EGF<SLK/R#K>dgHVUZYGD]xC(É4]-l¼z[GF>dgYGD]k”cU HVN ]"]/RwQHVU ]-j*N [email protected]½¾SLU DLF>P`dgDfYGHVUGU SDE)R#F"bqR'• N HVF<D]<TDLU TDfD]>K
deUŒ’SLF>dgDLNGF<[email protected]˜½¾DLU*w*deF<HVU
YGD\[email protected] SD\E)R#FAbgD]A]>deh"NGbqR#K>dgHVU ]AEGF<HVK<HVUn˜C½ dgYGSD\U½¾D]>KxYGHVU T\E)RV]AYGD
20%
hfDLK>K>F<DBDLU!SLw*dgYGDLU TDBNGU D\SLwQDLUK>N DLbebgD\K>F/R#U ]>deK>dgHVU%Œ’DLFJEGF<HVK<HVU2RwQDT\TDLK>K<D"­ P`NGF<D`n[–HVN ]–RwQHVU ]–w*N¤RwQDT
bgD]4gC (ɏ]rjN½ debSLK/R#deKŒŽHVF>KEGF<HV[)R#[GbgDOj*N DbgD]oTLF>deK<iLF<D]oYGDOjN)R#bedeK<S–R#EGEGbedgjN S]vR#NsfSLwQSLU DLhfDLUK<]rl¼z[GF>dgYGD]
EGNGdg]<]<DLU*KxRwQHVdeFxNGU [email protected]• N DLU [email protected] HVU¤U SLP`bedePQDR#[GbgDf]>NGFAbqR'EGF<HV[)R#[GdebedeK<[email protected] T<l DLhfDLUK\YGD]x]>K/R#K>dgHVU ]
YN‰¡zÕ\ncC½¾SLwQHVbeNGK>dgHVU‰[email protected]½¾SLU DLF>P`dgDYGD\bqRkEGF<HV[)R#[GdebedeK<SkYGH
D (v„phfHVU*K>F<SD]>NGFAbqRf­ P`NGF<D\„ n ÷ŠD]>KAYGHVU T

]>N ]<TDLEGK>de[GbgD-Y˜½¾yLK>F<D-[GdqR#dg]<SD`n
”
É HVNGF}wQSLF>d“­)DLFuTDLK>K<Dxl*¼*EcHVK>l i]<D`aGdebD]>KEcH`]<]>de[GbgDxYGDxTHVhE)R#F<DLFOb½¾SLwQHVbeNGK>dgHVU%RwQDT"b½¾SLU DLF>P`dgD-YN¬F/R#E
;
EHVF>[email protected]„ (HJ(‡ ~ (‘HJ(xa4j*NGdvF<DLEGF<S]<DLUK<DpbgDŠF/R#EGEcHVF>KfYNZU HVh"[GF<DˆY˜½¾SLwQSLU DLhfDLU*K<]fEcH`]<]<SY R#UKfR#NXhfHVdeU ]@„
]>K/R#K>dgHVU ”
] (‘JH (×]>NGFAbgDBU HVhB[GF<DfY˜½¾SLwQSLU DLhfDLUK<]BEcH`]<]<SY R#UK"R#N¤hfHVdeU ] ~ ]>K/R#K>dgHVU ]“(‘HJ(xacEcHVNGFxbgD¡zÕ±DLK
EHVNGFvbgDA{£Õ@nG‡rD–F/R#EGEcHVF>K}D]>KDLU¬Dt—cDLKvhfD]>NGF/R#[GbgDAE)R#FvbgD"¡zÕҔ*deb˜]>N¥fKYGDATHVhEGK<DLFE)R#F>hd˜bgD]SLwQSLU Dt
hfDLUK<]YN'¡zÕ«j*NGd ]<HVUK ~ (‘JH (̜’DLK (XðQta`THVhB[GdgDLU']<HVU*KSLPR#bgDLhfDLUK4„ (JH (xaTDj*NGdF<DLw*dgDLU*Ko»–hfD]>NGF<DLF4DLU
jN DLbgj*N Dx]<HVF>K<DxbqREGF<HV[)R#[GdebedeK<SBYGDxYGSTLbgDLU T/l DLFONGU D-]<K/R#K>dgHVU%YGDxEGbeN ]}Y R#U ]ub½¾SLwQSLU DLhfDLUKn|7UˆhfD]>NGF/R#U*K
TDAhfyLhfDAF/R#EGEHVF>KO]>NGFbgD]YGHVUGU SD]}l*¼*[GF>dgYGD]a debD]>KvEH`]<]<de[GbgDxYGDAwQHVdeF}jN)R#U Y¬bgD]YGDLNsphfD]>NGF<D]}]/HVUK
DLUÆRVTTHVF<Y˜a7R­ UZYGD'YGSLK<DLF>hdeU [email protected]<SLP`dgHVUZDLUÆSLU DLF>P`dgDpHVÈÅbgD]\TLF>deK<iLF<D]YGDk]<SLbgDTLK>dgHVUËR#EGEGbedgj*N S]R#Ns
SLwQSLU DLhfDLUK<]Ol¼z[GF>dgYGD]}U½ deU• N DLU TDLU*KOEGbeN ]bgD\¡zÕ\n
C RB­ P`NGF<Dx„ n ¦debebeN ]>K>F<DAbgDAF<S]>NGbeK/R#KuYNˆTR#bgTLNGb‘]<NGFbgD]}YGHVUGU SD]O¡zÕ DLKbgD]}YGHVUGU SD]}l*¼*[GF>dgYGD]n CD]
‘

THVNGF>[D]BTHVF>F<D]>EcHVU Y R#UKR#Ns¤w`R#bgDLNGF<]"YGHVUGU SD]BE)R#F"bqRˆ]>deh"NGbqR#K>dgHVUZ]<HVUK\SLPR#bgDLhfDLUK\hfHVUK>F<SD]a;DLbebgD]
]<HVUKTR#bgTLNGbgSD]f»mE)R#F>K>deFYGD]\Œ’HVU TLK>dgHVU ]Cg(ɏ]n£C½¾D]>K>dehkR#K>dgHVUËYGDŠb½¾SLU DLF>P`dgD¬YGD]@SLwQSLU DLhfDLUK<]kYNË¡zÕ
EGF<HwzdgDLUK–[email protected]>l H*YGD\6
‡ DJ|
‡ ?–{£Õ œ’TtŒJn)T<l)R#EGdeK>F<}
D ðQtaGj*NGd‘D]>K}YGHVU T"SLK/R#bgHVUGU SD-]>NGF}bqRB• N HVF<D]<TDLU TD`n
š U‹EDLNGKpTHVU ]>K/R#K<DLF¬]>NGFkbqR2­ P`NGF<D!„ n ¦‰j*N D%bqR2w`R#bgDLNGFŠYN‹F/R#K>dgHƄ (‘HJ(‡ ~ (HJ(¹hfD]>NGF<S!E)R#F'bqR

• N VH F<D]<TDLU TDfD]>KAEGF/R#K>dgj*N DLhfDLUKBK<[email protected]<DLhE ]-]>¼]>K<SLhkR#K>dgjN DLhfDLU*K-deUŒŽSLF>dgDLNGF<Dk»[email protected]]>NGF<SDE)R#F
bgD'¡zÕ@aDLK"TDAIJN ]<j*N½§»¬]/R#K>NGF/R#K>dgHVUÆYN2F/R#EGEHVF>Ka‘wQDLF<] 19 D€@n‡rDLK>K<DkTHVU ]>K/R#K/R#K>dgHVUÆTHVU­ F>hfDkbgDŒŽR#deK
10
jN D–bgD–bgHVK}YGDAYGHVUGU SD]l¼z[GF>dgYGD]U D–EcDLNGKE)RV]yLK>F<DONGK>debedg]/SxYdeF<DTLK<DLhfDLUKuEcHVNGFD]>K>dehfDLFbqRBEGF<HV[)R#[GdebedeK<S
YGDYGSTLbgDLU T<l DLhfDLU*K\YNÅ¡zÕ@n‘‡[email protected]<S]>NGbeK/R#K"D]<K-THVl SLF<DLUKBRwQDTb½¾HV[ ]<DLF>w`R#K>dgHVU‰YGD]-gC (ɏ]-l*¼*[GF>dgYGD]a‘jNGd
ŒŽRwQHVF>dg]<DLUK£bgD]EDLK>deK<D]EGF<HV[)R#[GdebedeK<S]7YGD4YGSTLbgDLU T/l DLhfDLUKnQC‘D4F/R#EGEcHVF>K„ (JH (‡ ~ (‘JH (X]<DLhB[GbgDv]<HVN ]J†D]>K>dehfS
E)R#FbgD4{£Õ@aj*NGdQŒRwQHVF>dg]<DoYGHVU TbgD] ~ (‘JH (ÅE)R#F‘F/R#EGEcHVF>K;R#Nsx„ (‘JH (xn D†b*D]>K‘YGHVU TU STD]/]/R#deF<D4Y˜½§R#EGEGF<H#ŒŽHVU YdeF
bgD]}SLK>N YGD]}]>NGFub½ deU• N DLU TD"YGD]}TLF>deK<iLF<D]OYGDxjN)R#bedeK<S"R#EGEGbedgj*N S]OR#Ns¬SLwQSLU DLhfDLUK<]Ol¼z[GF>dgYGD]n
C DOhfH¼QDLU¬bgDOEGbeN ]v]I|GFoYGD–hfD]>NGF<DLFrb½¾Dt¥kTRVTLdeK<SxYGDOYGSLK<DTLK>dgHVU¬YNˆ¡zÕÌDLU†YGD]<]/HVN ]YGDubqRB]/R#K>NGF/R#K>dgHVU
D]>K}YGD-THVU ]>K>F>NGdeF<D"NGUm]<HVN ]JF<S]/DR#N‰”GDLUmYGSLEGbgH¼R#U*KONGU D"TLdeU jN)R#U*K/R#deU DBYGD-]<K/R#K>dgHVU ]u]>NGEGEGbgSLhfDLU*K/R#deF<D]a
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
vv
,6-0/.1£„ n ¦
wQHVbeNGK>dgHVU YNXF/R#EGEcHVF>[email protected]„ (‘HJ(‡ ~ (‘HJ( DLUZŒ’HVU TLK>dgHVUXYGDpb½¾SLU LD F>P`dgD`a7YGSLK<DLF>hdeU SDmRwQDTpbgD]

,£
2
SLwQSLU DLhfDLUK<]Ol¼z[GF>dgYGD]a)RwQDT-bgD]}YGHVUGU SD]uYN!¡zÕ DLKORwQDTxbgD]}]>dehBNGbqR#K>dgHVU ]n
Ydg]>EH`]/SD]vR#N¬TDLU*K>F<DxYGD]oK>F>dqR#UGP`bgD]uSjNGdebqR#K<SLF/R#NsŠŒ’HVF>hfS]rE)R#FvbgD]]>K/R#K>dgHVU ]Dtsdg]>K/R#UK<D]a*deb˜D]>KrEcH`]<]>de[GbgD
Y˜½§R#[)R#dg]<]<DLFx]<DLU ]>de[GbgDLhfDLU*K-bqR'wVR#bgDLNGF-YGD\b½¾SLU DLF>P`[email protected]]/R#K>NGF/R#K>[email protected]½§RVTTDLEGK/R#U TD`n|U2Dt—cDLKacDLU¤EGF<H#
TSY R#UKfYGDpbqR!]<HVF>K<D`a7HVUËHV[GK>dgDLUKkNGUÆF<S]/DR#NËYGDpE)RV]@F<SYNGdeKfY˜½ NGUZŒŽRVTLK<DLNGF √ a£YGHVUKkb½¾Dt¥kTRVTLdeK<SˆYGD
2
YGSLK<DTLK>dgHVUˆwVR#NGK wQDLF<]
D†€\aGTHVhhfD"bgDAhfHVUK>F<D-bqRB­ P`NGF<D-„ n ^n CDA• Ns¬YGD]}F/R¼QHVU ]uTH`]>hdgj*N D]
17

5.10

R#NGP`hfDLUK/R#UKkF/R#EGdgYGDLhfDLUKbgHVF<]<j*N DŠb½¾SLU DLF>P`dgD¬YdehdeUN D`a7TD]@jN DLbgj*N D]@]>K/R#K>dgHVU ]]>NGEGEGbgSLhfDLU*K/R#deF<D]f]<Dt
F/R#dgDLUK']<N¥k]/R#UK<D]kEHVNGFfhfD]<NGF<DLFfbgDˆ]>EcDTLK>F<DmR#N†YGD]<]>N ]kYGD
D†€\aSLK<DLU Y R#U*KŠR#deU ]>dubqR¤PR#hhfD
5.1017
Y˜½¾SLU DLF>P`dgDpRVTTD]/]>de[GbgD¬»ˆb½ š [ ]<DLF>w`R#K<HVdeF<DpÉ7dgDLF>F<D¬[email protected]»mNGUXYGHVhkR#deU D'K>F<i]\deU*K<SLF<D]<]/R#UKf]<NGFBbgD'EGbqR#U
RV]>K>F<HVEGl¼]>dgj*N D`nM b½§R#dgYGD\Y˜½ NGU!]/HVN ]uF<S]<DR#NaDLU!hfD]>NGF/R#UKAbgD\]>EDTLK>F<DBYGD-ŒRVWHVUmdeU YGSLEcDLU Y R#UK<DYGDBbqR
THVhEH`]>deK>dgHVUËDLU†YGD]/]<HVN ]fYGD |D€@a7HVUXEcDLNGKhfD]>NGF<DLFb½¾Dt¥kTRVTLdeK<SmYGD¬YGSLK<DTLK>dgHVU YNXF<S]/DR#NË]>K/R#U
4
Y R#F<YÆ[email protected]¤THVhE)R#F<DLFkR#NsÆEGF<SYdgTLK>dgHVU ]K>l SHVF>dgjN D]fEHVNGFTHVUK>F/R#deU YF<DˆbqR%THVhEcH`]>deK>dgHVUËYGD]@F/R¼QHVU ]
TH`]>hdgjN D]n
üš¸ü‚
Vÿx| åK·Fÿ „™6I> Ž4ÿSK I I QÿSÿ%N I Î +N_ ]^4ÿ%
žbO
M ­ U\YGDK<DLF>hdeU DLF£TDoT/l)R#EGdeK>F<Do]<NGF‘b½¾SLK>N YGDrYGD4b½§RVTTDLEGK/R#U TDvYGDb½ š [ ]/DLF>wVR#K<HVdeF<DrÉ7dgDLF>F<DrMuNGPQDLFaU HVN ]
}
EGF<S]<DLUK<HVU ]ubgD]vEGF<DLhdgDLF<]}F<S]>NGbeK/R#K<]}Y˜½ NGU D"SLK>N YGDAEGF<SLbedehdeU)R#deF<D-wzdg]/R#UKu»@THVU*K>F/R#deU YF<[email protected]`]>deK>dgHVU
Äz
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
,.-F/.1 „ n ^
wQHVbeNGK>dgHVU%YGDAb½¾Dt¥kTRVTLdeK<S-YGD"YGSLK<DTLK>dgHVUmDLUpŒ’HVU TLK>dgHVU%YGDAb½¾SLU DLF>P`dgD`aGEHVNGFNGU%]<HVN ]JF<S]<DR#N

2
£
√
hˆaVDLK7EcHVNGF£bgDoF<S]/DR#NfYGD]>NGFJŒŽRVTDvY˜½¾MONGPQDLFn`C½¾Dt¥kTRVTLdeK<S}YGDYGSLK<DTLK>dgHVUk]ÎR#K>NGF<D
YGDrE)RV]
∼ 866
[DR#N THV1500×
NGEmEGbeN ]K<2ÚVK}
EcHVNGFbgD-]<HVN ]JF<S]<DR#Na»\NGU D-SLU DLF>P`dgDxwQHVdg]>deU D-YGD
17 D€@n
5.10
YGD]F/R¼QHVU ]OTH`]>hdgj*N D]}DLUK>F<D
1018
DLK
1019
D€@n
É R#F>hdbgD]xU HVh"[GF<DLN ]/D]BHV[ ]<DLF>w`R#[GbgD]-YNÆ¡zÕ±j*NGd]<HVUK-]/DLU ]>de[GbgD]"»ŠbqR¬THVhEcH`]>deK>dgHVUa˜U HVN ]"RwQHVU ]
£
SLK>N YdgSxbgD]vF/R#EGEcHVF>K<]}U(‘HVKœ‡ ~ (HJ(xaj*NGd˜F<DLEGF<S]<DLU*K<[email protected]ŒF/RVTLK>dgHVUˆY˜½¾SLwQSLU DLhfDLU*K<]OEcH`]<]<SY R#U*KuR#NˆhfHVdeU ]
]>K/R#K>dgHVU u
] (JH (±E)R#F>hdrbgD]SLwQSLU DLhfDLU*K<] ~ (‘HJ( YGSLK<DTLK<S]@E)R#[email protected]ŠF<S]<DR#NXYGDŠ]>NGFJŒŽRVTD`n‡rD]@w`R#F>dqR#[GbgD]
n
EDLNGwQDLU*KfyLK>F<D¬YGSLK<DLF>hdeU SD]'RwQDTpNGU Dp[HVUGU DŠEGF<STLdg]>dgHVUaYNƌR#deKYNXP`F/R#U YZU HVh"[GF<DˆY˜½¾SLwQSLU DLhfDLU*K<]
THVbebgDTLK<S]aDLKpK>F<i]ˆ]>dehEGbgDLhfDLU*KmTR#F¬debx]>N¥fKˆYGD¤THVhEGK<DLFˆE)R#F>hdxbgD]¬SLwQSLU DLhfDLUK<] ~ (‘JH ( œ’DLj
K (XðQta
THVh"[GdgDLU!]<HVUKuSLPR#bgDLhfDLU*Ku„ (JH (xa ð_(‘JH (xa)DLK<T`nenen
C D-TR#bgTLNGb;YGD]F/R#EGEcHVF>K<]}U(‘HJ(‡ ~ (HJ( D]>KuDt—˜DTLK>N S-DLUm]>dehBNGbqR#K>dgHVU!»@E)R#F>K>deF–YGD]vŒ’HVU TLK>dgHVU ]–Cg(ɏ]a
EHVNGFAYGD]–U H¼R#NsªYGD-ŒŽDLFADLKAYGD]–EGF<HVK<HVU ]xDLKATD»kYd“—˜SLF<DLUK<D]xSLU DLF>P`dgD]n‘‡rD]–EGF<SYdgTLK>dgHVU ]AK>l SHVF>dgj*N D]
EDLNGwQDLU*KyLK>F<DvYdeF<DTLK<DLhfDLUKoTHVhE)R#F<SD]oR#Ns\YGHVUGU SD]a`THVhhfDvbgDrhfHVUK>F<DvbqRu­ P`NGF<Dv„ n§Ñ`_n`É;HVNGFD]>K>dehfDLF
b½¾SLU DLF>P`dgD-YGD]}SLwQSLU DLhfDLU*K<]uYN%¡zÕ@aU HVN ]}]<HVhhfD]}SLwzdgYGDLhhfDLUK–THVUŒF<HVU*K<S]OR#N¬hfyLhfDxEGF<HV[GbgiLhfD"jN D
EGF<[email protected]”GjN DLbebgD"SLU DLF>P`dgDxNGK>debedg]<DL[F Ï
C D]F<S]>NGbeK/R#K<]debebeN ]>K>F<S]£]>NGF‘bqR­ P`NGF<D4„ n§Ñ`_}NGK>debedg]<DLUK»K>deK>F<DoY˜½¾DtsDLhEGbgDrNGU DoD]>K>dehkR#K>dgHVU\YGDob½¾SLU DLF>P`dgD
SLK/R#bgHVUGU SDO]<NGF4bqR–• N HVF<D]<TDLU TD`nÉ;HVNGFobgD]oK>F<HVdg]F/R#EGEcHVF>K<]rU(‘HJ(‡ ~ (HJ(‹j*N D}U HVN ]vRwQHVU ]rTR#bgTLNGbgS`a*debD]>K
EH`]/]>de[GbgDvYGDTHVhE)R#F<DLFbqR–hfD]>NGF<DYN'¡z՞RwQDTbgD]EGF<SYdgTLK>dgHVU ]£Œ’DLFDLK7EGF<HVK<HVUnz‡rDLK>K<DTHVhE)R#F/R#dg]<HVUkU½§R
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
,6-0/.1£„ n§Ñ`_
wQHVbeNGK>dgHVU‹YGDpb½§RVTTDLEGK/R#U TDˆdeU
THVU YdeK>dgHVU ]}2,
YGDx£ YGSTLbgDLU T<l DLhfDLU*[email protected]œ’DLU¬DtsdePQDR#UK
DLUŠ]>dehBNGbqR#K>dgHVUm»BE)R#F>K>deFvYGD]Cg(ɏ]THVF>F<D]>EcHVU
deUŒ’SLF>dgDLNGF<]u» ◦ n
ÄzÄ
]>K/R#U*K/R#U SDˆDLUZŒŽHVU TLK>dgHVU YGDpb½¾SLU DLF>P`dgD`a7EcHVNGFfYd“—cSLF<DLU*K<D]
~ a „ :a ð@HVN Ø ]>K/R#K>dgHVU ]X(HJ(utn CD]THVNGF>[cD]}]<HVU*K}TR#bgTLNGbgSD]
Y R#UK–»"YGD]rU H¼R#Ns¬YGDOŒ’DLFa*EcHVNGFvYGD]R#UGP`bgD]vÔSLUGdeK>l)R#Ns
45
E)RV]YGD}]<DLU ]Y R#U ]4bgD]7F<SLP`dgHVU ]4HVÈkbgD]THVNGF>[D]–÷tEGF<HVK<HVU‘÷}DLKO÷ΌŽDLF÷]<HVU*KK>F<i]7EcDLNk]<SLE)R#F<SD]aT`½¾D]>KJÞ»†YdeF<D
EHVNGF–bgD]uw`R#bgDLNGF<]AYN%F/R#EGEcHVF>KAU(‘HJ(‡ ~ (HJ( EGF<H*T/l D]AYGD HVN!YGD n˜‡ol)RVjN DBF/R#EGEHVF>KAEcH`]<]<iYGDBYGHVU T
0
1
NGU DAPR#hhfD-Y˜½¾SLU DLF>P`dgD-Y R#U ]bqRVj*N DLbebgDxdebD]>KEGbeN ]}]/DLU ]>de[GbgD`aGDLK}YGHVU T-DLU¬F<DLP`F<HVNGE)R#UKObgD]deUŒŽHVF>hkR#K>dgHVU ]
YGD]F/R#EGEHVF>K<]}„ ‡ ~ %a ð ‡ ~ DLK ‡ ~ aGHVUˆEcDLNGK}SLK>N [email protected]`]>deK>dgHVU!YGDLEGNGdg]
D†€IJN ]<j*N½§»
∼ 1018
∼ 1019
Ø
D€@n
M}­ UÅYGDfhfDLK>K>F<DŠDLUÅSLwzdgYGDLU TDfNGU DkSLwQDLU*K>N DLbebgDŠhfH*Yd“­)TR#K>dgHVUZYGDfbqR¬THVhEcH`]>deK>dgHVUXRwQDT'b½¾SLU DLF>P`dgD`a
U HVN ]ORwQHVU ]ATHVU ]>K>F>NGdeKONGU%D]>K>dehkR#K<DLNGFAj*NGd‘EcDLF>hfDLKOY˜½ deU Ydgj*N DLFObqRfEGF<HVEHVF>K>dgHVU¤YGDxU H¼R#Ns!YGDxŒŽDLFuE)R#F
F/R#EGEHVF>K7R#Ns-EGF<HVK<HVU ]aVY R#U ]‘NGU Drl*¼*EcHVK>l i]<Dr]>dehEGbgDoHVÈ@TD];YGDLNsB]<DLNGbgD];THVhEH`]/R#U*K<D]7]<HVUK;EGF<S]<DLU*K<D]n
OHVN ]"YGSt­ UGdg]<]<HVU ]BbqR2÷ΌF/RVTLK>dgHVUÅYGDŒŽDLF÷
a‘jNGdU HVN ]"bgDF/R#EGEcDLbgHVU ]-U k
D THVF>F<D]<EHVU YªE)RV]BYdeF<DTt
αFe
K<DLhfDLUK\»ˆNGU DhfD]>NGF<D'Dt—cDTLK>dewQD'YGDbqR¬EGF<HVEcHVF>K>dgHVUÅYGDkU H¼R#Ns‰[email protected]ŒŽDLF"Y R#U B
] bgD]-F/R¼QHVU ]\TH`]>hdgj*N D]a
THVhhfD-SLK/R#UK–YGSt­ UGdgDxdehEGbedgTLdeK<DLhfDLUK–E)R#F}b½¾Sj*N)R#K>dgHVU‰”
nToT
αFe (E) × PnFe (E) + [1 − αFe (E)] × Pnp (E)
(E) =
3ToT
αFe (E) × P3Fe (E) + [1 − αFe (E)] × P3p (E)
œ„ n
`

ÄzMv
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
VH È nToT
F<DLEGF<S]<DLU*K<D"bgD-F/R#EGEHVF>K}U(HJ(‡ ~ (‘HJ(ôhfD]>NGF<S\»@b½¾SLU DLF>P`dgD a Fe
[email protected]<HV[)R#[GdebedeK<S\YGD
E Pn (E)
3ToT (E)
YGSTLbgDLU T<l DLFkR#NÆhfHVdeU ] (‘HJ(ÒEcHVNGFBNGU DŠPQDLF>[cD'YGDkŒŽ[email protected]»mb½¾SLU DLF>P`dgD
DLK p
bqRmEGF<HV[)R#[GdebedeK<S¬YGD
n
E
Pn (E)
YGSTLbgDLU T<l DLFR#N‰hfHVdeU ] (‘HJ(ƒEcHVNGF-NGU DkPQDLF>[DkdeUGdeK>dgSDfE)R#F"EGF<HVK<HVUnCDkTR#bgTLNGbrYGD]-EGF<HV[)R#[GdebedeK<S]@YGD
n
YGSTLbgDLU T<l DLhfDLU*K Fe,p
D]>KBDt—cDTLK>N SŠDLU‰]>dehBNGbqR#K>dgHVUX»ˆb½§R#dgYGDkYGD]"Œ’HVU TLK>dgHVU ]\Cg(ɏ]aEHVNGF\T<l)RVj*N D
Pn (E)
deUK<DLF>w`R#bebgD-Y˜½¾SLU DLF>P`dgD`n)CR\ŒF/RVTLK>dgHVU%YGDAŒ’DLF
D]>K}YGHVU T"HV[GK<DLUN D"DLU¬F<S]/HVbewVR#U*K}b½¾SjN)R#K>dgHVUZœ„ n tn
`
αFe
CD]oF<S]>NGbeK/R#K<]rHV[GK<DLUN ]vRwQDTObgD]oYGHVUGU SD]rYNp¡zÕ¨]/HVUK4debebeN ]<K>F<S]o]>NGF4bgD]­ P`NGF<D]o„ n§Ñ aQ„ n§Ñ`Ñ"DLK4„ n§Ñ ~ n

C½¾SLwQHVbeNGK>dgHVUfYGDvbqRuŒF/RVTLK>dgHVUYGDoŒŽDLF7RwQDTvb½¾SLU DLF>P`dgDD]>K£TR#bgTLNGbgSDvEHVNGF£bgD]£K>F<HVdg]£F/R#EGEcHVF>K<]7THVF>F<D]>EHVU Y R#U*K
»
a a 0n OHVN ]7RwQHVU ]£YGS¯I<»}hfDLU*K>dgHVUGU SvbgDŒR#deK£jN½ deb*¼[email protected]#bqR#PQDvY˜½ NGU DowzdeUGP`K/R#deU DvYGD4EcHVNGF<TDLUK
n=456
DLUK>F<D4b½¾SLU DLF>P`dgD4{£ÕXDLKb½¾SLU DLF>P`dgD4YGHVUGU SD4E)R#F‘bgD]Ï!HVUK<Dt¯‡vR#F>bgHGa#DLK‘YGHVU T4U HVN ]RwQHVU ];SLPR#bgDLhfDLUK£TR#bgTLNGbgS
»@K>deK>F<D-deU YdgTR#K>d“ŒbgD]}ŒF/RVTLK>dgHVU ]–YGDxŒŽDLFuHV[GK<DLU*N D]–]<db½¾HVU%R#NGP`hfDLUK<DR#F>K>d“­)TLdgDLbebgDLhfDLU*K–b½¾SLU DLF>P`dgD\YGHVUGU SD
E)R#FbqR\• N HVF<D]<TDLU TDBYGD
œò­ P`NGF<Dx„ n§Ñ`ÑQvHVNmYGD
œò­ P`NGF<D-„ n§Ñ ~ tn
15%
25%
C‘R'EGF<DLhdgiLF<DF<DLhkR#F<j*N DjN D\b½¾HVU¤EcDLNGKOŒR#deF<DR#N2]>NVI>DLK–YGDTD]–F<S]>NGbeK/R#K<]a˜D]<KAj*N D\bqRkŒF/RVTLK>dgHVU2YGD
Œ’DLF–YdehdeUN D\deU YGSLUGdqR#[GbgDLhfDLUK"RwQDT"b½¾SLU DLF>P`dgD`adeU Ydgj*N)R#UKANGU!R#bebgiLPQDLhfDLU*KxYGD"bqRkTHVhEH`]>deK>dgHVU¤DLUK>F<D

DLK _B|D€@nG¡*NGdewVR#U*Krb½¾ST/l DLbebgDAY˜½¾SLU DLF>P`dgDONGK>debedg]/SD`abqR-ŒF/RVTLK>dgHVUpYGDuŒ’DLFoEDLNGKvYGSLE)RV]<]<DLFrb½ NGUGdeK<SOHVNpDLU THVF<D

yLK>F<D}U SLPR#K>dewQD`a*TDuj*NGd K>F/RVYNGdeKr]<dehEGbgDLhfDLUKrbgDvŒŽR#deKrjN D}bgD]EcHVdeUK<]rDtszEcSLF>dehfDLUK/R#Ns']/HVUK4E)RV]oTHVUK<DLU*N ]
DLUK>F<DkbgD]"THVNGF>[cD]-K>l SHVF>dgj*N D]Š÷ΌŽDLF÷fDLK'÷tEGF<HVK<HVU‘÷Vn;CD]"F<S]>NGbeK/R#K<]BTHVF>F<D]<EHVU Y R#U*[email protected]»pNGU DkSLU DLF>P`dgD'YGD
]<DLhB[GbgDLUKxyLK>F<D-bgD]OEGbeN ]–THVUw`R#deU TR#UK<][email protected]Œ’F/RVTLK>dgHVU¤YGDxŒŽDLFOSLwQHVbeN)R#UK
• N HVF<D]<TDLU TDBF<DLl)R#N ]<]<SD"YGD
15%
YGDAhkR#UGdgiLF<D-F<DLhkR#F<j*N)R#[GbgDLhfDLUKATHVl SLF<DLUK<DBDLUK>F<D DLKu_fYGDLEGNGdg] 18 D€‹IJN ]<j*N½§» 19 D€@n

10
10
DÞb–D]>KpSLwzdgYGDLUK¬jN D!TD]ŠF<S]>NGbeK/R#K<]¬YGSLEcDLU YGDLUKˆY˜½ NGUžT<l HVd“sžR#F>[GdeK>F/R#deF<DªYGDmbqRÅU HVF>hkR#bedg]/R#K>dgHVU¨DLU
SLU DLF>P`[email protected]>debedg]<SD`acDLKxjN½ debD]>KAYGHVU [email protected]`]<]>de[[email protected]˜½¾DLU2THVU TLbeNGF<[email protected]¤F<S]>NGbeK/R#K-]<TLdgDLUK>d“­)j*N D\­R#[GbgDY R#U ]
b½¾SLK/R#K}EGF<S]<DLU*Kn)CD]udeU TDLF>K>deK>N YGD]O]>NGFubqR\wVR#bgDLNGF–YGDAbqR\ŒF/RVTLK>dgHVU!YGDAŒŽDLFEGF<HwzdgDLUGU DLUK–hkRIJHVF>deK/R#deF<DLhfDLU*K
YGDAb½ deU TDLF>K>deK>N YGD-bedgSD"»b½¾D]>K>dehkR#K>dgHVUmYGDxb½¾SLU DLF>P`dgD`n
–HVN ]"THVU TLbeN DLF<HVU ]@YGHVU TkDLUÆ]<HVNGbedeP`U)R#[email protected] THVF<DfNGU DfŒ’HVdg]BjN½ debrD]>K-dehEcHVF>K/R#[email protected]<D]>K<DLFBbgD]"]>d“
h"NGbqR#K>dgHVU ]@YGD]-PQDLF>[cD]\[email protected]ŒŽRVWHVUÅEHVN ]/]<SD`aEcHVNGF"j*N½¾DLbebgD]"F<DLEGF<HzYNGdg]<DLUK\bgD]"YGHVUGU SD]\bgDEGbeN ]"­)YGiLbgDt
hfDLUKoEH`]<]<de[GbgD`n*¡*d bgD]hfHzYGiLbgD]4]½§RwQiLF<DLUKrdeU TR#E)R#[GbgD]rYGD}F<DLEGF<HzYNGdeF<DORwQDTODtsGRVTLK>deK>N YGDub½¾DLU ]<DLh"[GbgD–YGD]
YGHVUGU SD]@THVbebgDTLK<SD]BE)R#F\bgD]@DtszEcSLF>dgDLU TD]@SLK>N YdqR#[email protected]]BPQDLF>[D]R#K>hfH`]>EGl SLF>dgj*N D]a;TDLK>K<D'deUŒŽHVF>hkR#K>dgHVU
EHVNGF>F/R#deKryLK>F<DuNGK>debgDuEHVNGF4ŒŽR#deF<DuEGF<HVP`F<D]<]<DLFrU HVK>F<DuTHVhEGF<SLl DLU ]>dgHVU¬YGDubqRxEGl¼]>dgj*N DuYGD]oE)R#F>K>dgTLNGbgD]R#Ns
K>F<i]l)R#NGK<D]uSLU DLF>P`dgD]n
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
Äz¿
, -0/.1‘„ §n Ñ
6
wQHVbeNGK>dgHVUXYGDbqR¬ŒF/RVTLK>dgHVUÆ[email protected]ŒŽDLF
RwQDTkb½¾SLU DLF>P`dgD`aTR#bgTLNGbgSDp»pE)R#F>K>deF\YGD]"F/R#EGEcHVF>K<]
αFe

u
±Ð
2
U(‘HJ(‡ ~ (‘JH (©]>[email protected]]YGHVUGU SD]YN¡zÕ\n£C½¾D]<K>dehkR#K>dgHVUËYGD'b½¾SLU DLF>P`dgDmD]>K\Dt—˜DTLK>N SDˆRwQDTpbqR%hfSLK>l HzYGD
NGK>debedg]/R#UK7bgD]THVNGF>[cD]7Y˜½§R#K>K<SLUN)R#K>dgHVUŠYGHVUGU SD]7E)R#F7bgD]7]>dehBNGbqR#K>dgHVU ]aRwQDT}[email protected]/R#bgHVUGU)R#PQDu]>NGF7b½¾SLU DLF>P`dgD
hfD]>NGF<SDxE)R#FbqR\• N HVF<D]<TDLU TD`n
Äz)ú
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
, -F/.1‘„ §n Ñ`Ñ
.
wQHVbeNGK>dgHVUZYGDfbqRpŒ’F/RVTLK>dgHVUÅ[email protected]ŒŽDLF
RwQDTfb½¾SLU DLF>P`dgD`a;TR#bgTLNGbgSDŠ»pE)R#F>K>deF\YGD]"F/R#EGEcHVF>K<]
αFe
±Ð
2
U(‘HJ(‡ ~ ( HJ(Ù]>[email protected]]YGHVUGU SD]YN¡zÕ@n;C4½¾D]>K>dehkR#K>dgHVUËYGDŠb½¾SLU DLF>P`dgDˆD]>[email protected]—cDTLK>N SDˆRwQDTpbqRmhfSLK>l HzYGD
NGK>debedg]/R#UK7bgD]4THVNGF>[D]7Y˜½§R#K>K<SLU*N)R#K>dgHVU'YGHVUGU SD]7E)R#F7bgD]7]>dehBNGbqR#K>dgHVU ]aRwQDTNGUSLK/R#bgHVUGU)R#PQD}]>NGF7b½¾SLU DLF>P`dgD
n
hfD]>NGF<SDxE)R#FbqR\• N HVF<D]/TDLU TD`a F<DLl)R#N ]<]<SDxYGD
15%
úÀ0[±°¬&*õ’õf*GG£õkõ’Á*®²2õޓGÃÃQ$Üf‘Ã
Äz
, -0/.1‘„ §n Ñ ~
6
wQHVbeNGK>dgHVUXYGDbqR¬ŒF/RVTLK>dgHVUÆ[email protected]ŒŽDLF
RwQDTkb½¾SLU DLF>P`dgD`aTR#bgTLNGbgSDp»pE)R#F>K>deF\YGD]"F/R#EGEcHVF>K<]
αFe
2
±
Ð
U(‘HJ(‡ ~ (‘HJ(©]>[email protected]]YGHVUGU SD]YN¡zÕ\n£C½¾D]<K>dehkR#K>dgHVUËYGD'b½¾SLU DLF>P`dgDmD]>K\Dt—˜DTLK>N SDˆRwQDTpbqR%hfSLK>l HzYGD
NGK>debedg]/R#UK7bgD]THVNGF>[cD]7Y˜½§R#K>K<SLUN)R#K>dgHVUŠYGHVUGU SD]7E)R#F7bgD]7]>dehBNGbqR#K>dgHVU ]aRwQDT}[email protected]/R#bgHVUGU)R#PQDu]>NGF7b½¾SLU DLF>P`dgD
n
hfD]>NGF<SDxE)R#FbqR\• N HVF<D]<TDLU TD`a)F<DLl)R#N ]<]<SDxYGD
25%
ÄzW
[email protected]ÃQÃQ$ÜfÃQyÛ;& Á* &tÛ%&ÂÞGÃQ
!
"
#$%&
À¾ ÍÏÁ5À
Ñ
Á5Ä
ÇÓ ¾
(
Í
¾Y* ÀÒ(lÆÈÁÈÀ
ǾÔ
ÉgÕ ÍÏÀ¾
ÀÒ
ÄÂÃ À
ÏÀ Á5΂ÉÊÀ
(
Ò
Ì
ÏÀ¾
ÍÏÎÀÂÁ
Ð ÑÒnÓ]ÔÕÑÖ!ÓY×nØ6ÙÚ#ÛÝÜÝÞ߶à
õ3Ù{Ú_Þ'æA×ÑÖ!ÓIÞ߶Ö]ßùÒAÑnêFß>×/ÜÝÛÝæ*ð ßÜÖ1ßvÞß×æ!ð‘ÒnÜÝÞnӑ×/ÜÝÛÝæ*ð Ö]ßÒvÑåCÑnð_ßnÔÕßnð#ܕҟÖ!Ó:ÞÑnÒ/ßÙ!ÓNÖ1ßùÒnÓ]ޕñ/Ù×ßßÒnܧÚ#ÞÑnÒ/ßnð–3
܏Ñ>ß¹Ö1ߥԶÙ*ð~Û=ÿÞßÕÖ1ÑÜ Ù!ەêGêxÑ>ß šõ ßnÒ¥Ö!Û õŠÑnÞ>ßnð‡ÜßÒ¶ÔÕÑܕØIæAÖ]ßҗڑæ!Ó]ÞÕßÒÜÝÛGÔÕßnÞê>á ÑnðußnÞÝé*Û=ßÕÖ1ßÒ¶ÑåCÑnð_ßnÔÕßnð#ܕҟÒ/æ!ð‡Ü
êúÙdÒ/æ!Ó]Þ×ß¹êúÙÚ#ê;Ó]ҟەԊÚ6æ!ÞÜ Ù!ð‡Üß¶ÖWá ەð#×ßnÞÜÝÛÝÜÝÓ6Ö1ß{Ú6æ!ÓIޟêFÙ%ÔÕßÒnÓ]Þ>ß¹Ö!ÓYÒ Ú~ß×/ÜÝÞ>ßÕÖ á Ñnð_ßÞÝé!Û ß¹Ö]ßҟÞÙ0/æ!ð‘Ò¶×>æCÒ43
ÔÛ=ëÓ~ßÒ oõâá ßÒnÜÝەԹÙ*ÜÝÛ=æ!ðwÖ1ߟêá Ñnð_ßÞÝé!Û ß¹Ù*åCß׶êxßÒvÒnÛGÔ¥ÓIêúÙÜÝÛÝæ!ð~ÒtÒ/ßnÔÕänêxߟÒæ*ÓõTÞnÛGÞ¶Ö á ßnÞnÞßnÓIÞÒâ4Ò /CÒ܏ÑnÔ¹Ù*ÜÝÛÝë/ӑßÒ
Ö1яڑßnð_ÖÙ!ð‡ÜÖ1ßâêá Ù!ð]é*êF–ß EÑnð‡Û•Ü•ØIÙ*ê ûuÚ#Þnەð#×Û Ú6Ù!êxßnÔÕßnð‡ÜÙ!TÓ 3'Ö1ßÒÒÓ]ÒÖ1ß 45◦ Ö7ŸßÚ#ê„Ó1Òû‹êFÙ¹åÙ*êFßnÓIÞÖ]ßvê>á Ñnð_ßnÞ=é!Û ß
æ1ä܏ßnð~Ó~ߥÖ1яÚ~ßnð#Ö%Ö á Ó]ð_ßv^Ø /Ú6æܕØ6ÿÒ/ߥÖ]ßâ×>æ*ԗڑæCÒÛÝÜÝÛÝæ*ð õ3ÙÔÕÑܕØIæEÖ1d
ß ×GØ0×pٝÚ17 ґá Ù'õTÞÙ!ð#×nØ1ÛÝÜ,Ö1ßt×ßÒ,Ú#Þæ1änêxÿnÔÕßÒâßnð|×>æ!ÞnÞnÛúé1ß>Ù!ð‡ÜêFÙåÙ!êxßnÓIÞtÖ*ÓÕÒnÛúé!ð_Ù!ê_ì¹òô1ôô
Ô¹ÿ/ÜÝÞßÒ:Ö1ßdÔ¹Ù!ð~Û ÿnÞ>ß:ìNåCÑnÞnÛ .ßnÞdêFÙÕÚ#ÞæÚ#ÞnÛ=Ñ/܏Ñ:Ö á ÛhÒ/æ*ÜÝÞ'ænÚ#Û=ß%Ö]ßқÖ!ەÞß>×/ÜÝÛÝæ*ð‘Ò%Ö á Ù!ÞnÞەåCÑ>ß|Ö1ßÒÕÞ0Ù /æ!ð‘Ò%×>æC4Ò 3
ÔÛ=ëÓ~ßÒû,ßÜ,ßnð:Ó6ÜÝÛGê„ÛhÒ/Ù!ð‡ÜWêá Ñnð_ßÞÝé!Û €ß Ú17×>æ!Ô¥ÔÕߊÞѕñAÑnÞ>ßnð#×ß ,õ ß,Ú6Ù!ÞÙ!ÔÕÿÜÝÞ>ß S = S (θ)/S (38◦ ) û
38
1000
1000
ëÓ]۟Þß7Ú#ÞÑnÒ/ßnð‡Üß êFß|ÒnÛúé*ð#Ù!êŠëÓIÛvÒAßnÞÙ!ەÜÔÕßÒnÓ]Þ>Ñ ì“òôô1ôwÔ ÒÛâêúÙfé1ßnÞAä>ß5ÑÜ Ù!ÛÝÜÜ æ*Ô¹äÑ>ß Ù!åCß>× ÓIð Ù*ð]é!êxß
ûTßÒÜêá æ1äÒAßnÞnåÙ1änêxßÕÖ*ÓNÞ>ÑÒAßÙ!ÓwÖ1ßÒnÓ]ޕñ/Ù×ß¹ë/Ó]ÛßÒn܊×>æ!ÞnÞÑnêxÑ>ߛì%ê>á Ñð_ßnÞ=é!Û=߶ÔÕßÒÓIÞÑß{Ú6Ù*ޟêFAÙ Ó60æ 3
θ = 38◦
ÞßÒ/×ßnð#×ß )âæ!Ó1ÒtÙ!åæ!ð~ÒÚ#ÞæÚ6æCÒAÑùÓIð_ß{ÔÕÑܕØIæAÖ]ß{ÒەԥÛGêFÙ!ەÞßnû±ëÓIÛWÓ6ÜÝÛGê„ÛGÒ/ßtÑ=éÙ*êFßnÔÕßnð#Üêá Ñnð_ßÞÝé!Û €ß Ú17×>æ!Ô¥ÔÕß
ÞѕñAÑnÞ>ßnð#×ßnû±ÖÙ!ð‘ÒêFÙëÓ~ßnêGêxߗêFß S ßÒܱßÒÜÝÛGÔÕыé!Þ Û1×ß{Ù!Ó!ïvÒەԥÓ]êFÙ*ÜÝÛ=æ!ð‘Òð~ÓIÔ¹ÑÞÛ=ëÓ~ßÒûWÖæ*ð‡Ü§êxßÒ Ú_ÞÑÖ*ÛÝ×/ÜÝÛ=æ!ð‘Ò
38
ڑßnÓIåCßnð‡Ü 6ÜÝÞ>ߥ×>æ!ð‘ÒnÛÝÖ]ÑnÞÑßÒ×>æ!Ô¥ÔÕÂß . Ù1äêFßÒûuÚ6æ!ÓIÞâêxßÒâÙ!ð]é!êxßÒvەðAñAÑnÞÛ ßnÓIÞҟì 45◦ õ ßÒ Ò Ú‘ß>×/ÜÝÞßnÒÖWá ÑnðußnÞÝé*Û=ßÔÕßÒÓIÞÑnÒ Ú6Ù*Þ ê>á ã—äÒAßnÞnåÙ*Ü æ!ەÞßçÛ ßnÞnÞß èŠÓ!é1ßÞTÙ*åCß×{×ßÒÖ!Û õŠÑÞßnð#܏ßÒTÔ¹Ñ/ܕØIæAÖ1ßÒ
Òæ!ð‡Ü§Ú#Þ>ÑÒ/ßnð#܏ÑÒ
Äzc
ñA,/.
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
A¶ó<579?:\@u1Ò5‚?%·579‚@}1B )? ·c6B9?:\@u1Ò³\5gBm¶1{}¶71\5¸¹571¢?[Bm³\6 ·¶gB57º}6׳\5#B)6B·–Þ
Ý
Óº}95
C HVF<]<j*N½ NGU DAPQDLF>[cDxD]>KYGSLK<DTLK<SDAE)R#FbgDAF<S]<DR#N¬YGDx]>NGFJŒŽRVTD`aT`½¾D]>KJÞ»†YdeF<D"jN½¾DLbebgDAK<HVN T/l D-]>N¥k]/R#[email protected]
hfDLUK4YGDu]>K/R#K>dgHVU ]EcHVNGF]/R#K>dg]JŒR#deF<DuR#NsTLF>deK<iLF<D]4YNfUGdewQDR#NkYGDuYGSTLbgDLU T<l DLhfDLUK|( ~ œ’TtŒ˜T<l)R#EGdeK>F<D ~ ta`bgD]
YGHVUGU SD]YGDuK<HVNGK<D]vbgD]v]>K/R#K>dgHVU ]R¼R#UKuYGSTLbgDLU T/l SANGUZ(}Ñ\HVNpNGUZ( ]<HVU*KDLUGF<DLP`dg]>K>F<SD]DLKvRV]/]<H*TLdgSD]»

NGUˆSLwQSLU DLhfDLUKn
MwVR#U*KuY˜½¾D]>K>dehfDLF–bgD]uTR#F/RVTLK<SLF>dg]>K>dgj*N D]OYGD"[email protected]>[cD'œEcHVdeUKOY˜½ dehE)RVTLKaYdeF<DTLK>dgHVUa)SLU DLF>P`dgD`nenen›taNGU D
EGF<H*TSYNGF<D¬YGD¬]/SLbgDTLK>dgHVU YGD]f]>K/R#K>dgHVU ]D]>KkR#EGEGbedgj*N SDmR­ UËY˜½¾SLbedehdeU DLFfbgD]k]>K/R#K>dgHVU ]fj*NGdrU D¬YGHVdewQDLU*K
E)RV]vE)R#F>K>dgTLdeEDLF–»"b½¾SLwQSLU DLhfDLU*KuEcHVNGFYdewQDLF<]<D]F/R#dg]<HVU ]}j*N D–U HVN ]}DtsEGbedgTLdeK<HVU ]E)R#FbqR\]>NGdeK<D`nGCDxTLF>deK<iLF<D
(v„fD]>KSLPR#bgDLhfDLUKAR#EGEGbedgjN S-R­ UˆY˜½§RV]<]<NGF<DLF}jN DAb½¾SLwQSLU DLhfDLU*KuU D–EGF<HwzdgDLUKOE)RV]Y˜½ NGUmYGSTLbgDLU T<l DLhfDLU*K
Œ’HVF>K>NGdeKnCD]–SLwQSLU DLhfDLUK<]-R#deU ]>d£]<SLbgDTLK>dgHVUGU S]A]<HVU*K–DLU ]>NGdeK<DBF<DTHVU ]>K>F>NGdeK<]A]>NGdew`R#UK–NGU D"EGF<HzTSYNGF<D\jNGd
D]>KfdehEGbgSLhfDLU*K<SD%Y R#U ]'bgD%‡rÕ}Mx¡a7ŒŽHVNGF>UGdg]<]/R#U*KŠR#deU ]>d}NGU DmF<DTHVU ]>K>F>N TLK>dgHVUžR#NGK<HVhkR#K>dgjN D`+
n D†buDtsdg]>K<D
EGbeN ]>dgDLNGF<]hfSLK>l HzYGD]fEcHVNGFF<DTHVU ]>K>F>NGdeF<DˆbgD]kSLwQSLU DLhfDLUK<]ŠYNXF<S]<DR#N YGD¬]>NGFJŒRVTD`a£U HVN ]fEGF<S]<DLU*K<HVU ]
TDLbebgD-j*NGdD]>KNGK>debedg]<SD-Y R#U ]bgD\‡rÕ}MA¡a)YGHVUK}U HVN ]}Œ’DLF<HVU ]N ]ÎR#PQDAK<HVNGKOR#NˆbgHVUGPkYGDxTDLK>K<D-SLK>N YGD`n
C‘[email protected]<HzTSYNGF<D-YGDx]<SLbgDTLK>dgHVU!DLK}YGDxF<DTHVU ]>K>F>N TLK>dgHVU!D]>[email protected]]>NGdew`R#UK<Df”
ß
Q%K†ÿ%g ÿ%€O_I> <
C D]O]>K/R#K>dgHVU ]uj*NGd˜wQSLF>d“­)DLUKxR#NˆhfHVdeU ]}NGU D"YGD]uTHVU YdeK>dgHVU ]–]>NGdewVR#U*K<D]u]<HVU*KuSLbedehdeU SD]–YGDxbqR]<SLbgDTt
K>dgHVU2”
bqRª]>K/R#K>dgHVU«R2SLK<SˆYGSTLbgDLU T/l SD!E)R#FkK>F>deP`PQDLF'DtsK<DLF>U D‰”E)R#F'DtsGDLhEGbgDˆbgHVF<]ŠY˜½ NGU K>deFfbqRV]<DLFŠY R#U ]
?
b½§R#K>hfH`]>EGl iLF<D`arNGU]<deP`U)R#bOD]>KŠDLUwQH¼QS¤Y R#U ]'b½¾SLbgDTLK>F<HVUGdgjN D¤Y˜½ NGU D%TLNGwQD%]>dehBNGbeK/R#U SLhfDLUKˆR­ U
Y˜½¾D]>K>dehfDLF\bgDkYGSTR#bqR#PQD'K<DLhEHVF<DLbvDLUK>F<D'bgDK>F>deP`[email protected]{£ÕÙDLK"bgDkK>F>deP`[email protected]¡zÕ@n‘CD'K>F>deP`PQDLFBDtsK<DLF>U D
EcDLNGKR#N ]/]>d)EGF<HwQDLUGdeF}YGD]rYdgHzYGD]rbqRV]<DLFvdeU ]>K/R#bebgSD]vY R#U ]rbgD]rTLNGwQD]vEcHVNGFrbgD]rSLK>N YGD]vYGDObedeU SR#F>deK<S`n
[email protected]]>K/R#K>dgHVU%R\NGU%]>deP`U)R#bdgYGDLUK>d“­)SBTHVhhfDxEGF<HwQDLU)R#U*K–Y˜½ NGU%STLbqR#deF Û
?
[email protected]]>K/R#K>dgHVUmD]<KbgD-THVhE)R#P`U HVU%Y˜½ NGUmYGHVNG[GbgDLKOYGD-TLNGwQD] 5
?
[email protected]]>K/R#K>dgHVU%R\NGU DBR#beK>deK>N YGD-deU THVF>F<DTLK<DkœdeUŒŽSLF>dgDLNGF<DB» _`_`[email protected]>F<D]t

?
Ü ä›ß¯+
ë 9Jç¯ä ã/ê¾âëvñâì¿ûé#ê§ëŽßJàá+ûßJë4ëŽê:útàVã/éOàuû#ãÎàëoäçü 9Jä:9Jç¯áŽâìÎà#ê›èé#ßû߯ëoçJéO‰LßJëß¯áoñ`ßJé]‰tßJàá4ãÎê§à#ëŽêM›“ãÎê¾âßBûO9JçJä§ßJàçý߯âá’é#ëèé$ü ÿ
éà#ß;áŽâßJàá†ã/ê›àß ûߣç¯é]‰LßJë˜ëŽê§ïoé#ä¾á†ãÎàO9JïßJàá˜ëŽéâ‘ä§ß£â 9JëŽß>ã/éT0½*ßJëëŽê:útà#ãÎéOàOñâì\‰L߯àVãÎàáûßJëV9JçJä›ãÎê¾âëëŽìtàá‘ç>ã/â†ã/ç¯á 9¯âê§ë`9Jë‘ñVã/â<ûß
àìtp
ï ƒâßJé#ëŽß¯ëcìtëŽçJê§ä§ä›ã/áê§ìtàë7ãÎéáìtéâûß7ä›ãrä§ê:útà#ß ûßgƒVã/ëŽßtð#ßÞáëŽìÎàá‘ãÎê§ë`9JïßJàá‘ê¤ûßJàáêâÁ`ãAƒ#ä§ßJë
8½*ß}â 9JëŽß>ãÎb
é ûßOëŽéã⠛“ãÎçJ3
ß ûßu‚ä ü ä^;QæËñ`ìtëŽãë ¼nûßOáŽâìtê§ëvñ#ãÎê¾âßJ|
ë ûߖë’á†ã<áê›ìÎà#ëèéêcëŽìtà|
á û@9Jñ#ä§ì>Åî 9J߯|
ë ÿ®åæ"z
ï ¼ÞáŽâßJërçä ü é#à3
ß ûß
äçü ãÎéáŽâçß ÝßJè
ë ˆûìtOé ƒ#ä§ß¯á\ë ˆûß}ë’á†ã/áê§ìÎà#ëñ`߯âï߯áŽáß¯à+
á û ü 9Þá_é ûê§ß¯ârä§ßJp
ë éVéç¯áéVã<áê§ìtà#6
ë ûéëŽ:ê útà#ãÎä˜ïßJëŽéâ 9O߯®
à › ìtàç¯áê§ìte
à ûßuä›ã
ûê›ë’á†ã/à#çJß ÿo‚ä ü \ã àß ûߣä›.
ã út߯„â ƒ`4ß
cÀ’Ä Ú Á*õÃ`>f k*ÃQfV&‘ÃQò>fÛ;ÁÁfi‰fbÛ;& Á* & Û;ˆ&ÂÞGÃQ
Äzv
ÿ~eÒÇ.Q%Kÿ%g 4ÿ%*QQ%ÿÿ j6#ëÎ_])4ÿ%
©b$`2ê
C DªTLF>deK<iLF<D2YGDªUGdewQDR#N̄ËD]>KmR#EGEGbedgjN SÅ»Æb½¾SLwQSLU DLhfDLUKaOU HVN ]¬F/R#EGEDLbgHVU ]mbgD]ˆYGDLNsžTHVU YdeK>dgHVU ]
EH`]<]<de[GbgD]x”
b½¾SLwQSLU DLhfDLUK\THVUK>dgDLUKBK>F<HVdg]-]>K/R#K>dgHVU ]xwQHVdg]<deU D]"R¼R#U*K-E)RV]<]<[email protected]@UGdewQDR#N‰YGDYGSTLbgDLU T/l DLhfDLUKu(}Ñ
(‘JH (xnCD]K>F<HVdg]]>K/R#K>dgHVU ]˜ŒŽHVF>hfDLUKNGU-K>F>dqR#UGP`bgDdg]<HzTiLbgDHVN"Sj*NGdebqR#K<SLF/R#bnÕxR#U ]TDTRV]ab½¾SLwQSLU DLhfDLU*K
D]>KuSLK>dgj*N DLK<z
S (‘JH (xn
b½¾SLwQSLU DLhfDLUK-THVU*K>dgDLUK"j*N)R#K>F<DB]<K/R#K>dgHVU ]AR¼R#UKxE)RV]<]/S"bgDBUGdewQDR#NªYGD\YGSTLbgDLU T/l DLhfDLU“
K (}Ñ!œdeU Yd“Œ
?
Œ’SLF<DLhhfDLU*B
K (vlGF<D]>l HVbgYmHV*
N (‘JH (uvY R#U ]NGU D"THVU­ P`NGF/R#K>dgHVUm„*‡ œNGU D-]>K/R#K>dgHVUmR¼R#UKOK>F<HVdg]EGbeN ]

EGF<H*T/l D]rwQHVdg]>deU D]ΝtnG¡*d)bg
D ( ~ EGF<HwzdgDLUKYN'YGSLK<DTLK<DLNGFYGD}• N HVF<D]<TDLU TD`azb½¾SLwQSLU DLhfDLUKYGHVdeKrTHVU*K<Dt
UGdeF;j*N)R#K>F<Dr]>K/R#K>dgHVU ]£DLUBTHVU­ P`NGF/R#K>dgHVU„*‡ n#ÕAR#U ];TDrTRV]ab½¾SLwQSLU DLhfDLU*KD]>K;SLK>dgj*N DLK<6
S (vlGF<D]>l HVbgY˜n

?
ƒ!ÿ$#ÿ%$$#4ÿ4ÿKIM' IV4ÿ
™ U DTé!ÞÙ!ەð_ßuD]>KYGSt­ UGdgDTHVhhfDK>F<HVdg]]>K/R#K>dgHVU ]U HVUkR#bedeP`U SD]DLUfTHVU­ P`NGF/R#K>dgHVUkYGDvK>F>dqR#UGP`bgDdg]<HzTiLbgD
u
HVN%Sj*NGdebqR#K<SLF/R#bn˜‡rDLK>K<DBP`F/R#deU D\EDLF>hfDLKAY˜½¾Dt—˜DTLK>N DLFONGU DBEGF<DLhdgiLF<[email protected]]>K>dehkR#K>dgHVUªYGD-bqRkPQSHVhfSLK>F>dgD\YGDBbqR
PQDLF>[D–»xE)R#F>K>deFvYGD}bqRVj*N DLbebgD}w`R-YGSTHVNGbgDLFrbqRxEGF<HzTSYNGF<DOYGDO]<SLbgDTLK>dgHVUŠYGD]r]>K/R#K>dgHVU ]n*CD]oK>F<HVdg]4K<DLhE ]vYGD
YGSTLbgDLU T<l DLhfDLUK\YGD]A]>K/R#K>dgHVU ]xYGD\bqRkP`F/R#deU D\EcDLF>hfDLK>K<DLUKxE)R#FAK>F>dqR#UGP`NGbqR#K>dgHVU‰[email protected]#bgTLNGbgDLFxbqRŠYdeF<DTLK>dgHVU
R#deU ]>d‘j*N DxbgDxK<DLhE ]–Y˜½§R#F>F>dewQSDBYN¬Œ’F<HVUK–YGDxPQDLF>[cD`n)É7beN ]>dgDLNGF<]OhfSLK>l HzYGD]u]<HVU*KuEcH`]<]>de[GbgD]}EcHVNGFuT/l HVdg]>deF
bqRfP`F/R#deU D`abqRkEGF<H*TSYNGF<[email protected][email protected]"bqR'THVbebqR#[cHVF/R#K>dgHVU2D]>K–YGD\]<SLbgDTLK>dgHVUGU DLFAbqRkP`F/R#deU [email protected]¼R#UKxbgDBEGbeN ]
Œ’HVF>Ku]<deP`U)R#b‘K<HVK/R#brœ’]/HVhhfS"]>NGF}bgD]OK>F<HVdg]u]<K/R#K>dgHVU ]Νtn¡*db½¾SLwQSLU DLhfDLUKx]/R#K>dg]JŒR#deKOR#NsˆTHVU YdeK>dgHVU ]AY€
N (v„ a
SLK>dgjN DLK<H
S (‘JH (xa]<DLNGbgD]ObgD]–]<K/R#K>dgHVU B
] (JH (ô]/HVUK–NGK>debedg]<SD]uEcHVNGF–bgD"T<l HVd“s!YGD"bqRfP`F/R#deU D`nÕxR#U ]ObgD"TRV]AHVN
bg”
D (v„kD]>K}SLK>dgj*N DLK<Sx„*‡ aGbgD]}]>K/R#K>dgHVU ]uYGDxbqR\P`F/R#deU D]uYGHVdewQDLU*K–R#EGE)R#F>K<DLUGdeF–»NGU DxTHVU­ P`NGF/R#K>dgHVU%„*‡ n


GuK I>JُÿNOJI * +I WÆ+KÿbI‘ÿ%~KIM' IVÿ
É HVNGF7T/l)RVjN Dv]>K/R#K>dgHVUabgDoYGSTR#bqR#PQDrK<DLhEcHVF<DLb DLU*K>F<DobgDrYGSTLbgDLU T<l DLhfDLU*K4YGDobqR}]>K/R#K>dgHVUDLK;b½§R#F>F>dewQSD
δt
YN¬ŒF<HVU*K–YGD"PQDLF>[D"D]>KOTR#bgTLNGbgS`a)bgDxŒ’F<HVUKOYGD"PQDLF>[D"EGbqR#U%SLK/R#U*K–YGSLK<DLF>hdeU SBE)R#FubqRfP`F/R#deU D`n)C‘Rk]>K/R#K>dgHVU
D]>K;THVU ]/DLF>wQSD]>dG]<HVU
D]<K;THVhEGF>dg]DLU*K>F<D
DLK7D]>K7SLK>dgjN DLK<SDR#hB[GdeP`Ð D}]>d R#EGE)R#F>K>dgDLUK
δt
[−1µs ; +2µs]
δt
»
n|7UpYGDLl HVF<]vYGDOTD]4Œ’DLU yLK>F<D]abgD]v]>K/R#K>dgHVU ]r]<HVUKvTHVU ]>dgYGSLF<SD]THVhhfDuŒ’HVF>K>NGdeK<D]vDLKrU D
[−2µs ; +4µs]
]<HVUK}E)RV]}NGK>debedg]<SD]}Y R#U ]}[email protected]]<NGdeK<D-YGDAb½§R#U)R#be¼z]/D`n
Ä)Ä
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
GqK  I> Ù4ÿ%NOJI>JNK Q)ÿ%
™ U D]>K/R#K>dgHVU2D]>K-THVU ]>dgYGSLF<SDTHVhhfDfSLK/R#U*K-dg]<HVbgSD]>d[email protected]½§R¬R#N TLNGU DfwQHVdg]>deU Df»ŠhfHVdeU ]-YGD ¦`_`_
O

hfiLK>F<D]\R#NGK<HVNGF\Y˜½¾DLbebgDkHVNÆ]>d4DLbebgDfU½§RˆjN½ NGU D']<DLNGbgDfwQHVdg]>deU D'Y R#U ]-NGUÅF/R¼QHVUZYGDeð Ç hˆnCD]\]>K/R#K>dgHVU ]
dg]<HVbgSD]}]<HVU*KuSLbedehdeU SD]uYGD-b½§R#U)R#be¼z]<D-R#NmhfyLhfDxK>deK>F<D-j*N DAbgD]}]>K/R#K>dgHVU ]ŒŽHVF>K>NGdeK<D]n
Q$„™6I> ÿSK·‚QQ%ÿÿ
ì
C ½¾SLwQSLU DLhfDLU*K4D]>K£YGDoU HVNGwQDR#N]<HVNGhdg]R#Ns\TLF>deK<iLF<D]7YNu(v„ nQ‡rDLK>K<DrwQSLF>d“­)TR#K>dgHVUfD]>K;U STD]<]/R#deF<DR­ U

Y˜½¾SLbedehdeU DLFYGD]£THVU­ P`NGF/R#K>dgHVU ]YGDrK†¼zEDr„*‡ HVÈ@b½ NGU DrYGD]£j*N)R#K>F<Dv]>K/R#K>dgHVU ]7]<DLF/R#deK;Œ’HVF>K>NGdeK<DvDLK7SLbedehdeU SD

E)R#FbqRuEGF<HzTSYNGF<DrYGDr]<SLbgDTLK>dgHVUn`¡*d*b½¾SLwQSLU DLhfDLU*K7E)RV]<]<DobgD+(v„–NGU Dr]<DTHVU YGDoŒ’HVdg]adebzD]<KTHVU ]<DLF>wQSvTHVhhfD
SLK/R#UK}NGU%SLwQSLU DLhfDLUKuEGl*¼z]<dgjN D`n
ñA,¯®
ò 579‚@}1B )? ·c6B9?:\@u1
³\5ôó@õ¶1{}¶71\5¸¹571¢?
–HVN ]YGSTLF>dewQHVU ]bqRvEGF<HzTSYNGF<D]>K/R#U Y R#F<Y"YGD7F<DTHVU ]>K>F>N TLK>dgHVUBYGD]SLwQSLU DLhfDLUK<];YNxF<S]<DR#N"YGD]<NGFJŒŽRVTD
jNGdxD]>K¬NGK>debedg]/SDªE)R#F¬NGU D¤P`F/R#U YGDªE)R#F>K>dgD2YGD¤bqRZTHVbebqR#[cHVF/R#K>dgHVU MONGPQDLFˆDLKˆjN D¤U HVN ]%RwQHVU ]mT/l HVdg]>d
EHVNGF–K<HVNGK<D]–U H`]xR#U)R#be¼z]<D]nc‡rDLK>K<D\hfSLK>l HzYGDBEcDLF>hfDLKAYGDBF<DTHVU ]<K>F>NGdeF<[email protected]\NGU DBP`F/R#U YGDB­R#[GdebedeK<S\bgD]
cEcHVNGFYGD];SLU DLF>P`dgD]
HV[ ]<DLF>w`R#[GbgD]‘EGF>dehkR#deF<D]£YGDobqRPQDLF>[cDOœ’YdeF<DTLK>dgHVUaV‘T “ANGFaDLK£]>deP`U)R#b» _`_`_uhҜ
S1000

]>NGEcSLF>dgDLNGF<D]‘» |D†€\nV‡rDLK>K<D7EGF<H*TSYNGF<D4U½¾D]>K˜E)RV]‘YGSt­ UGdeK>dewQDoDLKc­ PQSD`aDLbebgDD]<K‘R#N"THVUK>F/R#deF<Do]>N ]<TDLEGK>de[GbgD

Y˜½¾yLK>F<D\R#hfSLbedgHVF<[email protected]#N%ŒNGF–DLK–»hfD]<NGF<DBYGD"b½¾SLwQHVbeNGK>dgHVU¤YGD"U HVK>F<DBTHVhEGF<SLl DLU ]<dgHVU¤YN¤YGSLK<DTLK<DLNGFADLK–YGD
bqR\EGl¼]>dgj*N D-YGD]PQDLF>[cD]uR#K>hfH`]>EGl SLF>dgj*N D]n
CD]}Yd“—˜SLF<DLUK<D]–SLK/R#EcD]}[email protected]<DTHVU ]>K>F>N TLK>dgHVU!]<HVUK}bgD]u]>NGdewVR#U*K<D]"”
[email protected] tI> <ŽÿSK I |_` „R=í C R\EcH`]>deK>dgHVUmYNˆT‘“ANGFuYGD-bqR\PQDLF>[D`a‘œ a a taGD]>K}TR#bgTLNGbgSDBTHVhhfD-SLK/R#UKu]<deK>N SD"R#N¬[)R#F>¼zTDLU*K>F<D
‘
xc yc zc
YGD]K>F<HVdg]f]>K/R#K>dgHVU ]'YGD¬bqR!P`F/R#deU D`aoRwQDT¬NGUËEcHVdgYG]EGF<HVEcHVF>K>dgHVUGU DLb–»!bqR¤F/RVTLdeU DˆTR#F>F<SDmYN ]>deP`U)R#buYGD
T<l)RVj*N DO]<K/R#K>dgHVUnG‡rDLK>K<D–EGF<DLhdgiLF<DAD]>K>dehkR#K>dgHVUpU DOEDLF>hfDLKrE)RV]rNGU DuP`F/R#U YGD–F<S]<HVbeNGK>dgHVU¬]>NGFrbqR-EcH`]>deK>dgHVU
YNˆ‘T “ANGFa‘œ’DLUK>F<D _`_DLKuÑ`_`[email protected]>F<D]ΝvhkR#dg]}]<DLF>KuYGDAEcHVdeUKuYGD"YGSLE)R#F>KuYGDxbqR\EGF<HzTSYNGF<D`n

[email protected] tI> <ŽÿSK It ÿ%g X‚IL $ Q ÿ
C RmEGF<DLhdgiLF<DpSLK/R#ED'[email protected]<DLF>hdeU LD FbqR%YdeF<DTLK>dgHVUZY˜½§R#F>F>dewQSDpYGD'bqRˆPQDLF>[cD`a;YGSt­ UGdgD'E)R#F\b½§R#UGP`bgD
‘
ÔSLUGdeK>l)R#b DLKAE)R#FOb½§R#UGP`bgDR#ÔdehBNGK>l)R#b aT HVU ]>dg]<K<[email protected]»kK>F<HVNGwQDLFAbqRŠYdeF<DTLK>dgHVU¤YGD\EGF<HVE)R#PR#K>dgHVU2YN%ŒF<HVU*K
θ
ϕ
cÀwv ï ÃQ>f‘V&Ã`fÛ;%õÞeÁQÁfi‰f
Ä)ÄÄ
GY DPQDLF>[D–R#deU ]>dj*N D}]<HVU'K<DLhE ]oY˜½§R#F>F>dewQSD–R#N']<HVbn*ÕxR#U ]4TDLK>K<D}EGF<DLhdgiLF<D–SLK/R#EcD`aQbgDŒ’F<HVUKrD]>KoRV]<]<dehdebgSO»
NGUŠEGbqR#Uˆ]/DAEGF<HVE)R#PQDR#UK}bgDAbgHVUGPfYGD–b½§RsGDxYGDxYGSLwQDLbgHVEGEDLhfDLU*KOYGD–bqR\PQDLF>[D-»BbqR\wzdeK<D]<]<DAYGD–bqR\beNGhdgiLF<D
nC‘D]\R#UGP`bgD] DLK aR#deU ]>dojN DbgDkK<DLhE ]"Y˜½§R#F>F>dewQSD
YN‰‘T “[email protected]#NÅ]<HVb4YGSt­ UGdg]/]/R#UKBbgDfEGbqR#U‰]<HVU*K
c
θ
ϕ
T0
YGSLK<DLF>hdeU S]A»kb½§R#dgYGD\YGD]uK<DLhE ]AYGD"YGSTLbgDLU T/l DLhfDLUK DLKAYGD]uEcH`]>deK>dgHVU ]@œ a a }YGD]–]>K/R#K>dgHVU ]aDLU
ti
xi yi zi
hdeUGdehdg]/R#UKObgD 2 ]>NGdew`R#UK"”
χ
χ2 =
2
√
N
ti − T0 + (u xi + v yi + 1 − u2 − v 2 zi )/c
X
i=1
σi2
œ`ðn 

R wQDT
a
a
DLK 2 D]>KubqRfwVR#F>dqR#U [email protected]]<]<HzTLdgSDB»fb½ deU TDLF>K>deK>N YGD
u = sin θ cos φ v = sin θ sin φ z = cos θ
σi
]>NGF}bgD-K<DLhE ]–YGD"YGSTLbgDLU T/l DLhfDLUK YGD-bqR]<K/R#K>dgHVUnÉ;HVNGF–TDLK>K<D"EGF<DLhdgiLF<DBD]<K>dehkR#K>dgHVUa 2 D]>K­GsGSDB»
ti
σi
z
a
V
H
È
U
v
]
D
>
]
o
K
q
b
"
R
Y
G
N
<
F
S
–
D
Y
Š
N
)
E
V
R
r
]
˜
Y
¾
½
S
/
T
)
l
#
R
U
>
K
e
d
e
b
g
b
V
H
G
U
)
U
#
R
Q
P
D
G
Y
D
v
]
>
]
e
d
`
P
)
U
#
R
N
˜
s
G
n
r
‡
L
D
>
K
<
K
u
D
`
w
#
R
g
b
L
D
G
N
FrTHVF>F<D]<EHVU Y
2
∆t
∆t = 25
12
»\bqR\wVR#F>dqR#U TDBY˜½ NGU DxwVR#F>dqR#[GbgDBR#bgSR#K<HVdeF<DxK>deF<SD-]<DLbgHVUˆNGU D-bgHVd˜NGUGd“Œ’HVF>hfD-]<NGF
n
[0 , ∆t]
C ½¾SLwQSLU DLhfDLUKfTHVU*K<DLU)R#UK'R#NÆhfHVdeU ]\K>F<HVdg]@]>K/R#K>dgHVU ]a;TDLKRIJN ]>K<DLhfDLU*K»mK>F<HVdg]\E)R#F/R#hfiLK>F<D]@bede[GF<D]

D]>KuK<HVNVI>HVNGF<]–EH`]<]<de[GbgD ø n˜‡rDBEGF<DLhdgDLFxRIJN ]>K<DLhfDLU*K–EcDLF>hfDLKAY˜½¾HV[GK<DLUGdeF–NGU D\F<S]<HVbeNGK>dgHVU¤Y˜½¾DLUwzdeF<HVU
3.5◦
]>NGF »![)RV]/]<D¬SLU DLF>P`dgD`aDLKfYdehdeUN DmjN)R#U YZb½¾SLU DLF>P`dgDmR#NGP`hfDLU*K<DˆEcHVNGFkR#K>K<DLdeU YF<D¬bgD] ◦ EcHVNGF _`_
θ
1

|D€@n4C½¾D]<K>dehkR#K>dgHVUYGD
R#deU ]<duHV[GK<DLUN D%D]>KkDLUK/RVT/l SD¤Y˜½ NGU DˆDLF>F<DLNGF']>¼]>K<SLhkR#K>dgj*N DˆYGDˆb½¾HVF<YF<D%YN
θ
YGDLP`F<S`adeU*K>F<H*YNGdeK<DOE)R#Fob½§R#EGEGF<HszdehkR#K>dgHVUpEGbqR#U D–jNGdD]>KŒR#deK<Du]<NGF4bqRxPQSHVhfSLK>F>dgDAYNkŒF<HVU*KrYGDuPQDLF>[D`n*C‘R
YGSLK<DLF>hdeU)R#K>dgHVUmYGD D]>KvY˜½§R#NGK/R#UK}EGbeN ]vEGF<STLdg]/D–j*N D D]>KoP`F/R#U Y˜abqRBF<S]<HVbeNGK>dgHVUˆD]<KvYGD–b½¾HVF<YF<DAYGD–„ ◦
ϕ
θ
»\[)RV]<]<DxSLU DLF>P`dgD-DLK}hfDLdebebgDLNGF<DBjN D-Ñ ◦ R#NˆYGD]/]>N ]}YGz
D ð`_|D†€\aRwQDT-NGU D–ŒR#de[GbgD-DLF>F<DLNGFu]<¼z]>K<SLhkR#K>dgj*N D`n
M bqRŠ]>NGdeK<[email protected]"RIJN ]>K<DLhfDLUKa˜[email protected]]>deP`U)R#b4» _`_`_ŠhfiLK>F<D]kœ
ƒ
uD]<KAD]<K>[email protected])R#FxdeUK<DLF>EcHVbqR#K>dgHVU
S1000

œbgHVPR#F>deK>lGhdgjN D;YGD]£]>deP`U)R#[email protected]];]<K/R#K>dgHVU ]£R#NGK<HVNGFYGD _`_`_OhˆnQ‡rDLK>K<Dow`R#bgDLNGF7]<DLF>K£YGD4EcHVdeUK7YGDvYGSLE)R#F>K

EHVNGFb½§RIJN ]>K<DLhfDLUK–YGDAbqR\ŒŽHVU TLK>dgHVUˆbqR#K<SLF/R#bgD-YGD-Ydg]>K>F>de[GNGK>dgHVUn
\dî+OVÿÿN4ÿSK IɟOþ¦ï
M ­ UXYGDpYGSLK<DLF>hdeU DLFfbgD
}
a£j*NGdvD]<K\NGU DpHV[ ]<DLF>w`R#[GbgD¬j*NGdrEcDLF>hfDLKY˜½¾D]>K>dehfDLFb½¾SLU DLF>P`dgDˆYGDŠbqR
S1000
PQDLF>[D`abqRmŒ’HVU TLK>dgHVUXYGDpYdg]>K>F>de[GNGK>dgHVUXbqR#K<SLF/R#bgDªœ’CÕA{"D]<[email protected] ]>K<SDp]>NGF\bgD]\]<deP`U)R#NsÆYGD]@]>K/R#K>dgHVU ]DLU
Œ’HVU TLK>dgHVU!YGD-bqRfYdg]<K/R#U TDB»b½§RsGD`n)C‘D]uE)R#F/R#hfiLK>F<D] a a
DLK
SLK/R#UKO­GsS]ODLKOTHVUGUN ]AY˜½§R#EGF<i]ObgD]
θ ϕ T0
zc
SLK/R#ED]uEGF<STSYGDLUK<D]abgD"]<deP`U)R#b;» _`_`_khˆa
aR#deU ]>d;jN DxbgD]–THzHVF<YGHVUGU SD]–YN!T‘“ANGFAR#N!]<HVba
DLK

S1000
xc
ùJñ`ìtéâçJßJä›ãvä§ßJë˜áŽâìtê§ëë’á†ã<áê§ìtà#ëàß ûìtêk‰L߯àáñ#ãÎë'þ¯áŽâß7ã/ä§ê¤úÎà9J߯ëJð`çJߏèé#êQßJë’á‘é#àߣçJìÎà_ûê¾áê§ìtàOâß>èé#ê§ëŽß£ñVã<â‘ä›ß}m†¾
Ä)Ä
yc
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
v
aG]<HVUKuYGSLK<DLF>hdeU S]OE)R#[email protected]]/R#K>dgHVU!YN
χ2
]>NGdew`R#UKB”
2
N th
X
Si (S1000 , xc , yc ) − Si
χ =
Sith (S1000 , xc , yc )
i=1
2
œ`ðn§ÑQ
VH È th
D]>KrbgDO]>deP`U)R#bK>l SHVF>dgj*N D–YGHVUGU S–E)R#FrbqR"C‘Õ–{ÅDLK
bgD–]>deP`U)R#bDLUGF<DLP`dg]<K>F<S–Y R#U ]rbqR
Si (S1000 , xc , yc )
Si
]>K/R#K>dgHVU ]>deK>N SDB»\NGU D-Ydg]>K/R#U TD
YGDxb½§RsGDxYGHVUGU SDxE)R#F"”
i
ri
2
p
ri2 = (xi − xc )2 + (yi − yc )2 + (zi − zc )2 − u(xi − xc ) + v(yi − yc ) + 1 − u2 − v 2 (zi − zc )
œ`ðn ~ 
‡ HVhhfD-U HVN ]}b½§RwQHVU ]OhfDLUK>dgHVUGU S\Y R#U ]}bgDxT<l)R#EGdeK>F<[email protected]ÑaGEGbeN ]>dgDLNGF<]OT<l HVd“sˆYGD-CÕA{X]<HVU*K}EcH`]<]>de[GbgD]a
r
bqRTHVbebqR#[cHVF/R#K>dgHVU%MONGPQDLFOR¼R#U*KOYGSLwQDLbgHVEGEcS"]<D]EGF<HVEGF<D]uŒŽHVU TLK>dgHVU ]n––HVN ]NGK>debedg]<DLF<HVU ]ObqR\Œ’HVF>hBNGbgD-YdeK<D
÷tE)R#F/R#[HVbgD-bgHVPVbgHVP)÷Va)j*NGd]½¾STLF>deK"”
S th (r) = S1000
r β+γ ln( 1000r m )
1000 m
œ`ðn „*
VH È-bgD]E)R#F/R#hfiLK>F<D] DLK YGSLEcDLU YGDLUKYGDbqRvU)R#K>NGF<D`aYGDb½§R#UGP`bgDÔSLUGdeK>l)R#b*DLKYGD7b½¾SLU DLF>P`dgD4YGD7bqR}E)R#F>K>dgTLNGbgD
β γ
EGF>dehkR#deF<D`n C‘[email protected] Y R#U TDBR#UGP`NGbqR#deF<D-YGDxTD]uTH*Dt¥kTLdgDLU*K<]uD]>KE)R#F/R#hfSLK>F>dg]<SDB»\b½§R#dgYGD-Y˜½ NGU¬EHVbe¼zU ÚVhfD
DLU
a*HV[GK<DLUN¬DLUp]>deh"NGbqR#K>dgHVU¬EcHVNGF4bgD]r[)RV]<]<D]rSLU DLF>P`dgD]nÉ;HVNGFrbgD]rSLwQSLU DLhfDLUK<]uTHVUK<DLU)R#UKK>F<HVdg]
1/ cos θ
]>K/R#K>dgHVU ]abgD]vw`R#bgDLNGF<]}YGD DLK ]/HVUK­GsGSD]}»@TD]w`R#bgDLNGF<]deUGdeK>dqR#bgD]a ]>dcb½¾SLwQSLU DLhfDLU*KOTHVUK>dgDLU*KOj*N)R#K>F<D
β
γ
]>K/R#K>dgHVU ]}HVN¬EGbeN ]a bgD]E)R#F/R#hfiLK>F<D] DLK ]<HVUKubqR#dg]<]<S]vbede[GF<D]uY R#U ]}b½§RI¯N ]>K<DLhfDLU*Kn)|7UmK<HVNGK<DxF>deP`N DLNGFa
β
γ
NGU Dul*¼*EcHVK>l i]<DA]>NGFrbqR-U)R#K>NGF<DAYGD–bqR"E)R#F>K>dgTLNGbgDAEGF>dehkR#deF<D–D]<KoU STD]<]ÎR#deF<DOEcHVNGFvYGSLK<DLF>hdeU DLFbgD]ow`R#bgDLNGF<]
YGD
DLK aTDLEcDLU Y R#[email protected]]\Yd“—cSLF<DLU TD]DLU*K>F<DŠEGF<HVK<HVUÆDLK\U H¼R#NsZYGDfŒŽDLFB]<HVU*K\YNÆhfyLhfDpHVF<YF<D'j*N D
β
γ
b½ deU TDLF>K>deK>N YGDuYN D–R#Ns\hfHzYGiLbgD]4l)RVYF<HVUGdgj*N D]4DLKr»AbqRx]>dehBNGbqR#K>dgHVUŠYGD}YGSLK<DTLK<DLNGFn‡rHVU TLF<iLK<DLhfDLUKazU HVN ]
NGK>debedg]<HVU ]bgD]E)R#F/R#hfiLK>F<D]–THVF>F<D]>EcHVU Y R#UKAR#NsŠEGF<HVK<HVU ]uEHVNGF}bqR\F<DTHVU ]<K>F>N TLK>dgHVUn
É HVNGFObgD]–SLwQSLU DLhfDLUK<]AEcH`]<]<SY R#U*K–]<DLNGbgDLhfDLU*KOK>F<HVdg]A]>K/R#K>dgHVU ]a NGU¤[email protected]¯N ]<K<DLhfDLUK–EGbqR#U!YN
;
ŒF<HVU*K}YGD–PQDLF>[DxD]>KDt—˜DTLK>N S-RwQDTAbgD]vU HVNGwQDLbebgD]w`R#bgDLNGF<]YGDAbqRBEH`]<deK>dgHVUˆYNpT‘“ANGFaEGNGdg]vNGUˆYGDLNsdgiLhfD
DLK}YGDLF>UGdgDLFuRI¯N ]<K<[email protected]‘Õ–{ZD]>KDt—˜DTLK>N S"R­ UˆY˜½¾HV[GK<DLUGdeF}NGU DxhfDLdebebgDLNGF<D"D]>K>dehkR#K>dgHVUmYN
n
S1000
É HVNGFbgD]SLwQSLU DLhfDLUK<]oTHVhEH`]/S]YGDvEGbeN ]YGDvK>F<HVdg]]>K/R#K>dgHVU ]aVdeb D]>K£EH`]<]<de[GbgDY˜½¾D]>K>dehfDLFbqRATHVNGF>[GNGF<D
;
YNpŒF<HVU*KuYGDxPQDLF>[cD`aGDLKOR#deU ]>dHV[GK<DLUGdeFONGU DxF<DTHVU ]>K>F>N TLK>dgHVUmEGbeN ]}EGF<STLdg]<D`n
cÀwv ï ÃQ>f‘V&Ã`fÛ;%õÞeÁQÁfi‰f
Ä)Ä¿
\dî+OVÿÿN+PTU <€ÿ Mÿ%KÆÿS KÆÿ
C R!hfHzYGSLbedg]/R#K>dgHVU YNXŒ’F<HVUKfYGD¬PQDLF>[cDpE)R#FNGUËEGbqR#UËD]>KNGU Dˆ[HVUGU DˆR#EGEGF<HszdehkR#K>dgHVUa4hkR#dg]kDLbebgD
‘
THVU YNGdeKŠ»!NGU DˆDLF>F<DLNGFŠ]>¼z]<K<SLhkR#K>dgjN Dˆ]<NGFbqRªYdeF<DTLK>dgHVUnoC‘D]E)R#F>K>dgTLNGbgD]ŠR¼R#UKŠNGUËP`F/R#U YXhfHVhfDLU*K
K>F/R#U ]>wQDLF<]<DA]<D–EGF<HVE)R#PQDLUKuDLU¬]½¾SLbgHVdeP`U)R#U*KuYGD–b½§RsDAYGD–bqRBPQDLF>[D`aGDLK]<HVU*KYGHVU TxDLUŠF<DLK/R#F<YpE)R#FvF/R#EGEcHVF>K
R#NfŒ’F<HVUKrEGbqR#UŠj*NGd]<DuEGF<HVE)R#PQD–bgD}[email protected]}b½§RsGD`n DÞbD]>KrYGHVU TuEcH`]<]>de[GbgDOYGD}hfHzYGSLbedg]<DLFvY R#U ]4NGUŠEGF<DLhdgDLF
K<DLhE ]bgD–ŒF<HVU*K}YGDAPQDLF>[DxTHVhhfD"]>EGl SLF>dgjN DxYGDAF/R¼QHVU
THVU ]>K/R#UKnc¡[email protected]<DLF>hdeU)R#K>dgHVU%]½¾Dt—˜DTLK>N DxDLU
R
hdeUGdehdg]/R#UKObgD 2 ”
χ
χ2 =
√
N
ti − T0 + (u xi + v yi + 1 − u2 − v 2 zi −
X
σi2
i=1
HVÈ
ri
D]>[email protected]]>K/R#U TD"»@b½§[email protected]]>K/R#K>dgHVU
i
2
ri2
)/c
2R
œ`ðn:ðQ
aGYGHVUKub½¾DtsEGF<D]<]>dgHVU%D]>K}YGHVUGU SDxE)R#F}b½¾Sj*N)R#K>dgHVUŜ`ðn ~ tn
,6-0/.1ðn
S]/HVbeNGK>dgHVUmR#UGP`NGbqR#deF<D-YGDAbqRBF<DTHVU ]>K>F>N TLK>dgHVU!¡zÕ DLUpŒ’HVU TLK>dgHVUˆYGD–b½§R#UGP`bgDxÔSLUGdeK>l)R#b EcHVNGF
 K<D2e
Yd“—cSLF<DLU*B
]xa h"NGbeK>deEGbedgTLdeK<S]\YGD]>K/R#K>dgHVU ]-Y R#U ]AbgD]-SLwQSLU DLhfDLU*K<]nCD]"TDLF<TLbgD]xU HVdeF<]-THVF>F<D]>EcHVU YGDLUθKBR#Ns
SLwQSLU DLhfDLUK<]A» ~ ]>K/R#K>dgHVU ]"œ
„f|D†€xta)bgD]uTR#F>F<S]uF<HVNGPQD]}F<DLEGF<S]<DLU*K<DLUK–bgD]}SLwQSLU DLhfDLUK<]x»@j*N)R#K>F<D
ð
]>K/R#K>dgHVU ]\œ „p|D†€
| E _k
|D†€xuDLK–bgD]uK>F>dqR#UGP`bgD]A[GbgDLN ]ObgD]ASLwQSLU DLhfDLUK<]xTHVUK<DLU)R#U*K“ðk]<K/R#K>dgHVU ]
HVN¬EGbeN ]Bœ£ò
| ≈ñ [email protected]|4Dð€-tn)Õ@ð±
½§R#EG F<D]-:‚ ðJð#†n

M TDLK>K<DxSLK/R#EcDxYGDObqRBEGF<HzTSYNGF<D`abqR\PQSHVhfSLK>F>dgDxYGD–bqR\PQDLF>[DAD]>K]>N¥k]/R#hhfDLU*K}[GdgDLU¬THVUGUN DxEcHVNGF
¨
NGK>debedg]<DLFfNGUËhfHzYGiLbgDˆYGDp• N TLK>N)R#K>dgHVU ]Š]>NGFfbgDpK<DLhE ]'YGDˆYGSTLbgDLU T<l DLhfDLU*KpYGSLK<DLF>hdeU S%DLU SLK>N YdqR#UKŠbqR
Ä)Äzú
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
<F SLEHVU ]/D¬YGD]kYGHVNG[GbgDLK<]kYGD¬]<K/R#K>dgHVU ]m‚:ðJð#†n7C‘RªwVR#F>dqR#U TD 2 NGK>debedg]/SDˆY R#U ]fb½§RIJN ]>K<DLhfDLUKŠD]>KfYGHVUGU SD
σi
E)R#F-”
σi2
= (0.4 + 1.2 cos θ) ×
"
1.2 T50
SiVEM
2
+ 600ns
2
#
œ`ðn 
Ø
HVÈ
D]<K4b½ deUK<DLF>w`R#bebgDAYGDuK<DLhE ]rU STD]<]ÎR#deF<DuEcHVNGFoj*N DubgDO]>deP`U)R#bdeUK<SLP`F<SxR#K>K<DLdeP`U D Y YGD–]/Rxw`R#bgDLNGFa
T50
50
DLK VEM D]>K;bqRuwVR#bgDLNGF£[email protected]€–|Ï©YN\]>deP`U)R#bhfD]>NGF<SrY R#U ]£bqRO]>K/R#K>dgHVU nV‡rDLK>K<DrE)R#F/R#hfSLK>F>dg]/R#K>dgHVUEcDLF>hfDLK
Si
i
GY D-F<DLU YF<D\THVhEGK<DBYGDBb½§R#NGP`hfDLUK/R#K>dgHVU2YGD"b½¾SLE)R#dg]<]<DLNGFAYNmŒ’F<HVUKAYGD-PQDLF>[cDBbgHVF<]<jN D"b½¾HVU!]½¾SLbgHVdeP`U DBYGD
b½§RsDxYGD-bqR\PQDLF>[D`n
‡ DLKkRIJN ]>K<DLhfDLUK'R#hfSLbedgHVF<DˆbqR!F<S]<HVbeNGK>dgHVUR#UGP`NGbqR#deF<D`a4THVhhfDˆbgDphfHVUK>F<D¬bqR%­ P`NGF<DNðn a7j*NGd}D]>K
r

THVhEGF>dg]<DˆDLU*K>F<D n DLK'_n§¦ ◦ EcHVNGFfbgD]'SLwQSLU DLhfDLUK<]pR¼R#UKŠjN)R#K>F<D%]>K/R#K>dgHVU ]kRwQDT
a7TDmjNGd
 
θ < 60◦
THVF>F<D]>EcHVU Y¬P`bgHV[)R#bgDLhfDLUKA»YGD]}SLU DLF>P`dgD]uYGDAPQDLF>[cD-THVhEGF>dg]<D]ODLUK>F<D ~ DLK _|D€@n‡rHVhhfD-HVUmEDLNGK

]½ ¼'R#K>K<DLU YF<D`aGbgHVF<]/jN DAbgDAU HVhB[GF<D-YGDA]>K/R#K>dgHVU ]uR#NGP`hfDLUK<D`a)bqRBF<DTHVU ]>K>F>N TLK>dgHVU!R#UGP`NGbqR#deF<D-D]<KvY˜½§R#NGK/R#U*K
EGbeN ]EGF<STLdg]<D`n
C R–YGDLF>UGdgiLF<DSLK/R#EcDoYGDrbqROF<DTHVU ]>K>F>N TLK>dgHVUPQSHVhfSLK>F>dgj*N DD]>Kb½§RIJN ]J
‘
K<DLhfDLUKv]>NGFrbgD]oYGHVUGU SD]vY˜½ NGUfŒ’F<HVUKvYGDOPQDLF>[DOYGHVUKvbgD}F/R¼QHVUpYGDOTHVNGF>[GNGF<DAD]>K4w`R#F>dqR#[GbgD`n*MON'THVNGF<]vYGD
bqRBEGF<HVE)R#PR#K>dgHVUˆYGDObqRBPQDLF>[cD`aTDLbebgDt†TLd]½¾SLK<DLU YpbqR#K<SLF/R#bgDLhfDLUKa DLKYGHVU TA]>NGdew`R#UK}]<HVU¬]>K/RVYGDAYGDAYGSLwQDLbgHVE
EDLhfDLU*KabgDOŒF<HVU*K}YGDAbqR"PQDLF>[cD–U DAEGF<S]<DLUK<DAE)RV]vbgDAhfyLhfDAF/R¼QHVUˆYGDATHVNGF>[GNGF<D`n C‘RB­ P`NGF<z
D ðn§ÑazdebebeN ]>K>F<D
bgD]U HVK/R#K>dgHVU ]}NGK>debedg]/SD]}EcHVNGFb½¾DtszEGF<D]<]<dgHVUmYN¬F/R¼QHVU!YGDxTHVNGF>[GNGF<D"YN¬Œ’F<HVUK"”
ï ^Ü>f€Û;@ÃQ&#;&"GG#G£õò
→
−
−
R(xi , yi , zi ) = R0 − →
ri · d u,v,w = R0 − (xi − xc )u − (yi − yc )v − (zi − zc )
RwQDT
zc
CD
­GsS-»b½§R#beK>deK>N YGD"YGHVUGU SDxE)R#F}b½¾D]<K>dehkR#K>dgHVUmYNˆ[)R#F>¼TDLUK>F<D`n
χ2
p
`œ ðn¤÷`
1 − u2 − v 2
»\hdeUGdehdg]<DLFO]½¾STLF>deKuR#bgHVF<]B”
χ2 =
√
N
ti − T0 + (u xi + v yi + 1 − u2 − v 2 zi −
X
σi2
i=1
2
ri2
)/c
2R(xi ,yi ,zi ,u,v)
œ`ðn§¦Q
‡ DfU HVNGwQDLbRIJN ]>K<DLhfDLUK\U½§R#hfSLbedgHVF<DpjN DkK>F<i]"EcDLN2bqRmF<S]<HVbeNGK>dgHVUZR#UGP`NGbqR#deF<D`ahkR#dg]Bdeb4EcDLF>hfDLK\YGD
r
F<SYNGdeF<Dxb½¾DLF>F<DLNGFu]>¼]>K<SLhkR#K>dgj*N D-]>NGF »j*N DLbgjN D]
◦ ]<DLNGbgDLhfDLU*Kn
θ
0.3
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä)ÄG
,6-0/.1ðn§Ñ ¡zT/l SLhkR\debebeN ]>K>F/R#UKvb½¾SLwQHVbeNGK>dgHVUˆYNŠF/R¼QHVUˆYGD–THVNGF>[GNGF<D–YNŠŒ’F<HVUKvYGD–PQDLF>[DAR#NŠTHVNGF<]YGDObqR
EGF<HVE)R#PR#K>2dgHVUn)Õ@½§R#EGF<D]"‚ ~ †n
ó
‘ÿ%KBI–î_O`ÿ%‹ÿ%<!ÿSKIɟuþ¦ï
C½§RIJN ]>K<DLhfDLUK¬YGDˆbqRÅCÕA{ƒD]>KŠDt—cDTLK>N S%NGU D!YGDLNszdgiLhfDmŒ’HVdg]pRwQDT%bqR‰hfyLhfDˆŒŽHVU TLK>dgHVUarR­ U«YGD
EGF<DLU YF<DˆDLUËTHVhEGK<D¬bqR!U HVNGwQDLbebgDˆD]>K>dehkR#K>dgHVUYGDpbqR¤PQSHVhfSLK>F>dgDˆTR#bgTLNGbgSDˆY R#U ]b½¾SLK/R#EcD¬EGF<STSYGDLU*K<D`n
‡rDLK>K<DoYGDLF>UGdgiLF<DSLK/R#EcD4EcDLF>hfDLK7Y˜½¾HV[GK<DLUGdeF£NGU DohfDLdebebgDLNGF<DrF<S]<HVbeNGK>dgHVU]>NGF;bqRuEH`]>deK>[email protected]‘“ANGF}œ
h'
≈ 40
RwQDTxNGU DAŒR#de[GbgDxDLF>F<DLNGFO]<¼z]>K<SLhkR#K>dgj*N D`n
ñA,†+
0,B ? :/¸¹ºz?:\@}1
³B5ôó@õ†¶71B5£·Lö}:<5
‡ HVUK>F/R#deF<DLhfDLU*KŠR#NXYGSLK<DTLK<DLNGFfYGD'• N HVF<D]<TDLU TD¬jNGdrhfD]>NGF<DŠb½¾SLU DLF>P`dgD¬YGDŠbqR%PQDLF>[DpYGDŠhkR#UGdgiLF<D
r
TR#bgHVF>dehfSLK>F>dgjN D`aVb½¾D]>K>dehkR#K>dgHVUfYGDrb½¾SLU DLF>P`dgD}RwQDTobgDrF<S]<DR#NYGDv]>NGFJŒRVTDo]½¾Dt—cDTLK>N DvYGDrhkR#UGdgiLF<DvdeU YdeF<DTLK<D
E)R#F7b½ deUK<DLF>hfSYdqR#deF<D–YN
Jn DÞbŒŽR#NGKYGHVU TK>F<HVNGwQDLF4bqR–F<DLbqR#K>dgHVU'THVF>F<DTLK<D}DLU*K>F<Db½¾SLU DLF>P`dgDuYGDvbqRAPQDLF>[cD`a
S1000
bgD–]<deP`U)R#b˜» _`_`_BhfiLK>F<D]}DLKvb½§R#UGP`bgDAÔSLUGdeK>l)R#b az]>NGFvK<HVNGK<DObqR\PR#hhfDAY˜½¾SLU DLF>P`dgDxhfD]<NGF<SDOE)R#Fb½ š ÉM\n
θ

ՖD4bqRw`R#bedgYdeK<SoYGDoTDLK>K<D4F<DLbqR#K>dgHVU
aYGSLEcDLU YBYdeF<DTLK<DLhfDLU*K£TDLbebgDrYNB]<EDTLK>F<DrY˜½¾SLU DLF>P`dgDoj*NGd*D]<K
E(S1000 , θ)
THVU ]>K>F>NGdeK»uE)R#F>K>deFYGD]£YGHVUGU SD]7YNk¡zÕ@nVMuN†YGD]<]<N ]7YGD4b½¾SLU DLF>P`dgD}YGDobqR–]/R#K>NGF/R#K>dgHVUfYGDrb½§RVTTDLEGK/R#U TD`a`bqR
YGSLK<DLF>hdeU)R#K>dgHVU'THVF>F<DTLK<DuYGDvb½¾SLU DLF>P`dgDOD]>K7bqRx]<HVNGF<TD}bqR–EGbeN ]7dehEHVF>K/R#U*K<DOY˜½ deU TDLF>K>deK>N YGDO]>NGF7bgD}]>EcDTLK>F<D`n
É7beN ]>dgDLNGF<]‘hfSLK>l H*YGD];HVUK;SLK<S7hdg]<D]R#N"EHVdeU*K;R#N"]<DLdeUBYGD7bqRTHVbebqR#[cHVF/R#K>dgHVU\MONGPQDLFaR­ UBYGDF<S]<HVN YF<D
bgD}EGF<HV[GbgiLhfD–Y˜½¾D]>K>dehkR#K>dgHVU¬YGD}b½¾SLU DLF>P`dgDxRwQDTObgD–¡zÕ@nzC‘RxEGF<DLhdgiLF<D–hfSLK>l HzYGDOj*NGd)wzdgDLUKrYdeF<DTLK<DLhfDLUKu»
Ä)ÄW
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
b ½¾D]>EGF>deKuD]>KOY˜½ NGK>debedg]<DLFObgD]O]>deh"NGbqR#K>dgHVU ]OEcHVNGFOYGSLK<DLF>hdeU DLF–[email protected]<DLbqR#K>dgHVU
n‡rDLEcDLU Y R#UKaTDLK>K<D
E(S1000 , θ)
K<DT<lGUGdgj*N [email protected]>F<HzYNGdeK-NGU [email protected]¼zEcHVK>l i]<D]>NGFAbqRpTHVhEH`]<deK>dgHVU2YN!F/R¼QHVUGU [email protected]`]>hdgj*N D`a˜DLK–EcHVNGF>F/R#deK
SLPR#bgDLhfDLUK–]<HVN—F>deFOY˜½¾DLF>F<DLNGF<]O]>¼]>K<SLhkR#K>dgj*N D]}YGSLEcDLU Y R#UKuK>F<HVEmŒ’HVF>K<DLhfDLU*KOYGDxb½§R#UGP`bgD-ÔSLUGdeK>l)R#ba hfDLKJ
K/R#UKuDLU%YGHVNGK<DxbqR\wVR#bedgYdeK<S"YGD-TDLK>K<DxEGF<HzTSYNGF<D`n
| U!F<SRVTLK>dgHVU2»kTD]–THVU ]>K/R#K/R#K>dgHVU ]abqRkhfSLK>l H*[email protected]<D\YNZ÷‡rHVU ]>K/R#U*K”D†U*K<DLU ]>deK†¼ª‡oNGK÷VaHVNò×GØ0×£aR
7
SLK<SvEGF<HVEH`]/SD-”VbqR–EGF<HVEGF>dgSLK<S}Y˜½ dg]<HVK>F<HVEGdgDOYGD]YdeF<DTLK>dgHVU ]4Y˜½§R#F>F>dewQSD]oYGD]7F/R¼QHVU ]oTH`]>hdgjN D]4D]>K7NGK>debedg]<SD
EHVNGFxYGSLK<DLF>hdeU DLF-NGU [email protected]<DLbqR#K>dgHVU

jNGd7]<HVdeKAEGF/R#K>dgj*N DLhfDLUK-deU YGSLEcDLU Y R#UK<D'YGD]xÏ%HVU*K<Dt¯‡vR#F>bgHGn
S1000 E
OHVN ]4wQDLF>F<HVU ]vjN DOTDLK>K<DuhfSLK>l HzYGDuEcH`]<]<iYGDuSLPR#bgDLhfDLUKYGD]4deU THVU*wQSLUGdgDLUK<]aGDLKoU HVN ]rEGF<S]<DLUK<DLF<HVU ]rbgD]
R#beK<DLF>U)R#K>dewQD]uj*NGd‘HVUKOSLK<SxDtszEGbgHVF<SD]–R#NmTHVNGF<]}YNˆK>F/RwVR#deb‘YGD-K>l i]<D`n
£üòû£üÞý
§
þåQVÿ%$ I> <ÿSK°‚Q%ÿ% MÿrI‘ÿ%Kÿô¹ <`ÿÉ©ZI>K
C ½¾SLU DLF>P`dgDoYGD£bqRvPQDLF>[D£EcDLNGK‘yLK>F<DD]<K>dehfSD4»rE)R#F>K>deF‘YGD7bqRrw`R#bgDLNGF‘YN

DLKYGDb½§R#UGP`bgD7ÔSLUGdeK>l)R#b a»
S1000
θ
b½§R#dgYGDYGD]‘]<deh"NGbqR#K>dgHVU ]n#CD7EGF>deU TLdeEcD4D]>K‘K>F<i]‘]>dehEGbgD}”R#EGF<i];RwQHVdeF]>dehBNGbgSNGU-bgHVK;YGD7PQDLF>[D]Y˜½¾SLU DLF>P`dgD]a
YGD DLKuYGDAEGF>dehkR#deF<D]\œòŒŽDLF}HVNˆEGF<HVK<HVUˆKÞ¼*EGdgj*N DLhfDLUKΝ}Yd“—cSLF<DLU*K<]a)HVUˆTR#bgTLNGbgD-bgD
hfH¼QDLU!»\b½§R#dgYGD
θ
S1000
YNfhfyLhfDEGF<HVP`F/R#hhfD–YGDvF<DTHVU ]>K>F>N TLK>dgHVUpjNGd)D]<K4R#EGEGbedgjN S–R#[email protected] SD]n*C‘RAE)R#F/R#hfSLK>F>dg]/R#K>dgHVUpYGDbqR
F<DLbqR#K>dgHVU
]½¾Dt—˜DTLK>N D"YGDxhkR#UGdgiLF<DBDLhEGdeF>dgjN D`aDLUmYGSLK<DLF>hdeU)R#UK–NGU DxŒŽHVU TLK>dgHVUmj*NGd‘]½§RIJN ]>K<D
E(S1000 , θ)
bgDAhdgDLNs¬EH`]<]<de[GbgD-R#NsŠF<S]>NGbeK/R#K<]OYGDAbqR]>dehBNGbqR#K>dgHVUn
ù ½§R#d£YGHVU TB]<deh"NGbgSBNGUmbgHVKAYGD-PQDLF>[cD]–RwQDT‡rHVF<]<d Ç RkDLKO|4RV]>¼ ¡*dehˆaEHVNGFAYGD]uU H¼R#Ns%YGD"Œ’DLF–DLKOYGD]
EGF<HVK<HVU ]uYGHVU*K}bgD]}SLU DLF>P`dgD]u]½¾ST<l DLbgHVUGU DLUKADLUK>F<D 17.6 DLK 20.4 D€@n
10
10
|7UBK>F/RVWR#UK7DLU\ST/l DLbebgD4bgHVPR#F>deK>lGhdgj*N DrbqRvŒŽHVU TLK>dgHVU
» ­GsGS`aHVU\THVU ]>K/R#K<Doj*N Db½¾SLwQHVbeNGK>dgHVU
S1000 (E) θ
YGD
RwQDT
D]>K¬EGF<HzT<l D2Y˜½ NGU D2bgHVd-YGDªEGNGdg]<]/R#U TD`auRwQDTªNGU DªbgSLPQiLF<D2SLwQHVbeNGK>dgHVU YGDªbqRÆEDLU*K<D
S1000
E
bgHVPR#F>deK>lGhdgjN D`n*CD]7F<S]>NGbeK/R#K<]4YGD]4]>dehBNGbqR#K>dgHVU ]EcDLNGwQDLUK4yLK>F<DuRIJN ]>K<S]7E)R#F7NGU DŒ’HVU TLK>dgHVUkYGDvbqROŒŽHVF>hfD-”
lnS1000 = α(θ) + β(θ) ln E + γ(θ)(lnE)2
SjNGdew`R#bgDLUK<DB»@NGU DAF<DLbqR#K>dgHVU%DLU*K>F<D
E
E = exp
DLK
S1000
−β +
œ`ðn§^Q
YGDxbqRBŒ’HVF>hfDk”
!
p
β 2 − 4γ(α − lnS1000 )
2γ
œ`ðn Q_ 

EHVNGFobqRVjN DLbebgDObgD]rTH*Dt¥kTLdgDLU*K<] a LD K ]<HVU*KobqR#dg]/]<S]4bede[GF<D]a*DLKrYGSLEDLU YGDLU*KvYGDub½§R#UGP`bgD–ÔSLUGdeK>l)R#bDLKvYN
α β γ
K†¼zEDxYGD-EGF>dehkR#deF<D`n
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
C‘RYGSLEcDLU Y R#U TD-DLU
θ
Ä)Ä%[
YGD]}THzDt¥kTLdgDLUK<]–HV[GK<DLUN DBD]>[email protected]]>NGdew`R#UK<Dk”EcHVNGFbgD]EGF<HVK<HVU ]B”
α(θ) = 4.69 − 17.6 cos(θ) + 26.7 cos2 (θ) − 12.16 cos3 (θ)
β(θ) = −2.14 − 10.1 cos(θ) + 10.9 cos2 (θ) + 3.9 cos3 (θ)
γ(θ) = 0.29 − 0.75 cos(θ) + 0.56 cos2 (θ) − 0.1 cos3 (θ)
œ`ðn

`
œ`ðn ÑQ

œ`ðn ~ 

DLKEHVNGF}bgD]}U H¼R#NsˆYGDAŒŽDLF-”
α(θ) = 6.01 − 22.1 cos(θ) + 32.1 cos2 (θ) − 14.1 cos3 (θ)
β(θ) = −0.02. + 2.23 cos(θ) − 1.2 cos2 (θ)
γ(θ) = 0.11 − 0.24 cos(θ) + 0.11 cos2 (θ)
œ`ðn *„ 

œ`ðn ðQ

œ`ðn 
Ø
‡ DLK>K<D!F<DLbqR#K>dgHVU‹EcDLF>hfDLKpYGD%F<DTHVU ]<K>F>NGdeF<D%b½¾SLU DLF>P`dgDªYGD]'PQDLF>[cD]Š]<deh"NGbgSD]ˆRwQDT!NGU D¤Ydg]>EcDLF<]>dgHVU
r
hfDLdebebgDLNGF<DxjN D Ñ Y R#N†YGD]/]>N ]vYGD 19 D†€ôRwQDTANGU DADLF>F<DLNGF]>¼]>K<SLhkR#K>dgj*N D–YGD”ð Y aDLUK>F<D-Ñ`_\DLK ~ _ Y
10

YGDYdg]>EDLF<]<dgHVU2DLUK>F<D 18 DLK 19 D€@aRwQDTNGU Df]<¼z]>K<SLhkR#K>dgj*N [email protected]½¾HVF<YF<DfYGDf¦ Y n'ArdgDLU2DLU*K<DLU YNa
10
10
TD]uw`R#bgDLNGF<]uU½ deU Ydgj*N DLUKxE)RV]Ob½¾DLF>F<DLNGFxjNGd;D]>KOTHVhhdg]/D\RwQDT\TDLK>K<DBhfSLK>l HzYGDB]>NGF–bqRF<DTHVU ]<K>F>N TLK>dgHVU
YGD]w*F/R#dg]uSLwQSLU DLhfDLUK<]n
C [email protected]­ P`NGF<Duðn ~ debebeN ]>K>F<D-bqR'THVhE)R#F/R#dg]<HVU¤DLUK>F<D\bgD
‘
hfH¼QDLU¤HV[GK<DLUN2»E)R#F>K>deFAYGD-b½§RIJN ]>K<DLhfDLU*K
S1000
YGDmbqRÅCÕA{ DLK¬»‰E)R#F>K>deFpYNž]>deP`U)R#bAYdeF<DTLK<DLhfDLUKmhfD]>NGF<S!» _`_`_ÅhfiLK>F<D]nvÉHVNGFˆHV[GK<DLUGdeFpbgD

S1000
YdeF<DTLK<DLhfDLUK\hfD]>NGF<S`aGI½§R#do]>dehBNGbgSfbgD]"hfyLhfD]"PQDLF>[cD]"Y R#U ]"NGU DkTHVU­ P`NGF/R#K>dgHVUZYN‰F<S]<DR#NÆYdeK<D%÷LDLU
SLK<HVdebgDQ÷'œ’TtŒ4T/l)R#EGdeK>F<DB„*taEcHVNGFObqRVj*N DLbebgD Ñk]>K/R#K>dgHVU ]A]<HVUK–EGF<S]<DLU*K<D]OE)R#F–deUK<DLF>w`R#[email protected]}ð`_kh Y R#U ]–bgD

F<StŒ’SLF<DLU*K>[email protected]'PQDLF>[D`n˜CD]<deP`U)R#b4» _`_`_Šh D]>K-R#deU ]>dTR#bgTLNGbgSDLUªhfH¼QDLUGU)R#U*K"bgD] ъ]>deP`U)R#NsªYGD]


]>K/R#K>dgHVU ]"»pTDLK>K<DfYdg]>K/R#U TD`n‘Õ–[email protected] ]a˜Ydg]>EH`]ÎR#UK"YGD _ŠPQDLF>[cD]-Yd“—˜SLF<DLUK<D]-E)R#FBSLU DLF>P`dgDfDLKxE)R#F abgD
θ

D]>K}DLUˆF<SR#bedeK<SxhfH¼QDLUGU S"]>NGF Ñ`_wVR#bgDLNGF<]n

S1000
É[email protected]]B[)RV]<]<D]\SLU DLF>P`dgD]a)I¯N ]<j*N½§»ˆj*N DLbgj*N D]\|4D€@a‘bgDŠ]>deP`U)R#brhfH¼QDLUXHV[GK<DLUNZE)R#F"b½§RIJN ]>K<DLhfDLU*K
YGDfbqRˆC‘Õ–{ D]>KB]<¼z]>K<SLhkR#K>dgj*N [email protected]]>NGEcSLF>[email protected]»mTDLbeNGdrHV[GK<DLU*NËRwQDT'bgDfF<S]<DR#NZDLUÅSLK<HVdebgD`n7‡rDLK\Dt—˜DLK
EGF<HwzdgDLUKYNkŒR#deKvjN DubqR-F<DTHVU ]>K>F>N TLK>dgHVU¬dehEcH`]<D–»-b½¾SLwQSLU DLhfDLU*KbgDOTLF>deK<iLF<"
D (v„ aR#bgHVF<]rj*N DuY R#U ]rbgD–TRV]
YNF<S]<DR#NkDLUkSLK<HVdebgD`aR#N TLNGU'TLF>deK<iLF<DU½¾D]>KoR#EGEGbedgj*N S`a`bgD}]>deP`U)R#bD]>K]>dehEGbgDLhfDLUKohfD]>NGF<Su» _`_`_AhˆnCD]

SLwQSLU DLhfDLUK<]»x[)RV]<]/DuSLU DLF>P`dgD–HVUKvYGHVU TODLU'hfH¼QDLUGU D–NGU
EGbeN ]oSLbgDLwQSOjN DubgDu]>deP`U)R#bEGF<SYdeK4E)R#FrbgD
S1000
F<S]<DR#NŠDLU¬SLK<HVdebgD`aTR#Fv]<DLNGbg]rbgD]r]<deP`U)R#NsŠ]>NGEcSLF>dgDLNGF<]R#Np]/DLNGdebYGD–YGSTLbgDLU T<l DLhfDLU*KO]<HVU*KvTHVhEGK/R#[Gdebedg]/S]n
C‘RˆEcDLUK<DŠbgHVPR#F>deK>lGhdgj*N D¬D]>[email protected]#bgDLhfDLUKbgSLPQSLF<DLhfDLUKfYd“—˜SLF<DLUK<D¬DLU*K>F<DŠbgD
RIJN ]>K<SŠDLKBbgD
S1000
S1000
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
S(1000) in VEM
Ä)Äc
S(1000) from fit LDF θ = 56°
S(1000) from star array θ = 56°
102
10
1
1
10
102
Energy in EeV
,.-F/.1ðn ~ |7wQHVbeNGK>dgHVUpYNk]>deP`U)R#b» _`_`_AhfiLK>F<D]oDLUfŒ’HVU TLK>dgHVUkYGD}b½¾SLU DLF>P`dgD`aHV[GK<DLUN¬RwQDT}b½§RI¯N ]>K<DLhfDLU*K
GY D\bqRkCÕA2 { œK>F>dqR#UGP`bgD]x[GbgDLN ]ΝOHVN2R [email protected]\F<S]<DR#NªDLUªSLK<HVdebgDˆœ’Ydg]<j*N D]OF<HVNGPQD]ttnC‘D]AF<S]>NGbeK/R#K<]xTHVF>F<D]J
EHVU YGDLU*KB»pYGD]-PQDLF>[cD][email protected]<HVK<HVU ]B]>deh"NGbgS]"E)R#F\‡rHVF<]>d Ç RpDLK-|4RV]>¼ ¡*dehˆa˜EHVNGF"NGU‰R#UGP`bgDÔSLUGdeK>l)R#boYGD
ð ◦n
Ø
EGF<HwQDLU)R#U*KOYNˆF<S]<DR#N%DLUˆSLK<HVdebgD`n
 j*NGdz]/DLF/ROR#EGEGbedgj*N SDR#NsBYGHVUGU SD]adebzTHVU*w*dgDLU*KY˜½ NGK>debedg]<DLF
Mu­ U\YGDrYGSLK<DLF>hdeU DLF£bgD4F<DLbqR#K>dgHVU
S1000 E
bqROF<DLbqR#K>dgHVUdg]<]>N DvYGDvbqROhfyLhfDT<l)R#deU D}Y˜½§R#U)R#be¼]<D`a`THVhEGF<DLU)R#UK4bgDvhfyLhfDvhfH*YGiLbgDvYGDCÕA{va`T`½¾D]>KJÞ»†YdeF<D
Y R#U ]U HVK>F<D-TRV]bqRE)R#F/R#[HVbgDxbgHVPVbgHVP n
C ½¾D]>K>dehkR#K>dgHVU‰YGD\b½¾SLU DLF>P`dgDf»

E)R#F>K>deF\YN
hfD]>NGF<S'E)R#F\bgD'F<S]<DR#NZYGD']<NGFJŒŽRVTD'E)R#FBb½ deU*K<DLF>hfSYdqR#deF<DˆYGD]@]>deh"NGbqR#K>dgHVU ]@EGF<S]/DLUK<D
S1000
b½ deU THVUwQSLUGdgDLU*KrY˜½¾yLK>F<D;IJN ]>K<DLhfDLUKoYGSLEDLU Y R#U*K<DuYGDTD]7]>dehBNGbqR#K>dgHVU ]nCD]F<DLbqR#K>dgHVU ]
 HV[GK<DLU*N D]
S1000 E
RwQDTrY˜½§R#NGK>F<D];EGF<HVP`F/R#hhfD]7YGDo]>dehBNGbqR#K>dgHVU ]7YNBYGSLK<DTLK<DLNGF7DLK£YGD]‘PQDLF>[cD]7R#K>hfH`]>EGl SLF>dgjN D]£THVU YNGdg]<DLU*K
»vYGD]‘D]>K>dehkR#K>dgHVU ]YGD7b½¾SLU DLF>P`dgDU HVK/R#[GbgDLhfDLU*K;Yd“—˜SLF<DLUK<D]nCD]˜deU TDLF>K>deK>N YGD];]>NGFbgDhfH*YGiLbgD7l)RVYF<HVUGdgj*N D
NGK>debedg]<SvY R#U ];bqRO]<deh"NGbqR#K>dgHVUfYGD]£PQDLF>[cD]deU*K>F<H*YNGdg]/DLUKNGU DvDLF>F<DLNGF7]>¼]>K<SLhkR#K>dgjN Dv]>NGF;bqROwVR#bgDLNGF7YN
S1000
»xNGU DOSLU DLF>P`dgDOYGHVUGU SD`azEGF>deU TLdeE)R#bgDLhfDLUKvEcHVNGFobgD]4P`F/R#U YG]R#UGP`bgD]4ÔSLUGdeK>l)R#NsªœŽR#N†YGDLbq»"YGDu$„ 𐐠ð`_ ◦ 7E)R#F
bgDA[GdqR#dg]uY˜½ NGU DxEGF<HVEcHVF>K>dgHVUmYGDxhBN HVU ]}EGF<SYdeK<D-Yd“—˜SLF<DLUK<DBY˜½ NGUˆhfHzYGiLbgD"»\NGU%R#NGK>F<D`n
õ
f‘ÃQ>fÁgffxÛ;‰õSi2Á[email protected]Û;2GÁ‰&‘mõ*¬ó >f#ö"GVõò
Õ ½§R#NGK>F<DpE)R#F>Ka£b½¾D]>K>dehkR#K>dgHVUËYGDŠb½¾SLU DLF>P`dgDˆD]>KTHVU YdeK>dgHVUGU SD¬E)R#[email protected]¬T<l HVd“sXYGDŠb½ l¼zEHVK>l i]/DŠŒŽR#deK<D
@
]>[email protected]`]<deK>dgHVUˆYN¬F/R¼QHVUGU DLhfDLUKATH`]>hdgj*N D`n ”¨K>deK>F<D-Y˜½¾DtsGDLhEGbgD`a)RwQDTAbgD]}]>deh"NGbqR#K>dgHVU ]uNGK>debedg]<SD]
Y R#U ]fTDLK>K<D¬SLK>N YGD`a7NGU DpPQDLF>[cDˆYGDpEGF<HVK<HVU Y˜½¾SLU DLF>P`dgD¬deUŒŽSLF>dgDLNGF<Dm» _¤|D†€¢D]<KF<DTHVU ]>K>F>NGdeK<D%»%NGU D

cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä)Äv
SLU DLF>P`dgD]>¼z]<K<SLhkR#K>dgjN DLhfDLU*KxEGbeN ]xSLbgDLwQSDYGD ð Y DLUªhfH¼QDLUGU D]>d£b½¾HVUªNGK>debedg]<D\bqR'F<DLbqR#K>dgHVU

S1000 E

T HVF>F<D]>EHVU Y R#U*K\R#N¤ŒŽDLFn˜MuN†YGD]/]>N ]-YGD _p|D†€\[email protected]• Ns¤[email protected]/R¼QHVU ]BTH`]<hdgjN D]-YGHVdeKBTDLEDLU Y R#U*K"yLK>F<D

YGHVhdeU S4E)R#FbgD]‘EGF<HVK<HVU ]aDLK;TDLbqR}hfyLhfDo]<dQb½¾HVU\]<NGEGEH`]<DNGU DoTHVhEcH`]>deK>dgHVU\hd“szK<D–œEGF<HVK<HVU ]?XU H¼R#Ns)
ìvêFÙâÒæ!ÓIÞ'×ßLaQbgD]7U H¼R#NsfSLK/R#UK4Ydg]<]/H*TLdgS]4R#NfTHVNGF<]4YGDbgDLNGF7EGF<HVE)R#PR#K>dgHVUnÉ;HVNGF4bgD]SLU DLF>P`dgD]oYGDvb½¾HVF<YF<D
YGD7b½¾|D€@aK†¼zEGdgjN DLhfDLU*K£Y R#U ]bqRvÔHVU DrYGD7w`R#F>dqR#K>dgHVU\YGD4b½§RVTTDLEGK/R#U TDoj*NGd*THVF>F<D]>EcHVU Y-P`bgHV[)R#bgDLhfDLU*K»bqR
EH`]>deK>dgHVUfYGDrbqR÷LT/l DLw*debebgDQ÷Y R#U ]bgD]>EDTLK>F<D`a`HVUf]½§R#K>K<DLU Yf»AHV[ ]<DLF>wQDLF7NGU DSLwQHVbeNGK>dgHVUkYGDvbqRATHVhEcH`]>deK>dgHVU
YGD] ‡r]n C½ l*¼*EcHVK>l i]<D"YGD"THVhEH`]>deK>dgHVU%U STD]<]/R#deF<D"EHVNGF}b½¾D]<K>dehkR#K>dgHVU%YGDxb½¾SLU DLF>P`dgD\RwQDT-bgD]OÏ%HVU*K<D
a
‡vR#F>bgHfYGHVdeKuYGHVU T-yLK>F<DAEGF>dg]<D"DLUˆTHVhEGK<D"Y R#U ]b½§R#U)R#be¼]<D-YGD]}YGHVUGU SD]OYGD-TDLK>K<DxPR#hhfD-Y˜½¾SLU DLF>P`dgD`n
,6-0/.1'ðn „
ŸOdg]>K<HVP`F/R#hhfDfYNªU HVhB[GF<DkY˜½¾SLwQSLU DLhfDLUK<]BYGHVUK-b½¾SLU DLF>P`dgDkD]>K-]>NGEcSLF>dgDLNGF<Df»jðp|D€@a˜E)R#F
dg]<]>N D–YGD]v]>deh"NGbqR#K>dgHVU ]a
2 n*C½¾SLU DLF>P`dgDxD]>KoD]<K>dehfSDA»"E)R#F>K>deFvYGDubqR"F<DLbqR#K>dgHVU
EHVNGFxYGD]AU H¼sin
R#Ns2θYGDBŒŽDLFkœ’TR#F>F<S]A[GbgDLN ]Ν–DLKxEHVNGFxYGD]AEGF<HVK<HVU ]kœ’Ydg]/E(S
jN D]x1000
U HV)deF<]ttn˜‡rDLK>K<DkYdg]>K>F>de[GNGK>dgHVU
D]>K}TR#bgTLNGbgSDB»@E)R#F>K>deF–YGD]}YGHVUGU SD]–YN!¡zÕôEGF>dg]<D]uDLU*K>F<D-bgDxEGF<DLhdgDLF4I>R#U*w*dgDLFOÑ`_`_#„'DLK}bgDxEGF<DLhdgDLFIJNGdeU
Ñ`_`_ n
Ø
deUK<DLF>w`R#bebgD–2 YGD
C Rˆ­ P`NGF<Dxðn „!F<DLEGF<S]<DLU*K<Dpb½ lGdg]>K<HVP`F/R#hhfD¬YNÅU HVhB[GF<D¬Y˜½¾SLwQSLU DLhfDLUK<]fYGHVU*[email protected]½¾SLU DLF>P`dgDŠD]<K\]>NGEcSt
‘
F>dgDLNGF<Dk»*ð¬|D†€\a‘E)R#F-deUK<DLF>w`R#bebgDkYGD
a˜TR#bgTLNGbgS'»pE)R#F>K>deF\YGD]"YGHVUGU SD]BYNÆ¡zÕ@nC½¾D]>K>dehkR#K>dgHVUZYGD
sin2 θ
b½¾SLU DLF>P`dgDD]>K-Dt—cDTLK>N SDf»Šb½§R#[email protected]'F<DLbqR#K>dgHVUœ`ðn _Qta‘Y R#U ]Ab½ l¼zEHVK>l i]<[email protected]<HVK<HVUÅHVN¤Œ’DLFncCD]-YGDLNs

Ydg]>K>F>de[GNGK>dgHVU ]}HV[GK<DLUN D]uU Dx]<HVUKE)RV]THVhE)R#K>de[GbgD]–RwQDTANGU D–ŒŽHVU TLK>dgHVUˆTHVU ]>K/R#UK<D`a)DLKEGF<S]<DLUK<DLU*K}NGU D
wVR#F>dqR#K>dgHVU2YGD\b½¾HVF<YF<DYGD  Y DLUK>F<[email protected]'wVR#bgDLNGF"R#Ns!P`F/R#U YG]xDLK-R#Ns%EcDLK>deK<]-R#UGP`bgD]AÔSLUGdeK>l)R#Nscn˜|7U
40 50
Ä=v)
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
]>NGEGEcH`]/R#UKpjN D!bgDm• NsYGD!F/R¼QHVU ]¬TH`]>hdgj*N D]pD]>KŠdg]<HVK>F<HVED Ë arbgD%U HVh"[GF<DªY˜½¾SLwQSLU DLhfDLU*K<]ˆYGSLK<DTLK<S]
Y R#U ]xb½§R#UGP`bgDk]/HVbedgYGD
R#NGK<HVNGFBYGD acE)R#F-NGUGdeK<SfYGDf]>NGFJŒRVTDEcDLF>EDLU YdgTLNGbqR#deF<DŠR#N‰YGSLK<DTLK<DLNGFB]½¾STLF>deK
dΩ
θ
a
V
H
È
D
>
]
;
K
G
N
U
r
D
T
V
H
U ]>K/R#UK<D`nV|[email protected]>dewVR#U*Kb½¾SLbgSLhfDLU*KY˜½§R#UGP`bgDo]/HVbedgYGD
dN
Φ0
dΩ = sin θdθ dϕ
(dS cos θ) dΩ ∝ Φ0
DLK4DLUk]<HVhhkR#U*Ko]>NGF aQHVU'HV[GK>dgDLUKrbqRAF<DLbqR#K>dgHVU
n š UŠ]½§R#K>K<DLU YfYGHVU T
dN
dN
ϕ
dS cos θ sin θdθ ∝ dS d(sin2 θ) ∝ Φ0
»"TD–j*N DubqRBYdg]>K>F>de[GNGK>dgHVUmR#UGP`NGbqR#deF<DAYGD]vSLwQSLU DLhfDLUK<]}DLU
]<HVdeKoEGbqR#K<D`aGR#N†YGD]<]<N ]rYGDOb½¾SLU DLF>P`dgDxYGD
sin2 θ
]/R#K>NGF/R#K>dgHVUmYGDxb½§RVTTDLEGK/R#U TD`n
‡rDLK>K<DEGF<HVEGF>dgSLK<SuU½¾D]>KF>dePQHVNGF<DLN ]<DLhfDLU*KowzF/R#dgD}j*N D]>dGb½¾D]>K>dehkR#K<DLNGFoYGD}b½¾SLU DLF>P`dgD`aDLUfb½¾HzTTLNGF<DLU TD}bgD
a*• N TLK>N D"R#NGK<HVNGF}YGDx]/RBw`R#bgDLNGFuR#K>K<DLU YN D-YGDAhkR#UGdgiLF<DAdgYGDLUK>dgj*N DxEcHVNGFK<HVN ]vbgD]uR#UGP`bgD]n)¡*NGdew`R#UK
S1000
b½§R#hEGbedeK>N YGDDLKAb½¾SLwQHVbeNGK>dgHVUÅRwQDT YGD]–• N TLK>N)R#K>dgHVU ]-YN
abqRŠYdg]>K>F>de[GNGK>dgHVU2DLU
EcDLNGKxyLK>F<D
θ
S1000
sin2 θ
]>deP`UGd“­)TR#K>dewQDLhfDLUKˆYd“—˜SLF<DLUK<D!Y˜½ NGU DmŒ’HVU TLK>dgHVU«THVU ]>K/R#UK<DZ‚ ð#†nv‡rHVhEGK<D!K<DLUN«YGD]f• N TLK>N)R#K>dgHVU ]¬YN
Ø
]>deP`U)R#b» _`_`_\h±j*NGd˜HVUK}SLK<S–hfD]>NGF<SD]]>NGFvbgD]YGHVUGU SD]}YNm¡zÕ@[email protected]]>K>F>de[GNGK>dgHVUˆDLU
R#K>K<DLU YN D

sin2 θ
EHVNGF>F/R#deK–RwQHVdeF}NGU DxdeU*K<DLU ]>deK<S-EGbeN ]}P`F/R#U YGD-Y˜½¾DLUwzdeF<HVU!Ñ`_ Y EcHVNGFbgD] EGF<HzT<l D]OYGD _ ◦ œ’TtŒv‚ tn
θ
Ø
Ø`Ø
‡rDLEcDLU Y R#UKacTHVhhfD"bgDBhfHVUK>F<D\bqRf­ P`NGF<H
D ðn „ abgD]–Ydg]>K>F>de[GNGK>dgHVU ]xHV[GK<DLUN D]xRwQDT\b½¾D]>K>dehkR#K>dgHVUªYGD
b½¾SLU DLF>P`dgD-dg]<]<N D-YGD]O]<deh"NGbqR#K>dgHVU ]OEGF<S]<DLUK<DLU*K–EGbeNGK<ÚVKONGU%STR#F>K–YGDx„_ ð`_ Y DLUK>F<DBb½ deUK<DLU ]>deK<[email protected]»@P`F/R#U Y
DLK–»fEDLK>deK n‡rD\THVhEHVF>K<DLhfDLU*K–EcDLNGKuEGF<HwQDLUGdeFxYGD"bqRkTHVh"[GdeU)R#dg]<HVUªYGDBEGbeN ]>dgDLNGF<]}ŒRVTLK<DLNGF<]aTHVhhfD
θ
U HVK/R#hhfDLUKxNGU [email protected] Y R#U TDfRwQDT\b½§R#UGP`[email protected]ÔSLUGdeK>l)R#b7YGDBbqR'F<S]<HVbeNGK>dgHVUªDLKxYGDBb½¾DLF>F<DLNGF-]>¼z]<K<SLhkR#K>dgjN D
]>NGFfb½¾SLU DLF>P`dgD`noC‘RªF<DTHVU ]>K>F>N TLK>dgHVU«YGD]fPQDLF>[cD]k]<deh"NGbgSD]'hfHVUK>F<Dmj*N D¬bqR2F<S]<HVbeNGK>dgHVUDLKfbgDˆ[GdqR#dg]'DLU
SLU DLF>P`[email protected] YGDLU*KAYGDBŒRVWHVU%dehEcHVF>K/R#UK<DYGDBb½§R#UGP`bgD\ÔSLUGdeK>l)R#ba˜R#deU ]>d7jN DBYGDBb½¾SLU DLF>P`[email protected]'PQDLF>[D`a
RwQDT-YGD]}wVR#F>dqR#K>dgHVU ]OYGDxEGbeN ]}YGz
D ð`_ Y n
É;HVNGFNGU D¬SLU DLF>P`dgD¬YGHVUGU SD`a7b½§R#K>K<SLUN)R#K>dgHVU‹YN
RwQDTpb½§R#UGP`bgD¬ÔSLUGdeK>l)R#bEGF<SYdeK<DpE)R#FbgD]]>d“
S1000
h"NGbqR#K>dgHVU ]@D]>KBEDLNGK\yLK>F<D'SLPR#bgDLhfDLU*K\K>F<HVE‰Œ’HVF>K<D`‚
n –HVN ]"SLK>N YdgDLF<HVU ]@[email protected]#EGF<i][email protected]>deKBbqR
hfSLK>l H*YGD"YN%‡rHVU ]>K/R#U3
K D†U*K<DLU ]>deK†¼m‡oNGKn
£üòû£ü0E
§
ŸIt
Q # ÿ ÷©Rø2©úùûïxþ
M ­ U'YGDu]½§R—F/R#U T/lGdeFrYGD]oEGF<HV[GbgiLhfD]rdeUGl SLF<DLUK<]}»xbqR-hfSLK>l H*YGD–Y˜½¾D]>K>dehkR#K>dgHVUpYGDub½¾SLU DLF>P`dgDARwQDTObgD]
u
]>deh"NGbqR#K>dgHVU ]acbqRkhfSLK>l [email protected]ŇrHVU ]>K/R#UKzDÞUK<DLU ]<deK†¼¤‡oNGKxNGK>debedg]/DBbqR'EGF<HVEGF>dgSLK<SY˜½ dg]<HVK>F<HVEGdgDYGD]AF/R¼QHVU ]
TH`]>hdgj*N D]EcHVNGF4D]>K>dehfDLFa*»xNGU D}SLU DLF>P`dgDuYGDF<StŒŽSLF<DLU TD`aQbqR"YGSLEDLU Y R#U TD–R#UGP`NGbqR#deF<DuYN
nQÕxR#U ]NGU
S1000
YGDLNszdgiLhfD}K<DLhE ]a`bgD]YGHVUGU SD]4l¼z[GF>dgYGD]a`hfD]>NGF<SD]4»xbqROŒ’HVdg]7E)R#F4bgDu¡zՋDLK4bgD{;Õ@a`]<HVU*KNGK>debedg]<SD]4EHVNGF
Ì à#ìtéM
ë ‰LßÞâŽâìtà#ëãÎé}çýVã/ñ#ê¾áŽâß‚üèé#ßï|þJïߘëŽê#ä‚ü ìÎàOëŽéñ#ñ`ìtëŽßèé#ßä›ã ûê§ë’áŽâêkƒ#éáê§ìÎàOã/àúÎé#ä›ãÎê¾âß^ûßJëMý‘ÝëßJë’ácïì¿ûé#ä:9JßñVã<â
éà|ûê§ñ2þtä§ß)ûßë‡4æçÿ~û ü ã/ïñ#ä§ê¾áé_ûßtðLä›ã ûê§ë’áŽâêkƒ#éáê›ìÎàûßJëãÎàOútä§ßJëU9¯à#ê¾á ýVã/éOàvì4ƒ#ëŽß¯â„‰¶9JëßJàáŽâßæ7߯áçæ ◦ ßJë’ááŽâ„¼Jëñ`ßJé}ãQ߯ç¯á 9Jßç
½ü ýîñ`ìÎá ýO¼JëŽ
ß û ü é¢
à é#éOà}ê§ëŽìÎáŽâìÎñ`ß4߯ë’á)ûìtàç;áìtéá)ÿ›eãÎê¾áâ†ãÎê§ëŽìÎà#àVãAƒ#ä§ßû#ã/à#ëä§ß7ç>ãÎë‘èé#ߣàìtéë‘ç¯ìtà#ëŽê¤û@9¯âìtàë
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä=v˜Ä
SLK/R#bgHVUGU DLFbqR"F<DLbqR#K>dgHVU
 DLK}R#deU ]>d˜HV[GK<DLUGdeFrNGU DxD]<K>dehkR#K>dgHVUpYGD–b½¾SLU DLF>P`dgDAEcDLN¬YGSLEDLU Y R#U*K<DxYGD]
S1000 E
]>deh"NGbqR#K>dgHVU ]n
CD\EGF>deU TLdeEDYGDBbqR'hfSLK>l H*YGDD]<[email protected]]>NGdew`R#UKf”c»'NGU D\SLU DLF>P`dgD\­GsGSD`acHVUª]>NGEGEH`]/D\j*N DBb½¾SLwQHVbeNGK>dgHVU
YN
DLU2ŒŽHVU TLK>dgHVUÆYGDkb½§R#UGP`bgDkÔSLUGdeK>l)R#bvD]>K"dgYGDLU*K>dgjN Dp»¬K<HVNGK<D]BbgD]\SLU DLF>P`dgD]nÉ;[email protected]<DLF>hdeU DLF
S1000
TDLK>K<DfSLwQHVbeNGK>dgHVUa‘bqR¬hfSLK>l H*YGD'6
‡ D>‡ôNGK>debedg]<[email protected]ŒŽR#deKBjN DbgD• NsªYGD]-F/R¼QHVU ]\TH`]>hdgjN D]BD]>K-dg]<HVK>F<HVEcD`a
DLK}YGHVU T-jN [email protected]]>K>F>de[GNGK>dgHVU%YGD]OSLwQSLU DLhfDLUK<]–DLU
D]>Ku]>NGEGEcH`]<SDxyLK>F<DxEGbqR#K<D`a EHVNGF}NGU D"SLU DLF>P`dgD
sin2 θ
]>NGESLF>dgDLNGF<DB»\b½¾SLU DLF>P`dgD"YGDx]/R#K>NGF/R#K>dgHVU!YGDAb½§RVTTDLEGK/R#U TD'œ
|4D€-tn
∼3−4
{ HVF>hfDLbebgDLhfDLUKacbqRkF<DLbqR#K>dgHVU¤DLUK>F<D\bgD
DLKub½¾SLU DLF>P`[email protected]]>K–YGHVU TB]<NGEGEH`]<SD\yLK>F<DxŒRVTLK<HVF>dg]/R#[[email protected]
S1000
NGU D–Œ’HVU TLK>dgHVUmYGD-b½§R#UGP`bgDxÔSLUGdeK>l)R#bDLKNGU DAŒŽHVU TLK>dgHVUmYGDxb½¾SLU DLF>P`dgDk”
S1000 (E, θ) = f (θ) × g(E).
œ`ðn `÷ 

C R\ŒŽHVU TLK>dgHVU
‘
a R#EGEDLbgSD"THVNGF>[DBY˜½§R#K>K<SLUN)R#K>dgHVUadeU YdgjN D"b½¾SLwQHVbeNGK>dgHVU%YGD-bqRwVR#bgDLNGFuYN
RwQDT
f (θ)
S1000
b½§R#UGP`bgDOÔSLUGdeK>l)R#baEcHVNGFvNGU DASLU DLF>P`dgD–­GsSD`nC½ l*¼*EcHVK>l i]<DxYGDOŒŽRVTLK<HVF>dg]ÎR#K>dgHVUp]<NGEGEH`]<DAj*N DOTDLK>K<DATHVNGF>[cD
F<D]>K<DdgYGDLUK>dgj*N DA»xK<HVNGK<D]obgD]4SLU DLF>P`dgD]nTOHVN ]4wQDLF>F<HVU ]4E)R#FobqR-]>NGdeK<D}j*N DLbYGDLP`F<SOYGDuTHVU­R#U TDARVTTHVF<YGDLF
»2TDLK>K<D%l¼zEHVK>l i]<D`n K/R#U*KpYGSt­ UGdgDª»2NGU D%THVU ]>K/R#U*K<D!EGF<i]abqR2Œ’HVU TLK>dgHVU
D]>K'E)R#FŠTHVU*wQDLUK>dgHVU
£
f (θ)
U HVF>hkR#bedg]<SD"»@]/R\w`R#bgDLNGFOR#NˆhdebedgDLNmYGDAb½ deU*K<DLF>wVR#bebgD
DLU
2 ajNGdTHVF>F<D]<EHVU Ym»
◦ ”
[0 , 0.75]
f (θ) =
S1000 (θ)
.
S1000 (38◦ )
sin θ
θ ' 38
œ`ðn Q¦ 

C‘R-Œ’HVU TLK>dgHVU
D]>KvT<l HVdg]<dgD–YGD–hkR#UGdgiLF<DADLhEGdeF>dgj*N D–THVhhfDxSLK/R#U*KYGDObqR-ŒŽHVF>hfD
a
g(E)
g(E) = K E α
HVÈ
DLK ]<HVU*KxYGD]–THVU ]<K/R#UK<D]a˜RwQDT wQHVdg]>deUªYGDBb½ NGUGdeK<S`ncMudeU ]>d;bqR'jN)R#U*K>deK<S
K
α
α
S1000 (E, θ)/f (θ) =
YGSLEDLU Y%NGUGdgj*N DLhfDLUK"YGDBb½¾SLU DLF>P`dgD`n˜|7U¤NGK>debedg]/R#UKxbqRkw`R#bgDLNGFAYGDBb½¾SLU DLF>P`[email protected]]>NGF<SD\E)R#FAbqR• N H#
g(E)
F<D]<TDLU TD%THVhhfD¤D]>K>dehkR#K<DLNGF¬YGD!bqR‰w*F/R#dgD!SLU DLF>P`dgD`arbqRªŒŽHVU TLK>dgHVU
EDLNGKpyLK>F<D¤YGSLK<DLF>hdeU SD¤E)R#F
g(E)
RI¯N ]>K<DLhfDLU*Ku]>NGFbgD]}YGHVUGU SD]ul¼z[GF>dgYGD]n
|7U2T<l HVdg]<dg]<]/R#UK"NGU [email protected]#bgDLNGF"[email protected]<StŒŽSLF<DLU TD
EcHVNGFxbgD
»
acHVU2YGSLK<DLF>hdeU DfEcHVNGF
S1000 (38◦ )
S1000 38◦
T<l)RVj*N DudeU*K<DLF>wVR#bebgDAYGD
j*N DLbebgDOYGHVdeKvyLK>F<DObqR-w`R#bgDLNGFrYN
EcHVNGFoj*N DubgDOU HVh"[GF<DAY˜½¾SLwQSLU DLhfDLUK<]
sin2 θ
S1000
R#N†YGD]<]>N ]}YGDxTDLK>K<Dxw`R#bgDLNGFu]<HVdeKbgD-hfyLhfD-jN DAb½ deU*K<DLF>wVR#bebgDBYGDxF<StŒŽSLF<DLU TD-TDLUK>F<S\R#NGK<HVNGFuYGD
◦n
θ ' 38
" õòÃ*&õ+Û*fÃQ&‘#qÛoeÁf˜&f‹&;qGfkõö
ö
õ ö
M ­ [email protected]<DLF>hdeU DLF7bgD4EGbeN ];EGF<STLdg]<SLhfDLU*K;EcH`]<]>de[GbgDobgD]£THVNGF>[D]£Y˜½§R#K>K<SLU*N)R#K>dgHVUk]>NGdew`R#UK;bgD4EGF>deU TLdeEcD
}
YGDubqRBhfSLK>l HzYGD"‡6DJ‡–aIL½§R#d˜YGSLwQDLbgHVEGEcS–NGU DAEGF<H*TSYNGF<DA]>dehEGbgDAj*NGdEcDLF>hfDLKY˜½¾HV[GK<DLUGdeFbqR"Œ’HVU TLK>dgHVU
f (θ)
Ä=v[v
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
»\E)R#F>K>deFuYGD]}YGHVUGU SD]aRwQDT-NGU D"D]>K>dehkR#K>dgHVUmYGD]}DLF>F<DLNGF<]O]>NGFbgD]hfD]>NGF<D]n
CDxEGF>deU [email protected]>l HzYGD-D]>KbgDx]<NGdewVR#U*K"”
"fV&‘Ã`>f•Û;-ÂÞfÃ`f-Ã&)i!&;õò®­ ”‹E)R#F>K>deFrYGD]oYGHVUGU SD]rYNfF<S]<DR#NŠYGDu]>NGFJŒRVTD`a`bgD]
Yd“—cSLF<DLU*K<]}R#UGP`bgD]ÔSLUGdeK>l)R#Ns
]/HVUK}]<SLE)R#F<S]DLU ðBdeUK<DLF>w`R#bebgD]YGD
DLUK>F<D DLK
n
θ ∈ [0◦ , 60◦ ]

sin2 θ
0 0.75
ÉHVNGFBT<l)RVj*N DfdeUK<DLF>w`R#bebgDkYGD
acHVU‰TR#bgTLNGbgDbqRŠŒŽHVU TLK>dgHVU‰TLNGh"NGbqR#K>dewQD
aj*NGd4THVF>F<D]>EcHVU Y‰R#N
sin2 θ
Σθ (S)
U HVh"[GF<DxY˜½¾SLwQSLU DLhfDLU*K<]–R¼R#UK}NGU
]>NGEcSLF>dgDLNGFu»BNGU DxTDLF>K/R#deU DAwVR#bgDLNGF nÉ£R#F}l¼zEHVK>l i]/D`adebŒR#NGK
S1000
S
]<DoEGbqRVTDLF4»uNGU DvSLU DLF>P`dgD]>N¥k]ÎR#hhfDLUK7P`F/R#U YGDvEcHVNGF£j*N DrbqROEGF<HV[)R#[GdebedeK<SYGDrYGSLK<DTLK>dgHVUfYGD]£PQDLF>[cD]7]<HVdeK
deU YGSLEDLU Y R#U*K<DYGD\b½§R#UGP`[email protected]ÔSLUGdeK>l)R#bœŽRVTTDLEGK/R#U TDk]/R#K>NGF<SDtnc|7U¤U D\THVU ]<dgYGSLF/R#UK-j*N DBbgD]ASLwQSLU DLhfDLU*K<]
R¼R#U*KNGU
]>NGEcSLF>dgDLNGF}»
a*HVU¬]½§RV]/]>NGF<DOj*N DOb½§RVTTDLEGK/R#U TD-D]>Kv]/R#K>NGF<SDx»"K<HVN ]
S1000
S1000 cut = 20 VEM
bgD]-R#UGP`bgD]n˜CDT/l HVd“sª[email protected]>K<Dw`R#bgDLNGF-D]>KAP`[email protected])R#FxbgD]-]>dehBNGbqR#K>dgHVU ]aj*NGd7THVhhfDU HVN ]xbgD\wQDLF>F<HVU ]
E)R#FbqR]>NGdeK<D`aEcDLF>hfDLK>K<DLUK–DLbebgD]OR#N ]<]>dYGDxTR#bgTLNGbgDLF–YGD]}THVNGF>[cD]uY˜½§R#K>K<SLUN)R#K>dgHVUn
ö
š UT/l HVdg]>deK'THVhhfDmF<StŒ’SLF<DLU TD%b½§R#UGP`bgD
aoTHVF>F<D]>EHVU Y R#U*KpR#NhdebedgDLNYGDmb½ deUK<DLF>w`R#bebgD%DLU
θ ' 38◦
acDLK-bgDf]>deP`U)R#b
nCRŠŒ’HVU TLK>dgHVU
]½¾HV[GK>dgDLU*KBYGHVU TfDLUÅYGSLK<DLF>[email protected])R#U*KBEcHVNGF
sin2 θ
S1000 cut = 20 VEM
f (θ)
T<l)RVj*N DxdeUK<DLF>w`R#bebgD"YGD
K<DLbebgD-jN D
2 bqR\wVR#bgDLNGFuYGD
cut n
sin θ
Sθ
Σθ (Sθ ) = Σ38◦ (S1000
)
M}­ U«YGD!YGSLK<DLF>hdeU DLF¬EGF<STLdg]<SLhfDLUKpbqR‰wVR#bgDLNGF
EHVNGF
OG#i2Á#>fÑÛ;X†>f‘ÃQò>f‘
­
Σ
Sθ
aQIL½§R#d7RI¯N ]<K<S-NGU DBŒ’HVU TLK>dgHVU¤YGDBbqRŒŽHVF>hfD
a EcH`]<]<SY R#U*KxYGDLNs
T<l)RVj*N D"deU*K<DLF>wVR#bebgDYGD
sin2 θ
σ(S) = AS B
E)R#F/R#hfiLK>F<D]ubede[GF<D]
DLK a ]>NGFubgD]}ŒŽHVU TLK>dgHVU ]OTLNGhBNGbqR#K>dewQD]
n)CD]–DLF>F<DLNGF<]O]<NGF}bgD]uE)R#F/R#hfiLK>F<D]
A
B
Σθ (S)
DLK
YGHVUGU SD]vE)R#Frb½§RI¯N ]>K<DLhfDLU*KEDLF>hfDLK>K>F<HVU*K}Y˜½¾SLwVR#beN DLFvb½¾DLF>F<DLNGF]>NGFrbqR\THVNGF>[D–Y˜½§R#K>K<SLUN)R#K>dgHVUn
δA
δB
C‘RO­ P`NGF<Dðn:ðxF<DLEGF<S]<DLUK<DbgD]7Œ’HVU TLK>dgHVU ]oTLNGh"NGbqR#K>dewQD]rHV[GK<DLUN D]o»–E)R#F>K>deFoYGD]YGHVUGU SD]EGF>dg]<D]4DLUK>F<D
[email protected]>K<DLhfDLU*K
bgD er ùR#UwzdgDLF"Ñ`_`_#„ŠDLKAbgD er ù`NGdeU2Ñ`_`_ aEHVNGFxYd“—˜SLF<DLUK<]AdeU*K<DLF>wVR#bebgD]xYGD
sin2 θ

Ø

DLUÅbgHVdvYGD'EGNGdg]<]/R#U TDpYGD]BŒ’HVU TLK>dgHVU ]
]/DkF<DLhkR#F<j*N DŠK>F<i]\TLbqR#deF<DLhfDLU*Kf]>NGF\bqR¬­ P`NGF<DpDLK\K>F/RVYNGdeK
Σθ (S)
]>dehEGbgDLhfDLUKObgDxŒR#deKuj*N DxbgD-]>EcDTLK>F<D-YGD]}F/R¼QHVU ]OTH`]<hdgjN D]OD]>K}EGF<HzT<l D"Y˜½ NGU D"bgHVd‘YGDxEGNGdg]<]ÎR#U TD"Y R#U ]
TDLK>K<DfPR#hhfDpY˜½¾SLU DLF>P`dgD`n;C½§RI¯N ]<K<DLhfDLUKYGD]-ŒŽHVU TLK>dgHVU ]@TLNGh"NGbqR#K>dewQD]@E)R#FBNGU DkbgHVdrYGDfEGNGdg]<]/R#U TDŠD]>K
DtsTDLbebgDLU*KaIJN ]<j*N½§»xNGU DuwVR#bgDLNGFvYGD
€A|ÏÅa*THVF>F<D]>EcHVU Y R#UK}»xNGU D–SLU DLF>P`dgDODLU*K>F<D
|4D€@az»
40 − 50
8 − 10
E)R#F>K>deFoYGDbqRVj*N DLbebgD}bgD}hkR#U jN DOYGDu]>K/R#K>dg]>K>dgj*N DU DuEDLF>hfDLKEGbeN ]rYGD}TR#bgTLNGbgDLFobgD]7ŒŽHVU TLK>dgHVU ]oTLNGhBNGbqR#K>dewQD]
RwQDTxEGF<STLdg]<dgHVUn
‡ HVU ]>dgYGSLF<HVU ]BNGUZR#UGP`bgDkÔSLUGdeK>l)R#b YGHVUGU S`aDLK\]>NGEGEH`]/HVU ]"j*N DfbgDfU HVhB[GF<D'Y˜½¾SLwQSLU DLhfDLU*K<]fR¼R#U*K
r
θ
b½¾SLU DLF>P`dgD a »
EGF<i]D]>K}YGDAbqR"ŒŽHVF>hfD” dN
DLKb½¾SLU DLF>P`dgD
−γ DLKjN DAbqR\F<DLbqR#K>dgHVUˆDLU*K>F<DxbgD S
E dE
1000
dE ∝ E
D]>KAYGDBbqRkŒŽHVF>hfDp”
1/α acTD\j*NGd7D]>K–b½ l¼zEHVK>l i]<DYGD\[)RV]<[email protected]\bqR'hfSLK>l H*YGDf‡6D>‡–ncMubgHVF<]AbgD
E ∝ (S1000 )
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä=v)¿
,6-0/.1ð:n ð
{)HVU TLK>dgHVU ]'TLNGhBNGbqR#K>dewQD]
EcHVNGFfYd“—cSLF<DLU*K<]fdeU*K<DLF>wVR#bebgD]'YGD
a£YGHVUGU)R#UK'bgD
sin2 θ
U HVh"[GF<D!Y˜2 ½¾SLwQSLU DLhfDLUK<]¬R#N†YGD]<]>N ]'YGD Σθ (S1000
nC‘) D]fŒŽHVU TLK>dgHVU ]k]<HVU*KkU HVF>hkR#bedg]<SD]¬»¤
bqRªwVR#bgDLNGFŠYGDˆF<St
Œ’SLF<DLU TDª”
n7C‘D¬THVhEHVF>K<SDL1000
hfDLU*K'DLUXbgHVd}YGDpEGNGdg]<]/R#U TD¬YGD]\ŒŽHVU TLK>dgHVU ]fTLNGhBNGbqR#K>dewQDmD]>K
Σ38◦ (20 VEM)
TLbqR#deF<DLhfDLUKu
wzdg]>de[GbgD`n
U HVh"[GF<D"Y˜½¾SLwQSLU DLhfDLU*K<]–R¼R#UK
S1000
»
dS1000
EGF<i]OD]>K}YGDAbqR\ŒŽHVF>hfD”
dN
dN dE
∝
∝ (S1000 )
dS1000
dE dS1000
CDAU HVh"[GF<DBY˜½¾SLwQSLU DLhfDLUK<]AYGHVUK}bgD"]>deP`U)R#b‘»
N (≥ S) ∝
Z

∞
S
h
(1−γ)
−1
α
i
.
_`_`[email protected]>F<D]}D]<Ku]>NGEcSLF>dgDLNGF}HVNmSLPR#b»
dN
dS1000 ∝ S
dS1000
(1−γ)
α
.
S
]½¾STLF>deKuYGHVU Tk”
œ`ðn Q^ 

| sESLF>dehfDLU*K/R#bgDLhfDLUKabgD]w`R#bgDLNGF<]4YGDub½ deU YdgTD YGDbqR-bgHVd)YGD}EGNGdg]<]ÎR#U TDORIJN ]>K<SD}]>NGFbgD]4ŒŽHVU TLK>dgHVU ]
£
B
TLNGh"NGbqR#K>dewQD]"]/HVUK-THVhEGF>dg]<D]-DLU*K>F<D
DLK
a‘RwQDTfDLU¤hfH¼QDLUGU D
n|UªEGF<DLU)R#UK"YGD]
−1.7
−2.0
B ' −1.8
wVR#bgDLNGF<]\YGD
THVhEGF>dg]/D]DLUK>F<D
DLK a;THVF>F<DLEcHVU Y R#UKk»mYd“—cSLF<DLU*K<D]D]>K>dehkR#K>dgHVU ]YGD'bqRˆF<DLbqR#K>dgHVU
α
0.9
1.
 a‘HVUÅHV[GK>[email protected]‰F<DLhEGbqRVWR#[email protected] R#U ]Bb½¾DjN)R#K>dgHVU ðn ^¬bqR¬EcDLUK<DkbgHVPR#F>deK>lGhdgjN D YNÅ]>EcDTLK>F<D

S1000 E
γ
Y˜½¾SLU DLF>P`dgD'”
aTD\jNGd;]<DLh"[GbgD\K<HVNGK–»fŒR#deKOTHVhE)R#K>de[GbgDRwQDT\bgD]OhfD]>NGF<D]
γ = 1 − Bα ∈ [−2.7 , −3.2]
YNk]>EcDTLK>F<DuY˜½¾SLU DLF>P`dgDOR#N†YGDLbq»BYG
D ð"|D€@nz‡rDLK>K<DOD]>K>dehkR#K>dgHVUŠ»-b½§R#dgYGDuY˜½ NGU'TR#bgTLNGbK>F<i]4]>dehEGbgDOY˜½¾HVF<YF<D
YGDˆP`F/R#U YGDLNGFpEDLF>hfDLKpY˜½§R#EGEGNG¼QDLFpbgDmT/l HVd“s‹Y˜½§RIJN ]>K<DLFkbgD]'ŒŽHVU TLK>dgHVU ]'TLNGhBNGbqR#K>dewQD]mRwQDT%NGU DmbgHVdAYGD
Ä=v ú
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
EGNGdg]<]/R#U TD`n
" õÃ&õHÛ;õžÃQ&‘# ÛoeÁf˜&f+­ MuEGF<i]fRwQHVdeFfE)R#F/R#hfSLK>F<S¬bqRmŒ’HVU TLK>dgHVUËTLNGhBNGbqR#K>dewQDmYGD
T<l)RVj*N D\deUK<DLF>w`R#bebgDfYGD
»'K>F/RwQDLF<]-bgD]–THzDt¥kTLdgDLUK<]
DLK
[email protected]#bgTLNGb£YGD\bqR'w`R#bgDLNGFAYGD
sin2 θ
A(θ)
B(θ)
Sθ
K<DLbebgD-j*N D
D
>
]
}
K
G
Y
V
H
G
U
U
S
>
]
e
d
h
G
E
g
b
L
D
f
h
L
D
*
U
u
K
)
E
#
R
"
F
”
cut
ö
Σθ (Sθ ) = Σ38◦ (S1000
)
Σ38◦ (S1000 cut )
Sθ =
A(θ)
1/B(θ)
œ`ðn§Ñ`_Q
C½¾D]>K>dehkR#K>dgHVUpYGD}b½¾DLF>F<DLNGFv]>NGF4T/l)RVjN D
D]>KrDt—cDTLK>N SD–»xE)R#F>K>deFoYGD]odeU TDLF>K>deK>N YGD]v]>NGFbgD]oE)R#F/R#hfiLK>F<D]
Sθ
DLK
YGHVUGU SD]rE)R#Fvb½§RIJN ]>K<DLhfDLUKYGD]oŒ’HVU TLK>dgHVU ]vTLNGhBNGbqR#K>dewQD]n)‡rDLK>K<D–hfSLK>l H*YGDOEDLF>hfDLK}R#deU ]>d
A(θ) B(θ)
Y˜½¾D]>K>dehfDLF–RwQDT-NGU D"P`F/R#U YGD-EGF<STLdg]>dgHVU%bgD]uTHVNGF>[cD]uY˜½§R#K>K<SLUN)R#K>dgHVUª]<DLbgHVUmbgD-EGF>deU [email protected]>l HzYGD
‡ DJ‡–aGDLKOY˜½¾SLwVR#beN DLFuj*N)R#UK>deK/R#K>dewQDLhfDLU*KObgD]}deU TDLF>K>deK>N YGD]O]>NGFbgD"TH*Dt¥kTLdgDLU*KOY˜½§R#K>K<SLUN)R#K>dgHVUn
6
C R%­ P`NGF<D*ðn debebeN ]>K>F<DpbgD]kTHVNGF>[D]Y˜½§R#K>K<SLU*N)R#K>dgHVU‹HV[GK<DLU*N D]k»!b½§R#dgYGD¬YGD¬TDLK>K<DphfSLK>l H*YGDˆEHVNGF
‘
Ø
Yd“—cSLF<DLU*K<D]w`R#bgDLNGF<]YN
YGDvF<StŒŽSLF<DLU TD`nQC‘ROŒŽHVU TLK>dgHVU
D]>K7K>F<i]7[GdgDLUkRIJN ]>K<SDvE)R#FNGU DrŒŽHVU TLK>dgHVU
S1000
f (θ)
DtszEcHVU DLUK>dgDLbebgDBYN¬K†¼zEDk”
`œ ðn§Ñ 
1
1
−
f (θ) = exp −η

◦
cos θ
cos(38 )
HVȬbgD-]<DLNGb˜E)R#F/R#hfiLK>F<D"bede[GF<D-D]>KbgDxTHzDt¥kTLdgDLUKOY˜½§R#K>K<SLU*N)R#K>dgHVU
η
œwQHVdeFuDLU TRVYF<Stn
]<DLhB[GbgDuyLK>F<DOTHVhE)R#K>de[GbgD–RwQDTubgD]oYGHVUGU SD]4EcHVNGFbgD]
]>NGEcSLF>dgDLNGF<]
C R-wVR#bgDLNGF
‘
η = 0.85 ± 0.03
S1000
»pÑ`_Š€–|ÏÆa˜THVF>F<D]>EcHVU Y R#[email protected]»ŠNGU DSLU DLF>P`dgD
|D†€\n;‡rHVhhfDfHVUªEDLNGKxbgDTHVU ]<K/R#K<DLF-]>NGFxbqR'­ P`NGF<D
∼4
ðn a4bgD]'w`R#bgDLNGF<]kdeUŒŽSLF>dgDLNGF<D]¬»ÅÑ`_2€–|Ï THVU YNGdg]<DLUKˆ»ªNGU D%w`R#bgDLNGFŠ]>deP`UGd“­)TR#K>dewQDLhfDLU*K¬EGbeN ]'ŒŽR#de[GbgD
Ø
YN‹THzDt¥kTLdgDLUK¬Y˜½§R#K>K<SLU*N)R#K>dgHVU¨TR#F'b½§RVTTDLEGK/R#U TD¤U½¾D]>KŠEGbeN ]p]/R#K>NGF<SD`nrC‘R2EGF<HV[)R#bedebedeK<SªYGD%YGSLK<DTLK>dgHVU
YGSLEDLU Y¬R#bgHVF<]YGD ”*bqR"­ P`NGF<D–„ n§^@Y R#U ]rbgDAT<l)R#EGdeK>F<Dx„@hfHVUK>F<Dxj*N D–bgHVF<]<j*N DOb½¾HVU¬YGD]<TDLU Y¬DLU†YGD]/]<HVN ]
θ
YGDfb½¾SLU DLF>P`dgDŠYGD']ÎR#K>NGF/R#K>dgHVUa‘bgD]BPQDLF>[D]BwQDLF>K>dgTR#bgD]@DLK"K>F<i]BdeU TLbedeU SD]@]<HVUK\bgD]-EGF<DLhdgiLF<D]»mU DfEGbeN ]
YGSTLbgDLU T<l DLFv»ATHVNGEfI] |GF7bgDvF<S]<DR#Nn*‡rD]4R#UGP`bgD]ÔSLUGdeK>l)R#Ns]/HVUK]<HVN ]JF<DLEGF<S]/DLUK<S]4Y R#U ]bqRxYdg]>K>F>de[GNGK>dgHVU
DLU
DLKoYGHVU T}bqR-THVNGF>[cD}Y˜½§R#K>K<SLU*N)R#K>dgHVUpTR#bgTLNGbgSDA»AE)R#F>K>deFrYGDuTDLK>K<DuYdg]<K>F>de[GNGK>dgHVUŠD]>K4EGbeN ]7ŒR#de[GbgDO»
sin2 θ
EDLK>deKoDLK4P`F/R#U Y n`C½§RI¯N ]<K<DLhfDLUKYGD}bqR–ŒŽHVU TLK>dgHVUpY˜½§R#K>K<SLUN)R#K>dgHVU¬THVU YNGdeKoU)R#K>NGF<DLbebgDLhfDLU*K}»ANGU Duw`R#bgDLNGF
θ
YGD EGbeN ]ŒŽR#de[GbgD`a j*NGd˜U½§[email protected])RV]}YGDx]/DLU ]EGl¼]>dgj*N D`n
η
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä=v
,6-0/.1$ðn
‡rHVNGF>[D]Y˜½§R#K>K<SLU*N)R#K>dgHVUŠHV[GK<DLU*N D]4E)R#F4bqR–hfSLK>l H*YGD}YNp‡rHVU ]>K/R#UK6D†U*K<DLU ]>deK†¼k‡oNGKEcHVNGFYGD]
Ø Y2N
YGD4F<StŒ’SLF<DLU TDoSLPR#bgDo»}Ñ`_aÑJðuDLK ~ _v€A|ÏÆa»PR#N T/l D4DLK _a ðDLK;Ñ`_v€A|Ï »YF<HVdeK<D`nCD
wVR#bgDLNGF<]3
S1000
]>deP`U)R#b*» _`_`_hžF/R#EGEcHVF>K<So»v]ÎRvwVR#bgDLNGFDLU
D]>K˜K>F/RVTSoDLUxŒŽHVU TLK>dgHVU" YGD 
n™OUBRI¯N ]<K<DLhfDLUK
◦
RwQDT-bqRŒ’ HVU TLK>dgHVU!DtszEcHVU DLUK>dgDLbebgDp`œ ðn§Ñ D]<θK}=deU 38
YdgjN S`nC‘Rw`R#bgDLNGFOYN%TH*Dt¥kTLdg1/
DLU*cos(θ)
K–Y˜½§R#K>K<SLU*N)R#K>dgHVU2D]>KuYGD
b½¾HVF<YF<DuYGD
aGDLK4U DO]<DLh"[GbgDO E)RV]4SLwQHVbeN DLFvDLUK>F<DAÑ`_BDLK ~ _x€A|ÏÆnQ|7U†YGD]<]/HVN ]rYGD–Ñ`_x€A|ÏÆaQbgD
0.85 ± 0.03
TH*Dt¥kTLdgDLU*KrY˜½§R#K>K<SLU*N)R#K>dgHVUˆD]>Ko]>deP`UGd“­)TR#K>dewQDLhfDLU*KEGbeN ]ŒR#de[GbgDBœEcDLUK<D–YGD}bqRBTHVNGF>[cD}EGbeN ]4ŒŽR#de[GbgDtaz]>deP`U D
jN DAbqR\EGF<HV[)R#[GdebedeK<SBYGD-YGSLK<DTLK>dgHVUmU½¾D]>KEGbeN ]udeU YGSLEDLU Y R#U*K<D"YGDxb½§R#UGP`bgD`n
"˜
ö
¡
ÃfsÛoeÁf˜&ò>f
š UžEcDLNGKˆTHVhEGF<DLU YF<D2EcHVNGF<jN HVdxNGU D¤ŒŽHVU TLK>dgHVU¨DtsEHVU DLU*K>dgDLbebgD‰]½§RIJN ]>K<D¤[GdgDLUÌ»ÆbqRZTHVNGF>[cD
˜Y ½§R#K>K<SLUN)R#K>dgHVU‰DLU¤THVU ]>dgYGSLF/R#U*KxbgD]–R#F>P`NGhfDLU*K<]-]>NGdew`R#UK<]@”)b½¾SLwQHVbeNGK>dgHVU2YN!EGF<H#­ b;bgHVUGP`deK>N YdeU)R#bYGD
bqR–PQDLF>[DvD]>K7[GdgDLUfYGSTLF>deK<DvE)R#F7bqROŒŽHVU TLK>dgHVUfYGDuxR#dg]<]/DLFJ†ŸudebebqRV]aVYGHVU*KU HVN ]7F/R#EGEcDLbgHVU ]7b½¾DtszEGF<D]/]>dgHVUˆ”
Ne (X) = Nmax
X − Xint
Xmax − Xint
Xmax −Xint
λ
exp
Xmax − X
λ
œ`ðn§Ñ`ÑQ
HVÈ
D]>KAbqR'EGF<H#Œ’HVU YGDLNGF-YN¤EcHVdeUK-YGD\EGF<DLhdgiLF<[email protected]<DLF/RVTLK>dgHVUÅYGDBbqR'E)R#F>K>[email protected]>dehkR#deF<D`a D]<K
bqRŠbgX
HVUGint
P`N DLNGF"Y˜½§R#K>K<SLUN)R#K>dgHVUaYGHVUK-bqRŠw`R#bgDLNGF-D]>[email protected]½¾HVF<YF<DfYGDe÷V_ŠP n¾TLh −2 Y R#U ]-YGD]xTHVU YdeλK>dgHVU ]
R#K>hfH`]>EGl SLF>dgj*N D]–]<K/R#U Y R#F<Y˜n)™OU D-PQDLF>[cD"]<DxEGF<HVE)R#PQDR#U*K-RwQDTBNGU!R#UGP`bgD-ÔSLUGdeK>l)R#b R#NGF/RfK>F/RwQDLF<]<S
NGU DfSLE)R#dg]<]<DLNGFBY˜½§R#K>hfH`]>EGl iLF<D
RwQDT
P n¾TLh −2 aDLKxE)θ R#F"THVU ]<Sj*N DLUK
X
/
cos
θ
X
'
875
ground
ground
EHVNGF
a£]/R%THVhEH`]/R#U*K<D¬SLbgDTLK>F<HVhkR#P`U SLK>dgjN Dm]<DLF/R%R#K>K<SLU*N SDˆY˜½ NGUƌŽRVTLK<DLNGF
X
ground − Xmax λE)R#FvF/R#EGEcHVF>K}»B]/HVUŠhkRsdeh"NGhÙYGDAYGSLwQDLbgHVEGEcDLhfDLUKuDLU
nzM£¼R#UK
Xmax −Xground / cos θ
∼ exp
Xmax
λ
ŒŽR#deKub½ l¼zEHVK>l i]<D"jN DAbgD-]<deP`U)R#b» _`_`[email protected]>F<D]O]<D–ŒŽRVTLK<HVF>dg]/D-DLUˆNGU DAŒŽHVU TLK>dgHVUmYGDxb½§R#UGP`bgDxÔSLUGdeK>l)R#b
DLKxNGU [email protected]ŒŽHVU TLK>dgHVU‰YGDb½¾SLU DLF>P`dgDmœ’TtŒ>n˜ DjN)R#K>dgHVU‹`œ ðn§Ñ >tabqRpj*N)R#UK>deK<S
EcDLNGK-yLK>F<DE)R#F/R#hfSLK>F>dg]<SD

f (θ)
DLhEGdeF>dgjN DLhfDLU*KOEcHVNGFNGU D-SLU DLF>P`dgDA­GsGSDf”
Xground
S1000 (θ)
∝ exp −
f (θ) =
K Eα
Λ cos θ
œ`ðn§Ñ ~ 
HVÈ'bqR\j*N)R#UK>deK<S YGS]>deP`U D–bqR"bgHVUGP`N DLNGF}Y˜½§R#K>K<SLUN)R#K>dgHVUˆEcHVNGFobgD–]>deP`U)R#b
nz|7U¬]<DOF/R#EGEHVF>K/R#U*K}»
b½§R#UGP`bgD–ÔSLUGdeK>l)R#bcΛYGD–F<StŒŽSLF<DLU TDœ
◦ tazHVUŠHV[GK>dgDLU*K[GdgDLU'bqR"ŒŽHVU TLK>dgHVU S1000 YGSt­ UGdgDxDLUjðn§Ñ RwQDT
f (θ)

[email protected]<DLbqR#K>dgHVU
n θ = 38
η = Xground /Λ
Ä=v%W
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
")#Ázõò>fËÃ!õޓÁgf‘*( ‘
ö
C R–YGDLF>UGdgiLF<DvSLK/R#EDvYGDrbqRuhfSLK>l H*YGDu‡6D>‡2THVU ]<dg]>K<D»}NGK>debedg]<DLF£b½¾SLU DLF>P`dgDvhfD]>NGF<SDrE)R#F£bqR}• N HVF<D]<TDLU TD
‘
DLK aGR­ UˆY˜½¾SLK/R#bgHVUGU DLFbqR"F<DLbqR#K>dgHVU
 n
THVhhfDAF<StŒ’SLF<DLU TD–EcHVNGF}YGSLK<DLF>hdeU DLFbgD]TH*Dt¥kTLdgDLU*K<]
K
α
S1000 E
C‘Rj*N)R#UK>deK<S\jNGdD]<KuTHVF>F<SLbgSD\»b½¾SLU DLF>P`dgD"{£Õ D]>K}bgD
a j*NGd‘D]>K}bgD"]>deP`U)R#b» _`_`_

S38 = S1000 (θ)/f (θ)
hfiLK>F<D]@jN½§R#NGF/R#[email protected]>[Dp]>dvDLbebgDkEcH`]<]<SY R#deK\NGUËR#UGP`bgDkÔSLUGdeK>l)R#b}YGD ◦ a;HVÈ
D]>KBbqRˆŒŽHVU TLK>dgHVU
38
f (θ)
Y˜½§R#K>K<SLUN)R#K>dgHVU¬j*N DU HVN ]rRwQHVU ]oYGSLK<DLF>hdeU S–EGF<STSYGDLhhfDLU*KnzÉ£R#Frl¼zEHVK>l i]<D`abgD
D]>KNGU DŒŽHVU TLK>dgHVU
S38
NGUGdgjN DLhfDLU*[email protected]½¾SLU DLF>P`dgD`a˜DLKxYGHVU TBb½§RIJN ]>K<DLhfDLU*K"YGD\bqRfŒŽHVU TLK>dgHVU
]>NGFAbgD]xYGHVUGU SD]AEcDLF>hfDLK-YGD
g(E)
YGSLK<DLF>hdeU DLFubqR\F<DLbqR#K>dgHVU
 EcHVNGFbgD]}SLU DLF>P`dgD]u]<NGESLF>dgDLNGF<D]–»\bqR]/R#K>NGF/R#K>dgHVUn
S1000
E
CD"bgHVKOYGD"YGHVUGU SD]–j*N DI½§R#dNGK>debedg]<S-EcHVNGFOTDLK>K<D\SLK>N YGD"THVF>F<D]<EHVU Y!»bqREcSLF>dgHzYGD"YN
06/02/2004
R#N
acDLK}THVUK>dgDLUKOEGF<i]}YGD
L
S
Q
w
L
S
U
L
D
f
h
L
D
*
U
<
K
u
]
*
l
*
¼
G
[
>
F
g
d
G
Y
D
]
G
n
C
@
R
<
F
D
T
V
H
U
>
]
>
K
>
F
N
L
T
>
K
g
d
V
H
ˆ
U
Q
P
SHVhfSLK>F>dgj*N D
06/05/2006
2.105
NGK>debedg]<DAbgD]vdeUŒŽHVF>hkR#K>dgHVU ]}YGD]}YGDLNsŠYGSLK<DTLK<DLNGF<]–»\bqR"Œ’HVdg]DLKEcDLF>hfDLK}Y˜½¾HV[GK<DLUGdeF}NGU DAdeU TDLF>K>deK>N YGD-YGDzð`_
hfiLK>F<D]u]>NGFubqREcH`]>deK>dgHVUmYN%‘T “ANGF–DLKuYGD
]>NGF}b½¾HVF>dgDLUK/R#K>dgHVUªYGDxb½§RsGD`n)C4½¾SLU DLF>P`dgDBD]>KuhfD]>NGF<SD-E)R#F
0.5◦
bgDxYGSLK<DTLK<DLNGFOYGD–• N HVF<D]<TDLU TDBDLUˆdeUK<SLP`F/R#U*KubgDxEGF<H#­ b‘bgHVUGP`deK>N YdeU)R#bHV[ ]<DLF>wQS`n
<] HVdeKhdgDLNsYGSLK<DLF>hdeU SO»xb½§R#dgYGD}YGDbqRAF<DTHVU ]>K>F>N TLK>dgHVU'PQSHVhfSLK>F>dgjN Dul¼z[GF>dgYGD`aQbgD
]>deP`U)R#b» _`_`_"hfiLK>F<D]uD]>KHV[GK<DLUNm»BE)R#F>K>deF}YGD]YGHVUGU SD]YGD]]>K/R#K>dgHVU ]vNGUGdgj*N DLhfDLUKn |7UˆDt—˜DLKazdebŒŽR#NGK

SLK/R#bgHVUGU DLFAbgD
j*N D\hfD]>NGF<D"bgD\F<S]<DR#NªYGD\]>NGFJŒRVTD\]/DLNGbaR­ UªYGD\EHVNGwQHVdeFANGK>debedg]<DLFxTDLK>K<D\F<DLbqR#K>dgHVU
S1000
EHVNGFK<HVNGK<D]ubgD]}YGHVUGU SD]}hfD]<NGF<SD]}E)R#FbgDB¡zÕ@n
ArdgDLUkj*N DvbgD
S1000
DÞbvD]<K\U STD]/]/R#deF<D¬Y˜½§R#EGEGbedgjN DLFkR#NsZSLwQSLU DLhfDLUK<]kl¼z[GF>dgYGD]NGUËDLU ]<DLh"[GbgDmYGDŠTLF>deK<iLF<D]kYGDp[HVUGU D
F<DTHVU ]>K>F>N TLK>dgHVUÅDLK-YGD]/SLbgDTLK>[email protected]ŒRVWHVU2]>dehdebqR#deF<Dk»Šb½§R#U)R#be¼]<DfYGD]-YGHVUGU SD]-YNZ¡zÕ\n˜C‘D]-TLF>deK<iLF<D]
dehEH`]/S]}]<HVUKbgD]O]>NGdew`R#UK<]-”
?
?
?
?
?
?
b½¾SLU DLF>P`dgD-{£Õ YGHVdeKuyLK>F<D-]>NGEcSLF>dgDLNGF<D"»
|4D€ DLKb½§R#UGP`bgDxÔSLUGdeK>l)R#b‘deUŒŽSLF>dgDLNGFO» ◦
0.3
60
bgD
YGHVdeKp]<DmK>F<HVNGwQDLFpY R#U ]ŠbgD!T<l)R#hEžYGDˆwzN D!YGD]'K<SLbgD]/THVED]!”oY R#U ]ŠbgD!TRV]'THVU*K>F/R#deF<D
Xmax
b½§RIJN ]>K<DLhfDLUKuYN¬EGF<H#­ b‘bgHVUGP`deK>N YdeU)R#b‘D]>KK>F<i]vEcDLNˆTHVUK>F/R#deUK–DLK}THVU YNGdeKu»\NGU DxD]>K>dehkR#K>dgHVUmYGD
b½¾SLU DLF>P`dgD-K>F<HVEmdeU TDLF>K/R#deU D`n
bqR!bgHVUGP`N DLNGF'YNXEGF<H#­ bubgHVUGP`deK>N YdeU)R#bOHV[ ]<DLF>wQSˆYGHVdeK'yLK>F<Dˆ]>N¥k]/R#hhfDLU*KfP`F/R#U YGD‰”£debvŒR#NGKNGU D
Yd“—˜SLF<DLU TD
P n¾TLh −2 DLUK>F<DŠbqRˆEGF<H#ŒŽHVU YGDLNGFR#K>hfH`]>EGl SLF>dgjN DŠhkRsdehkR#bgD
DLK
Xf − Xi > 300
Xf
hdeUGdehkR#bgD
HV[ ]<DLF>wQSD-E)R#F}bgD]K<SLbgD]<THVEcD]n
Xi
bgD 2 E)R#F}YGDLP`F<S-YGDxbede[cDLF>K<S-YGDAb½§RI¯N ]<K<DLhfDLUKOYNmEGF<H#­ bbgHVUGP`deK>N YdeU)R#bYGHVdeKuyLK>F<D-deUŒ’SLF>dgDLNGFO» _

χ
bqRAŒ’F/RVTLK>dgHVU'YGDubeNGhdgiLF<B
D (T<l DLF<DLU Ç HwkU DuYGHVdeKoE)RV]oDtsGTSYGDLF Y aR­ U'Y˜½¾HV[GK<DLUGdeFvNGU DuD]<K>dehkR#K>dgHVU
30
EGF<STLdg]<[email protected])R#U*K>deK<S"YGDAbeNGhdgiLF<D"YGD–• N HVF<D]/TDLU TD-F<DWLN D`n
[email protected]‘T “ANGF-YGHVdeK-yLK>F<D\[GdgDLUªF<DTHVU ]>K>F>NGdeKf”bqRpYdg]>K/R#U TDfR#Nª‘T “ANGF-YGDBbqRp]<K/R#K>dgHVUªR¼R#[email protected] ]–ŒŽHVF>K
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä=vL[
]>deP`U)R#b‘YGHVdeKuyLK>F<D-deUŒ’SLF>dgDLNGFO»¦`_`_hfiLK>F<D]n
R#N TLNGU D]>K/R#K>dgHVUfY R#U ]7bgDvF<StŒŽSLF<DLUK>dgDLb)YGDvbqROPQDLF>[cDvU DYGHVdeKyLK>F<DvEGbeN ]£EGF<H*T/l D}j*N DobqRA]<K/R#K>dgHVUkR¼R#UK
?
bgDxEGbeN ]vŒŽHVF>Ku]>deP`U)R#baGTD-j*NGdDtsGTLbeNGK}bgD]}YGSTLbgDLU T<l DLhfDLU*KuŒŽHVF>K>NGdeK<]n
CD}U HVhB[GF<DAY˜½¾SLwQSLU DLhfDLUK<]]/R#K>dg]JŒR#dg]/R#U*Kr»xK<HVN ]oTD]oTLF>deK<iLF<D]rU½¾D]>KoEGbeN ]4j*N DuYGD ~ Ñ`_#„pœ’]>NGF
•›ta
2.105
YGHVUK $„ ð}EcH`]<]<SY R#U*KNGU DrSLU DLF>P`dgDv]>NGEcSLF>dgDLNGF<Dr»u„–|4D€@n`‡rD];TLF>deK<iLF<D]7]<HVU*KdeU Ydg]>EcDLU ]/R#[GbgD];EcHVNGFU DoE)RV]

yLK>F<DoYGHVhdeU S]£E)R#F£YGD]£SLwQSLU DLhfDLUK<]£ŒŽHVF>K>NGdeK<]£HVNBK>F<i];hkR#bzF<DTHVU ]>K>F>NGdeK<]n‡rDLEcDLU Y R#UKa#debzTHVU*w*dgDLU*KY˜½¾yLK>F<D
K>F<i]4EGF>N YGDLU*Ku»-b½¾SLPR#F<YŠYGDOTD]vTLF>deK<iLF<D]vTR#Frdebg]oEcDLNGwQDLUKvdeU*K>F<H*YNGdeF<DAYGD]r[GdqR#dg]o]<NGFobqRBYdg]>K>F>de[GNGK>dgHVU¬YGD]
HV[ ]<DLF>wVR#[GbgD]K<DLbebgD]4j*N DvbgD
a`DLK4YGHVU T]>NGF7bgD
hfD]>NGF<SE)R#F7bgDF<S]<DR#NfYGDu]>NGFJŒŽRVTD`n‡rD]4TLF>deK<iLF<D]
Xmax
S1000
U Du]<HVU*KoTDLF>K/R#deU DLhfDLUKvE)RV]rYGSt­ UGdeK>d“Œ’]aDLK4debD]>K4U STD]/]/R#deF<DuY˜½¾SLK>N YdgDLFrbgDLNGFodehE)RVTLKr]>NGF4bgD]rHV[ ]<DLF>w`R#[GbgD]
YN¤¡zÕ@n ÕxR#U ]ubgDBT<l)R#EGdeK>F<DB„ a U HVN ]OhfHVUK>F<HVU ]Aj*N DxbgD]OHV[ ]/DLF>wVR#[GbgD]ubedgSD]–»fb½§RVTTDLEGK/R#U TD`aTHVhhfD"bgD]
C (ɏ]AHVN!bgD-F/R#EGEcHVF>KO„ (JH (‡ ~ (‘JH (xa˜]<HVU*K–]<N¥k]/R#hhfDLUKA]<DLU ]>de[GbgD]–EcHVNGFuhfDLK>K>F<D\DLU!SLwzdgYGDLU TDBYGD"K<DLbg]
Dt—cDLK<]n
,6-0/[email protected]ð¤n ÷ ‡rHVF>F<SLbqR#K>dgHVU\DLU*K>F<D7b½¾SLU DLF>P`dgD4{£ÕXDLKbgD
EHVNGFbgD]SLwQSLU DLhfDLUK<]l¼z[GF>dgYGD]
]/R#K>dg]JŒR#dg]/R#2 UK;bgD]£TLF>deK<iLF<D];YGDv]<SLbgDTLK>dgHVUnVC½§RIJN ]>K<DLhfSDL38
U*K=D]>K‘SF<1000
SR#bedg(θ)/f
]<SrEcHV(θ)
NGF;YGD];w`R#bgDLNGF<]£YGD
]>NGEcSLF>dgDLNGF<D]
»Ñ`[email protected]€–|4ÏÅaGYGDAhkR#UGdgiLF<D\»@]<DAEGbqRVTDLFOR#N†YGD]<]>N ]OYGDAb½¾SLU DLF>P`dgD"YGDx]/R#K>NGF/R#K>dgHVU!YGDAb½§RVTTSDLEG38K/R#U TD`n
C R"­ P`NGF<D”ðn¤÷-debebeN ]<K>F<DAbqR"THVF>F<SLbqR#K>dgHVUmDLU*K>F<DAb½¾SLU DLF>P`dgDx{£ÕÌDLKvbgD
‘
HV[GK<DLU*N¬EHVNGFvbgD]vSLwQSLU DLhfDLUK<]
S38
]/R#K>dg]JŒR#dg]/R#UK'bgD]'TLF>deK<iLF<D]pYGSTLF>deK<]ŠEGF<STSYGDLhhfDLUKn}C4½§RI¯N ]>K<DLhfDLU*K¬D]>KkF<SR#bedg]<S%EcHVNGF'YGD]'wVR#bgDLNGF<]pYGD
]>NGEcSLF>dgDLNGF<D]'»ÅÑ`_ª€A|ÏÅaoYGD¬hkR#UGdgiLF<D¤»2]<D¬EGbqRVTDLF¬R#N†YGD]<]>N ]'YGDˆb½¾SLU DLF>P`dgD!YGDm]/R#K>NGF/R#K>dgHVU«YGD
S38
Ä=v%c
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
b½§RVTTDLEGK/R#U TD`n)C‘RF<DLbqR#K>dgHVUmDLUK>F<D"bgD
S38
DLKb½¾SLU DLF>P`dgD"]½¾STLF>deKOYGHVU Tf”
E = 0.15 (S38 )
œ`ðn§Ñ#„*
1.08
RwQDTxbgD]}deU TDLF>K>deK>N YGD]O]>NGdew`R#UK<D]O]>NGFbgD]E)R#F/R#hfiLK>F<D]B”
DLK
n
K = 0.15 ± 0.02
α = 1.08 ± 0.04
{;deU)R#bgDLhfDLUKAHVUmEcDLNGKOSTLF>deF<D"bqRF<DLbqR#K>dgHVU¤DLUK>F<DBbgD"]>deP`U)R#b£» _`_`_hfiLK>F<D]–DLKub½¾SLU DLF>P`dgD\YGSLK<DLF>hdeU SD

E)R#[email protected]>l HzYGDB6
‡ DJ‡ô”
E = 0.15 (S1000 (θ))
1.08
× exp 1.08 η
1
1
−
cos θ cos(38 ◦ )
œ`ðn§ÑJðQ
C ½¾STR#F>KF<DLbqR#K>d“Œ)DLUK>F<Db½¾SLU DLF>P`dgDvR#deU ]>dQF<DTHVU ]<K>F>NGdeK<DE)R#F‘bqRhfSLK>l H*YGDv‡6DJ‡|?–{£ÕXDLK‘b½¾SLU DLF>P`dgD4hfD]>NGF<SD

E)R#FxbqRŠ• N HVF<D]<TDLU TD'D]>K-DLUªhfH¼QDLUGU D'YGDb÷ Y a‘RwQDTfNGU DfYdg]>EcDLF<]>dgHVU2Y˜½¾DLUwzdeF<HVU ~ ð Y n˜C4½¾D]>K>dehkR#K>dgHVU
YGD"b½¾SLU DLF>P`dgDf6
‡ D>‡žEcDLNGKAYGHVU T\yLK>F<[email protected] ]>dgYGSLF<[email protected]\U HVU![GdqR#dg]<SD`a˜RwQDT\NGU D"F<DLbqR#K>dewQDLhfDLU*K-P`F/R#U YGD
Ydg]>EcDLF<]>dgHVUn
G;’õ’òÁNÛ;%õ•i2Á@h‘VÛ;Yö
õ ö
C½ NGK>debedg]/R#K>dgHVUŠYGDvb½¾SLU DLF>P`dgD}hfD]>NGF<SD}E)R#FbgDYGSLK<DTLK<DLNGFoYGDr• N HVF<D]<TDLU TDuEHVNGFoSLK/R#bgHVUGU DLFbqRAF<DLbqR#K>dgHVU
 EDLF>hfDLK–Y˜½¾HV[GK<DLUGdeFONGU D"D]>K>dehkR#K>dgHVU!YGD-b½¾SLU DLF>P`dgD"YGD]uSLwQSLU DLhfDLU*K<]AYGSLK<DTLK<S]uE)R#FObgDB¡zÕ@a)YGD
S1000 E
hkR#UGdgiLF<DdeU YGSLEcDLU Y R#UK<D'»'bqR'ŒŽHVdg]-YGD]x]>dehBNGbqR#K>dgHVU ]"[email protected]>[cD]"R#K>hfH`]>EGl SLF>dgj*N D]aDLK-YGD]-]>dehBNGbqR#K>dgHVU ]
YNkYGSLK<DTLK<DLNGFnC‘D]EGF<SYdgTLK>dgHVU ]rYGD]4]>dehBNGbqR#K>dgHVU ]oYGSLEcDLU YGDLUKrDLUK>F<DOR#NGK>F<DOYGD]hfHzYGiLbgD]4Y˜½ deU*K<DLF/RVTLK>dgHVU ]
l)RVYF<HVUGdgjN Dr»vK>F<i]l)R#NGK<DoSLU DLF>P`dgD`aYGHVU*K‘bqR­R#[GdebedeK<SU½¾D]>KE)RV]DLU THVF<Do]/HVbedgYGDLhfDLUKSLK/R#[GbedgD–”bqRvhfSLK>l HzYGD
YN'6
‡ DJ‡ZEH`]<]/iYGDYGHVU T}b½¾SLU HVF>hfDORwVR#U*K/R#PQDuYGD}]½§R—F/R#U T/lGdeFoYGD}TDvEGF<HV[GbgiLhfDODLUfNGK>debedg]/R#UKoNGUGdgjN DLhfDLU*K
bgD]}YGHVUGU SD]n
‡rDLEcDLU Y R#UKacbqR'hfSLK>l H*[email protected]‰6
‡ D>‡ÌD]<KObgHVdeUªY˜½¾yLK>F<D\NGU D\E)R#U)RVTSD`a˜TR#FADLbebgD\F<DLEcH`]<[email protected]]>NGFAYGD]Al¼zEH#
K>l i]<D]uYGHVU*KOdeb‘ŒŽR#NGK–]½§RV]<]>NGF<DLF–YGD-bqRfwVR#bedgYdeK<S`aDLKODLbebgD\D]>K}U)R#K>NGF<DLbebgDLhfDLUKxbedehdeK<SDBE)R#FubqRŒŽR#de[GbgD]/]<D"YN
• Ns¬»\K>F<i]ul)R#NGK<D-SLU DLF>P`dgD`n

")#zõò>f Qà õÞeÁf#( ‘ ² h & Áf#( C4½¾D]>K>dehkR#K>dgHVUXYGDkbqRmF<DLbqR#K>dgHVU
S1000 E
D]>KxDt—cDTLK>N SD'»ŠE)R#F>K>deF"YGD]-YGHVUGU SD]-l*¼*[GF>dgYGD]ajNGd7F<DLEGF<S]/DLUK<[email protected]>N DLbebgDLhfDLUK\hfHVdeU ]-YGD _ Y YN

U HVh"[GF<DY˜½¾SLwQSLU DLhfDLU*K<]"YGSLK<DTLK<S]AE)R#FAb½ š ÉM\n˜C‘R'EGF<STLdg]>dgHVUªYGD\b½§RI¯N ]<K<DLhfDLUK-YN
YGSLEDLU Y¤YdeF<DTt
S38
K<DLhfDLUKAYNmU HVhB[GF<DBY˜½¾SLwQSLU DLhfDLU*K<]Al¼z[GF>dgYGD]OYGHVU*KOb½¾SLU DLF>P`dgD\D]>Ku]>NGEcSLF>dgDLNGF<[email protected]»
|D€@nC‘Dx• NsmYGD
∼4
F/R¼QHVU ]-TH`]<hdgjN D]-YGSTLF<HVdg]/]/R#UK-K>F<i]AF/R#EGdgYGDLhfDLU*K\RwQDTb½¾SLU DLF>P`dgD`a˜bqRŠPR#hhfDfY˜½¾SLU DLF>P`dgDf]<NGF–bqRVj*N DLbebgD
ö
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Ä=v%v
b ½§RI¯N ]>K<DLhfDLU*KxYGD"bqRkF<DLbqR#K>dgHVU

D]<K–THVU*K>F/R#deUK-]<DBbedehdeK<D\»
|4D€@nc‡rHVhEGK<D\K<DLUNªYGD]
S38 E
∼ 4 − 10
deU TDLF>K>deK>N YGD]xHV[GK<DLUN D]-]>NGF–b½§RIJN ]>K<DLhfDLUK-YGD\bqRkF<DLbqR#K>dgHVU

[email protected]>l H*YGDf‡6D>‡–ab½¾DLF>F<DLNGF
S1000 E
jN Dvb½¾HVUkTHVhhfDLKo]>NGF7b½¾SLU DLF>P`dgD}DLUkDtsK>F/R#EHVbqR#U*KbqRAF<DLbqR#K>dgHVU%œ`ðn§Ñ#„*7D]>KYGDb½¾HVF<YF<D}YGD ~ _ Y » _`_-|D€@n

‡rDLK>K<DxdeU TDLF>K>deK>N YGD\]>¼z]<K<SLhkR#K>dgjN D-SLK/R#U*KubedgSD"»bqR\ŒŽR#de[GbgD"]>K/R#K>dg]>K>dgj*N D"RVTLK>N DLbebgDLhfDLUKAYdg]>EcHVUGde[GbgD`a bgD
]<DLNGbQhfH¼[email protected]bqR}F<SYNGdeF<DoD]<K‘Y˜½§RVTTLNGh"NGbgDLF£EGbeN ];YGD4YGHVUGU SD]n#Ï!R#bel DLNGF<DLN ]/DLhfDLUKa`»vhfHVdeU ];Y˜½§R#K>K<DLU YF<D
EGbeN ]>dgDLNGF<]YdeÔR#deU D];Y˜½§R#UGU SD]abgDU HVh"[GF<DrY˜½¾SLwQSLU DLhfDLUK<]£l*¼*[GF>dgYGD];YGSLK<DTLK<S];U D]<DLF/R}E)RV]‘]>N¥k]/R#U*KEcHVNGF
HV[GK<DLUGdeF}NGU DxEGF<STLdg]>dgHVUm]ÎR#K>dg]JŒŽR#dg]ÎR#UK<D-]>NGFb½¾D]<K>dehkR#K>dgHVU%YGDAb½¾SLU DLF>P`dgD"YGDAb½¾HVF<YF<DBYGD _`_|D†€\n

 @h<{!Û;¤Â†GÃ`#>f C‘RkhfSLK>l HzYGDBYNª‡6DJ‡¨D]>Ku[)RV]<SDB]>NGFub½ l*¼*EcHVK>l i]<D\jN D"bqRfTHVNGF>[cD
Y˜½§R#K>K<SLUN)R#K>dgHVU‰D]>K–bqR'hfyLhfD\EHVNGFAK<HVNGK<D]AbgD]ASLU DLF>P`dgD]n š FacHVU¤]½§R#K>K<DLU Y¤»Š[email protected]*N DBTDLK>K<D\l¼zEcHVK>l i]<D
U DO]<HVdeKoEGbeN ]rw*F/R#dgDA»xl)R#NGK<DASLU DLF>P`dgD\”*DLUpDt—cDLKrbgD
SLwQHVbeN)R#U*KrbgHVPR#F>deK>lGhdgj*N DLhfDLUK–RwQDT–b½¾SLU DLF>P`dgD`a
Xmax
deb‘Dtszdg]<K<D-NGU D-w`R#bgDLNGFOYGD
K<DLbebgDBjN DBTDLbeNGd“†TLd;]/DxK>F<HVNGwQDBDLU†YGD]/]<HVN ]OYGD"b½§R#beK>deK>N YGDBYGD"b½ š [ ]<DLF>w`R#K<HVdeF<D
E
É7dgDLF>F<DkMONGPQDLFn‘ÕxR#U ]"TD'TRV]-bedehdeK<D`ajNGdr]<DEGF<HzYNGdeK"wQDLF<]
D€±EcHVNGF"NGU DkPQDLF>[DfYGDkEGF<HVK<HVU
∼ 1020.5
wQDLF>K>dgTR#bgD`acb½ l¼zEHVK>l i]<[email protected]\b½§R#K>K<SLUN)R#K>dgHVUÅDtszEcHVU DLUK>dgDLbebgDfR#EGF<i]AbgDBhkRszdehBNGh [email protected]*K"D]>K
K<HVK/R#bgDLhfDLUK}ŒR#N ]<]<D`n
Ï%yLhfDu]/R#U ]THVU ]>dgYGSLF<DLFobgD}TRV]HVÈfbgD
D]>K4]<HVN ]7bgDu]<HVbaVbgD]PQDLF>[cD]4YGDvEGbeN ]4l)R#NGK<DuSLU DLF>P`dgD}]<HVU*K
Xmax
EGbeN ]"÷ÞIJDLNGU D]÷VaGY R#U ]}bgDx]<DLU ]}HVȈDLbebgD]uHVUK}E)R#F<THVNGF>N%NGU D-SLE)R#dg]<]<DLNGF}Y˜½§R#K>hfH`]>EGl iLF<D"EGbeN ]vŒR#de[GbgD"R#EGF<i]
bgDuhkRsdeh"NGhˆa j*N DObgD]vPQDLF>[D]vYGD–EGbeN ]v[)RV]<]<D–SLU DLF>P`dgDxjNGdcHVUKbgDLNGF
EGbeN ]vl)R#NGKDLUˆR#beK>deK>N YGD`nGCD]
Xmax
PQDLF>[D]rYGD}[)RV]<]/D}SLU DLF>P`dgDOHVU*KoNGU D–CÕA{ªhfHVdeU ]oEDLU*K>N D–R#N'UGdewQDR#NŠYNŠ]<HVbaYNkŒŽR#deKrjN DubqRxEGbeNGE)R#F>KvYGD]
E)R#F>K>dgTLNGbgD]oR#dgDLUKrSLK<STLF<SD]o»–NGU DuR#beK>deK>N YGDuEGbeN ]7P`F/R#U YGD}DLK4HVU*K4YGHVU T}DLNfEGbeN ]4YGDK<DLhE ]7EcHVNGF4YdewQDLF>PQDLF
F/RVYdqR#bgDLhfDLUKuDLK}Yd“—cN ]<DLFn DÞbcDLUpF<S]>NGbeK<Dxj*N Dx]>dcb½¾HVU¬THVU ]>dgYGiLF<DxbqR\j*N)R#UK>deK<S
œŽRwQDT EGF<H*T/l D
S1000 /E α
α
YGD tabgD]}PQDLF>[D]}YGDA[)RV]/]<DxSLU DLF>P`dgD-YGHVdewQDLUKARwQHVdeF}NGUˆTHzDt¥kTLdgDLUK–Y˜½§R#K>K<SLUN)R#K>dgHVU%EGbeN ]P`F/R#U Yˆj*N DAbgD]
1
PQDLF>[D];»l)R#NGK<DrSLU DLF>P`dgD`On –HVN ]wQDLF>F<HVU ];E)R#F‘bqRu]>NGdeK<D4j*N DbgD]]>deh"NGbqR#K>dgHVU ];U*NGhfSLF>dgjN D];F<DLEGF<HzYNGdg]<DLUK7TD
THVhEHVF>K<DLhfDLU*KrR#K>K<DLU YNaQDLK7U HVN ]j*N)R#UK>d“­)DLF<HVU ]4bqR–wVR#F>dqR#K>dgHVUŠR#K>K<DLU YN D}YNfTH*Dt¥kTLdgDLU*K4Y˜½§R#K>K<SLUN)R#K>dgHVU
RwQDTxb½¾SLU DLF>P`dgD`n
Ü'Ü
‡rHVhhfDrU HVN ];b½§RwQHVU ]£hfDLU*K>dgHVUGU SvEGF<STSYGDLhhfDLUKa`bqROEGF<HVEGF>dgSLK<SvY˜½ NGUGd“ŒŽHVFJ
hdeK<[email protected]]<K>F>de[GNGK>dgHVUmDLU
YGD]SLwQSLU DLhfDLUK<]}U½¾D]<Kw*F/R#dgDAj*N DxY R#U ]vbgDxTRV]HVȬbgD
hfD]>NGF<S
sin2 θ
S1000
• N TLK>N DfYGDhkR#UGdgiLF<DdgYGDLU*K>dgjN DkEHVNGFxK<HVN ]"bgD]"R#UGP`bgD]xÔSLUGdeK>l)R#Ns˜n‘C‘RŠŒŽHVF>hfDYGDbqRpYdg]>K>F>de[GNGK>dgHVUÆYGD]
YGSLEDLU Y%YGDxb½§R#hEGbedeK>N YGD\YGD]}w`R#F>dqR#K>dgHVU ]OYGDxbqRkYdg]>EcDLF<]>dgHVUmYN
RwQDTBb½§R#UGP`bgD-ÔSLUGdeK>l)R#bnC‘R
sin2 θ
S1000
­ P`NGF<D}ðn§¦debebeN ]<K>F<D"bqRYdg]<K>F>de[GNGK>dgHVU¤DLU
R#K>K<DLU YN D\Y R#U ]ubgDBTRV]OHVÈ%b½¾HVUmEGF<DLU YmTHVF>F<DTLK<DLhfDLU*K-DLU
sin2 θ
THVhEGK<D}b½¾Dt—˜DLK4Y˜½ NGU D}w`R#F>dqR#K>dgHVU'RwQDT YGD]7• N TLK>N)R#K>dgHVU ]r]>NGFbgD
œ’TtŒ‚ 𝏝tnQC½ deU*K<DLU ]>deK<SA»AP`F/R#U Y
θ
S1000
Ø
‘ ò#£&fmf
sin2 θ
Äz¿
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
#R UGP`[email protected]]>KOEGbeN ]–dehEcHVF>K/R#UK<DYGD\DLUwzdeF<HVU ~ _ Y E)R#FAF/R#EGEcHVF>K-R#NsmEDLK>deK<]xR#UGP`bgD]AEcHVNGFObgD]ASLwQSLU DLhfDLU*K<]
Y˜½¾SLU DLF>P`dgDk]>NGEcSLF>dgDLNGF<Dk» ~ |4D€@a‘DLK"Y˜½¾DLU*w*deF<HVUZÑ`_ Y EcHVNGF-bgD]-SLwQSLU DLhfDLUK<]@R#N†YGD]<]>N ]BYGDeð¬|D†€\nD†b
D]>KTDLEDLU Y R#U*KuYd“¥kTLdebgD-Y˜½¾D]>K>dehfDLFbgD]}wQSLF>deK/R#[GbgD]v• N TLK>N)R#K>dgHVU ]uYN
]>NGFbgD]YGHVUGU SD]aGTR#FdebcŒŽR#NGK
S1000
Ydg]>EcH`]<DLF}YNˆ[cHVU¬hfH*[email protected]ÕA{ZEHVNGF}TR#bgTLNGbgDLFOb½¾DLF>F<DLNGF–RV]<]<HzTLdgSD-»\b½§RIJN ]>K<DLhfDLUKnc‡rD]• N TLK>N)R
K>dgHVU ]@EHVNGF>F/R#dgDLU*KkyLK>F<DŠEGbeN ]@ŒR#de[GbgD]jN Dpj*NGdvD]<[email protected]>deKfY R#U ]¬‚ ðV-œ’TtŒ>nv‚ ta;U HVN ]SLK>N YdgHVU ]fYGHVU T
Ø
Ø`Ø
b½¾Dt—cDLKxY˜½ NGU DYdg]>K>F>de[GNGK>dgHVUÅDLU
U
V
H
¤
U
G
E
q
b
#
R
<
K
@
D
G
N
G
U
g
d
*
j
N
L
D
f
h
L
D
U
K
c
E
V
H
G
N
x
F
e
d
e
b
e
b
N
>
]
>
K
<
F
DLFxbgD]xYd“—˜SLF<DLUK<]xKÞ¼*EcD]xYGD
sin2 θ
EGF<HV[GbgiLhfD]}deU*K>F>deU ]<ijN D]–»@bqR\hfSLK>l HzYGDB6
‡ DJ|
‡ ?–{£Õ@n
,.-F/.1ðn§¦ ÕOdg]<K>F>de[GNGK>dgHVU ]uDLU
R U ]}bgD-TRV]}HVȬbgD]v• N TLK>N)R#K>dgHVU ]OYN
]<HVU*K}EGF>dg]<D]
2 R#K>K<DLU YN D-Y #
S1000
LD U!THVhEGK<2 D`acEcHVNGF–bgD]xSLU DLF>P`dgDsin
]x]>NGEθSLF>dgDLNGF<D]x» ~ 
| D€¹œ’THVNGF>[D\[GbgDLN D–DLK–EHVNGF
|D†€¢œ’THVNGF>[cD
D<] KoDt—˜DTLK>N SO»AE)R#F>K>deFvYGD]4]>dehBNGbqR#EK>dgHV>U ]5nzÕ\½§R#EGF<i]–‚ ð#†n
F<HVNGPQDtnC½¾D]<K>dehkR#K>dgHVUpYGD]• N TLK/R#K>dgHVU ]vYN
S1000
Ø
M ­ U«Y˜½¾D]<K>dehfDLFˆb½ dehEHVF>K/R#U TD‰YGDªTDLKˆDt—cDLKˆ]>NGFpbgDªTHzDt¥kTLdgDLUKmY˜½§R#K>K<SLU*N)R#K>dgHVUaI½§R#dxTR#bgTLNGbgSªbgD]
u
THVNGF>[D]AY˜½§R#K>K<SLU*N)R#K>dgHVUÆ[email protected]>l HzYGDk‡6DJ‡ÌYGSTLF>deK<[email protected]<STSYGDLhhfDLUKa‘hkR#dg]xYGD"hkR#UGdgiLF<Df»'F<DLEGF<H#
YNGdeF<DxNGU DxYdg]>K>F>de[GNGK>dgHVU!DLU
jNGd˜U D"]<HVdeKE)RV]EGbqR#K<D`n
sin2 θ
Mu­ UY˜½¾D]>K>dehfDLFŠP`bgHV[)R#bgDLhfDLUKpbqRªhfHzYd“­)TR#K>dgHVU«YGDˆbqRÅTHVNGF>[D%Y˜½§R#K>K<SLUN)R#K>dgHVUa;I½§R#d}hfH*YGSLbedg]/S%YGD
hkR#UGdgiLF<D"R#F>[GdeK>F/R#deF<D-bqRBŒŽHVF>[email protected]]<K>F>de[GNGK>dgHVU%DLU
2 E)R#F}NGU D–ŒŽHVU TLK>dgHVUmYNˆKÞ¼*EcD”
sin θ
2
F (sin θ) =
HVÈ
]<d

exp ln(1 + δ)( sin22 θ − 1)
sin θ
0
]>d
sin2 θ < sin2 θ0
sin2 θ ≥ sin2 θ0
œ`ðn§Ñ 
Ø
T HVF>F<D]<EHVU [email protected]#N\YGSL[GNGK£YGDobqR}w`R#F>dqR#K>dgHVU\YGDobqROTHVNGF>[D`a#DLK D]<K‘bqRuwVR#bgDLNGF7R#K>K<DLdeUK<DvDLU
δ
sin2 θ =
n7ù)½§R#d}TR#bgTLNGbgSˆ]>[email protected]]YGHVUGU SD]YN¡zÕ@a;DLU*K>F<D¬bgD
LD KbgD
arbgD]THVNGF>[D]
sin2 θ0
0.75


1
01/01/2004
31/06/2006
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Äz¿Ä
,6-0/.1.ðn§^
éÙ!ӑ×nØ6ß˔r‡rHVhE)R#F/R#dg]<HVUžYGD]ŠTHVNGF>[D]pY˜½§R#K>K<SLU*N)R#K>dgHVU¨HV[GK<DLUN D]¬Y R#U ]'bgD%TRV]ŠY˜½ NGU D
Ydg]>K>F>de[GNGK>dgHV2 UˆDLU
EGbqR#K<D`aGDLKvY R#U ]vbgDATRV]HVȬbqR\Ydg]>K>F>de[GNGK>dgHVUˆD]>KYGDAbqR"ŒŽHVF>hfD
œ’TtŒ>nDjN)R#K>dgHVU
sin2 θ
F (θ)
DLK
n Ö*Þ'æ!ÛÝ܏ßm”SLwQHVbeNGK>dgHVU‰YNªTHzDt¥kTLdgDLUK"Y˜½§R#K>K<SLU*N)R#K>dgHVU
`œ ðn§Ñ [email protected]]Aw`R#bgDLNGF<]
RwQDT–Ø b½§R#hEGbedeK>N YGD œdeU YdgδjN =SDA0.2
DLU Y δ T=
HVF>F<0.4
D]<EHVU Y R#U*Ku»-b½§R#hEGbedeK>N YGDAYGD–wVR#F>dqR#K>dgHVUpYGDObqRBYdg]<K>F>de[GNGK>dgHVUη
δ
DLU
2 DLUK>F<D
◦ DLK
◦n
sin θ
θ = 60
θ=0
Y˜½§R#K>K<SLUN)R#K>dgHVU‰THVF>F<D]>EcHVU Y R#UK"»ŠYd“—˜SLF<DLUK<D]xw`R#bgDLNGF<]AYGD aRwQDT
œ’THVF>F<D]>EcHVU Y R#UKB»
δ
sin2 θ0 = 0.375
tn
θ = 38◦
CD]THVNGF>[cD]\Y˜½§R#K>K<SLU*N)R#K>dgHVU‹R#deU ]>dvHV[GK<DLUN D]k]<[email protected] ]>K>F<SD]]>[email protected]­ P`NGF<Djðn§^n£C½¾SLwQHVbeNGK>dgHVU YN
TH*Dt¥kTLdgDLU*K"Y˜½§R#K>K<SLUN)R#K>dgHVUZD]>KxSLPR#bgDLhfDLUKBF<DLEGF<S]<DLUK<SD'DLU¤Œ’HVU TLK>[email protected]½§R#hEGbedeK>N YGD aDLKxb½¾HVUªEcDLNGK
δ
THVU ]>K/R#K<DLF-jN D wVR#F>[email protected]/R#K>dgj*N DLhfDLUK-bedeU SR#deF<DLhfDLU*K\RwQDT n˜™OU2RI¯N ]<K<DLhfDLUK-bedeU SR#deF<[email protected]ŠTHVNGF>[cD
η
δ
YGHVUGU D
n*MudeU ]>da[GdgDLU\j*N½ deb*]/HVdeK;Yd“¥kTLdebgDvYGDEGF<SYdeF<DRwQDT4EGF<STLdg]>dgHVUj*N DLbebgD4YGDLwzF/R#deK;yLK>F<D
η = 0.85+0.38δ
[email protected]ŒŽHVF>hfDBDts RVTLK<D"YGDBbqRfYdg]>K>F>de[GNGK>dgHVUªDLU
YGD]–YGHVUGU SD]adeb]<DLhB[GbgD\j*N D-bqRfwVR#bgDLNGFAYN%THzDt¥kTLdgDLUK
sin2 θ
Y˜½§R#K>K<SLUN)R#K>dgHVUXYGHVUGU SDkE)R#FBbqR¬hfSLK>l HzYGDŠ6
‡ D>‡ EcHVNGF>F/R#deK\yLK>F<DkF/R#dg]<HVUGU)R#[GbgDLhfDLUK
EGbeN ]BSLbgDLwQSD
∼ 10%
jN DxTDxj*N DxU HVN ]uRwQHVU ]}deU Ydgj*N S`a ]<HVdeKEGbeNGK<ÚVK
R#N¬bedgDLNmYGD
n
η ' 0.95 ± 0.02
£üòû£üòû
§
η ' 0.85 ± 0.02
ÿ%€X‚I>_JQ%^ I> Ò¹ <`ÿ=ÍO©bIL$K
©bKÆ
‡ HVhhfD¬U HVN ]fb½§RwQHVU ]fhfDLU*K>dgHVUGU SmEGF<STSYGDLhhfDLU*Ka4b½ l*¼*EcHVK>l i]<DˆYGDpŒRVTLK<HVF>dg]/R#K>dgHVU YN]>deP`U)R#bO»
r
_`_`_!h DLUXNGU DŠŒŽHVU TLK>dgHVUËYGDŠb½¾SLU DLF>P`dgDˆDLKNGU D'Œ’HVU TLK>dgHVUYGD U½¾D]>KTDLF>K/R#deU DLhfDLU*KkE)RV]–IJN ]>K>d“­)SDˆ»
θ

l)R#NGK<DOSLU DLF>P`dgD`nzC½¾DtsK>F/R#EHVbqR#K>dgHVUˆYGDubqR-F<DLbqR#K>dgHVU
D]<KoYGHVU T–TDLF>K/R#deU DLhfDLUK}DLUK/RVT/l SDxY˜½ NGU D
EF D − S38
DLF>F<DLNGFu]>¼]>K<SLhkR#K>dgjN DAdehEcHVF>K/R#UK<D`ajN½ debŒŽR#NGK}SLw`R#beN DLFn
¡*dbgD]}hfHzYGiLbgD]uYGDB]>deh"NGbqR#K>dgHVU¤YGDxPQDLF>[cD]OR#K>hfH`]>EGl SLF>dgj*N D]OU D"]<HVU*K}E)RV]}K<HVK/R#bgDLhfDLU*KO­R#[GbgD]a E)R#F
Äz¿Mv
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
tD sDLhEGbgDDLUTDvj*NGd THVU TDLF>U DvbqRuF<DLEGF<HzYN TLK>dgHVUfEGF<STLdg]<DYGDrbqR–EGF<HVEcHVF>K>dgHVUfYGDvh"N HVU ]7EGF<HzYNGdeK<DY R#U ]£bgD]
PQDLF>[D]oœ’jNGdQU HVN ]bgD7F/R#EGEDLbgHVU ]YGSLEDLU Y"YdeF<DTLK<DLhfDLUK£YGD]˜deUK<DLF/RVTLK>dgHVU ];l)RVYF<HVUGdgj*N D]ΝtabqRvhfHzYGSLbedg]/R#K>dgHVU
YGDb½§R#K>K<SLUN)R#K>dgHVUZYGDbqRpPQDLF>[cDkRwQDTfbqRpEGF<H#ŒŽHVU YGDLNGF\R#K>hfH`]>EGl SLF>dgj*N DfEcHVNGF>F/R#deK-U SR#UGhfHVdeU ]\yLK>F<D[GdgDLU
hfH*YGSLbedg]/SD`aGY R#U ]bqRBhfD]>NGF<D-HVÈpb½§R#K>K<SLUN)R#K>dgHVU!EGF<HVEGF<DLhfDLUK–YdeK<DAU Dx]<DLh"[GbgDAE)RV]}YGDLwQHVdeF}YGSLEcDLU YF<D-YGD
bqR\EGl¼]>dgj*N Dxl)RVYF<HVUGdgjN D`n
MOdeU ]>da`bgHVF<]<j*N DbgD}]>deP`U)R#b» _`_`_xhfiLK>F<D]oD]>K4YGHVhdeU S}E)R#F4bqRxTHVUK>F>de[GNGK>dgHVU¬YGDbqRxTHVhEcH`]/R#U*K<DuSLbgDTt

K>F<HVhkR#P`U SLK>dgjN D`a#T`½¾D]>KJÞ»†YdeF<DoEHVNGFbgD]R#UGP`bgD]ÔSLUGdeK>l)R#Ns-deUŒ’SLF>dgDLNGF<]£»
abqRTHVNGF>[cD4Y˜½§R#K>K<SLU*N)R#K>dgHVU
∼ 45◦
EGF<SYdeK<DxE)R#F}bgD]}Ï!HVUK<Dt¯‡vR#F>bgHkEDLNGKOyLK>F<DxTHVU ]>dgYGSLF<SD"THVhhfDxF<DLbqR#K>dewQDLhfDLU*Ku­R#[GbgD`n
–HVN ]‘EGF<HVEcH`]<HVU ]7R#bgHVF<]7Y˜½ NGK>debedg]<DLF;bgD];THVNGF>[cD]£Y˜½§R#K>K<SLUN)R#K>dgHVUfEGF<SYdeK<D]£E)R#F;bgD]£]>deh"NGbqR#K>dgHVU ]R­ UYGD
YGSLK<DLF>hdeU DLFvNGU
jNGdc]<HVdeKoTHVF>F<SLbgSA»"b½¾SLU DLF>P`dgD–{£ÕÌ»"b½§R#dgYGD–YGD]oYGHVUGU SD]rl¼z[GF>dgYGD]aYGD}bqR"hfyLhfD
S38 MC
hkR#UGdgiLF<DBj*N D-EcHVNGFubqRfhfSLK>l H*YGD6
‡ D>|
‡ ?–{;Õ@nC½§Rw`R#UK/R#PQD\E)R#F–F/R#EGEHVF>KA»bqRhfSLK>l HzYGD6
‡ DJ‡¨D]>KOj*N D-bqR
wVR#bgDLNGFxYNªTHzDt¥kTLdgDLUK-Y˜½§R#K>K<SLUN)R#K>dgHVUÅ]<DLh"[[email protected] ]–­R#[GbgD\EcHVNGF–b½¾DtsK>F/R#EcHVbqR#K>dgHVU‰»fl)R#NGK<DSLU DLF>P`dgD`acbq»
HVȬU HVN ]u]/RwQHVU ]}j*N DxbqR\wVR#bgDLNGFOYGD YN%6
‡ D>‡‹U½¾D]>KE)RV][cHVUGU D`n
η
,.-F/.1'ðn _
‡rHVNGF>[D]-Y˜½§R#K>K<SLU*N)R#K>dgHVUÅHV[GK<DLU*N D]-EcHVNGF-YGD]-]>deh"NGbqR#K>dgHVU ]"EGF<HVK<HVU ]a˜EcHVNGFxYGD]-SLU DLF>P`dgD]

2
R#bebqR#UKuYGD »
|D€@n CD-THzDt¥kTLdgDLUK–Y˜½§R#K>K<SLUN)R#K>dgHVU!SLwQHVbeN D-bgDLUK<DLhfDLU*K–RwQDT-b½¾SLU DLF>P`dgD`a DLKuD]<KEGbeN ]
ŒŽR#de[GbgD-»K>F<4i]l)100
R#NGK<Dkœ
|4D€-vjN½§»EGbeN ][)RV]<]<DxSLU DLF>P`dgDŠœ
|D†€xtn
E = 100
E = 10
"&4"Û4eÁgf˜&‘f DÞb D]>KEcH`]<]>de[GbgDYGDu]>NGdew*F<D}bqRAhfyLhfD}YGSLhkR#F<T<l D–j*N D}TDLbebgD}YGDbqRAhfSLK>l HzYGD
YṄ6DJ‡–avT`½¾D]>KJÞ»†YdeF<DªYGD¤]>NGEGEcH`]<DLFŠNGU DmŒRVTLK<HVF>dg]/R#K>dgHVU¨YN«K†¼zEcD
auDLU
ö
S1000 (E, θ) = f (θ) × g(E)
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Äz¿¿
D ]>K>dehkR#UK}TDLK>K<DAŒŽHVdg]J†TLdcbqR"Œ’HVU TLK>dgHVU
RwQDT-bgD]]>deh"NGbqR#K>dgHVU ]n)C‘[email protected])R#U*K>deK<SxTHVF>F<SLbgSD"»Bb½¾SLU DLF>P`dgD-{£Õ
f (θ)
D]>KuR#bgHVF<]bgD
THVU ]>K>F>NGdeKOYGDxbqR\hfyLhfDAŒRVWHVUmjN½§[email protected]>l HzYGDB‡6D>‡|?–{;Õ@n
S38 MC = S1000 (θ)/f (θ)
ù ½§R#d YGHVU TrNGK>debedg]<SvbgD]£hfyLhfD]]>dehBNGbqR#K>dgHVU ]YGDrPQDLF>[D]R#K>hfH`]<EGl SLF>dgjN D]oDLK£YNYGSLK<DTLK<DLNGF4j*N DrEGF<STSt
YGDLhhfDLUK7EcHVNGF;bgDvTR#bgTLNGbGYGDrbqROF<DLbqR#K>dgHVU
 RwQDTvbgD]£Ï%HVU*K<Dt¯‡vR#F>bgHGaQDLKIL½§R#dzK>F/RVWSvEcHVNGF;EGbeN ]<dgDLNGF<]
S1000 E
SLU DLF>P`dgD]DLK£K†¼zED]YGDrE)R#F>K>dgTLNGbgD]4EGF>dehkR#deF<D]uœEGF<HVK<[email protected] H¼R#NkYGDrŒŽDLFΝ‘bgD
hfH¼[email protected]ŒŽHVU TLK>dgHVUfYGD
S1000
nVCD]THVNGF>[cD]Y˜½§R#K>K<SLUN)R#K>dgHVUŠ]<HVUKK>F<i][GdgDLUkRI¯N ]>K<SD]E)R#F7bqROŒ’HVU TLK>dgHVUkDtsEcHVU DLUK>dgDLbebg
D ðn§Ñ aQYGD
1/ cos θ

]<HVF>K<DBj*N½ NGU¤TH*Dt¥kTLdgDLU*K-Y˜½§R#K>K<SLUN)R#K>dgHVU2EcDLNGK–yLK>F<[email protected]­ UGd7DLKATHVhE)R#F<SfR#NsmF<S]>NGbeK/R#K<]AYGD\bqRfhfSLK>l HzYGD
‡ DJ‡–n
6
”ZK>deK>F<DvY˜½¾DtsDLhEGbgD`abqR­ P`NGF<|
D ðn _udebebeN ]>K>F<DrbgD]THVNGF>[cD]£Y˜½§R#K>K<SLUN)R#K>dgHVUkHV[GK<DLU*N D]£EHVNGF£YGD]EGF<HVK<HVU ]

»}EGbeN ]>dgDLNGF<]£SLU DLF>P`dgD]nVCDvTHzDt¥kTLdgDLUKY˜½§R#K>K<SLU*N)R#K>dgHVUfD]>KEGbeN ]ŒR#de[GbgD»}K>F<i];l)R#NGK<Dxœ
|D€-jN½§»
E = 100
EGbeN ][)RV]<]<DxSLU DLF>P`dgDŠœ
|D†€xtaTD-j*NGdTHVU­ F>hfD-b½§R¥fF>hkR#K>dgHVU%j*N DxbgDxTHzDt¥kTLdgDLUK–Y˜½§R#K>K<SLUN)R#K>dgHVU
E = 10
SLwQHVbeN D"TDLF>K/R#deU DLhfDLUK-RwQDT-b½¾SLU DLF>P`dgD`nCD]O]<deh"NGbqR#K>dgHVU ]OEDLF>hfDLK>K<DLU*K–YGDBjN)R#U*K>d“­)DLF–TDLK–Dt—cDLKa DLKAR#deU ]>d
Y˜½¾SLwVR#beN DLF}bgDxYGHVhkR#deU DBYGDAwVR#bedgYdeK<S"YGDxbqR\hfSLK>l H*YGD\6
‡ DJ‡–n
YGD]-SLwQSLU DLhfDLU*K<]BYGHVU*K
‡rHVhhfDU HVN ]xb½§RwQHVU ]-F<DLhkR#F<j*N SkR#NGE)R#F/Rw`R#UKa‘bqRpYdg]>K>F>de[GNGK>dgHVU‰DLU
sin2 θ
b½¾SLU DLF>P`dgDŠD]<K\D]>K>dehfSpRwQDTŠbgD]@Ï!HVUK<Dt¯‡vR#F>bgH¤EGF<S]<DLU*K<D'NGU D'F<DLhfHVUK<SD¬]<deP`UGd“­)TR#K>dewQDp»mP`F/R#U YXR#UGP`bgD`n
ÕAR#U ]pb½ l*¼*EcHVK>l i]<DªHVÈ«bqRŌ’F/RVTLK>dgHVU¨YGD%hBN HVU ]ˆEGF<HzYNGdeK<D¤E)R#FpbgD]¬]>deh"NGbqR#K>dgHVU ]mU½¾D]>KpE)RV]ˆTHVF>F<DTLK<D`a
b½¾SLwQDLUK>N DLb£YGS]/RVTTHVF<Y%RwQDT-bgD]uYGHVUGU SD]OD]>KOR#K>K<DLU YNkI¯N ]>K<DLhfDLU*K–»@P`F/R#U Y!R#UGP`bgD`n)C‘RfF<DLhfHVUK<SD\YGDxbqR
Ydg]>K>F>de[GNGK>dgHVU'DLU
»–P`F/R#U Y'R#UGP`bgDEcHVNGF>F/R#deKoyLK>F<D}DLUkE)R#F>K>dgDuYN Du»ANGU Du]>NGF<D]>K>dehkR#K>dgHVU'YGD}b½¾SLU DLF>P`dgD
sin2 θ
E)R#FxbgD]xÏ%HVU*K<Dt¯‡vR#F>bgH¬EHVNGFxbgD]
a˜THVU ]<SjN DLU TDY˜½ NGU
EGF<SYdeKxK>F<HVEªŒŽR#de[GbgD\E)R#FxF/R#EGEcHVF>K
θ & 45◦
S1000
R#Ns¬YGHVUGU SD]nÕxR#U ]uTDLK>K<D"l¼zEHVK>l i]/D`a debŒR#N YF/R#deK–R#bgHVF<]OEGbeN ]–YGDxh"N HVU ]AY R#U ]ObgD]u]>dehBNGbqR#K>dgHVU ]OEcHVNGF
yLK>F<DxDLU%RVTTHVF<YmRwQDT"bgD]}YGHVUGU SD]n
|[email protected])R#F/R#UK7bgD]£THVNGF>[D]£Y˜½§R#K>K<SLU*N)R#K>dgHVUkYGD]]<deh"NGbqR#K>dgHVU ]RwQDTvTDLbebgDQœ’]ΝYGDobqR}hfSLK>l HzYGD}6
‡ DJ‡–a#HVU
THVU ]>K/R#K<DuDt—˜DTLK>dewQDLhfDLUKj*N D}bgD]rÏ%HVU*K<Dt¯‡vR#F>bgH"EGF<SYdg]<DLU*KvNGU DOR#K>K<SLUN)R#K>dgHVUpEGbeN ]odehEHVF>K/R#U*K<D–»-P`F/R#U Y
R#UGP`bgD`nc¡*[email protected]­ P`NGF<q
D ðn ac]<HVU*KOF<DLEGF<S]<DLU*K<SD]AbgD]–wVR#bgDLNGF<]AYN¤TH*Dt¥kTLdgDLU*KAY˜½§R#K>K<SLU*N)R#K>dgHVU HV[GK<DLUN D]
η
`
RwQDTxbgD]}]<deh"NGbqR#K>dgHVU ]a)EHVNGFbgD]uR#UGP`bgD] ◦
L
D
u
K
E
V
H
G
N
F
n
0 < θ < 60◦
0◦ < θ < 45◦
C‘R'THVhE)R#F/R#dg]<HVU¤RwQDT\bgD]uw`R#bgDLNGF<]OHV[GK<DLU*N D]-RwQDT\bqRhfSLK>l HzYGD\YNª6
‡ D>‡žD]>KOdeUK<SLF<D]<]ÎR#UK<D¬” EcHVNGF
bgD]'R#UGP`bgD] ◦
abgD]ŠÏ%HVU*K<Dt¯‡vR#F>bgHÅ]<DLhB[GbgDLUKpEGF<SYdeF<D%NGU D%R#K>K<SLUN)R#K>dgHVU«wzF/R#dehfDLUKŠK>F<HVE
0 < θ < 60◦
dehEHVF>K/R#U*K<D`aohfyLhfD!DLUTHVU ]<dgYGSLF/R#UKpbgDmTRV]ŠHVÈ debuŒŽR#debebgD%EGF<DLU YF<DmNGU D!THVNGF>[cDmY˜½§R#K>K<SLUN)R#K>dgHVU 6
‡ D>‡
THVU ]>K>F>NGdeK<D}RwQDTvNGU DYdg]>K>F>de[GNGK>dgHVU
U HVU\EGbqR#K<D YGDvÑ`_ Y n‡rDLEcDLU Y R#UKabgHVF<]<j*N Dob½¾HVUf]/DobedehdeK<D}R#Ns
sin2 θ
◦ a`bgD}YGS]/RVTTHVF<YkD]>KhfHVdeU ]7•R#P`F/R#U*KaQbgD}THzDt¥kTLdgDLUKrY˜½§R#K>K<SLUN)R#K>dgHVUpYGHVUGU S}E)R#FbgD]4Ï%HVU*K<Dt¯‡vR#F>bgH
θ < 45
¯ç¶ü ßJë’á‘ä›ãrï|9¯á ý#ì¿ûßûé7ݒÝ;ìÎàxÝìÎà#ë’á†ãÎàáàáßJàëŽê¾áîOÝéáTœ
Äz¿)ú
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
,.-F/.1<ðn
éÙ!ӑ×nØ6ߪ”‘‡rHVhE)R#F/R#dg]<HVUÆYGD]BTH*Dt¥kTLdgDLU*K<]BY˜½§R#K>K<SLU*N)R#K>dgHVUÆHV[GK<DLU*N ]\RwQDTfbqRphfSLK>l HzYGD
`

u

2
YN%‡6JD ‡DLKuRwQDTxbgD]Ï%HVU*K<Dt¯‡vR#F>bgHfEcHVNGFbgD]}R#UGP`bgD]
◦ nCD]uTH*Dt¥kTLdgDLU*K<]uY˜½§R#K>K<SLUN)R#K>dgHVU!YGHVUGU S]
E)R#FxbgDf‡6JD ‡ôTHVF>F<D]>EcHVU Y R#UK\»ŠNGU DYdg]>K>F>de[GNGK>dgHVU‰DLU θ < 260 j*NGd;U½¾D]<KxE)RV]AEGbqR#K<D`a‘RwQDT
a˜]<HVUK
SLPR#bgDLhfDLUKOF<DLEGF<S]<DLUK<S]n
Ö!Þ'æ*ە܏ß'”GÏ!yLhfD-T<l
H`]<D-EHVsin
NGFbgDθ]uR#UGP`bgD]
θ < 45◦
n
δ = 0.2
EHVNGF–bgD]AEGF<HVK<HVU ]xD]>KxRV]<]<DLÔ@EGF<HzT<l [email protected]]AwVR#bgDLNGF<]AYN‰‡6D>‡–nÉ;HVNGFxbgD]–U H¼R#NsªYGDBŒ’DLFabgD]AEGF<SYdgTLK>dgHVU ]
YGD]x]>deh"NGbqR#K>dgHVU ]\]<HVU*K-DLU THVF<DbqR#F>[email protected]#N†YGD]<]>N ]a˜Y˜½¾DLU*w*deF<HVU
n˜CDkTH*Dt¥kTLdgDLU*K"Y˜½§R#K>K<SLUN)R#K>dgHVU
45%
YGHVUGU SxE)R#FbgD]}]>dehBNGbqR#K>dgHVU ]}E)RV]<]/DAEHVNGFbgD]EGF<HVK<HVU ]OYGD
»
DLUK>F<D DLK _`[email protected]|D€@a ]<HVdeKNGU D
∼ 1 ∼ 0.75


wVR#F>dqR#K>dgHVU%Y˜½¾DLUwzdeF<HVU
n
20%
É HVNGFob½¾D]>K>dehkR#K>dgHVUŠYGD}b½¾SLU DLF>P`dgD`a*deb)D]>K4YGHVU TuEH`]/]>de[GbgD}Y˜½ NGK>debedg]<DLFrNGU
;
j*NGd EGF<DLU '
Y DLU'THVhEGK<D
S38 MC
bqRAbgDLUK<DOSLwQHVbeNGK>dgHVUŠYN'TH*Dt¥kTLdgDLU*KrY˜½§R#K>K<SLU*N)R#K>dgHVUn*™OU'RIJN ]>K<DLhfDLUKrbedeU SR#deF<DuYGDbqR-THVNGF>[D
YGHVUGU D
η(E)
bgD]w`R#bgDLNGF<]}]>NGdew`R#UK<D]"”
E
η = 1.13 (±2%) − 9.6 10 (±20%) ×
Proton (θ < 60◦ )
1EeV
E
−4
η = 1.14 (±2%) − 9. 10 (±12%) ×
Fer (θ < 60◦ )
1EeV
−4
œ`ðn§Ñ ÷`
œ`ðn§Ñ`¦Q
œ`ðn§Ñ`^Q
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
DLKEHVNGF}bgD]OR#UGP`bgD]deUŒ’SLF>dgDLNGF<]–»
Äz¿
45◦
”
E
η = 1.0 (±3%) − 2.3 10 (±17%) ×
1EeV
E
η = 1.27 (±2%) − 3. 10−3 (±10%) ×
1EeV
−3
œ`ðn ~ _Q
Proton (θ < 45◦ )
œ`ðn ~ 

œ`ðn ~ ÑQ
Fer (θ < 45◦ )
C ½ l¼zEcHVK>l i]<DAYGD}bqR-ŒŽRVTLK<HVF>dg]/R#K>dgHVU¬YGDObqR-F<DLbqR#K>dgHVU

 D]>KoYGHVU TxR#EGEGF<HsdehkR#K>dewQDLhfDLUKuwQSLF>d“­)SD
S1000 E
]>NGFBYGD]BPR#hhfD]@Y˜½¾SLU DLF>P`dgD]fRV]<]<DLÔkP`F/R#U YGD]nC½¾D]<K>dehkR#K>dgHVUZYNZTHzDt¥kTLdgDLUK »ˆNGK>debedg]<[email protected] R#U ]BU HVK>F<D
η
hfSLK>l H*YGDrD]>KYGHVU TodeK<SLF/R#K>dewQD–”EcHVNGF;T/l)RVjN DrSLwQSLU DLhfDLUKa#b½¾SLU DLF>P`dgDvD]>KD]<K>dehfSD4E)R#FbqR}F<DLbqR#K>[email protected] SD
E)R#FbgD]£]>dehBNGbqR#K>dgHVU ]u`œ ðn _QtaVHVNBE)R#F£bqRuhfSLK>l H*YGD}6
‡ D>‡–aDLK£bgDrTH*Dt¥kTLdgDLU*K D]>K;D]<K>dehfSvRwQDTrbgD]F<DLbqR#K>dgHVU ]

η
TLd“†YGD]<]>N ]n˜‡rHVUGU)R#dg]<]/R#U*KxbqRfF<DLbqR#K>dgHVU2DLU*K>F<[email protected]
L
D
–
K

b
¾
½
L
S
U
L
D
>
F
`
P
g
d
D
;
{
@
Õ
ab½¾SLU DLF>P`dgD MC D]>KxTR#bgTLNGbgSD`n
S38 MC
E S3 8
¡*db½¾SLU DLF>P`dgDBR#deU ]>dHV[GK<DLUN D"D]<KuYd“—cSLF<DLU*K<DBYGD-EGbeN ]uYGD
j*N D-TDLbebgD-YGDBYGSLE)R#F>KaGbgD"THzDt¥kTLdgDLUK D]<K
50%
η
F<DTR#bgTLNGbgSO»AE)R#F>K>deFvYGD}TDLK>K<D}U HVNGwQDLbebgDuwVR#bgDLNGFrYGDb½¾SLU DLF>P`dgD–DLKbqRxEGF<HzTSYNGF<DuD]>K4deK<SLF<SD`nz‡rHVU TLF<iLK<DLhfDLUKa
bgDTH*Dt¥kTLdgDLU*KrY˜½§R#K>K<SLU*N)R#K>dgHVU'w`R#F>dgDK<DLbebgDLhfDLUKoEDLNŠRwQDT}b½¾SLU DLF>P`dgDOjN DvhfyLhfDO]>dGb½¾SLU DLF>P`dgDuD]<K>dehfSDbgHVF<]
[email protected]<DLhdgiLF<D-SLK/R#EcD-D]>KvŒR#N ]<]<DxY˜½ NGU¬ŒRVTLK<DLNGFuÑaGb½¾DLF>F<DLNGF}F<DLbqR#K>dewQDB]>NGF D]<KdeUŒ’SLF>dgDLNGF<D"»
n
η
5%
Õ[email protected]ŒRVWHVUmj*N DxEcHVNGF}bqR\hfSLK>l H*YGDBYN!‡6D>‡|?–{;Õ@aVI½§R#d
")#Ázõò>fËQäõޓÁgf‘*( ‘
NGK>debedg]<S}bgDuhfyLhfDubgHVKrY˜½¾SLwQSLU DLhfDLUK<]l*¼*[GF>dgYGD]vRwQDT–bgD]hfyLhfD]vTLF>deK<iLF<D]vYGD}[cHVUGU D}F<DTHVU ]<K>F>N TLK>dgHVUpj*N D
EGF<STSYGDLhhfDLUKnGC‘Rx­ P`NGF<3
D ðn Ñ"debebeN ]>K>F<DObqR-THVF>F<SLbqR#K>dgHVU¬DLUK>F<DObgD
aEHVNGFobgD]oSLwQSLU DLhfDLU*K<]}R¼R#U*K
S38 MC

n
θ < 45◦
YGHVUGU Df”
C½§RIJN ]>K<DLhfDLUK–[email protected]<DLbqR#K>dgHVU
MC 
ö
S38
EFD
E = 0.14 S38 MC
1.08
œ`ðn ~`~ 
R wQDTAbgD]deU TDLF>K>deK>N YGD]O]>NGFvbgD]E)R#F/R#hfiLK>F<D]"”
DLK
n‡rD]w`R#bgDLNGF<]]<HVUK
K = 0.14 ± 0.02 α = 1.08 ± 0.04
K>F<i]uEGF<HzT<l D]–YGDBTDLbebgD]–HV[GK<DLUN D]-RwQDT"bqRfhfSLK>l H*YGD‡6D>‡|?–{;Õ@aTDBj*NGd‘hfHVU*K>F<DBj*N D"THVhEGK<D\K<DLUN¤YGD
bqR-]>K/R#K>dg]>K>dgj*N D}l*¼*[GF>dgYGD–Ydg]>EcHVUGde[GbgD–RVTLK>N DLbebgDLhfDLU*KaNGU D}w`R#F>dqR#K>dgHVUŠYNŠTH*Dt¥kTLdgDLU*KY˜½§R#K>K<SLUN)R#K>dgHVUmRxEcDLN
Y˜½ deU• N DLU TDB]>NGFub½¾SLK/R#bgHVUGU)R#PQD\[email protected]• N HVF<D]/TDLU TD"Y R#U ][email protected]#hhfD"Y˜½¾SLU DLF>P`dgD\HVȈdebD]<K}EH`]/]>de[GbgD-YGD
bgD–ŒŽR#deF<D`n
C½¾STR#F>[email protected]<DLbqR#K>d“ŒODLUK>F<DŠb½¾SLU DLF>P`dgDˆR#deU ]>drF<DTHVU ]>K>F>NGdeK<DŠE)R#F\bgD
DLK\b½¾SLU DLF>P`dgD'hfD]<NGF<SDkE)R#[email protected]
S38 MC
• N HVF<D]<TDLU TD-D]>K}DLU¬hfH¼QDLUGU D-YGz
D ð Y a)RwQDTANGU D"Ydg]>EDLF<]<dgHVUˆY˜½¾DLUwzdeF<HVU ~ ð Y awVR#bgDLNGF<]}EGF/R#K>dgjN DLhfDLU*K
dgYGDLUK>dgj*N D]O»@TDLbebgD]uHV[GK<DLU*N D]–RwQDTxbqR\hfSLK>l HzYGD\6
‡ D>|
‡ ?–{;Õ@n
Äz¿W
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
,.-F/.1ðn Ñ ‡rHVF>F<SLbqR#K>[email protected]*K>F<D4b½¾SLU DLF>P`dgDr{£ÕXDLK‘bgD
MC EcHVNGF‘bgD];SLwQSLU DLhfDLUK<];l¼z[GF>dgYGD]£]/R#K>dg]JŒR#dg]/R#UK
]>NGEcSLF>dgDLNGF<D]A»pÑ`_
gb D]ATLF>deK<iL F<D2 ]xYGDB]<SLbgDTLK>dgHVUn˜C4½§RI¯N ]>K<DLhfDLU*K-D]>K–F<SR#bedgS]<38
S\EHVNGFxYGD]–wVR#bgDLNGF<]AYGD
S38 MC
€–|4ÏÅaGYGDAhkR#UGdgiLF<D\»@]<DAEGbqRVTDLFOR#N†YGD]<]>N ]OYGDAb½¾SLU DLF>P`dgD"YGDx]/R#K>NGF/R#K>dgHVU!YGDAb½§RVTTDLEGK/R#U TD`n
M ­ UËYGDˆTHVhE)R#F<DLFkbqR¤j*N)R#bedeK<SˆYGD¬bqRªTHVF>F<SLbqR#K>dgHVU‹DLUK>F<D¬b½¾SLU DLF>P`dgD%{£Õ¢DLKkbgD
u
»ªTDLbebgDˆYGD
S38 MC
bqR%hfSLK>l H*YGDm‡6D>‡–aIL½§R#dTHVhE)R#F<S¬bqR¤Ydg]<EDLF<]>dgHVUXDLUËSLU DLF>P`dgD¬{£Õ¢jN Dpb½¾HVUËHV[GK>dgDLU*Kk]>[email protected]]YGHVUGU SD]
l¼z[GF>dgYGD]aGE)R#F}deU*K<DLF>wVR#bebgD"YGD
aGEcHVNGFbgD"‡6DJ‡«DLKEcHVNGF}bgD
n
S38
S38 MC
C‘R%­ P`NGF<*
D ðn ~ debebeN ]>K>F<DˆbgD]@F<S]>NGbeK/R#K<]fHV[GK<DLUN ]']>[email protected]]SLwQSLU DLhfDLUK<]'l¼z[GF>dgYGD]f]/R#K>dg]JŒR#dg]/R#U*KkR#Ns

TLF>deK<iLF<D]oYGDO]<SLbgDTLK>dgHVUnzCDOTRV]rY˜½ NGU
RwQDTONGU'THzDt¥kTLdgDLUKY˜½§R#K>K<SLUN)R#K>dgHVUp­GsGSO»-bqRxw`R#bgDLNGF
S38 MC
η=1
D]>KBSLPR#bgDLhfDLU*[email protected]<S]<DLUK<S`a7»¬K>deK>F<DŠYGDkTHVhE)R#F/R#dg]/HVUn;C‘D]\Ydg]<EDLF<]>dgHVU ]\HV[GK<DLUN D]@HVU*K\NGU DŠR#hEGbedeK>N YGD
THVhE)R#F/R#[GbgD`aRwQDTvNGU DvbgSLPQiLF<D}R#hfSLbedgHVF/R#K>dgHVUkEHVNGF£bgDvTRV]7YN
RwQDTSLwQHVbeNGK>dgHVUkYGD E)R#F£F/R#EGEcHVF>K
S38 MC
η
R#NˆTRV]}HVÈ D]>K­GsGD`n
η
C½ NGK>debedg]/R#K>dgHVU¬YGDubqRxhfSLK>l HzYGDOTHVNGF>[cDOY˜½§R#K>K<SLU*N)R#K>dgHVUˆÏ%HVU*K<Dt¯‡vR#F>bgs
H ?¨{£Õ¨YGHVUGU D–YGHVU TOYGD]oF<S]>NGb“
K/R#K<]K>F<i]}EGF<HzT<l D]OYGD-TDLNsˆHV[GK<DLUN ]–RwQDT-bqRhfSLK>l H*YGDB6
‡ D>|
‡ ?–{£Õ@a)RwQDT"NGU D-jN)R#bedeK<S"YGDxb½¾SLK/R#bgHVUGU)R#PQD
R#N ]<]>d˜[cHVUGU DxY R#U ]}bqR\PR#hhfDBY˜½¾SLU DLF>P`dgD-HVȈb½¾HVUmYdg]>EcH`]<DAYGD-]>N¥k]ÎR#hhfDLUKOY˜½¾SLwQSLU DLhfDLU*K<]n
£üòû£ü
§
¹
QJ#ÿ%NILK `ÿ$ I> `ÿ%
–HVN ]‘EGF<S]<DLU*K<HVU ];[GF>dgiLwQDLhfDLUKYGDLNs\R#NGK>F<D];hfSLK>l HzYGD]Y˜½¾D]>K>dehkR#K>dgHVUYGDob½¾SLU DLF>P`dgDvYGD]SLwQSLU DLhfDLU*K<]
YNfF<S]<DR#NkYGDu]>NGFJŒŽRVTDujNGd EcDLNGwQDLUKryLK>F<D}NGK>debedg]<SD]4EHVNGF4F<DTHVU ]>K>F>NGdeF<DubgDu]>EDTLK>F<DuY˜½¾SLU DLF>P`dgDuYGD]F/R¼QHVU ]
TH`]>hdgj*N D]n
cÀg¿ – 4i2ftÛ!õޓÁgf‘*(
Äz¿V[
, -0/.1ðn ~
6
wQHVbeNGK>dgHVUˆYGD–bqR\Ydg]>EDLF<]<dgHVUŠF<DLbqR#K>dewQDxDLU¬SLU DLF>P`dgDxDLUŠŒŽHVU TLK>dgHVUpYN
a*EcHVNGFrbqRBhfSLK>l HzYGD
S38

j£
2
‡6DJ‡–aGDLK}EcHVNGFbgD
MC a RwQDT-HVNm]/R#U ]}SLwQHVbeNGK>dgHVU!YNˆTH*Dt¥kTLdgDLU*KOY˜½§R#K>K<SLU*N)R#K>dgHVU¤RwQDTxb½¾SLU DLF>P`dgD`n
S38
C‘RŠhfSLK>l H*YGDTHVU ]<dg]>K<Df»p]>NGEGEcH`]<DLFxjN [email protected]ŠF<DLbqR#K>dgHVU

EGF<SYdeK<D
")#Ázõò>f
ESD EF D
S1000 E
E)R#F7bgD]]>deh"NGbqR#K>dgHVU ]oD]>KTHVF>F<DTLK<D`a»–NGU DU HVF>hkR#bedg]ÎR#K>dgHVUfEGF<i]nQÉ;HVNGF4D]>K>dehfDLF7bgDvŒRVTLK<DLNGFYGDTHVUwQDLF<]<dgHVU
DLUfSLU DLF>P`dgDu»xR#EGEGbedgjN DLFr»–b½¾SLU DLF>P`dgD}YGHVUGU SD}E)R#F7bgD]Ï!HVUK<Dt¯‡vR#F>bgHGa*HVUfSLK/R#bgHVUGU D}b½¾SLU DLF>P`dgDO¡z՞RwQDTbgD]
SLwQSLU DLhfDLUK<]vl¼z[GF>dgYGD]n ‡rHVhhfDOU HVN ]ob½§RwQHVU ]vw*N'EGF<STSYGDLhhfDLUKaGbqRBYdg]>K>F>de[GNGK>dgHVU¬DLU
HV[GK<DLUN D
sin2 θ
»mE)R#F>K>deFfYGDŠb½¾SLU DLF>P`dgDˆYGHVUGU SDpE)R#FbgD]]<deh"NGbqR#K>dgHVU ]U½¾D]>KE)RV]@EGbqR#K<D`a7TDpj*NGd]<DLh"[GbgD¬deU Ydgj*N [email protected] D
SLwQHVbeNGK>[email protected]½§R#UGP`bgDoYGDbqR}Ydg]<EDLF<]>dgHVU\HVNBYGD4b½¾DLF>F<DLNGF;]>¼]>K<SLhkR#K>dgj*N D4]>NGF‘b½¾SLU DLF>P`dgD`n#|UBTHVU ]<Sj*N DLU TD`a
HVU']½§R#K>K<DLU Yp»-TDOjN DubqRxF<DLbqR#K>dgHVUpDLUK>F<D
DLK
EcHVNGF4bgD]
T/l)R#UGPQD–RwQDTub½¾SLU DLF>P`dgD`n|7UŠ]<D
ESD EFD
θ < 60◦
bedehdeK/R#UKR#NsBR#UGP`bgD];deUŒ’SLF>dgDLNGF<]» ◦ aEcHVNGFbgD]/jN DLbg]bgD]THzDt¥kTLdgDLUK<]7Y˜½§R#K>K<SLU*N)R#K>dgHVUkYGD];Ï%HVU*K<Dt¯‡vR#F>bgH
45
]<DLh"[GbgDLU*KODLUmRVTTHVF<Y%RwQDTxbgD"6
‡ D>‡–[email protected]>F<SLbqR#K>dgHVU

HV[GK<DLUN DBD]>KhfDLdebebgDLNGF<D`n
ö
ESD
EF D
C ½¾SLK/R#bgHVUGU)R#PQDkDLU*K>F<Dfb½¾SLU DLF>P`dgD'YGD]-]>deh"NGbeK/R#K>dgHVU ]\DLKxb½¾SLU DLF>P`dgDk{£Õ RwQDTfU HVK>F<DbgHVK"Y˜½¾SLwQSLU DLhfDLUK<]

l¼z[GF>dgYGD]}EcHVNGFbgD]uR#UGP`bgD]}deUŒŽSLF>dgDLNGF<]O» ◦ YGHVUGU Dk”
45
E = 0.78 (±0.03) × EMC
Hyp. proton
E = 1.02 (±0.04) × EMC
Hyp. fer
C½¾ST<l DLbebgD'Y˜½¾SLU DLF>P`dgDf{£Õ±D]>K-YGHVU TfDLUwzdeF<HVU
20%
œ`ðn ~ „*
œ`ðn ~ ðQ
DLU†YGD]<]<HVN ]-b½¾SLU DLF>P`dgDkÏ!HVUK<Dt¯‡vR#F>bgHˆEGF<HVK<HVUa‘DLK
Äz¿c
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
EGF/R#K>dgjN DLhfDLU*KudgYGDLU*K>dgjN D"»b½¾SLU DLF>P`dgD-Ï%HVU*K<Dt¯‡vR#F>bgHŒŽDLFn
É HVNGF%SLK/R#bgHVUGU DLFmbqR
F<DLbqR#K>dgHVU
 arHVU EcHVNGF>F/R#deKpSLPR#bgDLhfDLU*K¬EDLU ]/DLFŠ»Å]<SLE)R#F<DLF'bgD%bgHVKpYGD%YGHVUGU SD]Šl¼z[GF>dgYGD]pE)R#F
S1000 E
deUK<DLF>w`R#bebgDxY˜½§R#UGP`bgD]}ÔSLUGdeK>l)R#Ns˜aGDLK}YGDAYGSLK<DLF>hdeU DLFuY R#U ]}T<l)RVTLNGUˆYGD]}deUK<DLF>w`R#bebgD]bqR\F<DLbqR#K>dgHVU
S1000 (θ)

n7‡rDLK>K<DŠhfSLK>l HzYGDˆ»ˆb½§RwVR#U*K/R#PQDˆYGDŠU D'YGSLEcDLU YF<D¬Y˜½§R#N TLNGU Dpl¼zEcHVK>l i]<Dp]>NGFBbqR!THVhEcH`]>deK>dgHVUa
EFD
UGd4]>NGF-bqR¬Œ’HVF>hfD'YGDbqRˆTHVNGF>[cDfY˜½§R#K>K<SLU*N)R#K>dgHVU
n˜C‘Dk]<DLNGbEGF<HV[GbgiLhfDŠYGDkTDLK>K<DkhfSLK>l H*YGD'D]>K"j*N D
f (θ)
bqR!]>K/R#K>dg]>K>dgj*N D'l*¼*[GF>dgYGDˆU STD]<]/R#deF<DpEcHVNGF\F<SR#bedg]/DLFfTDLKRIJN ]>K<DLhfDLU*KfE)R#FdeUK<DLF>w`R#bebgD¬Y˜½§R#UGP`bgD¬ÔSLUGdeK>l)R#b
D]>K\[DR#N THVNGEXK>F<HVEZŒR#de[GbgD¬»mb½ l DLNGF<DˆRVTLK>N DLbebgDª”;deboŒŽR#N YF/R#deKkR#K>K<DLU YF<DpEGbeN ]<dgDLNGF<]YdeÔR#deU D]fY˜½§R#UGU SD]
EHVNGF-HV[GK<DLUGdeF"NGU DEGF<STLdg]>dgHVU‰]>N¥k]ÎR#UK<Dg•‘C4½¾SLK/R#bgHVUGU)R#PQDkDLUK>F<Dkb½¾D]>K>dehkR#K<DLNGF"YNÆ¡zÕ±DLK-b½¾SLU DLF>P`dgDk{;Õ
YGHVdeKu]½¾Dt—˜DTLK>N DLFODLUm]/DAF/R#EGEHVF>K/R#U*KA»@NGU!R#UGP`bgD-YGDxF<StŒŽSLF<DLU TD`aGŒR#NGK<D-YGD"j*N HVd˜[email protected]ŒR#de[GbgD]<]<DxYNˆU HVhB[GF<D
Y˜½¾SLwQSLU DLhfDLUK<]OdeUK>F<HzYNGdeKuNGU D-DLF>F<DLNGFu[DR#N THVNGEˆK>F<HVEmdehEHVF>K/R#U*K<D`n
ï zõò>f
ñA, Š
S1000 E
ºXó<9;6}óˆ³\6
Ü;G Û; Û;f)fÁ hMÜV0Û* &)fÀ’&i2gf
²
)µ'579?)·c5Ò571
B
¶71B5£·Lö}:<5
–HVN ]@RwQHVU ]@YGSTLF>deK\EGF<STSYGDLhhfDLU*KkYd“—˜SLF<DLUK<D]hfSLK>l HzYGD]@Y˜½¾D]>K>dehkR#K>dgHVUËYGDkb½¾SLU DLF>P`dgDpYGD]@SLwQSt
U DLhfDLUK<]arYGSLK<DTLK<S]'E)R#F'bgDmF<S]<DR#NYGD%]>NGFJŒŽRVTD`ar»ªE)R#F>K>deFpYGD]<j*N DLbebgD]kdeb–D]>KfEcH`]<]>de[GbgDmYGD%THVU ]>K>F>NGdeF<D
Yd“—cSLF<DLU*K<]u]>EcDTLK>F<D]}Y˜½¾SLU DLF>P`dgD`n
ù ½§R#dcNGK>debedg]<SObqRBEGF<H*TSYNGF<DAYGDOF<DTHVU ]>K>F>N TLK>dgHVUˆYGSTLF>deK<D–EGF<STSYGDLhhfDLUK}EcHVNGF}R#U)R#be¼z]<DLFrbgD]vYGHVUGU SD]
YNªEGF<DLhdgDLFI<R#UwzdgDLFBÑ`_`_#„mR#N ~ IJNGdebebgDLK"Ñ`_`_ nCD]-TLF>deK<iLF<D]BjNGd7HVUKBSLK<SkR#EGEGbedgjN S]BR#Ns¤SLwQSLU DLhfDLU*K<]
Ø
]<HVUKbgD]O]>NGdew`R#UK<]-”
b½¾SLwQSLU DLhfDLUK–YGHVdeKuyLK>F<z
D (v„
?
b½¾SLwQSLU DLhfDLUK-YGHVdeKAyLK>F<u
D (Xð%œ’TLdeU jˆwQHVdg]>deU D]AHVESLF/R#K>dgHVUGU DLbebgD“
] ?בT “ANGFOF<DTHVU ]<K>F>NGdeKAY R#U ]–NGU!K>F>d“
?
R#UGP`bgD-YGD-]<K/R#K>dgHVU ]uHVEcSLF/R#K>dgHVUGU DLbebgD]Ν
b½§R#UGP`bgDxÔSLUGdeK>l)R#b;D]>KdeUŒ’SLF>dgDLNGF–» ◦
?
60
b½¾SLU DLF>P`dgDxF<DTHVU ]>K>F>NGdeK<DBD]>K}]>NGEcSLF>dgDLNGF<D"» |D†€
3
?
Mu­ U¤Y˜½¾HV[GK<DLUGdeFxNGU¤[email protected] SD]xYGD\bqRkhfDLdebebgDLNGF<Dj*N)R#bedeK<[email protected]`]<]>de[GbgD`aIL½§R#d7SLbedehdeU [email protected]]–ESLF>dgHzYGD]
YNGF/R#UK}bgD]/jN DLbebgD]ub½§RVTjNGdg]>deK>dgHVU%YGD]}YGHVUGU SD]uSLK/R#deK}EcDLF>K>NGF>[cSD`n CD]}EcSLF>dgHzYGD]}DtsTLbeN D]u]<HVUK"”
YN ¦ ‡V_`¦ ‡VÑ`_`_#„ŠR#N!Ñ ‡ ‡VÑ`_`_#„%”EGF<HV[GbgiLhfD"Y R#U ]}b½§R#bePQHVF>deK>lGhfDBY˜½§RVTj*NGdg]>deK>dgHVUmYN!‡rÕuMA¡
?

Ø `
YNBÑ ÷J‡V_ ~ ‡VÑ`_`J_ ð–R#NB_ ÷J‡V_#„ ‡VÑ`_`J_ ðB”EGF<HV[GbgiLhfD4Y R#U ]‘b½§R#bePQHVF>deK>lGhfDoYGDYGSTLbgDLU T/l DLhfDLUKYGD]]>K/R#K>dgHVU ]
?
JçJߏèé#êQàü ê§ïñ#ä§ê›èé#ß;ñ#ãÎë‘èé#ߣä§ßJë˜â†ãJîìÎà#ëçJìtëŽïê›èé#߯ë˜ëŽìtê§ßJàá;ç¯ìtïñ`ìtë`9¯ë'ûß7à#ì>îã/éOàûß1›“߯âTœ
cÀeúsö" õÃ&;õ+Û£&Ë@܏Ã`#mgf Ágf‘*(
Äz¿v
YNmÑ`Ñ ‡V_#„‡VÑ`_`_JðkR#N%_ ~ V‡ _Jð ‡VÑ`_`_Jðm”EGF<HV[GbgiLhfD"YGD-THVhhBNGUGdgTR#K>dgHVU%F/RVYdgHGn
‡rD]@ESLF>dgHzYGD]THVUK>F>de[GN DLU*KŠTDLEcDLU Y R#UKfK>F<i]@EcDLNËDLUXK<DLF>hfD¬YGDpU HVh"[GF<DmY˜½¾SLwQSLU DLhfDLUK<]aYNZŒR#deKfYGD
b½§R#NGP`hfDLUK/R#K>dgHVU¤[email protected]/R#debebgD-YNˆF<S]<DR#NmRwQDT-bgDAK<DLhE ]n
?
C DªTR#bgTLNGb"YGDªb½§RVTTDLEGK/R#U TDÆRZSLK<SªDt—cDTLK>N SÅRwQDTªbqRZhfSLK>l HzYGD2YN¨THVhEGK/R#PQDÅY˜½ l DtsGR#PQHVU D œ’TtŒ>n
T<l)R#EGdeK>F<D%„*¬”4R#N‹THVNGF<]'YGDmbqRªEcSLF>dgH*YGDmR#bebqR#U*KŠYN er I>R#U*w*dgDLFpÑ`_`_#„ÆR#N ~ I¯NGdebebgDLKpÑ`_`_ deU TLbeN ]a4DLU

Ø
DtsTLbeN)R#U*KrbgD]vEcSLF>dgHzYGD]rTLd“†YGD]/]>N ]a*bgDOF<S]/DR#NpYGD–]>NGFJŒRVTDAYGDOb½ š [ ]/DLF>wVR#K<HVdeF<DAÉ7dgDLF>F<DxMONGPQDLF}RBRVTTLNGhBNGbgS
Ç h 2 n¾]>Fn¾]aHVÈ
Ç h 2
TDLbebeNGbgD6
] (Xð†]/DTHVU YGD]a*]/HVdeK4NGU DORVTTDLEGK/R#U TD–K<HVK/R#bgD–YGD
≈ 2.94 1010
1.35 1011
4280
]>F"R#UacEHVNGFBYGD]-SLwQSLU DLhfDLUK<]@R¼R#U*K\YGD]"R#UGP`bgD]
n‘|7U‰]<[email protected]/R#U*K\R#Ns2R#UGP`bgD]-deUŒŽSLF>dgDLNGF<]B»
θ < 60◦
Ç h 2 n¾]>Fn§R#UaG]<HVdeKuDLU*w*deF<HVU
MuAMA¡zM\n
◦ ab½§RVTTDLEGK/R#U TDBD]>K}YGD
11 Ç h 2 n¾]<Fn¾]aGHVN
45
0.9 10
2850
1.8×
−1 a`D]>KYGHVUGU SE)R#F4bgDvU HVh"[GF<DuY˜½¾SLwQSLU DLhfDLU*K<]vYGSLK<DTLK<S]E)R#F
C D]>EcDTLK>F<D}Y˜½¾SLU DLF>P`dgD`aDLU 2
m .s.sr.eV
deUK<DLF>w`R#bebgD-Y˜½¾SLU DLF>P`dgD`a)Ydewzdg]<SAE)R#F}b½§RVTTDLEGK/R#U TDBTHVF>F<D]>EHVU Y R#U*K<D`n
C ½§RVTTDLEGK/R#U TD¬YNÆF<S]<DR#NÆYGDŠ]>NGFJŒRVTD'[email protected]>F<Dp]/R#K>NGF<SDpR#NsÆR#bgDLUK<HVNGF<]YGD

D€@a;RwQDTŠbqR
E = 4
F<StŒ’SLF<DLU TD-Y˜½¾SLU DLF>P`dgD\YGHVUGU SD-E)R#FubgD]u]>dehBNGbqR#K>dgHVU ]n)ù ½§R#d;TR#beNGbgSxbgD"]<EDTLK>F<D-DLU%SLU DLF>P`dgDBR#N†YGD]<]>N ]OYGD
3
|D€@a˜RwQDTYGD]AdeUK<DLF>w`R#bebgD]xYGD
DLU
acK<HVNGKxDLUª]/RVT/l)R#UK"j*N D"bqR'EGF<DLhdgiLF<[email protected]`R#bgDLNGFa
0.1
log(E)
E ' 3.16
|D€@aF>dg]<j*N DY˜½¾yLK>F<DK>F<HVE"[)RV]<]<DuœŽRVTTDLEGK/R#U TDoF<SDLbebgD7EGbeN ]˜ŒR#de[GbgDtnCRvE)R#F>K>dgD4YN"]<EDTLK>F<DHVÈ"b½§RVTTDLEGK/R#U TD
D]>Ku»@THVNGEmI] |GF}]/R#K>NGF<SD"]<Dx]>deK>N D-YGHVU T"R#N†YGD]/]>N ]}YGD 18.6 D†€\n
10
CD]}]>EcDTLK>F<D]vEGF<S]<DLU*K<S]uY R#U ]TDLK>K<D-SLK>N YGDx]<HVU*KK>F/RVWS]uRwQDT"YGD]vdeU TDLF>K>deK>N YGD]O]>NGFvbgD–• NspTHVF>F<D]J
EHVU Y R#U*KNGUGdgj*N DLhfDLUK7»NGU DoDLF>F<DLNGF]>K/R#K>dg]>K>dgj*N D4]<NGFbgDU HVhB[GF<DoY˜½¾SLwQSLU DLhfDLU*K<]7Y R#U ]T/l)RVjN D4deUK<DLF>w`R#bebgD
Y˜½¾SLU DLF>P`dgD`nCD]-[)R#F>F<D]"Y˜½¾DLF>F<DLNGF<]"]<HVU*K"YGHVU Tf]<DLNGbgDLhfDLU*K"deU YdgTR#K>dewQD]aDLK"U DYGHVdewQDLUK"E)RV]"yLK>F<DfTHVU ]>d“
YGSLF<SD]fTHVhhfDˆF<S]>NGbeK/R#U*K'Y˜½ NGU DˆD]<K>dehkR#K>dgHVUËEGF<STLdg]<D`nCD]kdeU TDLF>K>deK>N YGD]']<NGFb½¾SLU DLF>P`dgDˆU D¬]<HVU*KfE)RV]
F<DLEGF<S]<DLUK<SD]abgD]4]>EcDTLK>F<D]hfHVU*K>F<S]4E)R#F4bqR-]>NGdeK<D}THVU ]<K>deK>N DLUKvYGD]7F<S]>NGbeK/R#K<]oEGF<SLbedehdeU)R#deF<D]n*CRxYGSLK<DLFJ
hdeU)R#K>dgHVUŠYGDOK<HVNGK<D]4bgD]r]<HVNGF<TD]vY˜½¾DLF>F<DLNGF<]v]>¼]>K<SLhkR#K>dgjN D]r]>NGFob½¾SLU DLF>P`dgD–D]<K4NGUkEGF<HV[GbgiLhfDATHVhEGbgDtsGD`a
jNGd)ŒŽR#deKvDLU THVF<Dub½¾HV[I>DLKvYGDuU HVhB[GF<DLN ]<D]SLK>N YGD]rR#Np]<DLdeUpYGD}bqRBTHVbebqR#[HVF/R#K>dgHVU¬MONGPQDLFa*DLKvj*NGd)U½¾D]>KrE)RV]
bgDA[GNGKuYGDxTDLK>K<DxEGF<S]<DLU*K<D"SLK>N YGD`n
C D]r]<HVNGF<TD]EGF>deU TLdeE)R#bgD]vY˜½¾DLF>F<DLNGFv]>¼z]<K<SLhkR#K>dgjN DuEHVNGFbgD]ohfSLK>l HzYGD]oj*NGd NGK>debedg]<DLU*Kob½¾SLU DLF>P`dgD–YGHVU
U SDuE)R#FrbqRx• N HVF<D]/TDLU TDAEGF<HwzdgDLUGU DLUK}YGDObqR-F<DLbqR#K>dgHVU¬Y˜½¾SLK/R#bgHVUGU)R#PQD

nzC½¾HVF<YF<DAYGD–P`F/R#U YGDLNGF
S38 EFD
YGDrTDLK>K<DDLF>F<DLNGFoD]>K7Y˜½¾DLUwzdeF<HVU
»u[)RV]/]<DvSLU DLF>P`dgD}DLK
EHVNGF7bgD]7SLU DLF>P`dgD]4R#NGK<HVNGFoYGD
|D†€\nJDÞb
30%
50%
100
D]>KoYGHVU T–»-b½ l DLNGF<DARVTLK>N DLbebgDuEGF<SLhkR#K>NGF<SAYGDOTHVU TLbeNGF<D–j*N)R#UK»xbqR-EGF<S]<DLU TD–HVN'U HVUŠY˜½ NGU DOTHVNGEGNGF<DAYGD
K†¼zED- Y R#U ]bgDA]>EcDTLK>F<D`nGÉ;HVNGF}bgD]vhfSLK>l H*YGD]}NGK>debedg]/R#UK}bgD]]<deh"NGbqR#K>dgHVU ]aGb½ deU TDLF>K>deK>N YGD-EGF>deU TLdeE)R#bgD
EGF<HwzdgDLUKfYNXhfHzYGiLbgDŠY˜½ deU*K<DLF/RVTLK>dgHVU l)RVYF<HVUGdgj*N DpNGK>debedg]<S`[email protected]]<DLh"[GbgDpEGF>N YGDLU*KkYGDpTHVU ]>dgYGSLF<DLFNGU D
DLF>F<DLNGFu]>¼]>K<SLhkR#K>dgjN D"]>NGF}b½¾SLU DLF>P`dgD"YGD-b½¾HVF<YF<D-YGD
a)THVhEGK<DxK<DLU*N%YGDxb½¾STR#F>KAR#EGE)R#F<DLUK–DLUK>F<D-bgD]
30%
Ä*ú
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
ST<l DLbebgD]OY˜½¾SLU DLF>P`dgD"{£Õ DLKO¡zÕ\n
C‘R-­ P`NGF<D3ðn „@debebeN ]>K>F<DObgD]r]>EDTLK>F<D]vHV[GK<DLU*N ]}RwQDTObqR-hfSLK>l HzYGDAYN¬‡6D>‡|?–{£ÕÌDLKRwQDTObgD]r]<deh"NGbqR

K>dgHVU ]u”Q»AE)R#F>K>deFrYN
œŽRwQDTuSLwQHVbeNGK>dgHVU'YGD £HVNkYdeF<DTLK<DLhfDLU*KvRwQDT}bqRAF<DLbqR#K>dgHVU
 YGSTLF>deK<D
S38 MC
η
S1000 E
DLU‰œ`ðn _Qtn

CD]O]>EDTLK>F<D]–HV[GK<DLUN ]A»E)R#F>K>deF–YGD-bqRfhfSLK>l H*[email protected]‡6DJ‡|?–{£ÕƒDLKuYGD"bqRhfSLK>l HzYGD
?–{;Õ ]/HVUK
S38 MC
EGF/R#K>dgjN DLhfDLU*K"dgYGDLU*K>dgjN D]a;RwQDTfY R#U ]-b½ deUK<DLF>w`R#bebgD
NGU DEcDLUK<DfbgHVPR#F>deK>l
E ∈ [1018.6 eV , 1019.6 eV]
hdgjN DkYGD
EcHVNGF-bgD'6
‡ D>‡ DLK
EcHVNGF-bgD
n;‡rD
γCIC = −2.81 (±0.09)
γS38MC = −2.78 (±0.15)
S38 MC
F<S]>NGbeK/R#K£U½¾D]>K;E)RV]7]>NGF>EGF<DLU)R#UKTR#F£bgD];F<DLbqR#K>dgHVU ]7Y˜½¾SLK/R#bgHVUGU)R#PQD}DLUK>F<Drb½¾SLU DLF>P`dgD{£Õ DLK£bgD
]<HVUK£EGF/R
S38
K>dgjN DLhfDLU*K}dgYGDLUK>dgj*N D]EcHVNGFbgD]YGDLNs'hfSLK>l HzYGD]nGCD]EcDLUK<D]}bgHVPR#F>deK>lGhdgjN D]}YGD]u]>EDTLK>F<D]HV[GK<DLU*N ]
RwQDTxbgD]OR#NGK>F<D]}hfSLK>l HzYGD]u]<HVU*K}SLPR#bgDLhfDLUK–K<HVNGK<D]uTHVhE)R#K>de[GbgD]ORwQDT"TD]w`R#bgDLNGF<]n
,.-F/.1Lðn „ ”ôPR#N T<l Dp”)]<EDTLK>F<DBDLU!SLU DLF>P`dgD\YGD]uF/R¼QHVU ]ATH`]>hdgjN D]ARwQDTBb½¾SLU DLF>P`[email protected]]>K>dehfSDBE)R#FubgD]
]>deh"NGbqR#K>dgHVU 2 ]x”»"E)R#F>K>deFvYN
œŽRwQDTASLwQHVbeNGK>dgHVUˆYGD oHVNŠYdeF<DTLK<DLhfDLU*KORwQDT–bqR"F<DLbqR#K>dgHVU

S38 MC
η
S1000 E
YGHVUGU SD"DLy
U ðn _a)DLK–»E)R#F>K>deF–YGD"[email protected]>l H*YGD‡6D>‡|?–{;Õ@n–” YF<HVdeK<D'”GbgD-hfyLhfDB]>EcDTLK>F<D-h"NGbeK>deEGbedgS\E)R#F
a3 YGDx]<HVF>K<D" jN½ NGU D-SLwQHVbeNGK>dgHVU%DLUˆbgHVdYGDxEGNGdg]<]/R#U TD −3 R#EGE)R#F/R#dg]<]<DAEGbqR#K<D`n
E
E
C HVF<]<j*N DAb½¾HVUm]<D–bedehdeK<D"R#NsˆR#UGP`bgD]deUŒŽSLF>dgDLNGF<]O»
YGDLP`F<S]abgDx]<EDTLK>F<DxHV[GK<DLU*N%D]>KEcDLNˆYd“—cSLF<DLU*K
45
YGD'TDLbeNGdTHVF>F<D]>EcHVU Y R#UKk»
nCRˆ­ P`NGF<Djðn ð%debebeN ]>K>F<DpTDLK>K<DpYd“—cSLF<DLU TDŠEcHVNGFBbgD]\hfSLK>l HzYGD]
θ < 60◦

[)RV]<SD]4]<NGF4bgD]4Ï!HVUK<Dt¯‡vR#F>bgHGaj*NGd)]<HVU*K4bgD}EGbeN ]r]>N ]<TDLEGK>de[GbgD]rYGDw`R#F>dgDLF4bgHVF<]<j*N D}b½¾HVU'E)RV]<]<DOYGD
θ < 60◦
»
nCDf]<EDTLK>F<DTHVF>F<D]>EcHVU Y R#UK»
D]>KxbgSLPQiLF<DLhfDLU*KBEGbeN ]"[)RV]xEcHVNGF-bgD]"]>dehBNGbqR#K>dgHVU ]
θ < 45◦
θ < 45◦
÷tEGF<HVK<HVU‘÷Va DLK}bgSLPQiLF<DLhfDLU*KuEGbeN ]}l)R#[email protected]>l HzYGD
MC ?–{£Õ@n
S38
CD]'­ P`NGF<D]p]>NGdew`R#UK<D]ŠF<DLEGF<S]<DLUK<DLU*KˆbqRÅTHVhE)R#F/R#dg]<HVUžYN‹]>EcDTLK>F<D%hfD]>NGF<SmE)R#Fpb½ š [ ]<DLF>w`R#K<HVdeF<D
cÀeúsö" õÃ&;õ+Û£&Ë@܏Ã`#mgf Ágf‘*(
Ä*ú‘Ä
,6-0/.1 ðn ð
θ < 45◦
‡rHVhE)R#F/R#dg]<HVUDLUK>F<D¬bgD¬]>EcDTLK>F<Dªœh"NGbeK>deEGbedgSmE)R#F 3 BHV[GK<DLUNËEcHVNGF
◦ DLKEcHVNGF
 aEcHV2 NGFrbqR-hfSLK>l HzYGD
?–{£ÕÌDLKrEHVNGFrb½¾D]>K>dehkR#K>dgHVU¬EYGHVUGU SD–E)R#FobgD]v]>dehBθNG<
bqR#K>dg60
HVU ]vEGF<HVK<HVUn
S38 MC
É dgDLF>F<DxMuNGPQDLF}RwQDTAbqR"hfSLK>l HzYGDxÏ!HVUK<D-‡vR#F>[email protected]<HVK<HVUˆDLK}RwQDTAbqRBhfSLK>l H*YGD"‡6D>‡|?–{£Õ@azTHVhE)R#F<S-R#Ns
7
hfD]>NGF<D]vYGD}b½¾DtsESLF>dgDLU TDAMuAMx¡zM¨DLKoTDLbebgD]vY˜½¾ŸOdeF<D]nzC‘R-U HVF>hkR#bedg]ÎR#K>dgHVUˆYNŠ]>EcDTLK>F<D–HV[GK<DLUN DxRwQDT–bqR
hfSLK>l H*YGD"6
‡ D>|
‡ ?–{£Õô]/DLh"[GbgD-yLK>F<DAdeUŒŽSLF>dgDLNGF<DxY˜½ NGU DxwzdeUGP`K/R#deU DxYGDAEHVNGF<TDLU*KO»@TDLbebgDxYGDxŸudeF<D]nGC½¾STR#F>K
DLUK>F<D\bgD"]>EcDTLK>F<D\MuNGPQDLFAÏ%HVU*K<[email protected]‡vR#F>bgHpDLK–bgD"]>EcDTLK>F<D\MuNGPQDLF-6
‡ DJ‡Ì]<DLhB[GbgDByLK>F<[email protected]#bgDLhfDLUKxYN!hfyLhfD
HVF<YF<D-YGDAP`F/R#U YGDLNGFOj*N DAb½¾STR#F>KuDLUK>F<DBMuAMx¡zM DLK}ŸOdeF<D]n
CDuYGS]ÎRVTTHVF<Y'RVTLK>N DLb]<NGF4bqRAU HVF>hkR#bedg]/R#K>dgHVU¬YNk]<EDTLK>F<DODLUK>F<DuDtsEcSLF>dgDLU TD]4NGK>debedg]/R#U*KobqRxK<DT<lGUGdgj*N D
YGDObqRB• N HVF<D]/TDLU TD-DLKvbgD]F<S]<DR#NsŠYGDxYGSLK<DTLK<DLNGF<]OR#Nˆ]<HVb˜]<DLh"[GbgDxEcHVNGFvb½ deU ]>K/R#UKOHV[ ]<DLF>wQSxYGDAhkR#UGdgiLF<D
]>dehdebqR#deF<DxE)R#F}b½ š [ ]<DLF>w`R#K<HVdeF<D-É7dgDLF>F<DBMuNGPQDLFn
Ä*úmv
,.-F/.1gðn
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
¡*EcDTLK>F<D}DLU'SLU DLF>P`dgDOYGD]7F/R¼QHVU ]rTH`]>hdgj*N D]hBNGbeK>deEGbedgSuE)R#F 3 aQHV[GK<DLUN¬RwQDTub½¾D]>K>dehkR#K>dgHVU

B
Ø
2
Ï%HVU*K<Dt¯‡vR#F>bgHŠEGF<HVK<HVU!DLKOb½¾D]>K>dehkR#K>dgHVUª‡6D>‡|?–{£Õ@a)EcHVNGFOYGD]AR#UGP`bgD]uÔSLUGEdeK>l)R#Ns%deUŒ’SLF>dgDLNGF<]A» ◦ nCD]
YGHVUGU SD]uYGDAb½¾DtsESLF>dgDLU TD"MuAMA¡zMôDLKuYGDAb½¾DtsESLF>dgDLU TDBŸudeF<D]u]<HVUK}SLPR#bgDLhfDLU*KOF<DLEGF<S]<DLU*K<SD]n 45
cÀeúsö" õÃ&;õ+Û£&Ë@܏Ã`#mgf Ágf‘*(
,6-0/.1gðn ÷
Ä*ú¿
¡*EcDTLK>F<DuDLUkSLU DLF>P`dgDOYGD]7F/R¼QHVU ]rTH`]>hdgj*N D]h"NGbeK>deEGbedgSOE)R#F 3 aQHV[GK<DLUN¬RwQDTub½¾D]>K>dehkR#K>dgHVU

2
Ï%HVU*K<Dt¯‡vR#F>bgH'EGF<HVK<HVU¤DLKub½¾D]<K>dehkR#K>dgHVUª‡6D>‡|?–{£Õ\aEcHVNGFOYGD]AR#UGP`bgD]uÔSLUGEdeK>l)R#NsmdeUŒŽSLF>dgDLNGF<]A» ◦ nCD]
YGHVUGU SD]uYGDAb½¾DtsESLF>dgDLU TD-MuxMA¡zMôDLKuYGDAb½¾DtsESLF>dgDLU TDBŸudeF<D]}]/HVUKuSLPR#bgDLhfDLU*KuF<DLEGF<S]<DLU*K<SD]n 60
Ä*úú
óô&€Û;& ¿Ü*Ã`%fÁf#(NÛ;k^ÜfÃ[email protected]‰À’&QÃ!$'&()*
É!è Ì
ɾfÇ ÆÈÁÈÇ*¿ÉÊÀ¾
Í
µsC Á5ÄÒ ÇÓÏÀ
Ô
À
Æ
(
ǂ¾Ô
ÉgÕ ÍÏÀ
"<
ÆÈÁÈÇ
Í(lÆ5ÉÊÇÓ
Z·[email protected]µ@ºXö}ºz?:\@}1´5‚?̺u1B:\B[@ )? ·[email protected]µ@:<5gB
˜
ArdgDLU'j*N D}b½¾Dtsdg]>K<DLU TDOYGD]4F/R¼QHVU ]vTH`]>hdgj*N D]o]/HVdeK4NGUfŒR#deKoSLK/R#[GbedcYGDLEGNGdg]vR#N'hfHVdeU ]oNGU']>dgiTLbgD`abgDLNGF
VH F>deP`deU D4Dts RVTLK<D4F<D]>K<D4deU TDLF>K/R#deU D`nCREGF>deU TLdeE)R#bgDrF/R#dg]<HVUBDLU\D]>Kj*N D`aTHVU*K>F/R#deF<DLhfDLUK4R#Ns-EGl HVK<HVU ]abgDLNGF<]
K>F/RIJDTLK<HVdeF<D]7]<HVU*K7YGSLw*dgSD];E)R#F£bgD]£T<l)R#hE ]7hkR#P`U SLK>dgj*N D]a`DLK£YGHVU TobgD]£YdeF<DTLK>dgHVU ]Y˜½§R#F>F>dewQSDYGD];F/R¼QHVU ]
TH`]>hdgjN D]uU D-EcHVdeUK<DLU*K–E)RV]OwQDLF<]}bgDLNGF<]–]<HVNGF<TD]nÕAD-EGbeN ]a)bgD]OT/l)R#hE ]OhkR#P`U SLK>dgj*N D]OPR#bqRVTLK>dgjN D]ADLK
DtszK>F/RPR#bqRVTLK>dgj*N D]R#N\]/DLdeU\YGD]<j*N DLbg]£]½¾Dt—˜DTLK>N D4bqROEGF<HVE)R#PR#K>dgHVUfYGD]E)R#F>K>dgTLNGbgD]7HVUK£NGUTHVhEHVF>K<DLhfDLU*K
T<l)RVHVK>dgj*N D`az]<NGFoYGD]rST<l DLbebgD]R#bebqR#UKYGD 8 » 20
‚ ÷†a*DtsTLbeN)R#U*KvbqRxEcH`]<]>de[GdebedeK<SuYGDOEGF<SYdeF<DObgD]
10 cm 10 cm Ø
K>F/RIJDTLK<HVdeF<D]£YGDrhkR#UGdgiLF<DrEGNGF<DLhfDLUKYGSLK<DLF>hdeUGdg]>K<D`nVC‘ROEGF<HVE)R#PR#K>dgHVUfYGD]F/R¼QHVU ]7TH`]>hdgj*N D];EDLNGK£YGHVU T
yLK>F<DTHVU ]>dgYGSLF<SDTHVhhfDNGU¤EGF<H*TD]/]>N ]xYd“—N ]<d“Œ‘IJN ]<jN½§»'l)R#NGK<DfSLU DLF>P`dgD¬”bgDLNGF-Ydg]>K>F>de[GNGK>dgHVUÅ]>E)R#K>dqR#bgD
D]>K£l HVhfHVPQSLU SLdg]<SDu]>NGF7YGDvP`F/R#U YGD]oST<l DLbebgD]a)÷LYdebeN)R#U*K÷b½ deUŒŽHVF>hkR#K>dgHVUk]>NGF7bqR–F<SLE)R#F>K>deK>dgHVUkYGD]4]<HVNGF<TD]n
‡rDLEDLU Y R#U*Ka˜bgHVF<]<j*N [email protected]½¾SLU DLF>P`dgDfYGD]xF/R¼QHVU ]"TH`]>hdgj*N D]"R#NGP`hfDLUK<D`a‘NGU DfR#UGdg]<HVK>F<HVEGdgDEcDLNGK"R#EGE)R
F/OR gK>F<D]<HVdeKxE)R#F<TDfj*N DBb½§R#EGEGF<HszdehkR#K>dgHVUZYd“—cN ]>[email protected]½¾D]>KxEGbeN ]xw`R#bqR#[GbgD`a˜]<HVdeKxE)R#F<TDfj*N D\bgDTH*Dt¥kTLdgDLU*K
YGDxYd“—N ]>dgHVUˆYGDLwzdgDLUK}]<N¥k]/R#hhfDLUK}P`F/R#U YˆEHVNGF}j*N DAYGD]deUGl HVhfHVPQSLU SLdeK<S]uY R#U ][email protected]]<K>F>de[GNGK>dgHVUmYGD]
]<HVNGF<TD]uEGNGdg]/]<DLUKAEGF<HzYNGdeF<D"NGU¤Dt—cDLK–hfD]>NGF/R#[GbgD`ncÉ7beN ]–E)R#F>K>dgTLNGbedgiLF<DLhfDLUKahfyLhfDB]>d;b½¾HVU¤]<[email protected]
F<SLP`dehfDYd“—N ]>d“Œ>abqRpYGDLU ]>deK<SYGD]x]<HVNGF<TD][email protected]/R¼QHVU ]-TH`]>hdgj*N D]-Y R#U ]AbqRŠPR#bqRszdgDD]<K–wzF/R#dg]<DLhB[GbqR#[GbgDt
hfDLUKAEGbeN ][email protected] R#U ]–bgD]OF<SLP`dgHVU ]–TDLU*K>F/R#bgD]AEGbeNGK<ÚVKxjN½¾DLU!ESLF>deEGl SLF>dgD`acTD\jNGdEHVNGF>F/R#deKxTR#N ]<DLF–NGU
bgSLPQDLFADtsGTi]xYN¤• Ns!DLUªYdeF<DTLK>dgHVU‰YNªTDLU*K>F<DPR#bqRVTLK>dgjN D`a˜DLKANGU2YGSt­)TLdeK-Y R#U ]AbqRŠYdeF<DTLK>dgHVU‰HVEGEcH`]<SD`n
Ֆ[email protected]¬[email protected]ŒŽ[email protected]˜½§RVYwQDTLK>dgHVUÅP`bgHV[)R#boYN‰Ydg]<j*N DPR#bqRVTLK>dgjN DfwQDLF<]-bgDl)R#bgH¬EcDLNGK
yLK>F<DAbqRTR#N ]<D-Y˜½ NGU D"R#UGdg]/HVK>F<HVEGdgDxhfD]>NGF/R#[GbgD`n
C‘RŠYdg]>K>F>de[GNGK>dgHVU‰R#UGP`NGbqR#deF<DYGD]–F/R¼QHVU ]xTH`]>hdgj*N D]xD]>KAYd“¥kTLdebgDf»'EGF<SYdeF<DBEGF<STLdg]/SLhfDLUKaTR#FxDLbebgD
YGSLEDLU Y¬YGD]THVU YdeK>dgHVU ]}YGDAEGF<HVE)R#PR#K>dgHVUa)jNGd˜]<HVU*KF<DLbedgSD]u»BbqRBŒ’HVdg]}R#NspTHVU­ P`NGF/R#K>dgHVU ]O»BP`F/R#U YGD-DLK
»ŠEDLK>deK<DST/l DLbebgDfYGD]-T<l)R#hE ]"hkR#P`U SLK>dgjN D]aDLK"»ŠbqRŠEH`]<deK>dgHVU2YGDbqx
R (‘DLF>F<DE)R#FxF/R#EGEcHVF>KBR#Nsª]>K>F>N Tt
K>NGF<D]}hkR#P`U SLK>dgj*N D]OhkRI>DLNGF<D]–THVhhfD-bgD]O[GF/RV]u]>EGdeF/R#NsmYGD"U HVK>F<D-PR#bqRsdgD`n)C‘RfEGF<S]<DLU TDBY R#U ]uU HVK>F<D
Ä*ú>c
wQHVdg]>deU)R#PQDpYGDŠU HVh"[GF<DLN ]<D]m÷L]<[email protected][GNGbebgD]÷Šj*NGdrEcDLNGwQDLUKEcDLF<[email protected]]<jN D'PR#bqRVTLK>dgj*N D¬DLK\EGF<HzYNGdeF<D
YGD]B÷LT/l DLhdeU SD]÷-bgD-bgHVUGP'YGD]<j*N DLbebgD]}bgD]uF/R¼QHVU ]–TH`]>hdgjN D]OYd“—N ]/DLUKuEGbeN ]}ŒŽRVTLdebgDLhfDLU*KOwQDLF<]ubgDxl)R#bgHGa
EDLNGwQDLU*KuSLPR#bgDLhfDLUK–yLK>F<DxF<D]>EcHVU ]/R#[GbgD]uY˜½§R#UGdg]/HVK>F<HVEGdgD]u]>EcSTLd“­)jN D]"‚ ¦#†n
Ø
|7Um]<NGEGEH`]/R#U*KuNGU D-Ydg]<K>F>de[GNGK>dgHVUml HVhfHVPQiLU D\YGD]u]<HVNGF<TD]OY R#U ]}bgD-Ydg]<j*N DxPR#bqRVTLK>dgj*N D"DLK}NGU D"Yd“Œò
ŒN ]<dgHVUfdg]<HVK>F<HVEcD`a`bqRxEcH`]>deK>dgHVUkDtsGTDLUK>F<SD–YGDbq”
R (DLF>F<DOY R#U ]4bqRAPR#bqRszdgD\œF/RVYdqR#bgDLhfDLUKDLKwQDLF>K>dgTR#bgDLhfDLU*KΝ
EGF<H*YNGdeK¬NGU D2R#UGdg]/HVK>F<HVEGdgD2YdeEHVbqR#deF<DªY˜½§R#hEGbedeK>N YGD
EGF<HVEcHVF>K>dgHVUGU DLbebgDÅR#N¨THzDt¥kTLdgDLUKmYGDªYd“—cN ]>dgHVU
α
a4beNGduhfyLhfD%YGSLEDLU Y R#U*KŠYGDmb½¾SLU DLF>P`dgDŤ‚ ÷V^#†n4™uU D!R#UGdg]<HVK>F<HVEGdgD%YdeEcHVbqR#deF<DˆEcDLNGKŠR#N ]/]>d}EGF<HwQDLUGdeF
D(E)
YN¬hfHVNGwQDLhfDLUKOF<DLbqR#K>d“Œ£YNˆ]>¼]>K<iLhfDx]<HVbqR#deF<DAE)R#FuF/R#EGEHVF>KuR#N¬EGbqRV]>hkRdeUK<DLF<]>K<DLbebqR#deF<D"K>F/R#U ]>EHVF>K/R#U*K}bgD]
deUGl HVhfHVPQSLU SLdeK<S]-hkR#P`U SLK>dgj*N D]"]>NGFxbgD]/jN DLbebgD]xYd“—cN ]<DLUK"bgD]xE)R#F>K>dgTLNGbgD]nÕxR#U ]ANGUªhfHzYGiLbgDf]>dehEGbedg]>K<D
¤‚ ÷V_‘HVÈpbgD]}]<HVNGF<TD]]<HVU*KuYdg]>K>F>de[GN SD]}YGDAhkR#UGdgiLF<DAl HVhfHVPQiLU D-Y R#U ]}NGUˆYdg]<jN DAY˜½¾SLE)R#dg]<]/DLNGF
aGDLK}HVÈ
2h
bgD]uF/R¼QHVU ]ATH`]>hdgjN D]–]<HVU*K–THVU­ U S]AY R#U ]–NGUml)R#bgHŠYGD"l)R#NGK<DLNGF ab½§R#UGdg]<HVK>F<HVEGdgD»NGU D\Ydg]>K/R#U TD
H
z
R#N†YGD]<]>N ]}YN¬EGbqR#UmPR#bqRVTLK>dgjN DŠœ
vD]>K}D]<K>dehfSD"»
n
z<h
δ ' 3D/cH × z/h
MƒNGbeK>F/RŠl)R#NGK<DSLU DLF>P`dgD`aNGU¤YdeEcÚ[email protected]"R#EGE)R#F/OR gK>F<Df]>d£bgD]AF/R¼QHVU ]BTH`]>hdgjN D]x]/DBEGF<HVE)R#PQDLU*KBYGD
ŒŽRVWHVUËj*N)RV]>dehfDLU*KfF<DTLK>debedeP`U Dm»!E)R#F>K>deFfYGD¬]<HVNGF<TD]fDtsGHVK>dgj*N D]F<DLbedgSD]k»!bqR!hkR#K>dgiLF<D¬U HVdeF<D¬]>NGEGEcH`]<SD
yLK>F<DkYdg]>K>F>de[GN SDpY R#U ]BbgDfl)R#bgHƤ‚ ÷ †n™OU D'R#UGdg]/HVK>F<HVEGdgD'YdeEcHVbqR#deF<DkEcHVNGF>F/R#[email protected]#bgDLhfDLUKyLK>F<DkEGF<HzYNGdeK<D

E)R#Fb½¾SLK/R#[Gbedg]<]<DLhfDLU*KrY˜½ NGU DuEGF<HVE)R#PR#K>dgHVUŠj*N)RV]>dehfDLU*KoYd“—cN ]>dewQDuDLU*K>F<D}NGU Du]/HVNGF<TD}YGHVhdeU)R#UK<Dœ’THVhhfD
b½§R#hkRV]YGDŠ€OdeF>PQH\[email protected] HVK>F<DpPR#bqRsdgD`n;ÕAD]@hBNGbeK>deEÚVbgD]fY˜½¾HVF<YF<DpEGbeN ]kSLbgDLwQS'EcDLNGwQDLUKfSLPR#bgDLhfDLU*KŠR#E
E)R#F/OR gK>F<DU)R#K>NGF<[email protected] R#U ]"TDLF>K/R#deU ]"hfH*YGiLbgD]nÉ;R#FBDtsDLhEGbgD`a˜NGUÅjN)RVYF>deEcÚVbgDfYGHVhdeU)R#UK\Y R#U ]-bqR
Ydg]>K>F>de[GNGK>dgHVU'R#UGP`NGbqR#deF<D}YGD]F/R¼QHVU ]4TH`]<hdgjN D]7EcHVNGF>F/R#deK7F<S]>NGbeK<DLF4Y˜½ NGUfDtsGTi]DLUEGF<HwQDLU)R#U TDuYNEGbqR#U
]>NGEcDLFPR#bqRVTLK>dgjN D`n
ò 5¸Ó@u1€?5£·Ìº}6
)µ'579?)·c5÷B[@}6"·c9;5Ò³\6´·cº$#@u1B1\5¸¹571¢? 9ë@€B¸¹:<?A6\5
B
Õ ½ NGUmEcHVdeUKxYGD-wzN D-PQSLU SLF/R#bacTR#F/RVTLK<SLF>dg]<DLF–bgD]xR#UGdg]<HVK>F<HVEGdgD]AYNmF/R¼QHVUGU DLhfDLUK"TH`]>hdgj*N D-EcDLF>hfDLK
@
Y˜½¾HV[GK<DLUGdeFBYGDEGF<STLdgDLN ]/D]-deUŒŽHVF>hkR#K>dgHVU ]B]>NGF"bqR¬Ydg]>K>F>de[GNGK>dgHVUÅYGD]"]<HVNGF<TD]a˜bqR¬]<K>F>N TLK>NGF<DkYGD]"T/l)R#hE ]
hkR#P`U SLK>dgjN D]BDLK-bqRŠw`R#bgDLNGF"YGD]"THzDt¥kTLdgDLUK<]\YGDYd“—N ]<dgHVU‰Dt—cDTLK>d“ŒŽ]n|U2E)R#F>K>dgTLNGbedgDLFab½§R#NGP`hfDLU*K/R#K>dgHVU
YGD]oTHzDt¥kTLdgDLUK<]vYGDOYd“—N ]>dgHVU¬RwQDTOb½¾SLU DLF>P`dgDAR"Y˜½ dehEcHVF>K/R#UK<D]vTHVU ]<Sj*N DLU TD]v]>NGF4bqR-EGl SLU HVhfSLU HVbgHVP`dgD
YGD]F/R¼QHVU ]OTH`]>hdgj*N D]PR#bqRVTLK>dgjN D]@œ’j*N DAU HVN ]U HVK<DLF<HVU ] ‡A]Νtn
a
‡rHVhhfD}bgD]U H¼R#NskYGD[)RV]<]/DSLU DLF>P`dgDu]<HVU*KEGbeN ]7ŒRVTLdebgDLhfDLUKrTHVU­ U S]oY R#U ]4bqRAPR#bqRszdgD`aQdebg]]½§RVTTLN
h"NGbgDLU*KuDLU¬EGbeN ]vP`F/R#U YGD-j*N)R#UK>deK<S-j*N DATDLNspYGD–l)R#NGK<D-SLU DLF>P`dgD`aGTHVU YNGdg]/R#U*K–R#deU ]>d»\NGUpF/R#dgYdg]<]<DLhfDLU*K
YNˆ]>EcDTLK>F<D`nzC‘DAbedgDLUmDLUK>F<DAbqR\EcDLUK<D
YNˆ]>EcDTLK>F<D–hfD]>NGF<SxYGD] -‡r]xa
aGDLKvbqR\EcDLUK<D-YN
a
γobs
∝ E −γobs
]>EcDTLK>F<Dx]<HVNGF<TD"YGD]F/R¼QHVU ]OTH`]<hdgjN D]a
−γs a YGSLEcDLU YˆYGDxbqRF<DLbqR#K>dgHVU%DLUK>F<D"bgD-THzDt¥kTLdgDLUKOYGD"Yd“Œò
∝E
Ä*ú>v
ŒN ]>dgHVUfDLK4b½¾SLU DLF>P`dgD`aT`½¾D]>Ko»-YdeF<DuYGDbqR–w`R#bgDLNGFoYGDb½¾DtsEcH`]/R#UK K<DLb)jN D
nQM‹]>N¥k]/R#hhfDLU*K
β
D(E) ∝ E β
l)R#NGK<DOSLU DLF>P`dgD`a
n*ÕAD}U HVhB[GF<DLNs'hfHzYGiLbgD]vYGDuEGF<HVE)R#PR#K>dgHVU¬Y R#U ]rbqRxPR#bqRsdgDuEcDLF>hfDLK>K<DLUK
γobs = γs + β
[email protected]<DLU YF<DTHVhEGK<[email protected]'wVR#bgDLNGF"YGD]AEGF>deU TLdeE)R#bgD]BHV[ ]<DLF>w`R#[GbgD]-YGD] ‡A]n˜CD]xw`R#bgDLNGF<]AF<Dj*NGdg]<D]-YGD
a
aEDLNGwQDLU*KOyLK>F<DxTDLEcDLU Y R#UK–RV]/]<DLÔ-Yd“—cSLF<DLU*K<D]OY˜½ NGUmhfH*YGiLbgD"»\b½§R#NGK>F<D`n
β
CD]ohfH*YGiLbgD]rdehEGbedgj*N)R#UKvNGU DOYd“Œ’N ]<]>dgHVUpY R#U ]obgDOYdg]<j*N D–DLK4bgDOl)R#bgHGadeU TLbeN)R#U*KrNGUŠwQDLUKvPR#bqRVTLK>dgj*N D
HVN'NGU DAYd“—cN ]>dgHVUpK>NGF>[GNGbgDLUK<D-]>NGP`PQiLF<DLU*K-¤‚ ÷V^˜NGU DOw`R#bgDLNGFYGD THVhEGF>dg]<DADLU*K>F<D-_¤n ÷"DLK_n§^aDLUˆRVTTHVF<Y
β
RwQDTvNGUf]>EcDTLK>F<Dv]<HVNGF<TD}RwQDT
aQTHVhhfD}R#K>K<DLU YNkY R#U ]£bgDTRV]7YGDvbqROK>l SHVF>dgDbedeU SR#deF<DYGDvb½§RVTTSt
γs ∼ 2.0
bgSLF/R#K>dgHVUˆYd“—N ]>dewQDAE)R#F}T/l H*T]nzÕ@½ NGUmR#NGK>F<DxTÚVK<S`abgD]vhfHzYGiLbgD]EGF<DLU)R#UKODLUˆTHVhEGK<DAbqR\F<StÞRVTTSLbgSLF/R#K>dgHVU
YGD] ‡r]YGDŠ[)RV]<]/DŠSLU DLF>P`dgD¬Y R#U ]fbgD'hdebedgDLNXdeU*K<DLF<]>K<DLbebqR#deF<DpŒRwQHVF>dg]<DLU*KfNGU Dpw`R#[email protected]
aDLU
β ' 1/3
a
RVTTHVF<[email protected]ŠK>l SHVF>dgD]>K/R#U Y R#F<Y2YGD\bqRpYd“—cN ]>dgHVUªY R#U ]-YGD]xT<l)R#hE ]"hkR#P`U SLK>dgjN D]-EGF<S]<DLUK/R#U*K-NGU D
K>NGF>[GNGbgDLU TDŠYGDŠKÞ¼*EcD%OHVbehfHVPQHVF<Hwcno¡*NGdew`R#UKTDLK>K<DŠl¼zEHVK>l i]<D`abgDp]>EcDTLK>F<D']<HVNGF<TDŠ]<DLF/R#deKfR#bgHVF<]\EGbeN ]
EDLU*K>NaDLU −2.4 nC‘RŠj*N D]>K>dgHVU%F<D]<K<DBHVNGwQDLF>K<DDLKAD]>KAYGDBP`F/R#U YGD\dehEcHVF>K/R#U TD\EHVNGFxYGSLK<DLF>hdeU DLFANGU D
E
K>l SHVF>dgD-YGDxb½¾HVF>deP`deU D-YGD]F/R¼QHVU ]OTH`]<hdgjN D]-¤‚ ÷Vцn
ÕxR#U ]"TDkTHVU*K<DtszK<D`adeboD]<K"deUK<SLF<D]/]/R#[email protected] HVK<DLF\jN DkbgD]-hfD]>NGF<D]\Y˜½§R#UGdg]<HVK>F<HVEGdgD]\EDLNGwQDLU*KBEcDLFJ
hfDLK>K>F<DOYGDOTHVUK>F/R#deU YF<D–bgD]hfHzYGiLbgD]vYGDuYd“—cN ]>dgHVU'YGD] ‡A]oR#deU ]>djN DubgDLNGFo]>EcDTLK>F<Du]/HVNGF<TD`aEGNGdg]<j*N D
a
bqR–wVR#bgDLNGFYGDvb½§R#UGdg]<HVK>F<HVEGdgDOYGSLEDLU YYNfTHzDt¥kTLdgDLUKrYGDYd“—N ]<dgHVUn`C½¾SLwQHVbeNGK>dgHVUkEGF<SYdeK<D}YGDvb½§R#UGdg]/HVK>F<HVEGdgD
RwQDTBb½¾SLU DLF>P`dgD\D]<[email protected]#[email protected]"Yd“—cN ]>dgHVU%NGK>debedg]/S`aDLKAYGDLw*F/R#deKASLwQHVbeN DLFxYGD"hkR#UGdgiLF<[email protected]]>dehdebqR#deF<[email protected]»
a T`½¾D]<Ku»YdeF<DxDLU β n
D(E)
E
Ä)
!
"
#$%&
Ä Á(('ÏÀ
&
Ä1†Ä ÆÈÇÉÊÁÈÀ
À Æ
É*) Í ¾ ÉÊÇÓ
Ð ÑÒnÓ]ÔÕÑÖ!ÓY×nØ6ÙÚ#ÛÝÜÝÞ߶à
+3ÙtÔÕÑܕØ6æAÖ1ߊÖ1ßTÞ>ß×>æ*ð‘ÒnÜÝÞnÓ6×AÜÝÛÝæ!ðaÖ1ßnÒÙ*ð~ÛhÒ/æ*ÜÝÞæÚ#Û ßÒTÖWá ÓIð_ßÖ*ÛhÒÜÝÞÛ änÓ6ÜÝÛÝæ!ð%Ù!ð]é!ÓIêFÙ!ÛGÞ>ß{Ö1ßÞÙ0/æ!ð‘Ò—×>æ!ÒÔ¥ÛÝë/ӑßÒ
ëӑ߶ð#æ!Ó]ÒTÚ#Þ>ÑÒ/ßð‡ÜßnÞæ!ð‘Ò¹ÖÙ!ð‘ÒêF߶×ØIÙÚ#ÛÝÜÝÞßÒÓIÛGåÙ*ð‡ÜWÚ6æCÒÒ/ÿÖ1ßÕÖ1ßҗÚ#Þ'ænÚ#ÞÛ Ñ܏ÑÒÒnÜ ÙÜÝÛhÒÜÝÛÝëÓ~ßÒ¶ëÓ]Û Ú‘ßnÓIåCßnð#Ü
6ÜÝÞßN×>æ*ԗÚ_ÞÛGÒ/ßÒvñ/Ù×ەêxßnÔÕßnð‡ÜŸÙ*ӓԹ0
æ /1ßnðƒÖ á ÓIð_ßNÙ!ð#Ù!êFæ>é!Û ß|ÛGð#ÙÜ=܏ßnð#Ö!Ó~ß|Ù*åCß×Nê>á ÑÜÝӑÖ1ßfÖ!ÓNÚ_Þ'æA×ßÒÒnÓ1Ò Ö1ß
Ö!Û õTÓ]ÒÛ=æ!ð ,+3Ù¥Ö!Û õTÓ1ÒnÛÝæ!ðŸÚ‘ßÓI}Ü 6ÜÝÞ>ßt×>æ!ð~ÒÛ=Ö1ÑnÞ>Ñ>ßv×>æ*ÔÔÕߊêFٟê;ەԥە܏ß{×>æ!ð#ÜÝÛGð~Ó~ßùÖ á Ó]ðŸÚ#ÞæA×ßnÒÒÓ]ÒùÖ]ߊԶÙ*Þ'×nؑß
Ù!êFÑ>Ù*Ü æ!ەÞß Ö7ŸÙ!ð‘Ò{êFßâ×>ÙCÒâ0æ 8%êFÙ¶Ô¹Ù!Þ×nØ6ߟÙ!êFÑ>Ù*Ü æ!ەÞßßÒnÜ Ù!ð~ÛGÒæ*ÜÝÞæڑßûەê§ßnÒnÜ_Ú6æ!ÒÒÛ änêxßâÖ]ßv×>Ù!êF×ÓIêFßÞvêúÙùÚ#Þæ1ä0Ù 3
änÛGê„ÛÝ܏ъÖ]ß,Ú#ÞÑnÒ/ßnð#×ßtÖ á Ó]ð_ß,ڑÙ!ÞnÜÝÛ=×ÓIêFßtßð:Ó]ðŸÚ6æ!ەð‡ÜÊÖ1ߗêá ßҏÚ6Ù×ߟßnð%ÞÑÒ/æ!ê„åÙ!ð‡Ü±ê>á Ñë/Ó6Ù*ÜÝÛ=æ!ð|Ö1ߊêFÙ¥Ö!Û õTÓ1ÒnÛÝæ!ðb3
ÙÖ!åCß×AÜÝÛÝæ!ðw×>æ*ÞÞ>ßÒ Ú‘æ!ð#ÖÙ!ð#܏ß
+3Ù:×>æ*ð~ð#Ù!ÛGÒÒ/Ù!ð#×ßÕÖ1߶×ßÜ=܏ߥÒæ!ê„Ó6ÜÝÛÝæ!ð:ڑßnÞnÔ¹ß/ܱÚ6Ù!Þ¶Ù!ð#Ù!êFæ>é!Û ß¶Ö1ßÕ×>Ù!êF×ÓIêFßnÞ¶êFÙâÚ#Þæ1äÙ1äÛGê„ە܏ÑdÖ1ßÞ>ß×>æ*ð‘4Ò 3
ÜÝÞÓIÛGÞ>ßÓIð_߶Ù!ԊÚ#ê„ÛÝÜÝÓ6Ö1ßÕÖ1߶Ö!Û .Ú -!êxß α û×>æ*ÞÞ>ßÒ Ú‘æ!ð#ÖÙ!ð#܊ì›ÓIðwåAÞÙ!ÛÖ*ÛR.Ú -*êFßÕÖ á Ù*ԗÚ_ê;ÛÝÜÝÓ6Ö]ß α Ò/æ!Ó1çÒ 30//Ù×ßnð#Ü
rec
ÖÙ!ð‘ÒtêxßvêFæ*ÜÖ1ßÖæ!ð~ð_ÑßÒÑÜÝӑÖ!Û=Ñ Ä[v
-,/.
¯
óôG#Ãhm õÁ!QeÛ£21o&f
ƒ:43'6B):[email protected]}1
>
5‚?ó<:\¶B·c5±µ\º}·9‚@}6B·MB̸Ó@5#}571
Õ R#U ]"NGUÆhdebedgDLNÆl HVhfHVPQiLU DŠDLK\dg]<HVK>F<HVEcD`a‘bgD'TH*Dt¥kTLdgDLU*KfYGDkYd“—cN ]>dgHVU
x
]/DkYGSt­ UGdeKf»¬E)R#F>K>deFYGD
D
b½¾SjN)R#K>dgHVUpYGDu{;dgT Ç ”
HVÈ
D]>K4bgD}wQDTLK<DLNGFvYGDLU ]>deK<SOYGDOTHVNGF/R#UKvTHVF>F<D]>EcHVU Y R#UK
J(r) = −DO n(r)
J(r)
R#Nˆ[GdebqR#UmU DLKOYNmU HVh"[GF<DBYGDxE)R#F>K>dgTLNGbgD]–K>F/RwQDLF<]/R#U*KONGU D"]>NGFJŒŽRVTD-NGUGdeK<S-EcDLU Y R#UK ]<DTHVU YGD`a)DLK

n(r)
[email protected] ]>deK<SxU*NGhfSLF>dgjN DBYGDAE)R#F>K>dgTLNGbgD]n
Y R#U ]kbgD%TRVYF<DmY˜½ NGU hfHzYGiLbgDˆE)R#F>K>dgTLNGbedgDLFpEcHVNGFfbgDˆhfHVNGwQDLhfDLU*K¬YGD]
™ U TR#bgTLNGb–YdeF<DTLKŠYGD
O
J(r)
E)R#F>K>dgTLNGbgD]œhkR#F<T<l DR#bgSR#K<HVdeF<[email protected][GF<DBE)R#F<THVNGF<]AhfH¼QDLUuEDLF>hfDLKAR#bgHVF<]AY˜½¾DtszEGF>dehfDLF
DLUˆŒŽHVU TLK>dgHVU
D
YGD]uYGHVUGU SD]OYNmEGF<HV[GbgiLhfDk” a)[email protected]*deK<D]/]<Dkœ’]>NGEGEcH`]<SD-THVU ]>K/R#U*K<D}YGD]uE)R#F>K>dgTLNGbgD]ODLKuNGU D-bgHVUGP`N DLNGF
c
λ
F<DLEGF<S]<DLUK/R#U*K-bgD\E)RV]xYGD\bqRkhkR#F<T/l DkR#bgSR#K<HVdeF<DHVN¤bgD\bede[GF<[email protected])R#F<THVNGF<]xhfH¼QDLU2YN¤EGF<HzTD]<]>N ]xR#bgSR#K<HVdeF<D`n
š UˆHV[GK>dgDLUKAR#bgHVF<]}bgD]F<DLbqR#K>dgHVU ]uF<S]>NGhfSD]uY R#U ]bgDAK/R#[GbgDR#N!]>NGdew`R#UK"”
YdehfDLU ]>dgHVU

Ñ YdehfDLU ]>dgHVU ]
@
~ YdehfDLU ]>dgHVU ]
9
Ï!R#F<T/l D"R#bgSR#K<HVdeF<D
C˜de[GF<D-E)R#F<THVNGF<]}hfH¼QDLU
D = λ c/2
D = λc
D = λ c/4
D = λ c/2
D = λ c/6
D = λ c/3
ïÚ#Þ>ßÒÒÛ=æ!ð‘ÒŸÖ!ÓY×æEß76 ×Û ßnð‡ÜTÖ1ßÖ!Û õTÓ1ÒnÛÝæ*ð
D
ÖÙ!ð~ÒùêxßҟÖ!Û õŠÑnÞ>ßnð‡Ü•ÒŸ×>Ù!Òßnð~åAÛGÒÙAé]ÑÒ
C ½¾Sj*N)R#K>dgHVU¬YGDOTHVU ]<DLF>w`R#K>dgHVUpbgH*TR#bgD–YNpU HVh"[GF<DAYGDOE)R#F>K>dgTLNGbgD]x”

R#bgHVF<]u»\b½¾Sj*N)R#K>dgHVU%YGDAbqRYd“—cN ]>dgHVU¬l)R#[GdeK>N DLbebgDk”
∂n(r,t)
∂t
+ ∇ · J(r) = 0
THVU YNGdeK
œ n 
Ø 
∂n(r, t)
− D 4n(r, t) = 0
∂t
GY HVUKŠbqR2]/HVbeNGK>dgHVU[GdgDLU‹THVUGUN D!EHVNGF'NGU DˆdeUVI>DTLK>dgHVU‹DLU
DLK
arD]>KkYGHVUGU SD%E)R#FŠbqR
r = 0
t = 0
PR#N ]<]>dgDLUGU DkœòŒŽHVU TLK>dgHVUmYGD"–F<DDLU%YNˆEGF<HV[GbgiLhfDx”
n (r, t) =
1
(4 π D t)
3/2
r2
exp −
4Dt
œ §n ÑQ
Ø
C ½¾Sj*N)R#K>dgHVU¤œ n§ÑQD]>KbqR-bedehdeK<DuTHVUK>deU*N DAYNkEGF<HzTD]<]>N ]oR#bgSR#K<HVdeF<D–]<HVN ]JeI<RVTDLUKnzÕxR#U ]bgDOTRV]4Y˜½ NGU D

Ø
hkR#F<T<l DkR#bgSR#K<HVdeF<D'»ŠK>F<HVdg]-YdehfDLU ]>dgHVU ][email protected])RV] a˜]½¾Dt—˜DTLK>N)R#UKB»ŠwzdeK<D]<]<[email protected] ]>K/R#U*K<D [email protected]<DLhE ]-D]>K
λ
c
YdeF<DTLK<DLhfDLUK–EGF<HVEHVF>K>dgHVUGU DLbR#NˆU HVhB[GF<DBYGD"E)RV]
YGDxbqRfhkR#F<T<l DŠ”
a)DLK
D]<KuYGHVUGU S"E)R#F
N
t = N λ/c
D
W˜À’Ä ‘ 21o&fXkõ;NÜG#ÃQ&#si2%Ügf
D = λc/6
Ä)¿
œ’TtŒ£K/R#[GbgDR#Ntn š UmEDLNGK}YGHVU TxF<SSTLF>deF<D'œ §n ÑQvTHVhhfDk”
Ø
p (r, N ) =
2πN λ2
3
−3/2
3r2
exp −
2N λ2
œ n~ 
Ø
.
jNGdYGHVUGU DubqRBYGDLU ]>deK<S–YGD}EGF<HV[)R#[GdebedeK<SxYGD}EGF<S]/DLU TD–Y˜½ NGU DOE)R#F>K>dgTLNGbgD–deUVIJDTLK<SD–DLU
DLK
R#EGF<i]
r=0 t=0
q
E)RV]n)‡rDLK>K<D-Ydg]<K>F>de[GNGK>dgHVU%D]>K}YGHVU TxPR#N ]/]>dgDLUGU DkœŽ»\P`F/R#U Y taGY˜½¾STR#F>KuKÞ¼*EcD
2 n š UˆEcDLNGK
N
N
σ = N3λ
U HVK<DLF–j*N DBTD"F<S]>NGbeK/R#KAEDLNGKASLPR#bgDLhfDLUKx]½¾HV[GK<DLUGdeF-»fE)R#F>K>deFAYN!K>l SHVF<[email protected]>F/R#b£bedehdeK<D`acDLUmU HVK/R#U*K
jN Drb½¾STR#F>K7K†¼zEcDrYGDrbqRAYdg]>K>F>de[GNGK>dgHVUfYGD]7YGSLEGbqRVTDLhfDLU*K<]4]>NGdew`R#UK7bgD]RsGD] a DLK R#EGF<i]£NGUE)RV]£w`R#NGK”
xy
< (ri )2x > = < (ri )2y > =< (ri )2z > = λ2
Z
z
u2x , y ,z p(u) dΩ =
λ2
3
œ n „*
Ø
VH È
D]>KobqR-YGDLU ]>deK<S–YGDuEGF<HV[)R#[GdebedeK<SAY˜½¾Dt—˜DTLK>N DLFoNGUŠE)RV]rY R#U ]4bqRBYdeF<DTLK>dgHVU a*j*NGd)w`R#NGK
Y R#U ]
p(u)
u
1/4π
bgDxTRV]l HVhfHVPQiLU D"DLK}dg]/HVK>F<HVED`n
Ä ú
óôG#Ãhm õÁ!QeÛ£21o&f
óôGÃ_h% õÁ¤ #%܏
8 @ ©$›³µ´O@˜$›)§œ°ª¢¬¡«À³"™Jšœ˜$›¢¤¡†§.™ š|¨ ªOž$³¿ž$¢¬«À¢¤¡†§6™ š|¨ ª§¶›šœ˜$·š6š°˜ ™± ©$˜ š.¨$³¿ª¡¢É·© «Àš.¢¬˜¿Ÿ šœn¡†§°š|šœ˜
Á ³µØ³¿˜¡^š·Æšœn¡†© §
¨$³@›™Jš|«À@˜$²@© šœ© ª ›© ¢¬´Å³@˜O¡© ˜u¨ ª†J·š¶››©$r›^™Jš|¯”³¿ªUUÇ š.³¿«À§œ³Å¡†@¢Àªš3ë
t=0
N
r=0
š·¡
λ
p (r, N ) =
2πN λ2
3
−3/2
3r2
exp −
2N λ2
æ:9$½ »Oé
.
8 @©$›V¨g@©$´@@˜›L§°²O³@«¬šœ¯"š°˜¡'°³¿«É·©$«¬šœª>«É³|™J¢É› ¡†ª¢Àž ©J¡†¢¬O˜”™Jš«À³.™J¢É›¡†³¿˜·š ¹6«F± @ª†¢À²@¢À˜ š@Á¿›@¢¬¡'«À³6˜ Oª¯"š™Jš Á@šœ˜
›@¯"¯”³¿˜¡L›© ªL¡†@©$›>«Àšœ›V³¿˜$²@«Àšœ›>«À³6™J¢É› ¡†ª¢Àž ©J¡†¢¬O˜
½ Í ³6™Jš°˜$›¢¬¡§™Jš^¨ ª†@ž$³@ž ¢¬«À¢¬¡§)¨O© ª'È© š ³¿¨ ¨³¿ª¡¢Àrš°˜ ˜ š
p
(r,
t)
r
¹
šœ›¡™J@˜$.™JO˜ ˜ §œš|¨$³¿ª.ë
[r , r + dr ]
p (r, t) = 4πr2
2πN λ2
3
3r2
.
exp −
2N λ2
−3/2
æ:9$½ 9é
0.22
N= 10 pas
0.2
N= 25 pas
N= 50 pas
0.18
0.16
0.14
0.3
N= 10 pas
N= 25 pas
N= 50 pas
0.25
0.2
0.12
0.15
0.1
0.08
0.1
0.06
0.04
0.05
0.02
0
-15
Probabilite
Probabilite
Ó<š·¡¡š6§œ°ª¢¬¡© ª†š+›š°ªU³|©J¡†¢¬«Àš6¨O© ª<˜$¿¡ª†š§°¡©$™ š6™Jš¶›^³¿˜ ¢É›@¡ª†@¨ ¢Àšœ›'«¬Oª†›†È©± ¢À«›°± ³¿²O¢¬ªU³|™Jš.°³¿«É·©$«¬šœª«É³B›¢¬²O˜ ¢¬¥°³¿¡¢¬¸
´_¢¤¡†§|› ¡U³Å¡¢É›¡¢ÉÈO©$š.™ š|«À³B´Å³¿«Àš°© ªª†šœ°@˜$›¡ª†© ¢¬¡š.™± ©$˜u™ ¢¬¨g×@«Àš|³µ´@šœ+˜ @¡ª†š+¯"§·¡†Ç J™Jš@½ 8 @©›^¨O© ´@O˜$›)§°²O³@«¬šœ¯"š°˜¡
ª†š°¯”³¿ªUÈ© š°ª)æ™ ³@˜$›>«À³6›§œª¢Àš^™Jšœ›L³¿˜³¿«À@²@¢Àšœ›!; éÈO©$š^°š·¡¡š<£‚@˜$·¡¢À@˜B™ š^™ ¢À›¡ª†¢¬ž$©J¡¢À@˜Bšœ›¡>šœ˜¡†@©J¡L¨O¢¬˜¡'›¢¬¯"¢À«À³@¢¬ª†š
¹X«É³.£‚O˜$n¡†¢¬O˜“™Jšª†§°¨³¿ª¡¢¬¡¢À@˜z™J©z¯"J™J© «Àš+™ š«É³X´_¢¤¡†šœ›†›š+™Jšœ›<¯"@«À§œ·©$«¬š¶›™ ³@˜$›«É³|¡†Ç §°Oª¢Àš°¢¬˜ §°¡¢ÉÈ© š+™Jš¶›²³ <O½
Í ³3ª†š°¨ ª†§œ›š°˜¡U³Å¡¢À@˜}²@ªU³¿¨ Ç$¢ÀÈ© š.™JšX·š¶›)£‚@˜$·¡¢À@˜$›^š¶› ¡™ @˜ ˜ §œš|¹¡†¢¤¡†ªšX™± š°¦_šœ¯"¨ «¬šX›© ª«É³B¥$²@© ª†š|·¢¬¸4™ šœ›†›O©$›°½
-10
-5
0
5
10
15
Distance / Libre parcours moyen
0
0
2
4
6
8
10
12
14
16
18
20
Distance / Libre parcours moyen
>]Cµä^>][email protected]
= â?>[email protected]†ànä°âB$á?CÅá7D:EBbßOâGFHC}ß@âB ä4Dçá`àBßOâI>[email protected]çá`à"ß@âP>[email protected]à·ä°âQBã†âRCS>[email protected]·äVUWBâ[email protected]ãM[ âRCU\[]Cµä[email protected]†ß“ß@â
â°á
[email protected]âßOâfdgCU$ãM[ â@â?>[email protected]†ànä°âB$á`âYFHCHßDÉäná[email protected]:KQUJá7D:EBxßih UWBgâ3ß@â·äãMENX5>jEÅäQC B$á`ânä
ß@Nâ=
FHCV10
>jEÅ_ä4DçN
á2D:E =
[email protected]ämlNU$âG=FHC550ce`bdLUWa @âßOâ"ß@JE Dçá4â[email protected]â?>[email protected]à·ä°âQBá4âmFHC“ß DÉäná[email protected]:KUJá2D:E BZß UnXoE¶ßUWF¬â3ßUnpÅâUã·á`â[email protected]>]Eµä4Dçá7D:EB
h C [email protected]â|äE U_äqF¬â·äXãSE [email protected]ânä[email protected]â·äUá`âXàrdgCF¬âRsut C [email protected]ämlU$âYFHC5vQEBã·á7D:EBeäWh à°á?CF¬âGpµâ@UäYF¬ânä[email protected]ß@â·ä|ß DÉäUá?CBãUânä F¬â
_
r `ia
XZEœß UWF¬â3ß@âmFHC}ß DÉänárC BgãUâGFwCq_ >AF0U_ä*>[email protected]¬âä°âXäMDçá7U]CBá^s}â[email protected] B √
λ N`
W˜Àwv ‘ 21o&fXfۑÃ`f­i2G#Ãhm õÁ*yff )܏
-,¯®
×:43'6B):\@u1
¯
5‚?̺}³C{}579G?:\@}1{z¸¹º·9c²\5±ºXó<¶7º
>
u:>·5±[email protected]}1
?[@
Ä
:[email protected] )? ·[email protected]µŠ5
C D]7F<S]>NGbeK/R#K<]oTLd“†YGD]<]>N ]HVU*KoSLK<SHV[GK<DLU*N ]oY R#U ]7bgDuTRV]Y˜½ NGU D}Yd“—cN ]>dgHVUfl HVhfHVPQiLU D`aTHVF>F<D]<EHVU Y R#U*K
»ŠNGUªEGF<H*TD]<]<N ][email protected]#F<T<l DkR#bgSR#K<HVdeF<D!œ’HVN2YGDCɏϪxR#N‰THVNGF<]-YN j*N DLb7bqR¬YdeF<DTLK>dgHVU‰YGD]-E)R#F>K>dgTLNGbgD]
T<l)R#UGPQDYGDrhkR#UGdgiLF<Drdg]<HVK>F<HVED`nQ¡*d R#EGF<i]7T<l)RVj*N DvdeUK<DLF/RVTLK>dgHVU'YGDobqRuE)R#F>K>dgTLNGbgD`aVbqROU HVNGwQDLbebgDYdeF<DTLK>dgHVUfD]>K
Ydg]>K>F>de[GN SD-]>NGdew`R#UKONGU DxbgHVd‘YGDxEGF<HV[)R#[GdebedeK<S"U HVUmNGUGd“ŒŽHVF>hfD-]>NGF}bqR]<EGl iLF<D`aGb½¾SjN)R#K>dgHVU!YGDxbqRYd“—cN ]>dgHVU
THVF>F<D]>EHVU Y R#U*K–»@TDxEGF<HzTD]<]>N ]D]<KhfH*Yd“­)SD`n
Õ R#U ]AbgDTRVYF<[email protected]½§R#U)R#bgHVP`dgDfSLwQH*j*N [email protected]“†YGD]<]>N ]acdeb7D]>KxdeU*K<SLF<D]<]/R#U*K"YGDTHVU ]>dgYGSLF<DLFxbgDTRV]xHVȪbqR
x
YGDLU ]>deK<SAYGDOEGF<HV[)R#[GdebedeK<S-EHVNGFvbqRBE)R#F>K>dgTLNGbgDxY˜½§RwQHVdeFu]ÎR"YdeF<DTLK>dgHVUmTDLU*K>F<SD-R#NGK<HVNGF}YGD Y R#U ]vNGUˆR#UGP`bgD
u
]<HVbedgYGD
D]>KOYGHVUGU SDxE)R#F"”
dΩ
œ ¤n ÷`
1 + αd · u
p(u) =
Ø
4π
VH È
D]>KANGUªwQDTLK<DLNGF-NGUGdeK/R#deF<[email protected]­Gs R#UK"NGU DYdeF<DTLK>dgHVU‰EGF>dewzdebgSLP`dgSDEHVNGFxbqRpYd“—cN ]>dgHVUn˜C‘RŠEGF<HV[)R#[GdebedeK<S
d
Y˜½§RwQHVdeFfNGU D¬YdeF<DTLK>dgHVU
D]>KYGHVU T¬hfH*YNGbgSDpE)R#FNGUXK<DLF>hfD¬DLUËTH`]<deUN ]fYGDpb½§R#UGP`bgD¬DLU*K>F<D
DLK a
u
u
d
YGHVUK\b½§R#hEGbedeK>N YGDŠD]<KBYGHVUGU SD'E)R#F"bqRˆTHVU ]<K/R#UK<D
œ}THVhEGF>dg]<DŠDLUK>F<D¬_ˆDLK tn š UÆF<DTHVUGU)R#deK\]/HVN ]
α

TDLK>K<DuŒŽHVF>hfD–bgD]rEGF<DLhdgDLF<]vK<DLF>hfD]-œhfHVU HVEÚVbgD“?ÌYdeEÚVbgDoY˜½ NGU¬YGSLwQDLbgHVEGEcDLhfDLUKu]>NGFvbgD]rl)R#F>hfHVUGdgj*N D]
]>EGl SLF>dgjN D]uYGD-TDxjNGd]/DLF/[email protected]@Ydg]>K>F>de[GNGK>dgHVU!R#UGP`NGbqR#deF<DBYGD]F/R¼QHVU ]OTH`]>hdgj*N D]E)R#F}bqR]>NGdeK<D`n
C [email protected] ]>deK<SAYGDxTHVNGF/R#UK azTHVF>F<D]<EHVU Y R#U*KOR#N¬[GdebqR#U¬U DLKYGD]E)R#F>K>dgTLNGbgD]K>F/RwQDLF<]ÎR#UK}NGU Dx]>NGFJŒRVTD
‘
Jz
SLbgSLhfDLUK/R#deF<D-EcDLF>EcDLU YdgTLNGbqR#deF<DB»\b½§RsGD
EcDLU Y R#UKONGU Dx]<DTHVU YGDxEcDLNGKu]½¾STLF>deF<D"R#N¬EGF<DLhdgDLF–HVF<YF<Df”
Oz
Jz =
1
4πτ
Z
θ=π/2 Z ϕ=2π
θ=0
dϕ cos θ sin θdθ
ϕ=0
Z
λ
0
→
−
d` [1 + α d · u(θ, ϕ)] −2`u(θ, ϕ) · ∇n(r)
HVȉœ a YGS]>deP`U DLUK}bgD]uTH*HVF<YGHVUGU SD]–]>EGl SLF>dgj*N D]}Y˜½§RsD
θ ϕ
|7UmDt—˜DTLK>N)R#UKOb½ deUK<SLP`F/R#K>dgHVUa bqRwVR#bgDLNGFuYGD
Jz
Oz
n
œ §n ¦Q
Ø
]½¾DtszEGF>dehfD-]/HVN ]bqR\ŒŽHVF>hfD”
1
∂n
1
Jz = − λ c
+ α c dz n(r)
6
∂z
3
œ §n ^Q
Ø
D]>[email protected]`]ÎR#UK<D-YN¬wQDTLK<DLNGF ]>NGFvb½§RsDxYGD]vÔQnGCDAhfyLhfDxTR#bgTLNGb‘EDLNGKyLK>F<DAhfDLU SAEcHVNGFbgD]
HVÈ
dz
d
R#NGK>F<D]}THVhEH`]ÎR#UK<D]–YGD TD-j*NGd˜EcDLF>hfDLKuY˜½¾HV[GK<DLUGdeFubqR\F<DLbqR#K>dgHVUƔ
J
→
−
1
J = −D ∇n(r) + α c n(r) d
3
Y R#U ]BbqRVj*N DLbebgD
v =
1
3
α c n(r) d
œ n _Q
Ø 
a£jNGdv]½ dgYGDLU*K>d“­)Dˆ»mNGU Dª÷twzdeK<D]<]<DŠY˜½§RVYwQDTLK>dgHVU‘÷Va7THVF>F<D]>EcHVU Y R#UK'R#N
Ä%W
óôG#Ãhm õÁ!QeÛ£21o&f
hfHVNGwQDLhfDLUK–YGD-YGSLF>dewQDxP`bgHV[)R#bgD-j*NGdD]>K}dehEcH`]<S-R#NspE)R#F>K>dgTLNGbgD]n
|7U!Dt—cDLKaF<DLhEGbqRVWR#UKAb½¾DtsEGF<D]<]>dgHVU¤YGD Y R#U ]Ob½¾SjN)R#K>dgHVU¤YGDBTHVU ]/DLF>wVR#K>dgHVU¤YNmU HVhB[GF<D\YGD"E)R#FJ
J
K>dgTLNGbgD]aGb½¾Sj*N)R#K>dgHVUmYGDAbqRYd“—cN ]>dgHVU ‡VRVYwQDTLK>dgHVU%]½¾STLF>deK"”
œ n 
Ø `
∂n(r, t)
→
−
− D 4n(r, t) + v · ∇n(r, t) = 0
∂t
C‘RŠ]<HVbeNGK>dgHVUªYGDBb½¾SjN)R#K>dgHVU2YGD\Yd“—N ]<dgHVUªRwQDTfRVYwQDTLK>dgHVU¤EHVNGFANGU D\deUVIJDTLK>dgHVUªYGD]–E)R#F>K>dgTLNGbgD]-DLU
DLK
D]>K}YGHVUGU SD-E)R#F}bqR\PR#N ]<]>dgDLUGU Df”
r=0
t=0
n (r, t) =
1
(4 π D t)
3/2
(r − v t)2
exp −
4Dt
!
œ n ÑQ
Ø 
ŒŽR#dg]ÎR#UK"R#EGE)R#F/ROgK>F<DkDtsEGbedgTLdeK<[email protected]@Y˜½¾DLU ]<DLh"[GbgD'»Šw*deK<D]<]/D YGD]xE)R#F>K>dgTLNGbgD]aRwQDTbqR
v
hfyLhfD-Ydg]>EcDLF<]>dgHVUˆj*N DxY R#U ]}bgDxTRV]dg]<HVK>F<HVEcD`n
|7UˆSTLF>dew`R#UK}THVhhfDAEGF<STSYGDLhhfDLU*Kub½¾Sj*N)R#K>dgHVUmTLd“†YGD]<]>N ]]<HVN ]]ÎR-ŒŽHVF>hfDADtszEGF>dehfSD"DLU'ŒŽHVU TLK>dgHVU
YGD]}j*N)R#UK>deK<S]uYGD-bqR\hkR#F<T<l DBR#bgSR#K<HVdeF<D DLK abqRYGDLU ]<deK<S-YGDAEGF<HV[)R#[GdebedeK<SBYGDAEGF<S]/DLU TD"R#EGF<i]
E)RV]
n λ
N
D]>K}YGHVUGU SDxE)R#F"”
p (r, N ) =
2πN λ2
3
−3/2
3 (r − N αd/3) 2
exp −
2N λ2
!
.
œ n ~ 
Ø 
q
jNGdD]<K}YGHVU TxNGU DxPR#N ]<]>dgDLUGU D"TDLUK>F<SD"]<NGF
a Y˜½¾STR#F>KuKÞ¼*EcD
2 n
N αd/3
σ = N3λ
CD}THzDt¥kTLdgDLUK j*N DU HVN ]4RwQHVU ]deUK>F<HzYNGdeKoY R#U ]4b½¾DtszEGF<D]/]>dgHVUkYGDvbqRxYGDLU ]>deK<SuYGDvEGF<HV[)R#[GdebedeK<S
α
p(u)
THVF>F<D]>EcHVU YF/RÆR#bgHVF<]¬YdeF<DTLK<DLhfDLU*K%»Åb½§R#hEGbedeK>N YGDªYNžYdeEÚVbgD!hfHzYNGbqR#UK¬bqRÆYdg]>K>F>de[GNGK>dgHVU¨R#UGP`NGbqR#deF<D
YGD] ‡r]a7DLKbgD¬wQDTLK<DLNGF
deU YdgjN DLF/R‰]/R¤YdeF<DTLK>dgHVUn4C4½¾SjN)R#K>dgHVU n ~ EDLF>hfDLK>K>F/RªYGDmjN)R#U*K>d“­)DLFkbqR
d
a
Ø 
Ydg]>EcDLF<]>dgHVUkK>l SHVF>dgj*N D]oYGD]ow`R#bgDLNGF<]F<DTHVU ]>K>F>NGdeK<D]vYGD]oE)R#F/R#hfiLK>F<D]rYNŠYdeEÚVbgD`abgHVF<]oYGDub½§R#U)R#be¼z]/DuY˜½ NGU
bgHVK}YGD-YGHVUGU SD]uTHVU*K<DLU)R#UK
YdeF<DTLK>dgHVU ]OY˜½§R#F>F>dewQSD]n
N
!
}
"
#$%&
À
(
|
ÇÓ ¾|ÆÈÁ5Í(lÆ5ÉÊÇÓ
ÎÁ Ä+ÏÀ
ÏÀ
Ò(('
ÊÀ
g
ÏÀ†¾
Ćà ÀÒ(
Ä É¾fÇÅÆÈÁ5Ç+*¿ÉÊÀ†¾
Ä
Ô
Æ'ÏÇ Ï
À
€
~
Ì
Ë
ÐÑÒnÓ]ÔÕÑÖ!ÓY×nØ6ÙÚ#ÛÝÜÝÞ߶à
+3ÙdÔÕÑܕØIæAÖ]ßC;6è—ç“Ú~ßnÞÔÕßܕûTìâÚ6Ù!ÞÜÝÛGÞ¥Ö
á Ó]ðfêFæ*ÜTÖ1ߥÖæ!ð~ðuÑ>ßÒ×>æ*ð‡Üßnð#Ù!ð#Ü êFÙ:Ö!ەÞß>×/ÜÝÛÝæ*ð5Ö á Ù!ÞÞnەåCÑ>߶Ö1ßnÒ
Þ'Ù0/æ!ð~Òd×>æCÒÔ¥Û=ëÓ~ßÒ¹ÒÓIÞ¹êFÙ|Ò Ú‡Ø6ÿÞß:×ÑnêFßnÒn܏ßûvÖ á ßÒnÜÝەÔÕßnÞ¹ê>á Ù!ԊÚ#ê„ÛÝÜÝÓ6Ö1ß%ß܊êFÙ Ö!ÛGÞ>ß×/ÜÝÛ=æ!ð Ö á ÓIð_ß%ÑnåCßnð‡ÜÝÓ~ßnê•êFß
Ù!ð~ÛGÒæ*ÜÝÞæÚ#Û ß¶Ö!Û Ú6æ!êFÙ!ەÞßÕæ!ÓwëÓ6Ù1Ö!ÞÓEÚ6æ!êFÙ!ەÞßtÚ#Þ>ÑÒAßnð‡Üß¹Ö1Ù!ð‘Ò¥×ßÒÕÖæ!ð~ð_ÑßÒ 9 ê•êFßâé1Ñnð_ÑnÞÙ!ê„ÛhÒA߶êFٟÚ#ÞæA×Ñ>Ö!Ó]Þ>ß
Ö á Ù*ð#Ù!ê /ÒAߥÖ1ßvÐTÙ2/CêxßnÛFéCØ|Ø6Ù1änÛÝÜÝӑßnê•êxßnÔÕßnð‡ÜTÓ6ÜÝÛGê„ÛGÒ/Ñ>ßûëÓIÛð_ߗڑßnÞnÔÕßÜWÚ6ÙCÒ¹Ö1ßÞ>ßnÔ¹æ!ð‡ÜßÞ¥Ö!ەÞ>ß×/܏ßnÔÕßnð‡Ü{Ù!Ó*ï
Ú6Ù!ÞÙ!ÔÕÿÜÝÞßnÒùÖ1ߊê>á Ù*ð~ÛhÒ/æ*ÜÝÞæÚ#Û ßu-Ù!ԊÚ#ê„Û•ÜÝӑÖ1ßtßÜÊÖ!ەÞß×AÜÝÛÝæ!ð `!+ ٟԹÑ/ܕØIæAÖ1¢ß ;‘è—ç“Ò‘á ÙÚÚ#ê„Û=ëӑßùÖ1Ù!ð‘Òêxßù×>ÙCÒ{0æ 8
êFßÖ1Ñ܏ß>×/܏ßnÓIÞ¶×æ!ð‘ÒÛ=Ö1ÑnÞ>ÑÚ6æCÒÒ/ÿÖ]ßÓ]ðuߟ×æ!Ó]å!ßnÞnÜÝÓIÞ>ß¹Ü æ*Ü Ù*êFߥæ!Ó¶Ú6Ù!ÞÜÝÛ=ßnê•êxߥÖ!ÓY×Û ßnê;û,ÒÙ!ð~
Ò 6ÜÝÞßڑæ!Ó]Þ¥Ù!Ó6Ü Ù!ð‡Ü
Ó]ð~Û ñ/æ!ÞÔÕß +3Ù:×>ÙnÚ6Ù×ÛÝ܏ѹÖ!Ó Ö1Ñ܏ß>×/܏ßnÓIÞÕì›Ô¹ßnÒÓIÞßnÞ¥ÓIð_ߥÙ!ð~ÛGÒæÜÝÞ'æÚ_Û=ߗÚ_ÞÑ×/ÛhÒAÑnÔÕßnð‡ÜÙ!åCß>×¹êFٛÔÕÑܕØIæAÖ1ߊڑßnÓ6¢
Ü 6ÜÝÞ>ß
×>Ù!ÞÙ×/܏ÑnÞnÛhÒAÑ>ßtÚ6Ù!ÞÓ]ðfÒAßnÓIê±ð_æ!ÔÕänÞßnûêFÙâÚ#ÓIÛhÒÒÙ!ð#×ß¹Ö1ßÞ>ß×>æ*ð‘ÒnÜÝÞnÓ6×AÜÝÛÝæ!ð~ûŠëÓ]ÛTÖ1яÚ~ßnð#Ö|Ö]ßêúÙ{ñ/æ!ÞnÔÕ߶Ö1ßêFÙ
ñæ!ð_×/ÜÝÛÝæ*ðNÖ á ßïÚ6æ!ÒÛÝÜÝÛÝæ!ðwÖ!ÓNÖ1Ñ܏ß×/܏ßÓ]Þ¹ßnð ëӑßnÒnÜÝÛ=æ!ð 7âÙ*ð‘Òâêx߶×>ÙCÒ¹Ö á Ó]ð_ߥ×æ!Ó]å!ßnÞnÜÝÓIÞ>ßvÚ6Ù!ÞÜÝÛ=ßnê•êxßûêFٟÚ#ÓIÛGÒÒÙ*ð#×߶Ö1ߥÞß×æ!ð‘ÒnÜÝÞnӑ×/ÜÝÛÝæ*ðþÖ!Ó Ö!Û .
Ú -!êxß¹Ö1Ñ7ڑßnð#ÖYÖ1ß
êá ^Ø /Ú6æ*ܕØ6ÿnÒ/ߟëӑßvê>á æ!ðâñÙ!ÛÝÜÊÒnÓ]ÞùêFÙÕÖ!ÛGÒnÜÝÞnÛ=äÓIÜÝÛ=æ!ðYÙ!ðIé!Ó]êFÙ!ەÞߥà3êúÙùÚ#ÓIÛhÒÒÙ!ð#×ßâÖ1ßvÞ>ß×>æ!ð‘ÒÜÝÞӑ×/ÜÝÛ=æ!ðwÖ!ÓaÖ!Û .Ú -!êxß
Ö!ÛGԥەð~Ó~ßêúæ*ÞÒ/ëӑ߹ê>á æ!ð Ù!ÓIÜ æ*ÞÛGÒ/߶êúÙ¥Ú#ÞÑnÒ/ßnð#×ßdÖ]ߥԹæAÖ1ßÒ{Ú#ê„Ó]Ò¹ÑnêxßnåCÑnÒûŠ×>æ!Ô¥ÔÕߥêxß¹ë/Ó6ÙÖ*ÞÛ .Ú -!êxßû{ÖÙ*ð‘ÒŸêFÙ
Ö!ÛhÒÜÝÞÛ änÓ6ÜÝÛÝæ*ðfÙ!ðIé!Ó]êFÙ!ەÞß
+ ßâ×>Ù!êF×Ó]ê Ö]ßùêFÙ¥ÒÛFé!ð~Û . ×>Ù*ð#×ߟÖ1ßÒvÙ!ԊÚ#ê;ÛÝÜÝӑÖ1ßÒvÖ á Ù*ð~ÛhÒ/æ*ÜÝÞæÚ#Û ßҊÞ>ß×>æ!ð~ÒnÜÝÞnÓ]ÛÝ܏ßÒÚ6Ù!ÞùêFÙ¥ÔÕÑܕØIæAÖ]—
ß ;6èŠç
ڑßnÞnÔ¹ß/ÜÖ1ߟ܏ßÒ܏ßnÞ7ß 6 ×Ù×ßnÔÕßnð‡Üê>á ^Ø /Ú6æ*ܕØ6ÿÒAßÖ á ÛGÒæÜÝÞ'æÚ_Û=ßÖ1ßvêúÙ%Ö!ÛGÒnÜÝÞnÛ änÓIÜÝÛ=æ!ð Ù!ð]é!ÓIêFÙ!ÛGÞ>ß Ä%c
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
–,/.
³
ò 579‚@}1B )? ·c6B9?:\@u1
³Zõ†6B1
³\:>µ%‚ó<5
º{}579´9‚@}6{}5£·[?6B·5
ºXó<5
?[@ ?
5‚? 6\1B:`Þ
ӑ@·c¸¹5±³\6´9;:<5ó
C RmYdg]>K>F>de[GNGK>dgHVUËR#UGP`NGbqR#deF<D'YN‰• NsÅYGDfF/R¼QHVU ]\TH`]<hdgjN D]\D]>KB]<NGEGEH`]<SD`aY R#U ]BTDLK>K<D'EGF<DLhdgiLF<D
‘
R#EGEGF<H*T/l D`a yLK>F<D-STLF>deK<D-]/HVN ]bqR\ŒŽHVF>hfD”
Φ(u) =
Φ0
(1 + αD · u),
4π
œ ÷zn 

HVȈbgDA• NsˆYd“—˜SLF<DLUK>dgDLba a)Y R#U ]}bqRYdeF<DTLK>dgHVU a THVU ]<dg]>K<D-DLUˆNGU DxE)R#F>K>dgD"dg]<HVK>F<HVEcD`a
a hfHzYNGbgSD
Φ
u
Φ0 /4π
Û D]>K
DLU
E)R#FBNGU DpTHVhEcH`]/R#UK<DpYdeEcHVbqR#deF<D`n£ÕAR#U ]\bqR%TRVYF<DpYGD'TDLK>K<D'E)R#F/R#hfSLK>F>dg]/R#K>dgHVUa
d
cos(u,
D)
D
NGU‰wQDTLK<DLNGFBNGUGdeK/R#deF<DkEcHVdeUK/R#U*K\Y R#U ]BbqRˆYdeF<DTLK>dgHVUZYNÅYdeEcÚVbgD`a‘DLK
D]<K-b½§R#hEGbedeK>N YGDpYN‰YdeEÚVbgD'E)R#F
α
F/R#EGEHVF>K"R#N2• Ns!dg]<HVK>F<HVEcD¬”
nCD\• NsªD]>K-YGHVU TkR#NGP`hfDLU*K<SkY˜½ NGUªŒRVTLK<DLNGF
Y R#U ]xbqR
0≤α≤1
1+α
YdeF<DTLK>dgHVUªYGD
DLKAF<SYNGdeKAYGD
Y R#U ]–bqRŠYdeF<DTLK>dgHVU2HVEGEcH`]<SD`nC4½§R#hEGbedeK>N YGDfYGD"b½§R#UGdg]/HVK>F<HVEGdgDD]>K
D
1−α
PQSLU SLF/R#bgDLhfDLUKAYGSt­ UGdgD-THVhhfD
œ ÷zn§ÑQ
Φmax − Φmin
δ=
Φmax + Φmin
DLK
]/HVUKF<D]>EcDTLK>dewQDLhfDLUKbgD]vw`R#bgDLNGF<]vhkRszdehkR#bgD]DLKvhdeUGdehkR#bgD]}YNŠ• Ns˜nC½§R#hEGbedeK>N YGD
Φmax Φmin
w`R#NGK\Dts RVTLK<DLhfDLUK
Y R#U ]BbgDpTRV]@YGDpTDLK>K<DŠE)R#F/R#hfSLK>F>dg]/R#K>dgHVUËYNXYdeEÚVbgD`n‡rDLK>K<DpSTLF>deK>NGF<DpYNŕ Ns
δ
α
EDLNGK}yLK>F<D-w*N DxTHVhhfD"bgD]}YGDLNs¬EGF<DLhdgDLF<]}K<DLF>hfD]uYN%YGSLwQDLbgHVEGEDLhfDLU*KOYGDxbqRYdg]<K>F>de[GNGK>dgHVU%R#UGP`NGbqR#deF<D
]>NGFrbgD]ol)R#F>hfHVUGdgj*N D]}]>EGl SLF>dgj*N D]A”*HVF<YF<D-_ŠœhfHVU HVEcÚVbgD|
 ?¨HVF<YF<D œ’YdeEÚVbgDtnCD–[GNGKYGD–bqR-hfSLK>l HzYGD

a DLK
»!E)R#F>K>deF'YGD]YGHVUGU SD]fhfD]>NGF<SD]E)R#FkNGU D¬DtszEcSLF>dgDLU TD
D]>KY˜½¾D]>K>dehfDLFfbgD]E)R#F/R#hfiLK>F<D]
Φ0 α
D
YGSLK<DTLK/R#UK£bgD];F/R¼QHVU ]£TH`]>hdgj*N D]n D†bŒŽR#NGK£YGHVU ToYGSLK<DLF>hdeU DLF7j*N)R#K>F<DoHV[ ]/DLF>wVR#[GbgD]v”bqRU HVF>hkR#bedg]/R#K>dgHVUfYN
• Ns!œhfHVU HVEcÚVbgDtaNGU DAR#hEGbedeK>N YGDfœ’YdeEcÚVbgDrDLKrbgD]YGDLNsŠR#UGP`bgD]YGSt­ UGdg]<]/R#U*K nÕxR#U ]vTDLK>K<DAEGF<DLhdgiLF<D
D
R#EGEGF<H*T/l D`abgDOYGSLK<DTLK<DLNGF}DLU¬j*N D]>K>dgHVUpD]>K]>NGEGEH`]/SOYGHVUGU DLFvNGU DADtszEcH`]>deK>dgHVU¬YGDObqR"K<HVK/R#bedeK<SxYNpTLdgDLbnC‘R
EGF<HV[)R#[GdebedeK<SpYGDkYGSLK<DTLK<[email protected]ÅSLwQSLU DLhfDLU*KfD]>KBYGHVU T'deU YGSLEDLU Y R#U*K<DpYGDfbqR%YdeF<DTLK>dgHVUÆY˜½§R#F>F>dewQSD`^
n DÞboD]>K
[GdgDLU'SLwzdgYGDLUKvjN½ NGUŠK<DLbYGSLK<DTLK<DLNGFrdgYGSR#bU½¾Dtszdg]>K<DOjN DOY R#U ]4b½¾D]<EGF>deKrYN'EGl¼]>dgTLdgDLUahkR#dg]rTDLKoDtsGDLhEGbgD
]>dehEGbgDxEcDLF>hfDLKOYGD-THVhEGF<DLU YF<D"ŒRVTLdebgDLhfDLUK–bgD-EGF>deU [email protected]>l HzYGD"[email protected]>deK<DBYGD"THVU YNGdeF<D
»\YGD]F<S]>NGbeK/R#K<]K<HVK/R#bgDLhfDLU*K<]ODtsEGF>dehkR#[GbgD]u]<HVN ]rŒŽHVF>hfD-R#U)R#be¼zK>dgjN D`a TDxj*NGdcU½¾D]>KEGbeN ]bgDxTRV]bgHVF<]/jN D
b½¾HVUmRVYGHVEGK<D-NGU¬hfHzYGiLbgDxEGbeN ]}F<SR#bedg]<K<D`n
HVÈ
|7UªSTLF>dew`R#UK-bgDB• Ns!]/HVN ]ObqR'ŒŽHVF>hfD¬œ ÷zn t acbgD]AE)R#F/R#hfiLK>F<D]-YNªYdeEÚVbgD]/HVUKAF<DLbedgS]"R#Ns%hfHVhfDLU*K<]

Üÿrà#ߣñ#ãÎëçJìÎàO›“ìÎà_ûâßã\‰tßJç7ä§ß7çJìßSƒçJê§ßJàá<ûß ûê„Qé#ëŽê§ìtàOå!âßJà#ç¯ìtàáŽâ 97ñâ 9Jç†9nûßJïï߯àáIkk
[À’Ä ï ÃQ>f‘V&Ã`fÛog&<f Û;0Ü$õˆQÈÃQ&&‘#ˆ) õ%'&<f;†i‰jÛ;& Ã*õ
YGDAbqRYdg]>K>F>de[GNGK>dgHVU
Φ(u)
Ä%v
E)R#F}bgD]F<DLbqR#K>dgHVU ]B”
Φ0 =
Z
Z
3
αD =
Φ0
Φ(u)dΩ et
uΦ(u)dΩ.
œ ÷zn ~ 
HVȈbqRf]<HVhhkR#K>dgHVU¤]½¾Dt—˜DTLK>N D"]>NGFuK<HVNGKObgD-TLdgDLbnCDBEGF>deU [email protected]>l zH YGDBD]<KuYGHVU T"YGDBYGSLK<DLF>hdeU DLF
»\E)R#F>K>deFuYGD]}YGHVUGU SD]uDtsEcSLF>dehfDLUK/R#bgD][email protected]#bgDLNGF}YGD]}deUK<SLP`F/R#bgD]
DLK GY St­ UGdgD]uE)R#F"”
I0
I0 =
Z
Φ(u)dΩ et
I=
R­ UˆYGDxF<DTHVU ]>K>F>NGdeF<D-bgDAwQDTLK<DLNGF–YdeEÚVbgDxTHVhhfDBSLK/R#UK
Z
I
uΦ(u)dΩ.
α D = 3I/I0
œ ÷zn „*
n
É HVNGFvTR#bgTLNGbgDLFvbgD]4hfHVhfDLU*K<]
DLK »xE)R#F>K>deFvYGD]oYdeF<DTLK>dgHVU ]vY˜½§R#F>F>dewQSDAYGD]oSLwQSLU DLhfDLU*K<]aYdew*dg]/HVU ]
I0 I
Œ’HVF>hfDLbebgDLhfDLU*KfbgD¬TLdgDLb}DLUXNGU D¬]<SLF>dgDpYGDpEGd“sDLbg]
arTHVNGw*F/R#U*KkT<l)RVTLNGU b½§R#UGP`bgD¬]<HVbedgYGD
n£|7U
{(i, j)}
δωi,j
U HVK/R#UK
bgD}U HVh"[GF<DAYGDuF/R¼QHVU ]vTH`]>hdgj*N D]oHV[ ]<DLF>wQS]vY R#U ]obqR-YdeF<DTLK>dgHVU¬YNkEGd“sGDLb
azbgDuU HVh"[GF<D
Ni,j
(i, j)
Yd“—cSLF<DLU*K>dgDLbY˜½¾SLwQSLU DLhfDLUK<]–YGSLK<DTLK<S]uY R#U ][email protected]<DTLK>dgHVU aGY R#U ]
a]½¾STLF>deKOR#bgHVF<]"”
dΩ
u
dN
(u)dΩ = Φ(u)E0 dΩ −→ Ni,j = Φi,j E0 δωi,j .
dΩ
œ ÷zn:ðQ
VH È
D]>KAb½¾DtsEH`]>deK>[email protected]ŠF<SLP`dgHVUÅYN2TLdgDLb
œ’DLUªh 2 n¾]Νta˜]>NGEGEcH`]<[email protected]>F<[email protected]>dgj*N DfEcHVNGFxK<HVNGK<D]
E0
dΩ
bgD]xF<SLP`dgHVU ]-YN‰TLdgDLbnCRŠwVR#bgDLNGF"YGD
U D–I>HVN)R#UKBR#N TLNGUÅF<ÚVbgD`acU HVN ]-NGK>debedg]<DLF<HVU ]
h n¾]a˜Y R#U ]
E0
E0 = 1 2
TDLKrDtsGDLhEGbgDx]>dehEGbed“­)SxR­ U¬Y˜½¾STLF>deF<DObgD]vF<S]<NGbeK/R#K<]r]/HVN ]oNGU DOŒŽHVF>hfDAjNGdEDLF>hfDLK>K>F/[email protected]]>N)R#bedg]<DLFvEGbeN ]
ŒŽRVTLdebgDLhfDLU*KNGU DvR#U)R#bgHVP`dgDrRwQDTrbgDEGF<HzTD]<]>N ];YGDhkR#F<T<l DR#bgSR#K<HVdeF<D`nD†bŒŽR#NGK;TDLEDLU Y R#U*K£PR#F<YGDLF£»}b½¾D]>EGF>deK
bgDxŒR#deKOj*N D
RbqRkYdehfDLU ]>dgHVU!YGD"h 2 n¾]a EcHVNGFOTHVU ]/DLF>wQDLFOb½ l HVhfHVPQSLU SLdeK<[email protected]]–SjN)R#K>dgHVU ]nM‘wQDTBTDLK>K<D
E0
STLF>deK>NGF<D`a
D]>KuYGHVU T"NGU D-w`R#bgDLNGF–R#EGEGF<HzT<l [email protected]ŒŽHVU TLK>dgHVU%j*N DxU HVN ]–T<l DLF<T/l HVU ]A»deUK<SLP`F<[email protected]”)DLU
Ni,j
]<HVhhkR#UK4]<NGF7K<HVNGK7bgD]7EGd“sDLbg]bqR–wVR#bgDLNGFYGD
aVU HVN ]HV[GK<DLU HVU ]YGHVU T}NGU DD]>K>dehkR#K>dgHVU'YGDvb½ deU*K<SLP`F/R#bgD`n
Ni,j
ÉHVNGF}NGU DxŒ’HVU TLK>dgHVU YGHVUGU SD-]<NGFbqR]>EGl iLF<D`aU HVN ]}NGK>debedg]/HVU ]}YGHVU TxbqRTHVF>F<D]>EcHVU Y R#U TD"]<NGdewVR#U*K<Df”
f
Z
f (u)dΩ −→
X
œ ÷zn 
Ø
fi,j δωi,j .
(i,j)
C D]–wQDLF<]>dgHVU ]xYdg]<TLF<iLK<D]xYGD]AhfHVhfDLUK<]
LD K aj*N DBb½¾HVUªU HVK<DLF/R
LD K a]<HVU*K-HV[GK<DLUN D]xP`F/ÖVTDR#Ns
I0
I
S0
S
F<DLbqR#K>dgHVU ]"”
X
X
œ ÷z¤n ÷`
I0 → S 0 =
HVÈ
ui,j
Ni,j
(i,j)
et
I→S=
D]<KbgDAwQDTLK<DLNGFuNGUGdeK/R#deF<D-Y R#U ][email protected]<DTLK>dgHVU%YN¬EGd“sGDLb
Ni,j ui,j ,
(i,j)
(i, j)
n
ÄgW
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
C R'YGDLF>UGdgiLF<DBSLK/R#EcD\THVU ]>dg]>K<DB»'T<l)R#UGPQDLF–bgD]A]<HVhhfD]–]>NGFuK<HVNGK<D]–bgD]–YdeF<DTLK>dgHVU ]@œ’]/HVhhfD]–]<NGFubgD]
‘
EGd“sDLbg]t4DLU¬]<HVhhfD]]<NGFrK<HVN ]}bgD]SLwQSLU DLhfDLUK<]n É;HVNGFbqR\F<SR#bedg]<DLFazdebcŒR#NGKrF<DLhkR#F<j*N DLF}j*N DObgHVF<]/jN D–b½¾HVU
]<HVhhfD"]<NGFuK<HVN ]ObgD]uSLwQSLU DLhfDLUK<]-R#EGE)R#F>K<DLU)R#U*K"»@NGU!hfyLhfD-EGd“sGDLb
acHVU!]/HVhhfD"DLUmF<SR#bedeK<S
(i, j)
Ni,j
Œ’HVdg]@bqRªwVR#bgDLNGFkTHVF>F<D]<EHVU Y R#U*K<D`n4É;HVNGFkNGU DpŒŽHVU TLK>dgHVU j*N DLbgTHVU jN D
]>NGFfbqR¤]<EGl iLF<D`a4R¼R#U*K
F (u)
Fi,j
THVhhfDkw`R#bgDLNGFY R#U ]\bqRmYdeF<DTLK>dgHVU
DLK
Y R#U ]\bqRmYdeF<DTLK>dgHVUXYGD'b½¾SLwQSLU DLhfDLUK a£R#bgHVF<]"bgD'bedgDLU
(i, j)
Fk
k
DLUK>F<DxNGU D"]<HVhhkR#K>dgHVUm]>NGF}bgD]uYdeF<DTLK>dgHVU ]uDLKu]>NGFbgD]uSLwQSLU DLhfDLUK<]–]½¾STLF>deK"”
X
Ni,j Fi,j =
X
œ ÷zn§¦Q
Fk ,
k
(i,j)
HVȬ[email protected]<DLhdgiLF<D-]<HVhhfD"]½¾Dt—cDTLK>N D"]>NGFbgD]EGd“sGDLbg]aGDLK}[email protected]]<DTHVU YGD-]<NGFbgD]uSLwQSLU DLhfDLUK<]n
|7U¬R#EGEGbedgjN)R#U*KTDLK>K<DuSLPR#bedeK<Sx»xb½¾Sj*N)R#K>dgHVU¤œ ÷zn¤÷`tazDLKvDLUkdgYGDLUK>d“­R#U*KbqRxŒŽHVU TLK>dgHVU
»xb½ dgYGDLUK>deK<S
F (u)
œ
"Y˜½ NGU DŠE)R#F>Ka£DLK»
Y˜½§R#NGK>F<DŠE)R#F>Ka;bqRˆwQDLF<]<dgHVUZYdg]/TLF<iLK<DŠYGD]@Sj*N)R#K>dgHVU ]¬œ ÷zn „*
F (u) = 1
F (u) = u
]½¾STLF>deKv­ U)R#[email protected]”
X
X
œ ÷zn§^Q
S0 =
1 =N
et
S=
k
uk ,
k
HVȬbgD]}]<HVhhfD]O]<HVUKuDt—˜DTLK>N SD]}]>NGFuK<HVN ]bgD]
SLwQSLU DLhfDLU*K<]OYNmbgHVK}YGD-YGHVUGU SD]n
N
CD]vj*N)R#K>F<D–]/HVhhfD]rYdg]/TLF<iLK<D]rTLd“†YGD]/]>N ]a
DLK a*EcDLNGwQDLUK}yLK>F<D–YdeF<DTLK<DLhfDLUKuTR#bgTLNGbgSD]}»"E)R#F>K>deF
S0 S
YGD]7YGHVUGU SD]aQDLK7bqRAYGSLF>dew`R#K>dgHVUkYGD]£E)R#F/R#hfiLK>F<D]4YNfYdeEcÚVbgD}YGSTHVNGbgD}R#bgHVF<]YGDvb½ dgYGDLU*K>d“­)TR#K>dgHVUŠDLU*K>F<D}TD]
]<HVhhfD]vDLK
DLK a*YGHVU*Kb½¾DtszEGF<D]<]<dgHVUpD]<KrYGHVUGU SDxDLUªœ ÷zn „*tnC½§R#hEGbedeK>N YGD-YN¬YdeEÚVbgDADLKv]/R\YdeF<DTLK>dgHVU
I0 I
]<HVUKuR#bgHVF<]OD]>K>dehfSDxP`F/ÖVTD"»\bqRF<DLbqR#K>dgHVU
a Y˜½¾HVȬb½¾HVUˆK>deF<Df”
αD = 3S/S0
α=
DLK
3
3 X
||
uk || ,
||S|| =
S0
N
œ ÷zn Q_ 

k
D=
S
.
||S||
œ ÷zn
`

C½¾D]>K>dehkR#K>dgHVU‹YGDˆTD]E)R#F/R#hfiLK>F<D]ŠD]>K[GdgDLU DLUK<DLU YN‹Y˜½§R#NGK/R#UKŠEGbeN ]EGF<STLdg]<Dˆj*N D¬bgDpU HVhB[GF<DˆK<HVK/R#b
Y˜½¾SLwQSLU DLhfDLUK<]-D]<K–P`F/R#U Y˜ncC½¾SLK>N YGDYGD\bqRkEGF<STLdg]>dgHVU2YGD\TDLK>K<[email protected]<DTHVU ]>K>F>N TLK>dgHVU2D]>K–EGF<S]<DLUK<SDY R#U ]AbqR
]>NGdeK<DxYGD-TDxT/l)R#EGdeK>F<D`n
CDŠbgDTLK<DLNGFfR#K>K<DLUK>d“ŒxR#NGF/Rm[GdgDLUXI] |GF\F<DLhkR#F<j*N SpbqRmTHVF>F<D]>EcHVU Y R#U TD¬DLUK>F<DŠb½¾STLF>deK>NGF<D¬YNZwQDTLK<DLNGF
YdeEÚVbgD
THVhhfD¤NGU Dª]<HVhhfDªYGD!wQDTLK<DLNGF<]
DLKpbgD¤wQDTLK<DLNGF¬EH`]<deK>dgHVU
THVF>F<D]>EHVU Y R#U*K!ȁbqR
αD
uk
r
Ydg]>K/R#U TD¤E)R#F<THVNGF>N D‰R#NžTHVNGF<]ˆY˜½ NGU D¤hkR#F<T/l D‰R#bgSR#K<HVdeF<D2»ÆK>F<HVdg]¬YdehfDLU ]>dgHVU ]aYGD¤E)RV]
n
λ = 3/N
‡rDLK>K<DuSj*NGdewVR#bgDLU TDOD]>KK>F<i]odeUK<SLF<D]<]ÎR#UK<D`azDLKwVR-EDLF>hfDLK>K>F<DOYGDuTR#F/RVTLK<SLF>dg]<DLFobqR-Ydg]>K>F>de[GNGK>dgHVU¬R#K>K<DLU YN D
[Àwvª%ÃQGeÛog&<fˆÃQ&&‘#¬) õòyff &)f;òÂÞ)_i2
ÄgWÄ
Y N¬wQDTLK<DLNGFOYdeEcÚVbgD`a DLUm]½§R#EGEGNG¼R#U*KO]>NGFbgD]uF<S]>NGbeK/R#K<]uYGSLhfHVU*K>F<S]uEGF<STSYGDLhhfDLU*KObgHVF<]}YGD-b½¾SLK>N YGD"YN
EGF<H*TD]<]<N ]‘YGDhkR#F<T/l DrR#bgSR#K<HVdeF<DrDLKYGDoYd“—N ]<dgHVUnϤR#bel DLNGF<DLN ]<DLhfDLUKa#bgDE)R#F/R#bebgiLbgDrD]>K[DR#N THVNGE\hfHVdeU ]
SLbgSLPR#UK7bgHVF<]<j*N DrbgDoYGSLK<DTLK<DLNGF4THVU ]<dgYGSLF<SoU½¾D]<K;EGbeN ]£E)R#FJŒR#deKaVT`½¾D]>K»–YdeF<DrY R#U ]£bgDrTRV]7HVÈ]/ROTHVNGwQDLF>K>NGF<D
YNˆTLdgDLbU½¾D]>K}EGbeN ]UGd‘NGUGd“Œ’HVF>hfDxUGd‘K<HVK/R#bgD`n)CD-TRV]}dgYGSR#boœ’THVNGwQDLF>K>NGF<D-K<HVK/R#bgDBDLKNGUGd“Œ’HVF>hfD}]<DLF/[email protected] T
SLK>N YdgSˆR#U)R#be¼*K>dgj*N DLhfDLUKfE)R#[email protected]¤]>NGdeK<D`a7»%K>deK>F<D¬Y˜½¾DtsGDLhEGbgDpEHVNGFbqR%hfSLK>l HzYGD`[email protected]]ÎRmPQSLU SLF/R#bedg]ÎR#K>dgHVU
]<DLF/RpEGF<S]<DLU*K<SDkYGDfhkR#UGdgiLF<D'jN)R#bedeK/R#K>dewQD`a‘bgD]-F<S]>NGbeK/R#K<]\SLK/R#UK\HV[GK<DLUN ]\E)R#FBR#debebgDLNGF<]BE)R#F"]>dehBNGbqR#K>dgHVU
UNGhfSLF>dgj*N D`n
–,¯®
°p5±9;ºB¨³xõ6\1B5Ò9ë@u6{}5£·[?6B·5µ?[@z?ºXó<5±[email protected]}1
³
6\1B:/ӑ@·c¸¹5
C DBEGF<DLhdgDLF–F/R¥fU DLhfDLU*KxjNGd;EDLNGK–yLK>F<D\R#EGEcHVF>K<SR#NmhfHzYGiLbgD\dgYGSR#b;YGSTLF>deK–EGF<STSYGDLhhfDLUK-D]>K–bqR
EGF>dg]<D'DLUXTHVhEGK<DpYGDŠbqRˆU HVUZNGUGd“ŒŽHVF>hdeK<SpYGD'bqR%THVNGwQDLF>K>NGF<D¬YNXTLdgDLbn£|7UXDt—cDLKaK<HVN ]@bgD]@YGSLK<DTLK<DLNGF<]
NGK>debedg]<S]\[email protected][ ]<DLF>wQDLF\bgD]BF/R¼QHVU ]TH`]>hdgj*N D]\]/HVUK\EGbeN ]\]/DLU ]>de[GbgD]@Y R#U ]\TDLF>K/R#deU D]YdeF<DTLK>dgHVU ]j*N D
Y R#U ]4Y˜½§R#NGK>F<D]–”QDLUŒŽHVU TLK>dgHVU'YGD}]/RxbqR#K>deK>N YGDu]>NG
F (DLF>F<D`aQNGU DODtszEcSLF>dgDLU TD}U D"÷twQHVdeK÷}E)RV]bgD]oYd“—cSLF<DLU*K<D]
F<SLP`dgHVU ]fYN TLdgDLbEcDLU Y R#UKfbqR¤hfyLhfDmYNGF<SD`nÕADpEGbeN ]a£b½¾Dt¥kTRVTLdeK<SmYGD¬YGSLK<DTLK>dgHVUEDLNGKfYGSLEcDLU YF<DˆYGD
b½§R#UGP`bgD¬Y˜½ deU TLdgYGDLU TDˆbgHzTR#b}jN D'ŒR#deKbqR!E)R#F>K>dgTLNGbgDmRwQDT¬bgD¬]/HVba;YGD¬]ÎR!U)R#K>NGF<D¬DLKfYGD¬]<HVUËSLU DLF>P`dgDˆHVN
DLU THVF<D"Y˜½§R#NGK>F<D]–Dt—cDLK<]ubedgS]–R#N¬ŒŽHVU TLK>dgHVUGU DLhfDLUKxYN!YGSLK<DTLK<DLNGFnCDBTR#bgTLNGbYGD"bqR]>NGFJŒRVTD"Dt—˜DTLK>dewQD"YGD
YGSLK<DTLK>dgHVUˆYN%¡zÕ YGDAb½ š [ ]<DLF>w`R#K<HVdeF<DxÉ7dgDLF>F<DxMONGPQDLF}YGSLwQDLbgHVEGESAEGF<STSYGDLhhfDLU*KOY R#U ]}TDLK>K<DAK>l i]/D–Dtsz
EGbedgjN DrYGDoŒŽRVWHVUYGSLK/R#debebgSDvjN DLbebgDvD]>K;b½ deU• N DLU TDYGD]7Yd“—˜SLF<DLUK<]£E)R#F/R#hfiLK>F<D]7EGl*¼z]>dgj*N D]£]>NGF;b½¾Dt¥kTRVTLdeK<S
YGDAF<DTHVU ]>K>F>N TLK>dgHVUn
¹E)R#F>K>deF¬YGD%TDLK>K<D¤THVUGU)R#dg]<]/R#U TD`arbqRƒ×>Ù!ÞÜß Ö1ßf×>æ*Ó]åCßÞnÜÝÓIÞ߉EcDLF>hfDLK¬YGD%THVUGU)ROgK>F<D!b½hßïڑæCÒÛÝÜÝÛÝæ*ð
HV[GK<DLUN D-E)R#FubgD-YGSLK<DTLK<DLNGFa
œ’DLU 2 ta Y˜½ NGU DxF<SLP`dgHVUmYNmTLdgDLbY R#U ]}bqRYdeF<DTLK>dgHVU a)»\NGU%R#UGP`bgD
E(u)
m .s
u
]<HVbedgYGD
EGF<i]n7C‘R!hfSLK>l HzYGD¬YGDpF<DTHVU ]>K>F>N TLK>dgHVU YNËYdeEcÚVbgDpEGF<STSYGDLUK<Dm]/D'PQSLU SLF/R#bedg]/DmR#NXTRV]fHVÈ
dΩ
Y˜½ NGU D-DtszEcH`]>deK>dgHVUˆU HVU%NGUGd“Œ’HVF>[email protected]#UGdgiLF<DB]>NGdewVR#U*K<Dk”GbgD-U HVh"[GF<D
Y˜½¾SLwQSLU DLhfDLU*K<]AHV[ ]<DLF>wQS]
Ni,j
Y R#U ]bqRYdeF<DTLK>dgHVUmYNˆEGd“sGDLb
]½¾STLF>deKhkR#deUK<DLU)R#U*K\”
”
(i, j)
œ ÷zn QÑ 

dN
(u)dΩ = Φ(u)E(u)dΩ −→ Ni,j = Φi,j Ei,j δωi,j .
dΩ
VH È
F<DLEGF<S]<DLUK<Dpb½¾DtsEcH`]>deK>dgHVU RV]<]<HzTLdgSD¬R#NÆEGd“sGDLb
n š Fa;bqR¤jN)R#U*K>deK<S¬jN DŠU HVN ] T/l DLF<T<l HVU ]'»
Ei,j
(i, j)
hfD]>NGF<DLFrD]<K4bgD• Ns
]<HVN ]>eI>RVTDLU*KazTHVF>F<D]<EHVU Y R#U*Kv»"NGUkU HVhB[GF<D–Y˜½¾SLwQSLU DLhfDLU*K<]
*a THVF>F>dePQS–YGD
Φ(u)
Ñi,j
ÄgWMv
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
b½¾DtszEcH`]>deK>dgHVUaGYGSt­ UGdE)R#[email protected]<DLbqR#K>dgHVU‰”
dÑ
1
Ni,j
(u)dΩ =
Φ(u)dΩ −→ Ñi,j =
.
dΩ
E(u)
Ei,j
œ ÷zn ~ 

C D]O]<HVhhfD]
DLK a)R#EGEGF<HzT<l)R#U*KObgD]deU*K<SLP`F/R#bgD]
LD K F<DLEGF<S]<DLUK/R#U*KAbgD]hfHVhfDLUK<]–YNˆ• Ns
S0
S
I0
I
Φ
HVUKOR#bgHVF<]}EHVNGF}U HVNGQw DLbebgD"DtszEGF<D]<]<dgHVU‰”
S0 =
X Ni,j
(i,j)
Ei,j
et
S=
X Ni,j ui,j
(i,j)
Ei,j
,
œ ÷zn *„ 

jNGd˜EcDLNGwQDLUKOyLK>F<DxF<SSTLF>deK<D]u]<HVN ]}bqR\Œ’HVF>hfDxYGD"]<HVhhfD]}]>NGFK<HVN ]ubgD]}SLwQSLU DLhfDLUK<]\”
S0 =
X 1
Ek
et
||S||
S0
et
S=
X uk
k
k
Ek
,
œ ÷zn Qð 

Y #R U ]7bqRVj*N DLbebgD
F<DLEGF<S]<DLU*K<D}b½¾DtsEH`]<deK>dgHVU'RV]<]<HzTLdgSD}»Ab½¾SLwQSLU DLhfDLUK nQÉ£R#FoTHVU ]>K>F>N LT K>dgHVUab½§R#hEGbedeK>N YGD
Ek
k
DLKAbqR'YdeF<DTLK>dgHVU2YNªYdeEcÚVbgD]<HVUK–­ U)R#bgDLhfDLU*K"YGSLK<DLF>hdeU S]xE)R#FxbgD]–hfyLhfD]xF<DLbqR#K>dgHVU ]x`w R#bqR#[GbgD]AEHVNGFAbgD
TRV]}HVȬb½¾DtsEH`]>deK>dgHVU%D]>KNGUGd“Œ’HVF>hfDk”
α=3
D=
S
.
||S||
œ ÷zn
Ø
CDBEGF>deU TLdeED\YGD"bqRfhfSLK>l H*YGD\F<D]>K<DBYGHVU T"bgDBhfyLhfD`abgD]OYdeF<DTLK>dgHVU ]AY˜½§R#F>F>dewQSD]xYGD]OF/R¼QHVU ]xTH`]>hdgj*N D]
]<HVUK%]<DLNGbgDLhfDLUK%EHVU YGSLF<SD]¬E)R#FmNGUžŒŽRVTLK<DLNGF%SLPR#bB»Æb½ deUwQDLF<]<DÅYGDªb½¾DtsEH`]>deK>dgHVU¨Y R#U ]mbqRXYdeF<DTLK>dgHVU
THVU ]>dgYGSLF<SD`a)R­ UmYGD-EGF<DLU YF<D-DLUmTHVhEGK<DxbqR\U HVUˆNGUGd“ŒŽHVF>hdeK<S"YGDAbqRfTHVNGwQDLF>K>NGF<D"YNˆTLdgDLbn
‡rDLK>K<D¬EGF<HzTSYNGF<DˆYGDpF<DTHVU ]>K>F>N TLK>dgHVU‹Y˜½ NGU YdeEÚVbgDˆR¤SLK<SˆEGF<HVEH`]<SDpEcHVNGFbqR¤EGF<DLhdgiLF<DpŒŽHVdg]E)R#F
¡zHVhhfDLF<]ˆ¤‚ ÷„V†n7CDŠK>F/Rw`R#deb}Dt—˜DTLK>N SˆR#NËTHVNGF<]YGD¬TDLK>K<DpK>l i]<D¬R%THVU ]<dg]>K<Sˆ»%F<DtŒ’HVF>hBNGbgDLFaHVEGK>dehdg]<DLF
DLKkPQSLU SLF/R#bedg]<DLF¬TDLK>K<D!hfSLK>l HzYGD!R#N«TRV]'HVȋbqRÅTHVNGwQDLF>K>NGF<D!YN«TLdgDLbAD]>K'E)R#F>K>dgDLbebgD`avDLK¬R#N‹TRV]ŠY˜½ NGU D
R#UGdg]<HVK>F<HVEGdgD"j*N)RVYF>deEcHVbqR#deF<D"j*N DLbgTHVU jN D`nc‡–½¾D]>KuTDLK>K<DxF<DtŒŽHVF>h"NGbqR#K>dgHVU¤jNGd‘U HVN ][email protected]>hdg]uYGDxhfHVU*K>F<DLF
jN D`aTHVUK>F/R#deF<DLhfDLUK4»uTD4j*NGd*D]>K;R¥fF>hfSoY R#U ]b½§R#F>K>dgTLbgDoHVF>deP`deU)R#baNGU DF<DTHVU ]<K>F>N TLK>dgHVU\YGD4b½§R#UGdg]<HVK>F<HVEGdgD
D]>KEcH`]<]>de[GbgDAhfyLhfD-DLU¬b½§R#[ ]<DLU TDBY˜½ NGU D-THVNGwQDLF>K>NGF<D"THVhEGbgiLK<D"YNmTLdgDLbn
–,†+
³
† ¶71\¶£·ºó<:\B)ºz?:\@}1
º}6
9;ºBž³\5±9ë@}6C{}5£·[?6"·c5±µ@º·b?:<5ó·ó<5Ù³\6´9;:<5ó
C R\hfSLK>l HzYGD-YGDxF<DTHVU ]>K>F>N TLK>dgHVU%EGF<S]<DLUK<SD"TLd“†YGD]<]>N ]U½¾D]>KE)RV]OR#EGEGbedgTR#[GbgD"YdeF<DTLK<DLhfDLUKAY R#U ‘
] bgD
TRV]OHVÈ%b½¾DtszEcH`]>deK>dgHVU¤Y˜½ NGU D"E)R#F>K>dgD\YN%TLdgDLb£D]>KOUNGbebgDp”)DLU!Dt—cDLKabgD]O]/HVhhfD]OYdg]<TLF<iLK<D]AYGSTLF>deK<D]AY #R U ]

[Àg¿ˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛ;mÃQ&&‘#*Ü;G*õ’õ!Û;& Ã*õ
ÄgW¿
b½¾SjN)R#K>dgHVU‰œ ÷zn „*oU½¾HVU*K}E)RV]YGDA]<DLU ]]>dcbgDLNGF}YGSLU HVhdeU)R#K<DLNGFO]½§R#UGU*NGbgD`n)‡rDLEDLU Y R#U*KaGbgDAF/R#dg]<HVUGU DLhfDLU*K

jNGdRkTHVU YNGdeK–»fbqRYGSLF>dewVR#K>dgHVU¤YGD]ŒŽHVF>h"NGbgD]œ ÷zn deU YdgjN DBjN D-TDLK>K<DBhfSLK>l HzYGD"D]>K}PQSLU SLF/R#bedg]ÎR#[GbgD`a
Ø
[GdgDLUŠj*N D}EGbeN ]rTHVhEGbedgjN SDAYN'EHVdeU*KvYGDuw*N D–R#bePQSL[GF>dgjN D`nC‘RB]>dehEGbedgTLdeK<S–YGDubqR-EGF<H*TSYNGF<D–EGF<STSYGDLU*K<D
D]>KOYN [email protected]#NmŒR#deKAjN DBbqRhkR#K>F>[email protected])R#UK
DLK DLUmŒŽHVU TLK>dgHVU!YGD]–E)R#F/R#hfiLK>F<D]xYN¤YdeEÚVbgD`a DLK a
I0
I
α
D
EDLNGK}yLK>F<DBYdeF<DTLK<DLhfDLUK–deUwQDLF<]<SD"EHVNGF}YGHVUGU DLF DLK
DLUpŒ’HVU TLK>dgHVU!YGD]hfHVhfDLUK<]–YGD a HVNmYGDAbgDLNGF
α
D
Φ
wQDLF<]>dgHVU ]uYdg]<TLF<iLK<D]
DLK aGdeU Ydgj*N SD]–Y R#U ]\œ ÷zn tnC‘RhfyLhfDBEGF<H*TSYNGF<D"EDLNGK–yLK>F<D"]>NGdewzdgD-Y R#U ]ObgD
S0
S
Ø
TRV]}Y˜½ NGU D-DtsEH`]>deK>dgHVU¬E)R#F>K>dgDLbebgDBYNmTLdgDLbn
CD]r]<HVhhfD]oYGSTLF>deK<D]rY R#U ]Aœ ÷zn SLK/R#U*KrTR#bgTLNGbgSD]v]>NGFbgD]rSLwQSLU DLhfDLUK<]vYGSLK<DTLK<S]a*debD]<KoTLbqR#deFoj*N D
Ø
bgD]rDtsEcH`]>deK>dgHVU ]
R#N¬YGSLU HVhdeU)R#K<DLNGF}U D–EcDLNGwQDLUKvE)RV]yLK>F<D–UNGbebgD]azhfyLhfDx]>dcYGD]rE)R#F>K>dgD]YN¬TLdgDLbcHVU*K
Ek
NGU D–DtsEcH`]>deK>dgHVU'U*NGbebgD`nGÉ;R#FuR#debebgDLNGF<]aR­ U¬Y˜½ dgYGDLU*K>d“­)DLF
DLK RwQDTAbgDLNGF<]vhfHVhfDLUK<]THVF>F<D]>EcHVU Y R#UK<]a
S0 S
deb–]>N¥fK¬YGD!bedehdeK<DLF¬b½ deUK<SLP`F/R#K>dgHVUôR#N«YGHVhkR#deU DªYNžTLdgDLb-HVÈ«b½¾DtsEH`]>deK>dgHVUžD]>KŠU HVU«UNGbebgD`n}É;HVNGF¬NGU
YGSLK<DTLK<DLNGFBR#N‰]<HVbYGHVUGU [email protected]ŒŽHVU TLK>dgHVUGU)R#UKBEDLU Y R#U*K-NGU DbgHVUGP`N DEcSLF>dgHzYGDYGDK<DLhE ]kœNGUÅR#UªhdeUGdehBNGh
5ta bqR\F<SLP`dgHVU%YNmTLdgDLb‘THVNGwQDLF>K<D"]<DLF/[email protected]<HVNVI>HVNGF<]uYGSLbedehdeK<SD-E)R#FOYGD]bedeP`U D]uYGD-YGSTLbedeU)R#dg]<dgHVU%THVU ]>K/R#U*K<D]a
T`½¾D]>Kv»BYdeF<D–deU YGSLEDLU Y R#U*K<D]}YGDOb½§RV]<TDLU ]>dgHVU¬YF<HVdeK<Dœ deP`lKvM–]<TDLU ]>dgHVU¬HVN M«E)R#FvbqR"]>NGdeK<DtnG‡rDTLd˜D]>K
a
a
NGU DxTHVU ]<SjN DLU TDBYdeF<DTLK<D-YGDxbqR\F<HVK/R#K>dgHVU!YGDAbqq
R (‘DLF>F<D-]<NGF}DLbebgDthfyLhfD"DLKOR#NGK<HVNGFuYN%¡zHVbgDLdebn
"˜)Û;f)f‘Á*Á^’&)V õ
ö
Í š¶›j·_@ªU™JO˜ ˜ §°š¶›s§¶ÈO©³Å¡Oª¢É³¿«Àšœ›b¨gš°ª†¯3š°¡¡†š°˜¡*™Jš
ªšœ¨§œªšœª<«É³B¨›¢¬¡¢À@˜u™Jšœ›)§°¡@¢À«Àšœ›^›©$ª)«É³3›¨ Ç$આš6°§°«Àšœ›¡š
¹u«F± ³¿¢É™Jšz™Jšz™ š°©J¦x³¿˜ ²O«¬š¶›"ë«À³q™ §œ·«À¢À˜$³¿¢É›O˜ ÁÈO©$¢'šœ›¡
¯"šœ›© ª§œšsš°˜•™ š°²@ª†§œ›“¹j¨$³¿ª¡¢Àª}™ šs«F± §œÈ©$³¿¡šœ© δª}°§°«Àšœ›¡š
您 ª†¿Ÿ š¶n¡†¢¬O˜s™Jš«0± §¶È©$³Å¡†š°© ª¡š°ª†ª†šœ›¡ª†š¶énÁš·¡6«F± ³@›†·šœ˜$›¢À@˜
™Jª†@¢¬¡š@Áµ˜ @¡§œš ›© ªL«À³¥$²O© ªš^·¢¬¸`·@˜¡†ªšOÁ¶È© ¢_šœ›¡L«F± ³¿˜$³¿¸
«¬O²@© š^™Jš)«À³6«À@˜ α²O¢¤¡†©$™Jš<¡šœªª†šœ›¡ª†š@½ 8 O©$›>©J¡¢À«¬¢É›š°ª†@˜$›L·š°¸
¨šœ˜$™ ³@˜O¡^«É³X˜ @¡†³Å¡†¢¬O˜“³¿˜$²@«À¿¸`›³¿¦_O˜ ˜ šè Ý æè¢À²@ǡݛ¸
·šœ˜$›¢À@˜gé%¨g@©$ª>§œ´_¢¤¡†š°ª>¡O©J¡š·O˜J£‚©$›¢¬O˜3³µ´Ošœ«F± §œ°ª¢¬¡© ª†š
™Jšx«F± ³¿¯"¨ «À¢¬¡©$™Jš*™ ©™J¢À¨g×@«Àš@½ Í ± ³@›†·šœ˜$›¢¬O˜r™Jª†@¢¬¡šZšœ›¡
·O¯3¨ ¡§°š¨›¢¬¡¢À´@šœ¯3šœ˜¡ ´@š°ªU›«0± Õ<› ¡+™ š ¹
Ç šœ© ª†šœ›œÁ
¹“¨³¿ª¡¢Àª6™J©Z¨g@¢À˜O¡.´@šœª˜³¿«0Á˜ @¡§ Á°@0ª†ªš¶›¨g24@˜$™$³¿˜¡+¹
«F± © ˜ š^™Jš¶›™Jšœ©J¦X¢À˜O¡†š°ªU›š¶n¡†¢¬O˜$›™ š«0γ± §¶·«À¢¬¨J¡†¢ÀÈ© š.æ™ §·¥$˜ ¢É›¸
›³@˜¡<«Àš+¨ «É³¿˜}™ ³¿˜$›<«ÀšœÈ© š°«›œ± š°Ægš¶n¡†© š«É³ª†¿¡U³Å¡†¢¬O˜z™Jš+«É³
‰ šœªª†š.³¿©J¡†@© ª™J©nŠ_@«Àš°¢À«çé<³µ´Ošœ«F± §œÈ©$³¿¡š°©$ª^°§°«Àšœ›¡šO½ Í ³
·Oªª†šœ›¨O˜$™ ³@˜$·šN³µ´OšœN«¬š¶›Z°_@ªU™J@˜ ˜$§°šœ›b›¨$Ç §°ª†¢ÀÈ© š¶›
æ Á é.™J©x¨ Ç_Ø_›¢É·¢Àš°˜Á³µ´@š¶«0± ³Å¦Jš3™ šœ› ™J¢Àª¢À²@§"´@šœª†›+«Àš
¾Lθ×@ϕ«Àš 8 @ªU™“·§œ«¬š¶› ¡†š@ÁJš¶› ¡™J@˜$˜ §°š.¨$³@ª6ë z
š°¡
◦
α (en◦ ) → ϕ
½
δ → 90 − θ
É;R#F;bqRO]>NGdeK<D`abqRuE)R#F>K>dgDrYN\TLdgDLbHV[ ]<DLF>wQSDoE)R#FbgDrYGSLK<DTLK<DLNGF}œ’DtszEcH`]>deK>dgHVU\U HVUUNGbebgD]<DLF/ROTHVU ]>dgYGSLF<SD
8¯ìtéOï|þJïß;éà#ßMáìtéâàO9Jß7ëŽêzä§ß û@9¯áßJç¯áßJéâ֛“ìÎà#ç¯áê§ìtàà#ßmáìtéâ‘߯á‘àé#ê¾áIkk
ÄgW)ú
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
THVhhfDOSLK/R#U*KTHVhEGF>dg]<DODLU*K>F<D–bgD]oYGSTLbedeU)R#dg]<HVU ]
DLK
œ’TtŒ‘DLU TRVYF<StaTHVF>F<D]>EcHVU Y R#UK}»
DLK
δmin δmax
θmin
n
θmax
C R‰hfSLK>l H*YGD%U DˆF<Dj*NGdgDLF>KŠE)RV]kU STD]<]ÎR#deF<DLhfDLUKpNGU D%DtsEcH`]>deK>dgHVU deU YGSLEcDLU Y R#UK<DªYGDmb½§RV]<TDLU ]>dgHVU
‘
YF<HVdeK<Dªœ’TD'j*NGdrD]<KBPQSLU SLF/R#bgDLhfDLU*KbgDŠTRV]@YGD]\DtszEcSLF>dgDLU TD]@F<SR#bedg]>K<D]@]<HVNGhdg]<D]»%YGD]\wVR#F>dqR#K>dgHVU ]YGD
bgDLNGFBDt¥kTRVTLdeK<SŠYGDkYGSLK<DTLK>dgHVUXR#NÅTHVNGF<]@YN‰K<DLhE ]ttahkR#dg]BNGUGdgjN DLhfDLU*K\bgDfŒŽR#deKBjN D']>dNGU‰[email protected]]>NGF
bqR¬]>EGl iLF<DfTSLbgD]<K<DkD]>K"HV[ ]/DLF>wQS'»pNGU DfYGSTLbedeU)R#dg]/HVU aR#bgHVF<]-K<HVN ]"bgD]BR#NGK>F<D]"EHVdeU*K<]\R¼R#UK\bqRphfyLhfD
δ
YGSTLbedeU)R#dg]<HVUp]<HVUKrSLPR#bgDLhfDLUKHV[ ]<DLF>wQS]AœŽRwQDTONGU DuDtsEH`]>deK>dgHVU'EHVNGw`R#UKvyLK>F<DuYd“—˜SLF<DLUK<D4j*N DLbj*N Du]<HVdeK
bgDLNGF M\n
a
C D]uhfHVhfDLUK<]–Y˜½¾HVF<YF<D-ÔSLF<HkDLKuNGU%YN¬• NsˆYGDxF/R¼QHVU ]OTH`]>hdgj*N D ] EDLNGwQDLU*KOYGHVU T"yLK>F<D-F<SSTLF>deK<]
Φ
]<HVN ]bqRBŒŽHVF>hfDf”
I0 =
et
I=
Z
Z
θmax
dθ sin θ
θmin
θmax
dθ sin θ
θmin
Z
Z
2π
dϕ Φ(u)
0
2π
œ ÷zn `÷ 

dϕ uΦ(u),
0
HVÈ'b½ deU*K<SLP`F/R#K>dgHVUˆD]>KrbedehdeK<SDAR#Ns'F<SLP`dgHVU ]HV[ ]<DLF>wVR#[GbgD]vYN¬TLdgDLbnzCDATR#bgTLNGbcYGD]rdeU*K<SLP`F/R#bgD]D]>KrDt—˜DTLK>N S
DLU¬F<DLhEGbqRVWR#UK
E)R#Fu]/HVUˆDtszEGF<D]<]<dgHVUŜ ÷zn A”
Φ(u)

Φ0 d
(2 + sαDz )
4
Φ0 d
Ix =
(1 − γ)αDx
4
Φ0 d
Iy =
(1 − γ)αDy
4
Φ0 d
(s + 2γαDz )
Iz =
4
I0 =
HVȬbgD]}THVU ]>K/R#U*K<D]
a
d s
DLK
p
]<HVU*K}bgD]}Yd“—˜SLF<DLU TD]a)]<HVhhfD]uDLKEGF<H*YNGdeK<]O]>NGdewVR#U*K<]"”
d = cos θmin − cos θmax
s = cos θmin + cos θmax
DLK
γ
œ ÷zn Q¦ 

D]>KuYGHVUGU S-E)R#Fb½¾DtszEGF<D]<]<dgHVU
œ ÷zn Q^ 

p = cos θmin × cos θmax
γ≡
s2 − p
.
3
œ ÷zn§Ñ`_Q
[Àeú2"&;fÃQNÛ;m#ÃQ>f‘#&Ã`f Qf
¡
ÃQGÃòÁ!Û!õò•i‰Á[email protected]Û
ÄgW
CDu]>¼]>K<iLhfD}Y˜½¾Sj*N)R#K>dgHVU ]Aœ ÷zn ¦Q4R#deU ]>dHV[GK<DLUNŠEDLNGKoyLK>F<DudeU*wQDLF<]<S}EcHVNGFoYGHVUGU DLFrbgD]F<DLbqR#K>dgHVU ]rDLUK>F<DObgD]

THVhEH`]/R#U*K<D]uYNˆwQDTLK<DLNGFOYdeEÚVbgD-DLUpŒŽHVU TLK>dgHVUmYGD]hfHVhfDLU*K<]B”
Ix
sIz − 2γI0
Iy
αDy =
sIz − 2γI0
sI0 − 2Iz
αDz =
sIz − 2γI0
αDx =
γ−p
γ −1
γ−p
γ −1
œ ÷zn§Ñ 

Õ DrŒŽRVWHVUf]>dehdebqR#deF<DOR#NfTRV]YGDTHVNGwQDLF>K>NGF<D}K<HVK/R#bgD}YNfTLdgDLbaQbgD]£E)R#F/R#hfiLK>F<D]oYNfYdeEcÚVbgD]<HVUKoD]>K>dehfS]
A
»ŠE)R#F>K>deF-YGD]xwQDLF<]>dgHVU ]-Ydg]<TLF<iLK<D]"YGD
DLK a˜j*NGd]<HVUKxK<HVNVI>HVNGF<]-YGHVUGU SD]"E)R#FxbgD]-SjN)R#K>dgHVU ]'œ ÷zn ðQtn
I0
I

C½¾HVF>dgDLUK/R#K>dgHVUŠ[email protected]ÚVbgDvD]>K£F<DTHVU ]>K>F>NGdeK<D}THVhhfDSLK/R#U*K7bqRAYdeF<DTLK>dgHVUfYN\wQDTLK<DLNGFoY R#U ]£bgDohfDLhB[GF<D}YGD
YF<HVdeK<D-YGDAb½¾SjN)R#K>dgHVUƜ ÷zn§Ñ ta DLKu]<HVUmR#hEGbedeK>N YGD THVhhfD-SLK/R#U*KO]/R\U HVF>hfD`n C4½§R#hEGbedeK>N YGD"YNmYdeEcÚVbgD

α
F<DTHVU ]>K>F>NGdeKu]½¾STLF>deKuYGHVU Tk”
αrec =
s
2
2
2
2
( γ−p
γ−1 ) (Sx + Sy ) + (sS0 − 2Sz )
(2γS0 − sSz )2
.
œ ÷zn§Ñ`ÑQ
q
‡ DLK>K<DBDtsEGF<D]<]>dgHVU!]<D-F<SYNGdeK-»
r
aTHVhhfD\R#K>K<DLU Y NacY R#U O
] bgDBTRV]–Y˜½ NGU D
αrec = S30 Sx2 + Sy2 + Sz2
DtszEcSLF>dgDLU TDxEcH`]<]<SY R#UKNGU D"THVNGwQDLF>K>NGF<D-K<HVK/R#bgD-YNmTLdgDLbrœ
a
a
a
t n
s = 0 d = 2 p = −1 γ = 1/3
–, Š
³
X6\:\B[B)º}1\95±³\5Ò·579‚@}1B )? ·c6B9?:\@u1
˜
5‚?Ì51¥ 9;º}9;:?¶±³B5ôó<º ¸¹¶‚?²[email protected]³\5
C ½¾SLK>N YGD}YGDrb½¾Dt¥kTRVTLdeK<S}YGDvF<DTHVU ]>K>F>N TLK>dgHVU'Y˜½ NGUfYdeEÚVbgD}RwQDTbqROhfSLK>l H*YGDu¡zM–É œ¡zHVhhfDLF<]MONG[GbedeU

É;R#F>deÔHVKΝpD]<K¬EGF<S]<DLUK<SD2Y R#U ]mTDLK>K<DªE)R#F>K>dgD2RwQDTªbqRÆhfyLhfDªYGSLhkR#F<T/l D‰j*NGd"RÆP`NGdgYGS¤bgD¤K>F/Rw`R#deb"YGD
K>l i]<D¤”£bgD]@F<S]>NGbeK/R#K<]fHVUK'SLK<S¬HV[GK<DLUN ]P`F/ÖVTDmR#NsZ]>deh"NGbqR#K>dgHVU ]kUNGhfSLF>dgj*N D]kDLKfHVU*KfSLK<S¬THVU­ F>hfS]
DLU ]>NGdeK<D"K>l SHVF>dgj*N DLhfDLUKn;‡rD\T/l HVd“s%YGD\EGF<S]<DLUK/R#K>dgHVU2D]<K–YGHVU T\EGF>deU TLdeE)R#bgDLhfDLUK"hfHVK>dewQSBE)R#FAb½§RV]>EcDTLK
÷tlGdg]>K<HVF>dgjN DQ÷vYGDvb½¾SLbqR#[cHVF/R#K>dgHVUkYGDrbqR–hfSLK>l HzYGD`n`CD]7F/R¥fU DLhfDLUK<]oDLK7PQSLU SLF/R#bedg]/R#K>dgHVU ]4YGDrbqR–EGF<HzTSYNGF<D
]<DLF<HVUK"YGHVU TfEGF<S]<DLUK<S]\]>N TTD]<]>dewQDLhfDLU*Ka‘DLU2E)R#F>K/R#U*KBYN‰TRV]-bgDEGbeN ]-]>dehEGbgDfDLKBDLUÅR#bebqR#UKBwQDLF<]xNGU D
hfH*YGSLbedg]/R#K>dgHVU¤EGbeN ]–F<SR#bedg]>K<D\DLKOEGbeN ]ATHVhEGbgDtsGD`n‡rDLK>K<D\SLK>N [email protected]ŒŽR#deK–b½¾HV[IJDLKAY˜½ NGU D\EGNG[GbedgTR#K>dgHVU2Y R#U ]
è{ÒnÜÝÞæ!ð#æ*Ô/›Ù!ð#Ö¥è{ÒnÜÝÞæڇØT/ÒnÛÝ×Òk¤‚ ÷ ~ †n
ÄgWW
ü‚üÞý
¸
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
žb$+
ÿ~4ÿ%N_ K I> <x^ Q$])4ÿ%¬ÿ%ÿ « |_` „QÿK I> `‘ÿ
M ­ U!Y˜½¾D]>K>dehfDLFAbqRkEGF<STLdg]>dgHVUªYGD"b½¾D]<K>dehkR#K>dgHVU2YGD]OE)R#F/R#hfiLK>F<D]xYN¤YdeEcÚVbgD`aI½§R#d;PQSLU SLF<SBNGU!P`F/R#U Y
u
U HVh"[GF<[email protected]<]"YGDYGHVUGU SD]a˜THVF>F<D]>EcHVU Y R#UKBT/l)RVTLNGUÅ»ŠNGU DYdg]>K>F>de[GNGK>dgHVU‰YGDYdeF<DTLK>dgHVU ]BY˜½§R#F>F>dewQSD]
hfH*YNGbgSD}E)R#F4NGUkYdeEcÚVbgD}YGD}TR#F/RVTLK<SLF>dg]>K>dgj*N D]rTHVUGUN D]aR#N jN DLbg]IL½§R#dR#EGEGbedgjN SbqRxhfSLK>l H*YGDOYGDF<DTHVU ]J
DLK DLKF<DTHVU ]<K>F>NGdeF<D}R#deU ]>d*bgDrYdeEcÚVbgD`n#É;HVNGF7SLK>N YdgDLF£b½ deU• N DLU TD
K>F>N TLK>[email protected]£TR#bgTLNGbgDLF£bgD];]/HVhhfD]
S0 S
YNmU HVhB[GF<DBY˜½¾SLwQSLU DLhfDLU*K<]x]>NGF}bqREGF<STLdg]<dgHVU!YGD"F<DTHVU ]>K>F>N TLK>dgHVUaQI½§R#d‘NGK>debedg]/S"YGD]ObgHVK<]uYGD"YGHVUGU SD]AYGD
K/R#debebgD]}Yd“—˜SLF<DLUK<D]a)EHVNGw`R#UKuTHVU*K<DLUGdeFOYGD _`_`_OIJN ]<j*N½§»\NGUˆhdebebedgHVUmY˜½¾SLwQSLU DLhfDLU*K<]n

ÕAD!ŒŽRVWHVU¨THVU TLF<iLK<D`a}bqRÆPQSLU SLF/R#K>dgHVUÌY˜½ NGU¨bgHVKˆYGDªYGHVUGU SD]ˆTHVU ]>dg]<K<D2»ÆK>deF<DLF%R#bgSR#K<HVdeF<DLhfDLUK
N
YdeF<DTLK>dgHVU ]v]>NGF4bgD–TLdgDLba*T`½¾D]<KJÞ»†YdeF<D
E)R#deF<D]vY˜½§R#UGP`bgD]
F<DLEGF<S]<DLU*K/R#UKbgD]rYdeF<DTLK>dgHVU ]Y˜½§R#F>F>dewQSD
N
(θi , ϕi )
YGD]pF/R¼QHVU ]%TH`]>hdgjN D]a}RwQDTªNGU D¤EGF<HV[)R#[GdebedeK<S
auHVÈ
D]>KˆYGHVUGU SªE)R#F
P(u) = Φ(u) × E(u)
Φ(u)
b½¾SjN)R#K>dgHVU¨œ ÷zn "DLK
YGSLEDLU YZYGD'b½¾DtszEcSLF>dgDLU TDpTHVU ]>dgYGSLF<SD`n£C‘R%U HVF>hkR#bedg]/R#K>dgHVUXYGD'b½¾DtsEcH`]>deK>dgHVU
E(u)

U DoIJHVN)R#U*KuR#N TLNGU¬F<ÚVbgD–Y R#U ]vbqRBF<DTHVU ]>K>F>N TLK>dgHVU%YNˆYdeEÚVbgD`a
YGS]<deP`U DLF/R\E)R#FrbqR\]>NGdeK<DAb½¾DtsEcH`]>deK>dgHVU
E(u)
F<DLbqR#K>[email protected] R#U ]AbqRkYdeF<DTLK>dgHVU aYGHVU*KAbqRkw`R#bgDLNGFxD]>K–E)R#FAYGSt­ UGdeK>dgHVU2THVhEGF>dg]/D\DLU*K>F<D_'DLK nCD\K>deF/R#PQD
u

R#bgSR#K<HVdeF<D'YGD]@SLwQSLU DLhfDLU*K<]]>NGdewVR#U*KBbqRmbgHVdoYGD'EGF<HV[)R#[GdebedeK<S
NGK>debedg]<D'bqR¬hfSLK>l HzYGDŠY˜½§RVTTDLEGK/R#U TDt
Φ(u)
F<S¯IJDTLK>dgHVU ”
DLK
]<HVU*K¬K>deF<S]pNGUGd“Œ’HVF>hfSLhfDLU*Km]>NGFpbqRÆ]>EGl iLF<D`arEGNGdg]pNGU‹U HVhB[GF<D‰R#bgSR#K<HVdeF<D
D]>K
θi
ϕi
z
K>deF<SŠNGUGd“Œ’HVF>hfSLhfDLU*KfDLUK>F<D
DLK
n¡*d D]<KBdeUŒŽSLF>dgDLNGFk»
a
Φmin
Φmax
z
P(θi , ϕi ) = Φ(θi , ϕi ) × E(θi , ϕi )
b½¾SLwQSLU DLhfDLUK–D]>K}THVU ]<DLF>wQS`a)DLK}TDI¯N ]<j*N½§»@TDxj*N DAbgDxbgHVK}YGD-YGHVUGU SD]uTHVU*K>dgDLUGU D
THVNGEGbgD]
n
N
(θi , ϕi )
M K>deK>F<DpY˜½ debebeN ]>K>F/R#K>dgHVUa£NGU D'ŒŽHVU TLK>dgHVUZY˜½¾DtsEcH`]>deK>dgHVUX]>dehdebqR#deF<Dˆ»mTDLbebgDpYGDŠb½ š [ ]<DLF>w`R#K<HVdeF<D¬É7dgDLF>F<D
MuNGPQDLFB]<DLF/RŠNGK>debedg]<SDE)R#F-bqRˆ]>NGdeK<D`aY R#U ]-bgDTRV]BYN‰]>deK<Dk¡*N Y2]/DLNGb4DLK"Y R#U ]xbgDfTRV]"HVÈ2bgD]"YGDLNsª]>deK<D]
]<HVUKkEGF<S]<DLUK<]noC‘R¤hfSLK>l HzYGDm]<DLF/R‰R#EGEGbedgjN SD¬EGbeN ]kbgHVdeU‹R#N TRV]kYGD¬b½¾DtszEcH`]>deK>dgHVU‹Dt—cDTLK>dewQDmR#K>K<DLdeU*K<D
E)R#Fub½ š ÉMƒYGDLEGNGdg]xÑ`_`_#„ a)hkR#dg]OEcHVNGFubqRkEGF<S]<DLUK<D\SLK>N YGD`a)U HVN ]ONGK>debedg]<DLF<HVU ]–bqRfŒ’HVU TLK>dgHVU!Y˜½¾DtszEcH`]>deK>dgHVU
K>l SHVF>dgjN DBTHVF>F<D]>EcHVU Y R#UK-»YGD]–SLU DLF>P`dgD]OHVÈ%bqREGF<HV[)R#[GdebedeK<S\YGD"YGSLK<DTLK>dgHVU¤D]>KuK<HVNVI>HVNGF<]OSLPR#bgD\»NGUa
deU YGSLEDLU Y R#hhfDLU*KxYGD"b½§R#UGP`bgD\Y˜½ deU TLdgYGDLU TDBYN!F/R¼QHVU ]–TH`]>hdgj*N D`nCRf]>NGFJŒRVTDBDt—˜DTLK>dewQD"YGDBYGSLK<DTLK>dgHVU
D]>K}R#bgHVF<]}YGHVUGU SD-]>dehEGbgDLhfDLU*KuE)R#FbgDAEGF<HzYNGdeK}[email protected]]>NGFJŒRVTD-R#Nˆ]<HVbch"NGbeK>deEGbedgSD-E)R#F
azHVÈ
D]>K
cos θz
θz
b½§R#UGP`bgD\ÔSLUGdeK>l)R#bbgH*TR#bncC½¾DtsEcH`]>deK>dgHVUªY˜½ NGUªEHVdeU*KAYGHVUGU SY R#U ][email protected]}œ’YGSTLbedeU)R#dg]<HVUa acDLKxRV]/TDLU ]>dgHVU
δ
YF<HVdeK<D`a M\acDLU2TH*HVF<YGHVUGU SD]BSjN)R#K<HVF>dqR#bgD]tAD]>K-YGHVU TfHV[GK<DLU*N DfDLU2deUK<SLP`F/R#U*K"[email protected]>F/RI>DLK"[email protected]
a
Y R#U ]bgDˆTLdgDLbwzNXbgH*TR#bgDLhfDLU*K'E)R#FbgDmYGSLK<DTLK<DLNGFa4YGSLEcDLU Y R#UK'YGDpbqRªbqR#K>deK>N YGDˆYN ]>deK<D¬DtsESLF>dehfDLU*K/R#bn
ÕxR#U ]"bgD'TRVYF<D'YGDkU HVK>F<DŠR#EGEGF<HszdehkR#K>dgHVUa;b½¾DtsEH`]>deK>dgHVUÆD]>KBTHVU ]<dgYGSLF<SD'THVhhfDpSLK/R#UK\NGUGd“ŒŽHVF>hfD'DLU
RV]<TDLU ]>dgHVUˆYF<HVdeK<D`n
ù ½§R#dYGHVU T TR#bgTLNGbgSmœU*NGhfSLF>dgjN DLhfDLU*KΝxb½¾DtsEcH`]>deK>dgHVUªF<DLbqR#K>dewQDfTHVF>F<D]>EcHVU Y R#UKB»ŠNGU2HVN2YGDLNs¤]>deK<D]
YGDuYGSLK<DTLK>dgHVU ] YGDOK†¼zED–MONGPQDLFa*]>deK>N S]v»"NGU DubqR#K>deK>N YGDAYGD
◦ EcHVNGF4bgD–]>N YŠDLK
◦
λsud ' −35
λnord ' +39
[Àeú2"&;fÃQNÛ;m#ÃQ>f‘#&Ã`f Qf
¡
ÃQGÃòÁ!Û!õò•i‰Á[email protected]Û
ÄgWV[
,6-0/.1÷zn
|£sEH`]<deK>dgHVUªF<DLbqR#K>[email protected]½ š [ ]<DLF>w`R#K<HVdeF<DÉ7dgDLF>F<DfMONGPQDLFkœEHVNGFxNGUªHVN2YGDLNsª]>deK<D]xDtszEcSLF>d“
s)42 DLUkŒ’HVU TLK>dgHVUpYGDObqR"YGSTLbedeU)R#dg]<HVUpDLUpTH*HVF<YGHVUGU SD]vSj*N)R#K<HVF>dqR#bgD]nÉHVNGFT<l)RVTLNGU DAYGD–TD]oYGDLNs
hfDLUK/R#N}
THVU­ P`NGF/R#K>dgHVU ]ab½¾DtsEcH`]>deK>dgHVU'D]>KoTR#bgTLNGbgSDuEcHVNGFNGUŠR#UGP`bgDuÔSLUGdeK>l)R#bYGDOYGSLK<DTLK>dgHVU'hkRsdehkR#bYGD _ ◦ HVN
Ø
¦`_ ◦ n
EHVNGFxbgDU HVF<Y˜n‘C‘RŠ­ P`NGF<Dmœ ÷zn –EGF<S]/DLUK<Dfb½¾SLwQHVbeNGK>dgHVUÆ[email protected]½¾DtszEcH`]>deK>dgHVU2F<DLbqR#K>dewQD'YGD]-YGDLNsªTHVU­ P`NGF/R

K>dgHVU ]xYGDBb½ š [ ]/DLF>wVR#K<HVdeF<DfɏdgDLF>F<DMuNGPQDLF-DLU!ŒŽHVU TLK>dgHVU2YGDBbqRŠYGSTLbedeU)R#dg]<HVUn‘C‘R'Œ’HVF>[email protected]]–ŒŽHVU TLK>dgHVU ]
YGSLEDLU Y‰Œ’HVF>K<DLhfDLU*KYGDfbqRmwVR#bgDLNGF\hkRszdehkR#bgDŠYGD'b½§R#UGP`bgDkÔSLUGdeK>l)R#bvbgHzTR#b‘IJN ]<j*N½§»¬bqRVj*N DLbebgDkbgD]\SLwQSLU Dt
hfDLUK<]EcDLNGwQDLUK–yLK>F<D-R#U)R#be¼z]<S]"”GY R#U ]bgDxTRV]YGDxb½ š ÉMBa)»\b½ l DLNGF<DBRVTLK>N DLbebgDxdebDtszdg]<K<D–YGDLNsphfSLK>l HzYGD]
YGD4F<DTHVU ]>K>F>N TLK>dgHVU'YGD];SLwQSLU DLhfDLU*K<]aVbqROEGF<DLhdgiLF<D]½§R#EGEGbedgj*N)R#UK4R#Ns"PQDLF>[cD];[email protected] TLbedeU SD]}œ
ta
θz ≤ 60◦
K/R#U Ydg]\j*N DfbqR%]<DTHVU YGD'EDLF>[email protected]'F<DTHVU ]>K>F>NGdeF<DŠbgD]BSLwQSLU DLhfDLU*K<]kR¼R#U*K
n;ÕxR#U ]"bgDŠTRVYF<D
θz ≥ 70◦
YGDfTDLK>K<DfSLK>N YGD`aGI½§R#dYGHVU TkTR#bgTLNGbgSfbgD]xŒŽHVU TLK>dgHVU ]"Y˜½¾DtsEH`]>deK>dgHVUÅY R#U ]-b½ l¼zEcHVK>l i]<D
a
DLK
θz = 60◦ 70◦
aR­ UªYGD\hfDLK>K>F<[email protected]¤SLwzdgYGDLU TD\b½ deUK<SLF<yLKxEcHVNGFAb½¾SLK>N YGD\YGD]xR#UGdg]<HVK>F<HVEGdgD]"Y˜½§R#NGP`hfDLUK<DLFxb½§R#UGP`bgDYGD
80◦
YGSLK<DTLK>dgHVU\hkRsdehkR#b˜”EGbeN ] D]>K‘P`F/R#U Y˜aVEGbeN ];bqRuEHVF>K>dgHVUYGDrTLdgDLbHV[ ]/DLF>wVR#[GbgDrD]>K£SLK<DLU YN D`n`ÕAR#U ];bgDrTRV]
θz
HVÈkbgD]rYGDLNs']>deK<D]rYGD}b½ š [ ]<DLF>w`R#K<HVdeF<DAÉ7dgDLF>F<DOMONGPQDLFv]<HVUKrEGF<S]<DLUK<]abqR-K<HVK/R#bedeK<S–YNŠTLdgDLbD]<KoHV[ ]<DLF>w`R#[GbgD
RwQDT
azR#NGP`hfDLU*K<DLF
EcDLF>hfDLKr]>dehEGbgDLhfDLU*K}Y R#U ]vTDOTRV]vY˜½§RVTTLNGh"NGbgDLFuEGbeN ]rY˜½¾SLwQSLU DLhfDLU*K<]}DLK
θz = 60◦
θz
YGDx]<DAF/R#EGEGF<H*T/l DLFOYNmTRV]}dgYGSR#b‘Y˜½¾DtszEcH`]>deK>dgHVUˆNGUGd“ŒŽHVF>hfD`n
C D]‘Yd“—˜SLF<DLUK<D]‘ŒŽHVU TLK>dgHVU ]Y˜½¾DtsEH`]<deK>dgHVU"F<DLbqR#K>dewQDoR¼R#UK£SLK<SK/R#[GNGbgSD]a<IL½§R#d*YGHVU T4SLK>N YdgSbqREGF<STLdg]>dgHVU
YGD4F<DTHVU ]>K>F>N TLK>[email protected];Yd“—˜SLF<DLUK<D]£w`R#bgDLNGF<]£YGD4b½§R#hEGbedeK>N YGD DLK£YGDob½¾HVF>dgDLUK/R#K>dgHVUmœ a YN\YdeEcÚVbgD`a
α
θ d ϕd
DLUªŒ’HVU TLK>dgHVUZYGDfbqR¬K/R#debebgD
YN‰bgHVK\YGDkYGHVUGU SD]nÕAR#U ]\T<l)RVj*N DkTRV]aNGU‰P`F/R#U Y‰U HVh"[GF<DŠYGDfbgHVK<]\YGD
N
ÄgWc
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
YGHVUGU SD]2œ 4 'D]>KŠPQSLU SLF<SªDLKŠF<DTHVU ]<K>F>NGdeKavEDLF>hfDLK>K/R#U*KmYGD¤THVhE)R#F<DLF¬bqRÅYdg]<K>F>de[GNGK>dgHVU¨YN¨YdeEÚVbgD
10
D]>K>dehfS-»TDLbebgD-YGSt­ UGdgD"DLUˆDLUK>F<SD"[email protected]]>deh"NGbqR#K>dgHVUn
ü‚ü0E
¸
ƒ¤ÿ%+O_g „ÿ*IL IV ê JQÿ%NRY‹Kÿ
1000
" = 0.10
(G alactic C enter)
PAO South
#
= 70°
max
800
600
400
N = 5000
200
25000
10
5
5.10
5
0
-1
-0,5
0
0,5
1
!"/"
,.-F/.1J÷zn§Ñ Ֆdg]>K>F>de[GNGK>dgHVUYGDb½¾DLF>F<DLNGF£F<DLbqR#K>dewQDr]>NGF‘b½§R#hEGbedeK>N [email protected]ÚVbgDoF<DTHVU ]>K>F>NGdeK;EHVNGF£Yd“—cSLF<DLU*K<D]
K/R#debebgD] 2 YN'bgHVKrYGDOYGHVUGU SD]a*Y R#U ]obgDuTRV]rYN']<deK<Du]>N Y'YGDub½ š ÉMžEcHVNGF4NGUŠR#UGP`bgDOÔSLUGdeK>l)R#bhkRszdehBNGh
YGD ◦ N
[email protected]ÚVbgD\EGF<S]<DLUKxY R#U ]xT<l)RVj*N D\bgHVKxYGD\YGHVUGU SD]xD]<K–HVF>dgDLU*K<[email protected]<]AbgDBTDLU*K>F<[email protected]#bqRVTLK>dgjN D`a˜DLK
EH`]/70
]<iYGDANGU D-R#hEGbedeK>N YGD"YGD _ Y œ
tn

α = 0.1
– 0i‰fÙÛ;‰õÞei•Ü£õò&<ÛåÛ£& Û£FÜ5õ ” K>deK>F<DY˜½¾DtsGDLhEGbgD`abqRŠYdg]>K>F>de[GNGK>dgHVU2YGD]-R#hEGbedeK>N GY D]
F<DTHVU ]>K>F>NGdeK<D]xD]>K–F<DLEGF<S]<DLUK<SD]<NGFObqRk­ P`NGF<Ds÷zn§Ñ'EHVNGFAYd“—cSLF<DLU*K<D]xw`R#bgDLNGF<]–YGD acY R#U ]–b½ l¼zEcHVK>l i ]<D
N
˜Y ½ NGU%bgHVKAYGD\YGHVUGU SD]xTHVUK<DLU)R#UKxNGU¤YdeEcÚVbgDBYGD _ Y Y˜½§R#hEGbedeK>N YGDpœ
OHVF>dgDLU*K<[email protected]<]ObgD\TDLUK>F<D
α = 0.1

PR#bqRVTLK>dgjN D`n;C‘RmŒŽHVU TLK>dgHVUÆY˜½¾DtsEH`]<deK>dgHVUÅNGK>debedg]<SDŠD]>K\TDLbebgDŠYGDkb½ š ÉM¢RwQDTŠbgDk]<deK<D']>N YÅDLK
n
θz = 70◦
C‘RÅjN)R#U*K>deK<S¤K>F/RVTSD!D]<K'b½¾DLF>F<DLNGFpF<DLbqR#K>dewQD¤]>NGFŠb½§R#hEGbedeK>N YGD¤F<DTHVU ]>K>F>NGdeK<D`aT`½¾D]>KJÞ»†YdeF<D!bqR2w`R#F>dqR#[GbgD
n;CD]Ydg]>K>F>de[GNGK>dgHVU ]fYGD
]<HVU*K\K<HVNGKf»mŒR#[email protected])R#K>de[GbgD]'RwQDTŠNGU D
δα/α = (αvrai − αrec )/αvrai
δα
E)R#F/R#hfSLK>F>dg]/R#K>dgHVU!PR#N ]<]>dgDLUGU D`a hfyLhfDBY R#U ]}bgD-TRV]OY˜½ NGUmF<DLbqR#K>dewQDLhfDLU*KOŒR#de[GbgDxU HVhB[GF<DBY˜½¾SLwQSLU DLhfDLU*K<]n
ÕxR#U ]fbgDmTRV]ŠYGD]fbgHVK<]ŠYGDmYGHVUGU SD]ŠYGDˆEcDLK>deK<DˆK/R#debebgD`aobqR2Ydg]<K>F>de[GNGK>dgHVUU½¾D]>KkE)RV]'TDLU*K>F<SD%]>NGFp_a4DLK
bqRÆYdg]>EDLF<]<dgHVU¨D]>KpP`F/R#U YGD`nu‡rDLEDLU Y R#U*Ka}bqRÅwVR#bgDLNGF¬hfH¼QDLUGU D2DLK¬bqRÆYdg]>EcDLF<]>dgHVUƒœ’STR#F>KˆK†¼zED¤YGD¤bqR
PR#N ]<]>dgDLUGU DvYGDxTD]uYdg]>K>F>de[GNGK>dgHVU ]OYdehdeU*N DLUK–bgHVF<]<j*N DAbgDxU HVh"[GF<D-Y˜½¾SLwQSLU DLhfDLU*K<]xR#NGP`hfDLUK<D`n
[Àeú2"&;fÃQNÛ;m#ÃQ>f‘#&Ã`f Qf
¡
ÃQGÃòÁ!Û!õò•i‰Á[email protected]Û
ÄgWv
,6-0/.1÷zn ~
|wQHVbeNGK>dgHVU YGDpbqR!wVR#bgDLNGFhfH¼QDLUGU DŜ[GdqR#dg]Ν\DLKfYGDpbqR¤Ydg]<EDLF<]>dgHVUËYGDpbqR¤Ydg]<K>F>de[GNGK>dgHVUYGD
LD U2ŒŽ2 HVU LT K>dgHVUÆYNÅU HVh"[GF<DŠY˜½¾SLwQSLU DLhfDLUK<] aEHVNGFBNGUÅYdeEcÚVbgDkYGD _ Y HVF>dgDLUK<S'wQDLF<]"bgD'TDLUK>F<D

δα/α
N
PR#bqRVTLK>dgjN D`n
C [email protected] Y R#U TD"DLU
‘
YN¬[GdqR#dg][email protected]]>EcDLF<]>dgHVUˆYGD
D]>KF<DLEGF<S]<DLU*K<SD-]>NGFbqR\­ P`NGF<D“÷zn ~ n
N
δα/α
C‘R-Ydg]>EcDLF<]>dgHVUkYGDub½¾DLF>F<DLNGF4F<DLbqR#K>dewQDO]>NGF4b½§R#hEGbedeK>N YGD–YNkYdeEcÚVbgDuD]>KEGF<HVEcHVF>K>dgHVUGU DLbebgDx» √ bgHVF<]<j*N D
1/ N
D]>KvP`F/R#U YŠDLKvbqR"wVR#bgDLNGFvhfH¼QDLUGU D-jN)R#U*K}»BDLbebgDAD]>KrEGF<HVEcHVF>K>dgHVUGU DLbebgD"»
”zU HVK>F<D–D]<K>dehkR#K<DLNGF}YGD
N
1/N
b½§R#hEGbedeK>N YGDfYN2YdeEÚ[email protected]*wQDLF>PQDkYGHVU [email protected][GdgDLUªwQDLF<]x]/R'w`R#bgDLNGF"R#K>K<DLU YN D`'n D†b7D]>KxSLPR#bgDLhfDLU*K"F/RV]<]>NGF/R#U*K
YGDAU HVK<DLFujN DAbqRwVR#bgDLNGF}hfH¼QDLUGU D-F<D]>K<DAK<HVNVI>HVNGF<]}deUŒŽSLF>dgDLNGF<D"»\bqRYdg]<EDLF<]>dgHVUn
C RB­ P`NGF<D“÷zn „debebeN >] K>F<DAb½¾SLwQHVbeNGK>dgHVUmYGD–[email protected]]>EcDLF<]>dgHVU¬YGD
‘
DLUpŒ’HVU TLK>dgHVUˆYGD EcHVNGF}Yd“—˜SLF<DLUK<D]
δα/α
α
wVR#bgDLNGF<]}YGD
œŽ»\PR#N T<l DtaDLKuDLUpŒŽHVU TLK>dgHVUmYGD
EHVNGF}Yd“—˜SLF<DLUK<D]OwVR#bgDLNGF<]uYGD œŽ»Y F<HVdeK<Dtn
N
N
α
Õ D-TD]OTHVNGF>[cD]}EcDLNGKOyLK>F<D"K>deF<SD-bqRfbgHVd‘PQSLU SLF/R#bgD
A
a)EDLF>hfDLK>K/R#U*K–Y˜½¾STLF>deF<DBbgD
σ(δα/α) ∝ (α N )−1
F/R#EGEHVF>Ku]<deP`U)R#b]>NGF}Ydg]>EcDLF<]>dgHVUmY˜½ NGU DxhfD]>NGF<D"YGHVUGU SD-THVhhfDf”
√
[σ(δα/α)]−1 = Kα α
√
N,
œ ÷zn§Ñ ~ 
VH ÈU HVN ]4RwQHVU ]oYGSt­ UGd bqR—Ú#ÓIÛhÒÒÙ!ð#×ߊÖ]ßÞ>ß×>æ*ð‘ÒnÜÝÞnÓ6×AÜÝÛÝæ!ðˆYN'YGSLK<DTLK<DLNGFa
n‡rDvE)R#F/R#hfiLK>F<DOTR#F/RVTLK<SLF>dg]<D
Kα
THVhEGbgiLK<DLhfDLUK"bqRˆTR#E)RVTLdeK<[email protected]½¾DtszEcSLF>dgDLU TDfTHVU ]>dgYGSLF<SD'»ŠhfD]>NGF<DLF-b½§R#hEGbedeK>N YGDkY˜½ NGU D'R#UGdg]<HVK>F<HVEGdgD
YdeEHVbqR#deF<Dx]<HVN ]>eI>RVTDLU*K<DxY R#U ]vbgD]rYGHVUGU SD]nGC‘R\EGNGdg]<]/R#U TDAYGDAF<DTHVU ]>K>F>N TLK>dgHVUˆYGSLEcDLU Y¬YGD–bqR-ŒŽHVF>hfDxYGD
bqR–Œ’HVU TLK>dgHVUpY˜½¾DtsEH`]>deK>dgHVU'F<DLbqR#K>dewQD`a*DLKoY R#U ]4bqR"TRV]4Y˜½ NGU D–THVNGwQDLF>K>NGF<DuU HVU'NGUGd“Œ’HVF>hfD`azYGDb½¾HVF>dgDLUK/R#K>dgHVU
Ä[G
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
YNBYdeEcÚVbgD`nÕxR#U ];U HVK>F<DrDtsDLhEGbgDrYN\YdeEcÚVbgDrHVF>dgDLUK<SrwQDLF<]‘bgDrTDLUK>F<DoPR#bqRVTLK>dgj*N D`a#HV[ ]/DLF>wQS4E)R#F;bgDo]<deK<D4]>N Y
YGD4b½ š ÉM\abqRuEGNGdg]<]/R#U TDvYGDoF<DTHVU ]>K>F>N TLK>dgHVUfD]>K£Y˜½¾DLUwzdeF<HVU
a`THVhhfDrHVU\EcDLNGKbgDrwQHVdeF£]>NGFbgD]
Kα = 0.33
RI¯N ]<K<DLhfDLUK<]}K>F/RVTS]O]>NGF}bqR\­ P`NGF<Du÷zn „ n MwQDTxNGU DxEGNGdg]<]/R#U TD"YGDxF<DTHVU ]>K>F>N TLK>dgHVU
a)b½¾DLF>F<DLNGF
Kα = 0.33
F<DLbqR#K>dewQD-]>NGFb½§R#hEGbedeK>N YGDBY˜½ NGUmYdeEÚVbgD"YGD
D]>K}Y˜½¾DLUwzdeF<HVU
RwQDT
5 SLwQSLU DLhfDLUK<]n
10%
30%
N = 10
,.-F/.1 ÷zn „
|7wQHVbeNGK>dgHVU YGD'bqR%Ydg]<EDLF<]>dgHVUXYGD
aDLUƌŽHVU TLK>dgHVUXYGD a;EHVNGFYd“—˜SLF<DLUK<D]@K/R#debebgD]
δα/α
YNÅbgHVKYGDŠ2 YGHVUGU SD]ˆœ=éÙ!ӑ×nØ6ß#@[email protected]ŌŽHVU TLK>dgHVUX
YGD a‘EcHVNGFYd“—cSLF<DLU*K<αD]@wVR#bgDLNGF<]YGD
œÖ*Þ'æ!ÛÝ܏ß#tn;™ONU
N
α
RI¯N ]<K<DLhfDLUKuYGD-T/l)RVjN D"THVNGF>[D-]<NGdewVR#U*K}b½¾SjN)R#K>dgHVUƜ ÷zn§Ñ ~ vD]>K}SLPR#bgDLhfDLUK–F<DLEGF<S]<DLU*K<S`n
Õ D"bqR'hfyLhfDBŒŽRVWHVU¤j*N D"EGF<STSYGDLhhfDLU*KaIL½§R#d£SLK>N YdgS
A
bqR-EGF<STLdg]<dgHVUpYGDObqR"F<DTHVU ]>K>F>N TLK>dgHVUˆYGDubqRBYdeF<DTLK>dgHVU¬YN¬YdeEÚVbgD–DLUŠTR#F/RVTLK<SLF>dg]ÎR#UKbgD]vYdg]>K>F>de[GNGK>dgHVU ]YGD]
R#UGP`bgD]uF<DTHVU ]<K>F>NGdeK<]uE)R#FOF/R#EGEHVF>K<]AR#NsmR#UGP`bgD]udeUGdeK>dqR#Ns˜[email protected]­ P`NGF<H
D ÷z:n ðfF<DLEGF<S]<DLUK<DBbgD]–Ydg]>K>F>de[GNGK>dgHVU ]
YGD\b½¾DLF>F<DLNGF-]>NGFAbqR'YGSTLbedeU)R#dg]/HVUªF<DTHVU ]>K>F>NGdeK<DEcHVNGFAYd“—˜SLF<DLUK<D]-wVR#bgDLNGF<]xYGD a˜R#deU ]>d7j*N D\b½¾SLwQHVbeNGK>dgHVU
N
YNp[GdqR#dg]DLKrYGD–bqR\Ydg]>EcDLF<]>dgHVU¬YGD RwQDT nzC‘D]vF<S]>NGbeK/R#K<]THVU TDLF>U)R#UK}b½§RV]/TDLU ]>dgHVUˆYF<HVdeK<Dœ Mx]<HVU*K
a
δ
N
]>dehdebqR#deF<D]rDLKoU DO]<HVUKrE)RV]4F<DLEGF<S]/DLUK<S]rdgTLdn*™OU D}bgHVd]>dehdebqR#deF<DA»-TDLbebgD–]>NGF4b½§R#hEGbedeK>N YGDAYN'YdeEcÚVbgDOEDLNGK
yLK>F<D–Œ’HVF>hBNGbgSD-EcHVNGF}bqR\F<DTHVU ]>K>F>N TLK>[email protected])R#dg]<HVUŔ
– 0i‰fÙÛ;2õrÛ£*Ã`f Û£&ÙÛ;0Ü$õ
σdec =
œ ÷zn§Ñ#„*
1
√ .
Kdec α N
Y #R U ]bqRVj*N DLbebgD
F<DLEGF<S]/DLUK<D-bqREGNGdg]<]/R#U TDxYGDxF<DTHVU ]>K>F>N TLK>dgHVU!YGDAbqRYGSTLbedeU)R#dg]<HVUn É;HVNGFub½¾DtsGDLhEGbgD
Kdec
YGD\bqRf­ P`NGF<Ds÷zn:ða˜[email protected]ªYdeEcÚVbgDY R#U ]–bqR'YdeF<DTLK>dgHVUÅYN¤TDLU*K>F<[email protected]#bqRVTLK>dgjN DHV[ ]/DLF>wQSBE)R#[email protected]]>deK<[email protected]]>N Y
YGDAb½ š ÉM\a bqR\EGNGdg]/]/R#U TD"R#UGP`NGbqR#deF<D"D]<K}YGD
−1
−1 aGHVNˆ]>d˜b½¾HVUmEGF<StŒ’iLF<D
◦n
Kdec ' 0.28 rad
Kdec ' 200
[Àeú2"&;fÃQNÛ;m#ÃQ>f‘#&Ã`f Qf
¡
ÃQGÃòÁ!Û!õò•i‰Á[email protected]Û
Ä[Ä
,6-0/.1_÷z:n ð Ֆdg]>K>F>de[GNGK>dgHVU ]£YGD7b½¾DLF>F<DLNGF;F<DLbqR#K>dewQDo]<NGFbqR}YGSTLbedeU)R#dg]<HVUBF<DTHVU ]>K>F>NGdeK<DAœ=éÙ!ӑ×nØ6ß#ta#DLK;SLwQHVbeNGK>dgHVU
YN‰[GdqR#dg]"2 DLK\YGDfbqRmYdg]>EDLF<]<dgHVU‰THVF>F<D]>EcHVU Y R#UKDLU2ŒŽHVU TLK>dgHVUÅYNÅU HVh"[GF<DŠY˜½¾SLwQSLU DLhfDLUK<]ˆœÖ!Þæ!ÛÝ܏ߝtnCD
[email protected]]>[email protected]>dgDLUK<SpwQDLF<]\bgD'TDLU*K>F<DŠPR#bqRVTLK>dgjN D`n£C‘R
YdeEÚVbgDŠDLUÆDLU*K>F<SDpEH`]/]<iYGDfNGU D¬R#hEGbedeK>N YGD
α = 0.1
Œ’HVU TLK>dgHVUmY˜½¾DtsEcH`]>deK>dgHVUˆTHVF>F<D]>EcHVU Y%R#Nˆ]>deK<Dx]>N YmYGDAb½ š ÉM\aRwQDT
◦n
θmax = 70
ÕADbqRphfyLhfDŒRVWHVUaEcHVNGF-bqRˆF<DTHVU ]>K>F>N TLK>dgHVUÆYGDfb½§RV]<TDLU ]>dgHVUÅYF<HVdeK<D`a‘bqR¬EGNGdg]<]ÎR#U TDfHV[GK<DLUN DŠD]>K
aGT`½¾D]>KJÞ»†YdeF<D −1
nGÉ;R#F–DtsDLhEGbgD`a ]<d
œ’THVhhfDxdeb˜EcHVNGF>F/R#deKuyLK>F<D
KRA ' 0.38 rad−1
KRA ' 150◦
α = 0.05
]>NGP`PQSLF<[email protected])R#F-bgD]-YGHVUGU SD]-Y˜½¾MuxMA¡zM EHVNGF"bgD]AF/R¼QHVU ]"TH`]<hdgjN D]BY˜½ NGU DSLU DLF>P`dgD
D€ZÍ7‚¤÷ 
Ø
∼ 1018
[email protected] SD]xYGD 5 SLwQSLU DLhfDLU*K<]a˜bqR'EGF<STLdg]>dgHVUªYGD\F<DTHVU ]>K>F>N TLK>dgHVU‰R#UGP`NGbqR#deF<DD]>KxYGD
10
b½¾HVF<YF<D-YGD ◦ EcHVNGF DLK
E
MBn
◦ HVNGF
12
δ
9.4
a
"&f‘ÃQtۍ*ÃQfV&‘ÃQò>f CR!EGNGdg]<]/R#U TDpYGDpF<DTHVU ]>K>F>N TLK>dgHVU YNËYdeEcÚVbgD`a£YGSt­ UGdgD¬E)R#F
Kα
EHVNGF}b½R#hEGbedeK>N YGDBDLKOE)R#F\œ
a
rEHVNGFObqRfYdeF<DTLK>dgHVUaYGSLEDLU YmU HVU%]<DLNGbgDLhfDLU*K–YGD"[email protected]ŒŽHVU TLK>dgHVU
Kdec KRA
Y˜½¾DtszEcH`]>deK>dgHVU2hkR#dg]-SLPR#bgDLhfDLU*K\[email protected]ˆYdeF<DTLK>dgHVUÅYN2YdeEcÚVbgDfY R#U ]-bgDTLdgDLbnMu­ U‰YGDTR#F/RVTLK<SLF>dg]<DLF\TDLK>K<D
YGDLF>UGdgiLF<DfYGSLEDLU Y R#U TD`aGIL½§R#dTR#bgTLNGbgSbqRŠEGNGdg]/]/R#U TDYGDF<DTHVU ]>K>F>N TLK>dgHVU‰YGDfb½ š ÉM œNGU2HVN2YGDLNsª]>deK<D]Ν
DLU
EHVNGFYd“—˜SLF<DLUK<D]kHVF>dgDLU*K/R#K>dgHVU ]'YNXwQDTLK<DLNGFfYdeEcÚVbgD`nC‘R%­ P`NGF<N
D ÷zn hfHVU*K>F<D¬bqR!w`R#F>dqR#K>dgHVU YGD
Ø
Kα
Œ’HVU TLK>dgHVUmYGDAbqRfYGSTLbedeU)R#dg]<HVUmYNmYdeEcÚVbgDxwzF/R#daGY R#U ]bgD-TRV]}Y˜½ NGUmHVN%YGDLNsp]>deK<D]u]>dehdebqR#deF<D]O»\b½ š ÉMBn
Õ R#U ]‘bgDrTRV]£YGD4bqRuTHVNGwQDLF>K>NGF<DvE)R#F>K>dgDLbebgD`abqRuEGNGdg]<]/R#U TDrYGDoF<DTHVU ]>K>F>N TLK>dgHVUD]>K‘hkRsdehkR#bgDrbgHVF<]<jN DobgD
x
YdeEÚVbgD}D]<KYdeF>dePQS}Y R#U ]NGU DF<SLP`dgHVUkEGF<H*T/l DuYGDvb½¾Sj*N)R#K<DLNGFoDLKEGbeN ]£ŒR#de[GbgDbgHVF<]<j*N DbgDYdeEcÚVbgD}D]>KHVF>dgDLU*K<S
wQDLF<]}NGUˆEcÚVbgD`a b½§R#hEGbedeK>N YGD\[email protected]`R#F>dqR#K>dgHVUmEcHVNGwVR#U*K–R#K>K<DLdeU YF<DBNGU¬ŒRVTLK<DLNGF
nc‡rDLK>K<D"YGSLEcDLU Y R#U TD
∼2
EDLNGK-]<DTHVhEGF<DLU YF<DŠj*N)R#bedeK/R#K>[email protected]ªF<DLhkR#F<jN)R#U*K\j*N [email protected]<]<j*N [email protected]ÚVbgDkD]>K-HVF>dgDLUK<SkwQDLF<]-NGU
YGD]4EcÚVbgD]aQbqR"F<SLP`dgHVUpYN'TLdgDLbbqR-hdgDLNs'HV[ ]/DLF>wQSDuE)R#Fob½¾DtszEcSLF>dgDLU TDATHVF>F<D]>EHVU Y"IJN ]>K<DLhfDLUK»-bqR-E)R#F>K>dgD
Ä[–v
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
HVȬ[email protected]#K>dgHVUmDLUmTH`]>deU*N ]uYNp• Ns¬wVR#F>dgDAbgDAEGbeN ]ubgDLUK<DLhfDLU*Kn
C HVF<]<j*N Dpb½¾DtsESLF>dgDLU TDpEcDLNGKHV[ ]<DLF>wQDLFb½¾DLU ]<DLh"[GbgDmYNËTLdgDLba£bqRmEGNGdg]/]/R#U TD¬YGDpF<DTHVU ]>K>F>N TLK>dgHVUËU D
YGSLEDLU YfE)RV]4YGD}b½¾HVF>dgDLU*K/R#K>dgHVUŠYN'wQDTLK<DLNGFoYdeEcÚVbgD}DLK4D]>KoYGHVUGU SD}E)R#F
Y R#U ]bgDOTRV]4YGDb½ š ÉM
Kα ' 0, 6
THVhEGbgDLKn‡rDLK>K<Dxw`R#bgDLNGFuD]>KODLU%RVTTHVF<Y%RwQDT"TDLbebgD-D]>K>dehfSD-E)R#FA¡zHVhhfDLF<]"‚¤÷„QjNGdHV[GK>dgDLU*KA_a ðkRwQDT
Ø
bqR¬hfyLhfDfŒ’HVU TLK>dgHVUZY˜½¾DtszEcH`]>deK>dgHVUn£‡rDfF<S]>NGbeK/R#K\U½¾D]>K"E)RV]\]>NGF>EGF<DLU)R#U*K\TR#FBbgD]BYGDLNs2hfSLK>l HzYGD]B]/HVUK
]>K>F>dgTLK<DLhfDLUK\dgYGDLUK>dgj*N D]@Y R#U ]"bgD'TRV]BYGD'THVNGwQDLF>K>NGF<DkK<HVK/R#bgD`n;C‘RmYd“—cSLF<DLU TDŠDLUK>F<DŠTD]BYGDLNs2w`R#bgDLNGF<]
YGD
EcDLNGKB]/DTHVhEGF<DLU YF<D'DLU2F<DLhkR#F<j*N)R#[email protected]*N DU HVK>F<DkD]>K>dehkR#K>dgHVUÆD]>K"HV[GK<DLU*N DŠ»pE)R#F>K>deF\YGDbgHVK<]
Kα
√ YGD'bqR
YGD'YGHVUGU SD]@THVhEGF<DLU)R#U*KEGbeN ]\Y˜½¾SLwQSLU DLhfDLU*K<]fj*N DkY R#U ]p¤‚ ÷„`–”bgDŠTHVhEHVF>K<DLhfDLU*KfDLU
1/ N
Ydg]>EcDLF<]>dgHVUÆYGD
U½¾D]>[email protected]#bqR#[GbgDpj*N½§»ˆE)R#F>K>deFY˜½ NGUZU HVh"[GF<D¬Y˜½¾SLwQSLU DLhfDLU*K<]fhdeUGdeh"NGh œ’TtŒu­ P`NGF<D
δα/α
÷zn§ÑQtaDLK–bqRkEGNGdg]<]/R#U TD\YGD"F<DTHVU ]>K>F>N TLK>dgHVU2Dt—˜DTLK>dewQD\YGSLK<DLF>hdeU SDf»E)R#F>K>deFxY˜½ NGU%EcDLK>deK–bgHVK–YGD\YGHVUGU SD]
D]>KK<HVNVI>HVNGF<]bgSLPQiLF<DLhfDLUKŠ]>NGESLF>dgDLNGF<D%»¤TDLbebgDˆHV[GK<DLUN DmRwQDTˆbqR!w`R#bgDLNGFkRV]>¼zhEGK<HVK>dgjN Dm»!P`F/R#U Y
n
N
C½§RVTTHVF<YmRwQDTk¤‚ ÷„VD]>K}YGHVU T-K<HVNGKO»BŒŽR#deK}]ÎR#K>dg]JŒŽR#dg]ÎR#UKn
0,7
0,6
K
PAO (N&S)
!
reconstruction power
0,5
0,4
0,3
PAO (S)
K
!
0,2
0,1
0
-90°
-60°
-30°
0°
30°
60°
90°
dipole declination
,.-F/.1÷zn
‡rHVhE)R#F/R#dg]/HVUŠYGDubqRAEGNGdg]<]/R#U TDOYGD}F<DTHVU ]>K>F>N TLK>dgHVUpYGD}b½§R#hEGbedeK>N YGD–YN'YdeEcÚVbgDuDLUŒŽHVU TLK>dgHVU
Ø TL2 bedeU)R#dg]<HVUmYN%YdeEÚVbgDAdeUGdeK>dqR#baY R#U ]bgD-TRV]}Y˜½ NGUmHVNˆYGDLNsˆ]>deK<D]uYGDAb½ š ÉM\n
YGDAbqRYGS3
Õ NfEcHVdeUKoYGD}w*N D}DtsEcSLF>dehfDLUK/R#ba*NGU D}EGbeN ]P`F/R#U YGDuEGNGdg]<]/R#U TD}YGDuF<DTHVU ]>K>F>N TLK>dgHVUŠdehEGbedgjN DOjN½ NGU
–
YdeEÚVbgD\Y˜½§R#hEGbedeK>N YGDYGHVUGU [email protected]]<DLF/RkF<DTHVU ]>K>F>NGdeKxEGbeN ]–F/R#EGdgYGDLhfDLUKa˜T`½¾D]>KJÞ»†YdeF<Dj*N½ deb;]/DLF/Rk]>deP`UGd“­)TR#K>d“Œ
[email protected]¤EGbeN ]AŒR#de[GbgD\U HVh"[GF<DfY˜½¾SLwQSLU DLhfDLU*K<]n<ON)R#UK>deK/R#K>dewQDLhfDLU*Ka;b½§RIJHVNGKxYN2]>deK<D\U HVF<YªYGD\b½ š [ ]<DLFJ
wVR#K<HVdeF<DkÉ7dgDLF>F<D'MONGPQDLF-EcDLF>hfDLK>K>F/RˆYGD'YGSLK<DTLK<DLF"NGU D'R#UGdg]/HVK>F<HVEGdgDkYdeEcHVbqR#deF<D'RwQDT'NGU DEGNGdg]<]ÎR#U TDk]>NGF
b½§R#hEGbedeK>N YGD
THVhEGF>dg]<DkDLUK>F<D
DLK
Œ’HVdg]AEGbeN ]AP`F/R#U YGD`a‘HVN2Y˜½ NGU¤ŒŽRVTLK<DLNGF-YGDLNsªDLUªhfH¼QDLUGU D
Kα
1.3
2.7
[Àeú2"&;fÃQNÛ;m#ÃQ>f‘#&Ã`f Qf
ÃQGÃòÁ!Û!õò•i‰Á[email protected]Û
¡
Ä[G¿
œ EHVNGF}NGU D"YdeF<DTLK>dgHVU%deU THVUGU*N D"YN%YdeEcÚVbgDtn¡zDLbgHVU%b½¾SjN)R#K>dgHVUZœ ÷zn§Ñ ~ ta)TDLbqRdehEGbedgj*N D-j*N DxbgDxU HVhB[GF<D
Y˜½¾SLwQSLU DLhfDLUK<]vF<Dj*NGdg]4EcHVNGFrRwQHVdeFobqR"hfyLhfDu]>deP`UGd“­)TR#U TDA]<DLF/R"DLUK>F<D
DLK
ŒŽHVdg]EGbeN ]4ŒŽR#de[GbgD`az]<HVdeK4„
1.7 7.3
Œ’HVdg]£hfHVdeU ]DLUkhfH¼QDLUGU D`aj*N DvY R#U ]bgDTRV]HVÈNGUf]/DLNGb ]>deK<DD]>KYdg]>EcHVUGde[GbgD`nQ|7UfTHVU ]>dgYGSLF/R#U*Koj*N DvbgD]>deK<D
U HVF<YŠYGHVNG[GbgDLF/[email protected]#bgDLhfDLU*Kb½§RVTTDLEGK/R#U TD-YGDOb½ š ÉM\aGDLKYGHVU TObgDOK/R#NsŠY˜½§RVTj*NGdg]>deK>dgHVUˆYGD]vSLwQSLU DLhfDLUK<]a
b½ š [ ]<DLF>wVR#K<HVdeF<DkÉ7dgDLF>F<DkMONGPQDLF"THVhEGbgDLK-EcHVNGF>F/RphfD]>NGF<DLF-NGU D'R#UGdg]<HVK>F<HVEGdgDkYdeEcHVbqR#deF<DfDLU2hfH¼QDLUGU D'¦
Œ’HVdg]AEGbeN ]"F/R#[email protected] [email protected]]>deK<Dk]>N Yª]<DLNGbuœ’HVN‰DLU*K>F<D ~ n „mDLK „ n DLU2Œ’HVU TLK>dgHVU‰[email protected]ˆYdeF<DTLK>dgHVUÅYN
 Ø
YdeEÚVbgDtn
‡ HVU TDLF>U)R#UKobqRAEGF<STLdg]>dgHVU'R#UGP`NGbqR#deF<D`ab½¾SLwQHVbeNGK>dgHVU'YGDvbqRAEGNGdg]<]ÎR#U TDYGDF<DTHVU ]>K>F>N TLK>dgHVU'D]<K7debebeN ]>K>F<SD
r
]>NGFobqR"­ P`NGF<D“÷zn¤÷BDLU'Œ’HVU TLK>dgHVUˆYGD–bqR\YGSTLbedeU)R#dg]<HVUˆYNpYdeEÚVbgD`aEcHVNGFoNGU¬HVN¬YGDLNsŠ]<deK<D]rDtsEcSLF>dehfDLUK/R#Ns˜n
C‘ROEGNGdg]<]/R#U TDrYGDoF<DTHVU ]<K>F>N TLK>dgHVUfYGDobqROYGSTLbedeU)R#dg]<HVU
aYGDrb½§RV]<TDLU ]<[email protected]<HVdeK<D
DLK£bqR}EGNGdg]/]/R#U TD
Kdec
KRA
R#UGP`NGbqR#deF<DBK<HVK/R#bgD
]<HVUK–F<DLEGF<S]<DLU*K<SD]n‡rDLK>K<D\YGDLF>UGdgiLF<D\D]>KAHV[GK<DLUN D»kE)R#F>K>deF–YGDBb½§R#UGP`bgD DLU*K>F<D
Ktot
γ
bgDAwQDTLK<DLNGFOYdeEcÚVbgDAF<DTHVU ]>K>F>NGdeKODLKbgDxwzF/R#d˜wQDTLK<DLNGFa wQSLF>d“­R#U*KubqR\F<DLbqR#K>dgHVUƔ
œ ÷zn§ÑJðQ
cos γ = cos θd cos θrec + sin θd sin θrec cos(ϕd − ϕrec ).
0,7
0,7
PAO South
0,6
0,5
K
-1
angular power (rad )
-1
angular power (rad )
0,6
RA
0,4
K
tot
0,3
K
dec
0,2
0,1
0
-90°
0,5
PAO (North & South)
K
dec
K
tot
K
RA
0,4
0,3
0,2
0,1
-60°
-30°
0°
30°
60°
90°
0
-90°
-60°
-30°
0°
30°
60°
90°
dipole declination
dipole declination
É7NGdg]<]ÎR#U TD2YGDªF<DTHVU ]>K>F>N TLK>dgHVU R#UGP`NGbqR#deF<D‰YGDªb½ š [ ]/DLF>wVR#K<HVdeF<DÅÉ7dgDLF>F<D‰MONGPQDLFaORwQDT2NGU
=œ éÙ!ӑ×nØ6ß#HV2 N YGDLNs¨œÖ!Þæ!ە܏ß#@]>deK<D]kDtszEcSLF>dehfDLUK/R#Ns˜a4DLUZŒŽHVU TLK>dgHVU YGD¬bqR¤YGSTLbedeU)R#dg]<HVU‹YN YdeEcÚVbgD`n7C‘R
EGF<STLdg]>dgHVU¤R#UGP`NGbqR#deF<DBK<HVK/R#bgD
a)DLU¤YGSTLbedeU)R#dg]<HVUa
aDLK–DLU¤RV]<TDLU ]>dgHVU%YF<HVdeK<D`a
a]<HVUK–F<DLEGF<St
Ktot
Kdec
KRA
]<DLUK<SD]n
,6-0/.1B÷zn¤÷
C HVF<]<j*N DubqR"THVNGwQDLF>K>NGF<DxYN'TLdgDLbcD]>KrE)R#F>K>dgDLbebgD`a*bqR-EGNGdg]<]/R#U T D–YGDuF<DTHVU ]<K>F>N TLK>dgHVU¬R#UGP`NGbqR#deF<DAYGSLEcDLU Y
YGDvb½¾HVF>dgDLUK/R#K>dgHVUpYNfYdeEÚVbgD`nQCRAwVR#F>dqR#K>dgHVU'YGD
D]>K7bqR–EGbeN 4
] dehEHVF>K/R#U*K<D`aQEGNGdg]<j*N D
w`RoI¯N ]<j*N½§»
KRA
KRA
Ä[ú
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
]½§R#UGUNGbgDLFoEHVNGF
nC‘RAF/R#dg]<HVUŠDLUkD]>K4j*N DbgHVF<]<j*N DbgD}YdeEcÚVbgDuD]>KrR#deU ]>dR#bedeP`U S`aQbqR"Ydg]>K>F>de[GNGK>dgHVU
δd = ±90◦
YGD]YdeF<DTLK>dgHVU ]}Y˜½§R#F>F>dewQSD-YGD]vF/R¼QHVU ]}TH`]>hdgj*N D]D]>K}]>¼zhfSLK>F>dgjN DxR#NGK<HVNGF}YGDAb½§RsGD–YGD]vEcÚVbgD]DLKU½¾D]>K
YGHVU TfEGbeN ]"hfH*YNGbgSD'DLUÆRV]<TDLU ]>dgHVUÆYF<HVdeK<D`n‘C‘RmTH*HVF<YGHVUGU SD MÒYNÅYdeEcÚVbgDU½§RmYGHVU TfEGbeN ]\YGDf]<DLU ]a
a
[email protected]]<]/R#U TD"THVF>F<D]>EHVU Y R#U*K<D"D]>K}YGHVU T"UNGbebgD`–n ” b½ deUwQDLF<]/D`a [email protected]#K>dgHVUmDLU M D]>KhkRsdehkR#bgD
a
[email protected]ÚVbgD¬Y R#U ]@bgDpEGbqR#UËYGD'b½¾Sj*N)R#K<DLNGFn7C‘R!EGNGdg]<]ÎR#U TDpK<HVK/R#bgD¬YGD'F<DTHVU ]<K>F>N TLK>dgHVUR#UGP`NGbqR#deF<D`a
YGHVUGU SD\E)R#F
aEGF<S]<DLUK<[email protected] D\bgSLPQiLF<DYGSLEcDLU Y R#U [email protected]ŠYGSTLbedeU)R#dg]<HVU2YNªYdeEÚVbgD`a˜YGD\b½¾HVF<YF<DYGD
Ktot
Y Y R#U ]bgD-TRV]}YNˆ]<deK<D-]>N Y¬YGDAb½ š [ ]<DLF>w`R#K<HVdeF<D"É7dgDLF>F<D"MuNGPQDLFn
±15
‡ HVhhfDvR#K>K<DLU YNabqR}EGF<STLdg]>[email protected]<DTHVU ]>K>F>N TLK>dgHVUfR#UGP`NGbqR#deF<DrY R#U ];bgD4TRV];Y˜½ NGU DoTHVNGwQDLF>K>NGF<DrK<HVK/R#bgD
r
YNªTLdgDLbD]>KAdeU YGSLEDLU Y R#U*K<[email protected]ŠYdeF<DTLK>dgHVU‰YNªYdeEÚVbgD`a˜]ÎR#NŒoEcHVNGFxbqR'YGSLK<DLF>hdeU)R#K>dgHVUÆ[email protected]½§RV]<TDLU TLdgHVU
YF<HVdeK<DuYNŠYdeEÚVbgD}j*NGd)E)R#FoYGSt­ UGdeK>dgHVU'U½§R-EGbeN ]oYGD}]/DLU ]bgHVF<]<j*N D
nON)R#UK>deK/R#K>dewQDLhfDLU*Ka Y R#U ]
δd = ± 90◦
bgD}TRV]4Y˜½ NGUŠYdeEÚVbgDuYGB
D ð Y Y˜½§R#hEGbedeK>N YGD–Y R#U ]4NGUkbgHVK4YGDuYGHVUGU SD]rTHVUK<DLU)R#U*K 5 SLwQSLU DLhfDLUK<]a*b½ š ÉM
10
THVhEGbgDLKuEcH`]<]<iYGDxNGU D"EGF<STLdg]>dgHVUªR#UGP`NGbqR#deF<D"K<HVK/R#bgDBY˜½¾DLUwzdeF<HVU
a◦ »THVhE)R#F<DLFARwQDT
EcHVNGFubgD
6.4
' 11◦
]>deK<Dx]>N Y¬]<DLNGbn
–,†ñ
³
Š·º}:I?5¸¹571¢? º}1\ºXó4# ? :<?A6\5´³B6

9º€BÙ³B5
ó<º
9ë@}6C{}5£·[?6"·c5
ºXó<5Ê6\1B:`Þ
?[@ ?
ӑ@·c¸¹5
M EGF<i];RwQHVdeF;YGSLK/R#debebgSrbgDEGF>deU TLdeEcDrYGD4bqR}hfSLK>l H*YGD`aVDLK;RwQHVdeF£EGF<S]<DLUK<SoNGU DoSLK>N YGDrYGDr]/REGNGdg]<]ÎR#U TDoYGD
O
F<DTHVU ]>K>F>N TLK>dgHVUp]<HVN ]NGU'RV]<EDTLKŒßïڑÑnÞnەԹßð‡Ü Ù!êyŒAœ’T`½¾D]>KJÞ»†YdeF<DO[)RV]<S}]>NGF4bgD]F<S]>NGbeK/R#K<]rYGDu]>dehBNGbqR#K>dgHVU ]
UNGhfSLF>dgj*N D]ΝtaU HVN ]BR#bebgHVU ]BRVYGHVEGK<DLF"NGUªEcHVdeUK\[email protected]*N DfEGbeN ]Ž-Œ Ù!ð#Ù*ê /*ÜÝÛ=ëӑߌ"Y R#U ]"TDLK>K<Df]<DTLK>dgHVU‰EHVNGF
YGSTLF>deF<Dxb½¾Dt¥kTRVTLdeK<S"YGDAF<DTHVU ]<K>F>N TLK>dgHVUn
‡ HVhhfD¬Œ’F<Sj*N DLhhfDLUK'DLUEGl¼]>dgj*N D`a7bqR¤hfHzYGSLbedg]/R#K>dgHVUYGSLK/R#debebgSD!DLKfF<SR#bedg]>K<D%Y˜½ NGUËEGl SLU HVhfiLU D
r
EDLNGK"F/R#F<[email protected]>F<DfDtszEGF>dehfSDkK<HVK/R#[email protected]]<HVN ]xŒŽHVF>hfDkR#U)R#be¼zK>dgjN D`nëOHVN ]\R#bebgHVU ]"YGHVU T'YGSTLF>deF<DfbgD]
F<S]>NGbeK/R#K<]4j*NGd EcDLNGwQDLUKryLK>F<D}HV[GK<DLUN ]v»AE)R#F>K>deFoYN'TRV]bgDEGbeN ]o]>dehEGbgDuR#N j*N DLb]½§R#EGEGbedgj*N DbqRAhfSLK>l HzYGD"”
bgD¬TRV]fY˜½ NGUËYGSLK<DTLK<DLNGFkEcH`]<]<SY R#U*KNGU DˆTHVNGwQDLF>K>NGF<D¬K<HVK/R#bgDˆDLKNGUGd“ŒŽHVF>hfDˆYN TLdgDLbn7ÕxR#U ]fTDphfHzYGiLbgD
]>dehEGbed“­)S`a bgD-wQDTLK<DLNGF–YdeEcÚVbgD-F<DTHVU ]>K>F>NGdeK–]½¾STLF>deKO]<dehEGbgDLhfDLUKA]<HVN ]}bqRŒ’HVF>hfDBY˜½ NGU D"]<HVhhfDB]>NGF}K<HVN ]
bgD]wQDTLK<DLNGF<]}EcH`]>deK>dgHVUˆYGD]OSLwQSLU DLhfDLUK<]aTHVhhfDxU HVN ]bgDxF/R#EGEcDLbgHVU ]}dgTLd”
αD =
3 X
uk ,
N
k
D†b£D]>KxYGHVU [email protected]>debgDDLKAdeUK<SLF<D]<]ÎR#UK"[email protected]]<[email protected]]<DLF>wzdeF-»'TDLK>K<DSLK/R#EcDYN¤E)R#F/R#bebgiLbgDDtszdg]<K/R#UK"RwQDTfbgDBEGF<HzTD]J
[Àqåmo òi‰fkf õ¨Ü˜À’&‘€Û;& ÃQGeÛ;mõªÃQ&‘*&#ˆ õ¤&<f;†i‰
Ä[
>] N ]@YGD'hkR#F<T/l DˆR#bgSR#K<HVdeF<Dˆ»%K>F<HVdg]YdehfDLU ]>dgHVU ]jN DŠU HVN ]fRwQHVU ]SLwQH*j*N SŠEGF<STSYGDLhhfDLUKn|7UËDt—˜DLKa
bgD"wQDTLK<DLNGF-YdeEcÚVbgD\F<DTHVU ]>K>F>NGdeKxEH`]/]<iYGD"hkR#K>l SLhkR#K>dgj*N DLhfDLUKBbgD]–hfyLhfD]AEGF<HVEGF>dgSLK<S]-jN DBbgD\wQDTLK<DLNGF
EH`]>deK>dgHVU THVU ]<STLNGK>d“Œ4R#Ns
YGSLEGbqRVTDLhfDLUK<]AR#bgSR#K<HVdeF<D]OYGDxbgHVUGP`N DLNGF
Y˜½ NGU DxE)R#F>K>dgTLNGbgD`nCD
r
N
λ = 3/N
TRV]"Y˜½ NGU DhkR#F<T/l DŠR#bgSR#K<HVdeF<D'dg]<HVK>F<HVEcDfTHVF>F<D]>EcHVU Y‰YGHVU TkR#NÆTRV]"Y˜½ NGUª• Nsªdg]<HVK>F<HVEcDm”T/l)RVjN DkE)RV]a
THVF>F<D]>EHVU Y R#U*[email protected]»mT<l)RVj*N D'YdeF<DTLK>dgHVUZY˜½§R#F>F>dewQSD`a;EH`]/]<iYGDNGU DfEGF<HV[)R#[GdebedeK<SpYGD']<DEGF<HzYNGdeF<DkdeU YGSLEcDLU
Y R#UK<D\YGD-bqRYdeF<DTLK>dgHVUncC‘RkYdg]>K>F>de[GNGK>dgHVU!YGD D]>KOR#bgHVF<]OdgYGDLUK>dgj*N D\»fbqRfYdg]>K>F>de[GNGK>dgHVU¤YGD
Y R#U ]ObgD
r
αD
TRV]}Y˜½ NGU¬• NspK<HVK/R#bgDLhfDLUK–dg]<HVK>F<HVED`n
C½§R#U)R#bgHVP`[email protected]<HV[[email protected]ŠhkR#F<T<l DŠR#bgSR#K<HVdeF<DfD]>KxSLK/R#[GbedgDE)R#FxbgD]xF<DLbqR#K>dgHVU ]"YGDTHVF>F<D]J
EHVU Y R#U TD-]<NGdewVR#U*K<D]"”
r
⇔
λ
⇔
œ ÷zn§Ñ 
Ø
œ ÷zn§Ñ ÷`
αD
3
N
œ ÷zn§Ñ`¦Q
Y R#U ]ObqRVj*N DLbebgD
D]>KAbgDBTHzDt¥[email protected]“—cN ]>dgHVUabqRkF<DLbqR#K>dgHVUªDLU*K>F<DTD\THzDt¥kTLdgDLUK-DLKAbgDBbede[GF<D
Ddiffusion
E)R#F<THVNGF<]}hfH¼QDLU R¼R#UKOSLK<SxYGSLhfHVU*K>F<SD"EGF<STSYGDLhhfDLU*Kn
λ
õ&’ )܏ C‘R2YGDLU ]>deK<SmYGD¬EGF<HV[)R#[GdebedeK<S%YNËwQDTLK<DLNGF'YdeEcÚVbgD¬F<DTHVU >] K>F>NGdeKaoY R#U ]fbgDˆTRV]'HVÈËbgD
• Ns¤[email protected]/R¼QHVU ]"TH`]>hdgj*N D]-D]>KAdg]<HVK>F<HVEcD`a˜D]>K-YGHVU T`a]>NGdew`R#UKxb½¾Sj*N)R#K>dgHVU §n Ña˜E)R#F/R#hfSLK>F>dg]/SDE)R#FxNGU D
Ø
PR#N ]<]>dgDLUGU D"»\K>F<HVdg]}YdehfDLU ]<dgHVU ]uYGHVUGU SD-E)R#FubqR\F<DLbqR#K>dgHVU‰”
p (αrec Drec , N ) =
6π
N
3/2
N
exp − ||αrec Drec ||2
6
œ ÷zn§Ñ`^Q
»
DtszEGF>dehkR#U*K'bqR‰EGF<HV[)R#[GdebedeK<SmEcHVNGFpjN DmbgDˆwQDTLK<DLNGFpYdeEÚVbgDmF<DTHVU ]>K>F>NGdeKˆR#deKkEcHVNGFkw`R#bgDLNGF
αrec Drec
EGF<i]n
3
d (αrec Drec )
C DfwQDTLK<DLNGFYdeEcÚVbgDfF<DTHVU ]>K>F>[email protected]`]<]<iYGDfYGHVU T'NGU Š
D HVF>dgDLUK/R#K>dgHVUZNGUGd“Œ’HVF>hfSLhfDLU*[email protected]<SLE)R#F>K>dgDŠ]<NGF"bqR
]>EGl iLF<D%”;]<D]\K>F<HVdg]@THVhEH`]ÎR#UK<D]]<HVU*K\YGHVU TpYdg]>K>F>de[GN SD ]]>NGdew`R#UK\NGU DŠbgHVdoPR#N ]/]>dgDLUGU DpYGDkhfH¼QDLUGU D
q n
UNGbebgD-DLKuY˜½¾STR#F>KuK†¼zED
3
σ=
N
«E)R#F>K>deFrYGDOb½¾Sj*N)R#K>dgHVUªœ ÷zn§Ñ`^Qta*debcD]>K4EcH`]<]>de[GbgDOYGDOTR#bgTLNGbgDLFvbqR-YGDLU ]<deK<S–YGDuEGF<HV[)R#[GdebedeK<SxYNkhfHzYNGbgD
YNwQDTLK<DLNGFoYdeEcÚVbgD ”`DLUfdeU*K<SLP`F/R#UKv]>NGF7K<HVNGKbgD]4R#UGP`bgD]4bgD]<DTHVU YhfDLhB[GF<DuYGD-œ ÷zn§Ñ`^QtaQbqR–EGF<HV[)R#[GdebedeK<S
”
α
Ä[W
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
,.-F/.1L÷zn§¦
Ֆdg]>K>F>de[GNGK>dgHVU!YGD"b½§R#hEGbedeK>N [email protected]%YdeEcÚVbgD-F<DTHVU ]>K>F>NGdeKAY R#U u
] bgDBTRV]OY˜½ NGU%• Nsˆdg]<HVK>F<HVEcD"DLU
_f_`_`[email protected] DLhfDLUK<]–T<l)RVTLNGUtnC‘R
Œ’HVU TLK>dgHVUfY˜½¾DtsEcH`]>deK>dgHVUfTHVF>F<D]>EcHVU YkR#[email protected]@]>deK<D]YGDvb½ š [ ]<DLF>w`R#K<HVdeF<Ø D}É7dgDLF>F<DuMuNGPQDLFn`™OUkRI¯N ]<K<DLhfDLUK
YGD¬bqRªYdg]>K>F>de[GNGK>dgHVU‹E)R#FkbqR¤Œ’HVU TLK>dgHVU‹YGSTLF>deK<DmY R#U ]fb½¾Sj*N)R#K>dgHVU œ ÷zn§Ñ`^QD]>KfF<DLEGF<S]/DLUK<S`aob½§RVTTHVF<Y D]>K
DtsTDLbebgDLU*Kn
DLUK>F<SDkœ I½§2R#d‘]>dehBNGbgS“ð`__`_`[email protected]<SR#bedg]/R#K>dgHVU ]OY˜½ NGUˆbgHVKuYGDxYGHVUGU SD]OYGD
YGDAF<DTHVU ]>K>F>NGdeF<D
αrec
»
dαrec
EGF<i]}]½¾STLF>deKOR#bgHVF<]B”
p (αrec , N ) =
HVȬbqRj*N)R#UK>deK<S
σα
σα−3
D]>K}YGHVUGU SDxE)R#F}bqR\F<DLbqR#K>dgHVU
r
α2rec
2 2
α exp − 2
π rec
2σα
σα =
q
3
N
œ ÷zn ~ _Q
n
C RBYdg]>K>F>de[GNGK>dgHVU¬YGD–b½§R#hEGbedeK>N YGDOF<DTHVU ]<K>F>NGdeK<D`aGY R#U ]obgD–TRV]rY˜½ NGU'• Ns'dg]<HVK>F<HVED–]>NGdeKvYGHVU TuNGU D–bgHVd
‘
]>dehdebqR#deF<Dx»"TDLbebgDAYNphfHzYNGbgDOYGD–bqR"w*deK<D]<]/DOYGD]rhfHVbgSTLNGbgD]}Y˜½ NGUpPR#Ԝ’TtŒDLU TRVYF<SxYNŠT/l)R#EGdeK>F<D tajNGd
Ø
K<DLU Y¬wQDLF<]NGU D-Ydg]<K>F>de[GNGK>dgHVUˆPR#N ]<]>dgDLUGU D"»P`F/R#U Y
œ’HVN%»\EcDLK>deK
taGY˜½¾STR#F>KuK†¼zEcD
n
N
σα
σα
C D\TR#bgTLNGb£EGF<STSYGDLUK-D]>KADLU¤K<HVNGK<D"F>deP`N DLNGFANGUGdgj*N DLhfDLUKxw`R#bqR#[GbgDBEcHVNGF–bgDBTRV]AY˜½ NGU [email protected]>K>NGF<D
K<HVK/R#bgDBDLK–NGUGd“ŒŽHVF>hfD`nc‡rDLEcDLU Y R#[email protected]%EcDLNGKubgDBwQHVdeFA]>NGFubqRf­ P`NGF<Du÷zn§¦abgHVF<]<j*N D"bqRkTHVNGwQDLF>K>NGF<D
D]>KK<HVK/R#bgDxDLKuj*N DAbqRBŒ’HVU TLK>dgHVU%Y˜½¾DtszEcH`]>deK>dgHVU¬U DAEcH`]<]<iYGDAE)RV]}YGDAK>F<HVE¬ŒŽHVF>K<D]w`R#F>dqR#K>dgHVU ]aGb½¾DtsEGF<D]<]>dgHVU
R#U)R#be¼*K>dgj*N D-j*N DAU HVN ]uRwQHVU ]OHV[GK<DLUN DBD]>KNGU D-DtsGTDLbebgDLUK<DBR#EGEGF<HszdehkR#K>dgHVUn
‡rDBF<S]>NGbeK/R#KAEDLF>hfDLK>K>F/RpYGD\TR#bgTLNGbgDLFxbgDBYGDLP`F<[email protected]@THVU­R#U TD]>K/R#K>dg]>K>dgj*N D\jN DBb½¾HVU¤EDLNGK-RVTTHVF<YGDLF
[Àqåmo òi‰fkf õ¨Ü˜À’&‘€Û;& ÃQGeÛ;mõªÃQ&‘*&#ˆ õ¤&<f;†i‰
Ä[[
» b½ l*¼*EcHVK>l i]<DxYGD–b½ dg]<HVK>F<HVEGdgDAYNŠ• NsŠHV[ ]<DLF>wQS–E)R#FvNGU DADtszEcSLF>dgDLU TDxYGHVUGU SD`n OHVN ]vYGSLwQDLbgHVEGEcDLF<HVU ]}TD
TR#bgTLNGbE)R#F}bgDx]>NGdeK<DxDLKuYdg]/TLNGK<DLF<HVU ]}bgD]}THVU TLbeN ]>dgHVU ]OjNGd˜EcDLNGwQDLUK–DLUˆyLK>F<DxK>deF<SD]n
õ&’„i‰VÛ;&;õÁ ÜG‰&)fRÛ£0Ü$õ CHVF<]/jN D'bqRmYdg]<K>F>de[GNGK>dgHVUXR#UGP`NGbqR#deF<DpYGD]\F/R¼QHVU ]@TH`]>hdgj*N D]@D]>K
hfH*YNGbgSD¬E)R#FkNGU D%R#UGdg]<HVK>F<HVEGdgD%YdeEHVbqR#deF<D`a7bgDˆwQDTLK<DLNGF'YdeEcÚVbgD¬F<DTHVU ]>K>F>NGdeK']½§R#EGE)R#F<DLUK<D¤R#NËwQDTLK<DLNGF
EH`]>deK>dgHVU HV[GK<DLU*N D}R#EGF<i]£NGU Dr]<NGdeK<DrYGD
E)RV]£YGDrbgHVUGP`N DLNGF
aV]>NGdewVR#U*K7NGU D4hkR#F<T/l DuR#bgSR#K<HVdeF<D
r
N
λ = 3/N
RwQDT¤RVYwQDTLK>dgHVUnrCR2E)R#F/R#hfSLK>F>dg]/R#K>dgHVUžYN• Ns ]<HVN ]'bqR¤ŒŽHVF>hfD%|jnOœ ÷zn kD]>KŠYGHVU Tmb½§R#U)R#bgHVP`N D¤YGD

b½¾DtszEGF<D]<]<dgHVU%œ n¤÷`7YGDbqR–EGF<HV[)R#[GdebedeK<S–YGDvŒR#deF<DNGUfE)RV]Y R#U ]4bqRxYdeF<DTLK>dgHVU nQÕADU HVNGwQDR#NaQbgD}E)R#F/R#bebgiLbgD
Ø
u
[email protected]\EGF<HV[GbgiLhfDYGD\bqRkhkR#F<T/l DfR#bgSR#K<HVdeF<DkRwQDTfRVYwQDTLK>dgHVUªU HVN ]AEcDLF>hfDLKxY˜½¾HV[GK<DLUGdeF-bqRkŒ’HVF>[email protected]\bqR
YGDLU ]>deK<SAYGDAEGF<HV[)R#[GdebedeK<SxYGDAF<DTHVU ]>K>F>NGdeF<DANGUpwQDTLK<DLNGFuYdeEÚVbgD
»
EGF<i]az]ÎRVT<l)R#UK
αrec Drec d3 (αrec Drec )
jN [email protected]]
SLwQSLU DLhfDLUK<]\]<HVUKBHV[GK<DLUN ]@»ŠE)R#F>K>deF"Y˜½ NGU DkYdg]>K>F>de[GNGK>dgHVUÆR#UGP`NGbqR#deF<DkTR#F/RVTLK<SLF>dg]<SDkE)R#F-NGU
N
wQDTLK<DLNGFuYdeEÚVbg¹
D åAÞ'Ù*ÛrY˜½§R#hEGbedeK>N YGD DLKuYGDxYdeF<DTLK>dgHVU
”
α
p (αrec Drec , N ) =
D
6π
N
3/2
N
exp − ||αrec Drec − α D||2
6
œ ÷zn ~ 

C DBwQDTLK<DLNGF-YdeEÚVbgDBF<DTHVU ]>K>F>NGdeK-D]>KAYGHVU [email protected]]>K>F>de[GN [email protected]"hkR#UGdgiLF<D\PR#N ]<]>dgDLUGU DR#NGK<HVNGFxYN¤wQDTLK<DLNGF
YdeEÚVbgD¤wzF/R#da}RwQDTªbgD¤hfyLhfDªSTR#F>KmK†¼zEDªj*N DªY R#U ]¬bgDªTRV]pdg]<HVK>F<HVEcD`n}C½§R#U)R#bgHVP`dgD2DLU*K>F<DªbqRÆwzdeK<D]<]<D
Y˜½§RVYwQDTLK>dgHVUˆY R#U ]obgDOEGF<HV[GbgiLhfDxYGDOhkR#F<T<l D-R#bgSR#K<HVdeF<DADLKobgD–wQDTLK<DLNGFYdeEcÚVbgDOwzF/R#dcD]>KrYGHVU T–SLK/R#[GbedgDAE)R#F
bqR\F<DLbqR#K>dgHVU%YGDxTHVF>F<D]>EcHVU Y R#U TDk”
3v
⇔ αD
c
)E R#F>K>deFxYGD"b½¾DtsEGF<D]<]>dgHVUXœ ÷zn ~ tadeb;D]>KOEH`]/]>de[GbgDBYGD\TR#bgTLNGbgDLFAbgD]OYdg]<K>F>de[GNGK>dgHVU ]xYN%hfHzYNGbgD
αrec

YN2wQDTLK<DLNGFBYdeEcÚVbgDF<DTHVU ]>K>F>NGdeKaYGDb½§R#UGP`bgDfK<HVK/R#b DLUK>F<D
DLK
a‘R#deU ]<d4jN DfbgD]BR#UGP`bgD]
DLK
γ
D
Drec
θrec
YGSt­ UGdg]<]/R#U*K–bqR'YdeF<DTLK>dgHVU¤F<DTHVU ]>K>F>NGdeK<D`mn OHVN ]ASLE)R#F>P`U DLF<HVU ]-R#N!bgDTLK<DLNGFAbgDBYGSLK/R#debŒŽRV]<K>dgYdgDLNs%YGD]
ϕrec
TR#bgTLNGbg]}DLK}U HVN ]}THVU TDLU*K>F<DLF<HVU ]–]<NGFbgD]F<S]>NGbeK/R#K<]uHV[GK<DLU*N ]n
C½ deUK<SLP`F/R#K>dgHVUp]>NGF7K<HVN ]7bgD]oR#UGP`bgD]YNf]<DTHVU YhfDLhB[GF<D}YGDb½¾Sj*N)R#K>dgHVU%œ ÷zn ~ ;EcDLF>hfDLKY˜½¾HV[GK<DLUGdeF4bqR

Ydg]>K>F>de[GNGK>dgHVUBYGD
nÉ;HVNGFYGD]‘wVR#bgDLNGF<];]>N¥k]/R#hfDLU*K‘P`F/R#U YGD];YN-U HVh"[GF<DrY˜½¾SLwQSLU DLhfDLUK<]}œ

αrec
N 1/α2
[email protected]]>K>F>de[GNGK>dgHVU%YGDxb½§R#hEGbedeK>N YGD"F<DTHVU ]>K>F>NGdeK<D-]½¾STLF>deKB”
”
1
p (αrec , N ) ' p
2πσα2
α
rec
α
(αrec − α)2
exp −
2σα2
!
œ ÷zn ~ ÑQ
p
j NGdrK<DLU Y‰wQDLF<]@NGU D'PR#N ]<]>dgDLUGU DpY˜½¾STR#F>[email protected]†¼zEcD
bgHVF<]<j*N DŠbgDkU HVhB[GF<D¬Y˜½¾SLwQSLU DLhfDLUK<]fD]<K
σα = 3/N
P`F/R#U Y˜n
Ä[c
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
C DŠTHVU TDLEGKYGDfEGNGdg]<]/R#U TDpYGDkF<DTHVU ]>K>F>N TLK>dgHVU ˆ
R SLK<S'deUK>F<[email protected]<]\YGD'b½¾SLK>N YGDŠYGDkbqRmEGF<STLdg]>dgHVU
YGD"bqRkhfSLK>l HzYGD`nM–HVN ]–F/R#EGEDLbgHVU ]–b½¾DtsEGF<D]<]>dgHVUªYGSt­ GU dg]<]/R#U*K
ajNGd£TR#F/RVTLK<SLF>dg]<D\bqRfEGNGdg]/]/R#U TDBYGD\bqR
Kα
hfSLK>l H*YGDB»\hfD]>NGF<DLF}b½§R#hEGbedeK>N YGD"YN%YdeEÚVbgD”
[σ(δα/α)]−1 = Kα α
œ ÷zn ~`~ 
√
N.
‡ HVhEGK<DK<DLUNÅYGDb½¾DtszEGF<D]/]>dgHVU‰[email protected]½¾STR#F>K-KÞ¼*EcD
r
j*N [email protected] VH N ]BRwQHVU ]"HV[GK<DLUN D`a‘U HVN ]xEcHVNGwQHVU ]BR#bgHVF<]
σα
EGF<SYdeF<D¬bqR¤w`R#bgDLNGFfK>l SHVF>dgj*N DˆYGD
Y R#U ]fbgDˆTRV]bgD¬EGbeN ]'>] dehEGbgDˆHVÈXbqRªTHVNGwQDLF>K>NGF<D%D]>KK<HVK/R#bgDmDLK
Kα
NGUGd“Œ’HVF>hfDk”
œ ÷zn ~ „*
1
1
(Kα )E=1 =
σα
√
= √ ≈ 0.58
3
N
‡ DLK>K<D"w`R#bgDLNGFOD]<KOYGHVU TBK<HVNGK–»fŒŽR#deK–DLU!RVTTHVF<Y¤RwQDT\TDLbebgDBHV[GK<DLU*N D\DLU!]>deh"NGbeK/R#K>dgHVU¤U*NGhfSLF>dgjN [email protected] R#U ]
r
[email protected]]xY˜½ NGU [email protected]Œ’HVU TLK>dgHVU2Y˜½¾DtsEcH`]>deK>dgHVU2THVF>F<D]>EcHVU Y R#UKBR#Ns¤YGDLNs¤]>deK<D][email protected]½ š [ ]/DLF>wVR#K<HVdeF<DÉ7dgDLF>F<DkMuN
PQDLF-”GbqRB­ P`NGF<D'œ ÷zn vhfHVUK>F<D"TLbqR#deF<DLhfDLU*KOj*N DxbqR\EGNGdg]<]/R#U TD"YGDxF<DTHVU ]>K>F>N TLK>dgHVU%D]<KdeU YGSLEDLU Y R#U*K<DBYGD
Ø
b½¾HVF>dgDLUK/R#K>dgHVUpYNkYdeEcÚVbgD}DLKEGF/R#K>dgj*N DLhfDLUKvSLPR#bgDu»AbqRAw`R#bgDLNGFK>l SHVF>dgj*N D
nC½¾Sj*N)R#K>dgHVU!œ ÷zn ~ ÑQ7F<D]>K<D
0.58
YGHVU TuNGU DODtsTDLbebgDLU*K<DxR#EGEGF<HsdehkR#K>dgHVU¬EHVNGFoNGU D}ŒŽHVU TLK>dgHVUŠY˜½¾DtsEcH`]>deK>dgHVUŠYGHVU*KobgD]ow`R#F>dqR#K>dgHVU ]r]/HVUKrEDLN
dehEHVF>K/R#U*K<D]o]<NGFbqRA]>EGl iLF<D`aQK/R#U*Koj*N DvbqRxTHVNGwQDLF>K>NGF<DOF<D]>K<DK<HVK/R#bgD`nC‘D}TRV]4Y˜½ NGU D}THVNGwQDLF>K>NGF<DOE)R#F>K>dgDLbebgD
U½§R}E)RV]£SLK<SrSLK>N YdgSR#U)R#be¼*K>dgj*N DLhfDLUK7YNGF/R#UKTDoK>F/Rw`R#debGYGD4K>l i]/D`aTDLEcDLU Y R#UKa#bqRuwVR#bedgYdeK<SvYGD];F<S]>NGbeK/R#K<]
YGSLhfHVUK>F<S]-Y R#U ]–[email protected]]AYN¤hfH*[email protected]]>dehEGbedg]>K<[email protected]\THVNGwQDLF>K>NGF<[email protected]<HVK/R#[email protected]“Œ’HVF>hfD\EcDLNGK–yLK>F<[email protected]¤E)R#F>K>dgD
SLK<DLU YN D"R#NmTRV]EGbeN ]uTHVhEGbgDtsD-HVÈ
U½¾D]>KEGbeN ]OTHVU ]>K/R#UKn
E(u)
‡ HVhhfDrbgD4hfHVU*K>F<DLUK7bgD]­ P`NGF<D]÷zn§ÑODLK÷zn ~ a#hfyLhfDrY R#U ]£bgDoTRV]£Y˜½ NGU DrTHVNGwQDLF>K>NGF<DvE)R#F>K>dgDLbebgDrDLK;U HVU
r
NGUGd“Œ’HVF>hfD`abqRxYdg]>K>F>de[GNGK>dgHVUpYGDb½§R#hEGbedeK>N YGD–YNkYdeEÚVbgDuF<DTHVU ]>K>F>NGdeK4F<D]<K<D`a*»AP`F/R#U Y a`NGU DPR#N ]/]>dgDLUGU D
N
√ n4C‘R‰wVR#bgDLNGFpYGD
TDLUK>F<SD!]>NGFŠbqRªwzF/R#dgDmw`R#bgDLNGF'YN«YdeEÚVbgD%DLKŠYGHVU*K'b½¾STR#F>KŠK†¼zEcD%YdehdeU*N D!DLU
1/ N
bqRpEGNGdg]<]/R#U TD'YGDF<DTHVU ]>K>F>N TLK>dgHVUZDLKB]/RˆYGSLEDLU Y R#U TDŠRwQDT'b½¾HVF>dgDLUK/R#K>dgHVUZYNÅYdeEcÚVbgDk]<HVU*K"TDLEDLU Y R#U*K
Yd“—cSLF<DLU*K<D]kEcHVNGFkT/l)RVjN DˆŒ’HVU TLK>dgHVU‹Y˜½¾DtszEcH`]>deK>dgHVUnC½¾SLK>N YGD!YGD¬bqR¤EGF<STLdg]<dgHVUYGD¬F<DTHVU ]>K>F>N TLK>dgHVU«D]>K
YGHVU T–F<SR#bedg]<SDx»-E)R#F>K>deFYGD]]>dehBNGbqR#K>dgHVU ]aGDLKvR#EGEGNG¼QSDxTHVU TDLEGK>N DLbebgDLhfDLU*KuE)R#FrbgD]TR#bgTLNGbg]}R#U)R#be¼*K>dgj*N D]
EGF<S]<DLUK<S]udgTLd˜hfyLhfD`n
C RAF<DTHVU ]<K>F>N TLK>dgHVUpR#UGP`NGbqR#deF<D–YNkYdeEcÚVbgDuD]>K4TR#F/RVTLK<SLF>dg]/SD}E)R#F4bqR-Ydg]>EDLF<]<dgHVUkYGD]4Ydg]>K>F>de[GNGK>dgHVU ]vYGD
‘
b½§R#UGP`bgD aEGF<HwQDLU)R#UKAYGD-TDLbebgD]}YGD
DLK
n
γ
θrec
C‘RYdg]<K>F>de[GNGK>dgHVUmYGDxb½§R#UGP`bgD
p (γ, N ) = 2π
6π
N
γ
−3/2 Z
0
∞
ϕrec
]½¾HV[GK>dgDLUKODLUmdeUK<SLP`F/R#UKOb½¾SjN)R#K>dgHVUZœ ÷zn ~ v
 ]>NGF
”

αrec
dαrec α2rec sin γ exp
−
N 2
α + α2rec − 2α αrec cos γ
6
œ ÷zn ~ ðQ
[Àqåmo òi‰fkf õ¨Ü˜À’&‘€Û;& ÃQGeÛ;mõªÃQ&‘*&#ˆ õ¤&<f;†i‰
Ä[v
HVÈ U HVN ]pRwQHVU ]ŠYGSLwQDLbgHVEGES%bgDˆEGF<HzYNGdeK']<TR#bqR#deF<D!Y R#U ]'b½§R#F>P`NGhfDLUK¬YGDˆb½¾DtsEHVU DLU*K>dgDLbebgD`nvCHVF<]<j*N ˆ
D bgD
U HVh"[GF<D"Y˜½¾SLwQSLU DLhfDLU*K<]OD]<K}P`F/R#U Y˜a‘œ
taDLK}THVhEGK<D-K<DLUN%YGD]OR#EGEGF<HsdehkR#K>dgHVU ]
LD K
N 3/α
sin γ ≈ γ
[email protected]<HV[)R#[GdebedeK<SBY˜½§RwQHVdeFONGUmR#UGP`bgD »
EGF<i]}]½¾STLF>deK"”
2
cos γ ≈ 1 − γ /2
γ
dγ
γ2
1
p (γ, N ) ' √ 2 γ exp − 2
2 π σγ
2σγ
œ ÷zn ~ 
Ø
R wQDT 2 1 q 3 n‡rHVhhfDBEGF<STSYGDLhhfDLUKa˜TDLK>K<DBYdg]>K>F>de[GNGK>dgHVU%K<DLU YmwQDLF<]uNGU D"PR#N ]<]>dgDLUGU D\Y˜½¾STR#F>K
σγ = α N
K†¼zED
bgHVF<]<j*N D
D]>KAP`F/R#U Y˜n˜ÕxR#U ][email protected]]xHVȪbgD\w*F/R#dYdeEcÚ[email protected]]>KxHVF>dgDLUK<[email protected]ˆYGD\b½§RsGDYGD] a
σγ
N
z
b½¾SjN)R#K>dgHVU%œ ÷zn ~ U D}EDLNGK7E)RV]7yLK>F<DuR#EGEGF<HszdehfSDuE)R#F7NGU DvPR#N ]<]>dgDLUGU D`a`b½§R#UGP`bgD SLK/R#UKE)R#FYGSt­ UGdeK>dgHVU
Ø
γ
THVhEGF>dg]DLUK>F<D DLK nVÕAR#U ]4TDrTRV]aE)R#FTHVU*wQDLUK>dgHVUabqR–wVR#bgDLNGF7YGDvbqR–Ydg]>EcDLF<]>dgHVUNGK>debedg]<SDD]>K£K<HVNVI>HVNGF<]
0 π
n
σγ
C‘[email protected]]<]ÎR#U TDxYGDxF<DTHVU ]>K>F>N TLK>dgHVU%YGD-b½§R#UGP`bgD
(Ktot )E=1 =
a
γ Ktot
aGD]>K}YGHVU T-YGHVUGU SD-E)R#FbqRF<DLbqR#K>dgHVU‰”
1
1
√
= √ ≈ 0.58 rad −1 .
3
σγ α N
œ ÷zn ~ ÷`
C RAEGNGdg]<]ÎR#U TD}YGDvF<DTHVU ]>K>F>N TLK>dgHVUpYGDvbqRxYdeF<DTLK>dgHVU'YNkYdeEcÚVbgD`aQYGSt­ UGdgDE)R#F–œ
‘
a
taQ]½¾HV[GK>dgDLU*KoYGD
θrec ϕrec
bqR\hfyLhfD–ŒŽRVWHVUˆDLUˆ]<HVhhkR#U*Kub½¾Sj*N)R#K>dgHVU‰œ ÷zn ~ v]>NGF
aGDLUˆSTLF>dewVR#U*K}b½¾SLbgSLhfDLUK–YGDAwQHVbeNGhfD-THVhhfD

αrec
n
3
2
d (αrec ) Drec = αrec dαrec sin θrec dθrec dϕrec
| UXdeUK<SLP`F/R#U*KkbqR!EGF<HV[)R#[GdebedeK<S‰œ ÷zn ~ \]<NGF
7
DLK]>NGF
a7RwQDT¬b½¾DtsEGF<D]<]>dgHVU̜ ÷zn§ÑJðQ\YGD
a

αrec
ϕrec
cos γ
EHVNGFObgD]}EcDLK>deK<]–R#UGP`bgD]@œ
DLK
taHVUmEcDLNGKuhfHVUK>F<DLF
sin θrec ' θrec
cos(θrec − θd ) ' 1 − (θrec − θd )2 /2
jN DAbqRYdg]<K>F>de[GNGK>dgHVU!YGDxb½§R#UGP`bgD
aGEcHVNGF}bgD]}P`F/R#U YGD]uwVR#bgDLNGF<]OYGD
œ
oK<DLU YmSLPR#bgDLhfDLU*K
θrec
N N 3/α2
wQDLF<]xNGU DPR#N ]<]>dgDLUGU DkTDLUK>F<SDf]>NGF a˜Y˜½¾STR#F>K-K†¼zEcD
dgYGDLUK>dgj*N Dk»¬[email protected]ˆYdg]>K>F>de[GNGK>dgHVU
θd
σθ = σ γ
YGDAb½§R#UGP`bgD nGC‘RfYdg]>K>F>de[GNGK>dgHVU%YGDAb½§R#UGP`bgD
EGF<S]<DLU*K<D-YGHVU TxbgDxhfyLhfD"THVhEHVF>K<DLhfDLU*KOj*N D-TDLbebgD-YGD
γ
θrec
b½§R#UGP`bgDxK<HVK/R#b a RwQDT-bqR\hfyLhfDxEGNGdg]<]ÎR#U TD-YGDxF<DTHVU ]>K>F>N TLK>dgHVUƔ
γ
(Kθ = Kdec )E=1 = (Ktot )E=1 ≈ 0.58 rad−1 .
œ ÷zn ~ ¦Q
C RxYdg]>K>F>de[GNGK>dgHVUŠYGDb½§RV]<TDLU ]>dgHVUŠYF<HVdeK<DF<DTHVU ]>K>F>NGdeK<D`azYGSt­ UGdgDE)R#F
‘
]½¾HV[GK>dgDLU*KoDLUk]<HVhhkR#U*K4b½¾Sj*N)R
ϕrec
K>dgHVUƜ ÷zn ~ ]<NGF
DLKO]>NGF
n |7U!THVU ]>dgYGSLF/R#UKAbgD]uhfyLhfD]–R#EGEGF<HszdehkR#K>dgHVU ]xj*N DxEGF<STSYGDLhhfDLUKa

αrec
θrec
DLKrDLUŠEGF<DLU)R#UK
azbqR"EGF<HV[)R#[GdebedeK<SAYGDOhfD]<NGF<DLF
»
EGF<i]
cos (ϕd − ϕrec ) ≈ 1 − (ϕd − ϕrec )2 /2
ϕrec dϕrec
]½¾STLF>deK7K<DLU YSLPR#bgDLhfDLUKowQDLF<]£NGU DYdg]>K>F>de[GNGK>dgHVUfPR#N ]<]>dgDLUGU DuTDLUK>F<SDuR#NGK<HVNGF4YGD
RwQDTNGUSTR#F>KKÞ¼*EcD
ϕd
n
1
σϕ =
sin θd
σγ
Ägc
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
C HVF<]<j*N D}bgD–YdeEcÚVbgDuD]>KvR#bedeP`U SO]>NGFob½§RsD aœ
HVN
tabqR"Ydg]>EcDLF<]>dgHVUŠYGD
D]<K4deU­ UGdgD`a
z θd = 0
θd = π
ϕrec
THVF>F<D]>EcHVU Y R#UK"R#N!ŒŽR#deKxj*N D"b½§RV]/TDLU ]>dgHVU2YF<HVdeK<D\U D\]/HVdeK–EGbeN ]AYGSt­ UGdgDY R#U ][email protected]]nC‘RŠEGNGdg]<]/R#U [email protected]
F<DTHVU ]>K>F>N TLK>dgHVUYGDrb½§RV]<TDLU ]>[email protected]<HVdeK<DrYGSLEcDLU [email protected] TvYGD4bqROYdeF<DTLK>dgHVUfYN\YdeEcÚVbgD`aV»}K>F/RwQDLF<]7bqR}F<DLbqR#K>dgHVU¬”
(KRA )E=1 = (Ktot )E=1 × cos δd ≈ cos δd × 0.58 rad −1 .
œ ÷zn ~ ^Q
C‘R!­ P`NGF<Dy÷zn¤÷!F<DLU THVUK>F<SD!EGF<STSYGDLhhfDLUKpdebebeN ]>K>F<DˆbqR¤EGNGdg]<]/R#U TD%YGD¬F<DTHVU ]>K>F>N TLK>dgHVUžR#UGP`NGbqR#deF<D%YN
YdeEÚVbgD'HV[GK<DLUN DpDLUÅ]<deh"NGbqR#K>dgHVUZEHVNGFBbgD]BYGDLNs‰]<deK<D]BYGDkb½ š [ ]<DLF>wVR#K<HVdeF<DŠÉ7dgDLF>F<DpMONGPQDLFnDÞboD]>KBdeUK<St
F<D]<]/R#U*K}YGDOF<DLhkR#F<j*N DLFuj*N D–bgD]vF<S]>NGbeK/R#K<]uR#U)R#be¼zK>dgjN D]YGSLhfHVU*K>F<S]uY R#U ]}TDLK>K<Dx]<DTLK>dgHVUˆYGSTLF>dewQDLU*K}K>F<i]
[GdgDLUfb½¾SLwQHVbeNGK>dgHVU'YGD}bqRAEGNGdg]<]/R#U TD}YGDvF<DTHVU ]<K>F>N TLK>dgHVUpR#UGP`NGbqR#deF<D–RwQDT}bqRAYGSTLbedeU)R#dg]/HVU'YNkYdeEcÚVbgDdeUGdeK>dqR#b
THVhhfDxF<DLEGF<S]<DLU*K<S"]>NGF}TDLK>K<DA­ P`NGF<D`n
CD]BF<S]>NGbeK/R#K<]\EGF<STSYGDLUK<]EcDLF>hfDLK>K<DLUKYGHVU TŠYGD'THVhEGF<DLU YF<[email protected]'YGSTLF>deF<DpRwQDTŠEGF<STLdg]>dgHVUÆbgD]
TR#F/RVTLK<SLF>dg]>K>dgj*N D]rYGDubqRxF<DTHVU ]>K>F>N TLK>dgHVU¬YNŠYdeEÚVbgD`a*Y R#U ]4bgDOTRV]oY˜½ NGU D–THVNGwQDLF>K>NGF<DuK<HVK/R#bgD–DLK4NGUGd“ŒŽHVF>hfD
hkR#dg]R#N ]<]<dQY R#U ]bgDTRV]Y˜½ NGU D4THVNGwQDLF>K>NGF<DK<HVK/R#bgD4EH`]/]<SY R#UK‘YGD7ŒŽR#de[GbgD]‘wVR#F>dqR#K>dgHVU ]£RwQDT4bqRYGSTLbedeU)R#dg]<HVUn
C½§R#U)R#bgHVP`dgD!DLUK>F<DˆbgDˆEGF<HV[GbgiLhfD%YGDmbqR¤hkR#F<T<l D¤R#bgSR#K<HVdeF<D`abqR‰Yd“—N ]>dgHVUDLKkbqR¤F<DTHVU ]>K>F>N TLK>dgHVU«Y˜½ NGU D
R#UGdg]<HVK>F<HVEGdgDkR#EGEcHVF>K<DNGU‰STLbqR#deF/R#PQDkdeUK<SLF<D]<]/R#U*K\»Šb½¾SLK>N YGD'YGD]-TR#F/RVTLK<SLF>dg]>K>dgj*N D]"YGDU HVK>F<DhfSLK>l HzYGD`a
DLKEDLF>hfDLKOYGDxYGSLF>dewQDLFOYGD–ŒRVWHVUmSLbgSLPR#UK<D"bgDxTHVhEHVF>K<DLhfDLU*K–YGD]E)R#F/R#hfiLK>F<D]OF<DTHVU ]>K>F>NGdeK<]n
ÕxR#U ]4bgD–TRV]rY˜½ NGU DOTHVNGwQDLF>K>NGF<D–E)R#F>K>dgDLbebgDADLKoU HVUŠNGUGd“ŒŽHVF>hfD`ab½§R#U)R#bgHVP`dgDADLUK>F<D–hkR#F<T/l DAR#bgSR#K<HVdeF<DADLK
F<DTHVU ]>K>F>N TLK>dgHVU%Y˜½ NGU%YdeEÚVbgD"U Dx]½¾DtszEGF>dehfD"EGbeN ]}]>dehEGbgDLhfDLU*Ka)U DxEcDLF>hfDLK>K/R#UKOEGbeN ]uY˜½ NGK>debedg]<DLFObgD]}]<H#
beNGK>dgHVU ]THVUGUN D]oY˜½ NGUfEGF<HV[GbgiLhfD}YGDYd“—cN ]>dgHVUfEcHVNGFHV[GK<DLUGdeF4TDLbebgD]Y˜½ NGUfEGF<HV[GbgiLhfDuYGDvF<DTHVU ]>K>F>N TLK>dgHVU
Y˜½§R#UGdg]<HVK>F<HVEGdgD`n
–,W¯
† ¶71\¶£·ºó<:\B)ºz?:\@}1
³
ü2£üÞý
¸
ó
º}6
9;ºBž³xõ6\1B5±º}1\:\B[@ )? ·[email protected]µ\:/5Ò?A6\º}³B·:>µ[email protected]ó<º}:<·5
Q%ÿ%__`JQr+KQÿrK†ÿ]^ IV$ Y‹K†ÿ
–HVN ]–RwQHVU ]xYGSTLF>deK–bgDBEGF>deU TLdeEcD\YGDBbqRkhfSLK>l H*YGDf¡zM–É«R#deU ]<d;j*N D"bqRkEGF<STLdg]>dgHVUªYGDBF<DTHVU ]>K>F>N TLK>dgHVU
jN½ debSLK/R#deK4EcH`]<]>de[GbgD}Y˜½§R#K>K<DLdeU YF<D–Y R#U ]bgD}TRV]oHVÈkbgDv• NsfYGDF/R¼QHVU ]oTH`]<hdgjN D]rTHVU ]>dg]>K<D}DLUkNGU D}E)R#F>K>dgD
dg]<HVK>F<HVEcD¬hfH*YNGbgSDˆE)R#FkNGU DmR#UGdg]<HVK>F<HVEGdgD%YdeEHVbqR#deF<D`n+D†bOD]>KkYGHVU T¬U)R#K>NGF<DLb–YGDˆ]<D¬EcH`]<DLFfbqRªj*N D]>K>dgHVU
]>NGdew`R#UK<Df”Gj*N DxYGDLw*dgDLU*KOTDLK>K<D"R#U)R#be¼]<Dx]>d˜b½§R#UGdg]/HVK>F<HVEGdgD"YNp• Ns¬U½¾D]>KEGbeN ]EGNGF<DLhfDLU*K–YdeEcHVbqR#deF<DÏ
‡rHVhhfDoU HVN ]bgD4wQDLF>F<HVU ];E)R#FbqRO]>NGdeK<D`aY R#U ];bgDoTRV];HVÈBb½¾DtszEcSLF>dgDLU TDvTHVU ]>dgYGSLF<SDrHV[ ]<DLF>wQD4bqRuK<HVK/R#bedeK<S
YNˆTLdgDLbaGbqR\EGF<S]<DLU TD-Y˜½ NGU DBR#UGdg]<HVK>F<HVEGdgD"Y˜½¾HVF<YF<DxEGbeN ]uSLbgSLwQSDxU DxEcDLF>K>NGF>[DAE)RV]bqRF<DTHVU ]>K>F>N TLK>dgHVU%YN
YdeEÚVbgD`n£|7UXDt—˜DLKa‘bgD]Yd“—˜SLF<DLUK<]@hfHzYGD]@YNXYGSLwQDLbgHVEGEDLhfDLU*KkDLUZl)R#F>hfHVUGdgjN D]]>EGl SLF>dgj*N D]\ŒŽHVF>hfDLUK
[À‚Wˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛoe&)fˆf)Ü;P’&‘Û;#0܏õ ò#
ÄgcÄ
NGU D-[)RV]<D\HVF>K>l HVPQHVU)R#bgD\EHVNGFObgD]}ŒŽHVU TLK>dgHVU ]OYGSt­ UGdgD]A]>NGFObqRk]<EGl iLF<D`n)CDBhfHVU HVEÚVbgDpœ
ta bgD\YdeEcÚVbgD
l=0
œ
tabgD4jN)RVYF>deEcÚVbgD
aDLK<T`nenen]/HVUKYGHVU TdeU YGSLEDLU Y R#U*K<];DLU*K>F<D4DLNsxDLKEcDLNGwQDLUKyLK>F<D4F<DTHVU ]>K>F>NGdeK
l=1
(l = 2)
]<SLE)R#F<SLhfDLUKn
Õ R#U ]7bgD}TRV]HVÈfbqRATHVNGwQDLF>K>NGF<DOD]>K7E)R#F>K>dgDLbebgD`aQbgD]4Yd“—cSLF<DLU*K<]4hfHzYGD]7U D}]<HVU*KEGbeN ]7deU YGSLEcDLU Y R#UK<]obgD]
x
NGU ]7YGD]4R#NGK>F<D]nQC‘R–EGF<S]/DLU TDYGDhfH*YGD]£EGbeN ]4SLbgDLwQS]Y R#U ]7bgDYGSLwQDLbgHVEGEcDLhfDLUKoYNf• NsBw`RAYGHVU Tv[GdqR#dg]<DLF
bqRˆw`R#bgDLNGF\YNXYdeEcÚVbgDkF<DTHVU ]<K>F>NGdeK<DpE)R#FBU HVK>F<DŠhfSLK>l HzYGD`n)DÞbvD]>K\TDLEcDLU Y R#UKEcH`]<]>de[GbgD'YGDŠPQSLU SLF/R#bedg]<DLF
U HVK>F<DˆhfSLK>l HzYGDˆEcHVNGFk¼ÆdeU TLbeNGF<DmbqR¤F<DTHVU ]<K>F>N TLK>dgHVU«]<deh"NGbeK/R#U SD!Y˜½ NGUYdeEcÚVbgDmDLKkY˜½ NGU‹jN)RVYF>deEcÚVbgD`n
OHVN ]7NGK>debedg]/DLF<HVU ]4E)R#FbqR-]>NGdeK<DbgDK<DLF>hfDOYGDF<DTHVU ]>K>F>N TLK>dgHVU'YN'YdeEÚVbg
D Ù!ӛò er æ!Þ'Ö*ÞßLa*Y R#U ]bgD}TRV]oHVÈfbgD
wQDTLK<DLNGF;YdeEcÚVbgD4D]>K;D]>K>dehfSv»b½§R#dgYGDrYGD4bqR}EGF<DLhdgiLF<D4wQDLF<]>[email protected] HVK>F<DhfSLK>l HzYGD`a#T`½¾D]<KJÞ»†YdeF<DoDLU"ŒŽR#dg]/R#U*K
b½ l¼zEHVK>l i]<D–j*N D}bgDu• NskYGD}F/R¼QHVU ]TH`]<hdgjN D]rDLKohfHzYNGbgSOE)R#F4NGUŠEGNGFrYdeEcÚVbgD`n*CRxF<DTHVU ]>K>F>N TLK>dgHVÈ
U Ù!Ó
æ*Þ'Ö!Þ>ßLaGj*N)R#UK»"DLbebgD`aYGS]>deP`U DLF/R"bqR"F<DTHVU ]>K>F>N TLK>dgHVUˆYNŠYdeEcÚVbgD–DLUkŒR#dg]/R#UKvb½ l*¼*EcHVK>l i]<D–jN DObgD}• Ns
2me
D]>KrTHVhEcH`]<S–Y˜½ NGUpK<DLF>hfDOYdeEcHVbqR#deF<DADLKvY˜½ NGUŠK<DLF>hfDAjN)RVYF>deEcHVbqR#deF<D`nCD–EGF>deU TLdeEDAYGDObqR-PQSLU SLF/R#bedg]ÎR#K>dgHVU
[email protected]>l H*YGD\¡zMOÉ R#N%TRV]}YNmj*N)RVYF>deEÚVbgD-D]<K}YGSTLF>deK}E)R#F}[email protected]]>NGdeK<D`n
< α dipole reconstruit >
Reconstruction of the dipole amplitude for quadrupolar anisotropy
-1
10
N= 500
N= 1000
N= 2500
N= 5000
N= 10000
N= 25000
N= 50000
N= 100000
N= 250000
N= 500000
-2
N= 1000000
10
-1
10
α quadrupole
,6-0/.1^÷zn§^
€4R#bgDLNGFBhfH¼QDLUGU DŠYGDfb½§R#hEGbedeK>N YGDpYN‰YdeEcÚVbgDfF<DTHVU ]>K>F>NGdeKRwQDT'bqRphfSLK>l HzYGD'YGDfF<DTHVU ]J
K>F>N TLK>dgHVUŠR#2 N er HVF<YF<D`abgHVF<]<j*N D}bgD• NsfD]>KohfH*YNGbgSuE)R#F4NGU'EGNGFoj*N)RVYF>deEÚVbgDO]>¼*hfSLK>F>dgj*N D`nC4½¾SLwQHVbeNGK>dgHVU
RwQDTxb½§R#hEGbedeK> N YGDBYNmjN)RVYF>deEcÚVbgDxEcHVNGF}Yd“—cSLF<DLU*K<D]uw`R#bgDLNGF<]}YGD
D]>KF<DLEGF<S]<DLU*K<SD`n
N
M ­ UY˜½ debebeN ]>K>F<DLF£[email protected]÷LTHVUK/R#hdeU)R#K>dgHVU‘÷uYNYdeEÚVbgDrE)R#F7bgD];hfH*YGD]£EGbeN ]£SLbgDLwQS]7bgHVF<]/jN DobqR–THVNGwQDLF>K>NGF<D
}
D]>K-E)R#F>K>dgDLbebgD`a)IL½§R#d4]<deh"NGbgS'NGU‰DLU ]<DLh"[GbgDŠYGDfbgHVK<]BYGDkYGHVUGU SD]\YGDkYd“—˜SLF<DLUK<D]BK/R#debebgD]a‘THVF>F<D]<EHVU Y R#U*K
ÄgcMv
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
»pNGU2• NsªhfHzYNGbgSfE)R#FBNGU‰EGNGF"j*N)RVYF>deEÚVbgD`a;DLKI½§R#dF<DTHVU ]>K>F>NGdeK\bgDkYdeEcÚVbgDk»¬b½§R#dgYGDkbqRphfSLK>l HzYGDŠR#N
HVF<YF<D`n)C‘RBŒ’HVU TLK>dgHVU%Y˜½¾DtszEcH`]>deK>dgHVUmTHVF>F<D]>EcHVU YˆR#Nm]>deK<Dx]<N Y¬YGDAb½ š [ ]<DLF>wVR#K<HVdeF<D"É7dgDLF>F<D-MONGPQDLFn CD]
1er
F<S]>NGbeK/R#K<]\]<HVUK\EGF<S]<DLUK<S]@]>NGFBbgD]-­ P`NGF<D]¬œ ÷zn§^Q-DLKpœ ÷zn _Qtn;C‘RmwVR#bgDLNGFBhfH¼QDLUGU DpYGDkb½§R#hEGbedeK>N YGDpYN

YdeEÚVbgD%F<DTHVU ]>K>F>NGdeK¬]>NGFŠEGbeN ]>dgDLNGF<]ŠF<SR#bedg]/R#K>dgHVU ]¬Y˜½ NGU«hfyLhfD!bgHVKŠYGD!YGHVUGU SD]p¼ËD]>KŠF<DLEGF<S]<DLUK<SªDLU
Œ’HVU TLK>dgHVUa*YNkU HVhB[GF<DOY˜½¾SLwQSLU DLhfDLU*K<]DLK4YGD}b½§R#hEGbedeK>N YGDAYNkj*N)RVYF>deEÚ[email protected]œ’YGHVUKobqR-YGSt­ UGdeK>dgHVUp]>dehdebqR#deF<D
»[email protected]ÚVbgDv]<DLF/RODtszEGbedgTLdeK<SDrE)R#F;bqRu]<NGdeK<Dtn#É;HVNGF7T<l)RVj*N DrF<SR#bedg]/R#K>dgHVUa#bqROYdeF<DTLK>[email protected])RVYF>deEcÚVbgD
D]>K}Yd“—˜SLF<DLUK<D`a)YGDAhkR#UGdgiLF<D\»\hfH¼QDLUGU DLFObgD]EH`]/]>de[GbgD]}Dt—˜DLK<]}Y˜½¾HVF>dgDLUK/R#K>dgHVUn
< α dipole reconstructed >
Reconstruction of the dipole amplitude for quadrupolar anisotropy
-1
10
αquad = 5 (%)
αquad = 10 (%)
αquad = 15 (%)
αquad = 20 (%)
αquad = 25 (%)
αquad = 30 (%)
αquad = 35 (%)
αquad = 40 (%)
αquad = 45 (%)
αquad = 50 (%)
3
10
4
10
5
10
6
10
Number of events
,.-F/.1'÷zn _
€4R#bgDLNGF–hfH¼QDLUGU DYGD\b½§R#hEGbedeK>N YGDYN¤YdeEcÚVbgDBF<DTHVU ]>K>F>NGdeK"RwQDT\bqR'hfSLK>l H*[email protected]<DTHVU ]J

2
K>F>N TLK>dgHVUŠR#N er HVF<YF<D`aQbgHVF<]<j*N D}bgD}• NsD]<K4hfHzYNGbgS}E)R#F4NGU'EGNGFoj*N)RVYF>deEÚVbgDO]>¼zhfSLK>F>dgjN D`n*C½¾SLwQHVbeNGK>dgHVU
[email protected] HVh"[GF<DŠY˜½¾SLwQSLU DLhfDLUK<]\EHVNGFBYd“—cSLF<DLU*K<D]"w`R#bgDLNGF<]"YGDb½§R#hEGbedeK>N YGD'YNÅj*N)RVYF>deEÚVbgDkD]>K-debebeN ]J
K>F<SD`n‘‡rD]xF<S]>NGbeK/R#K<]-HVU*K-SLK<SHV[GK<DLUN ]"EHVNGFxNGU D\ŒŽHVU TLK>dgHVU‰Y˜½¾DtsEH`]>deK>dgHVU2THVF>F<D]<EHVU Y R#U*KBR#N2]>deK<D]>N Y
YGDAb½ š [ ]<DLF>w`R#K<HVdeF<D"É7dgDLF>F<D-MONGPQDLFn
š UmEcDLNGKuF<DLhkR#F<j*N DLFA]>NGFubqR­ P`NGF<DŠœ ÷zn§^QujN D-bgHVF<]<jN D"bgD"U HVhB[GF<DBY˜½¾SLwQSLU DLhfDLU*K<]xD]>KuK>F<i]OP`F/R#U Y˜a
b½§R#hEGbedeK>N YGDxYNˆYdeEÚVbgD
K<DLU Yˆ»\yLK>F<DAEGF<HVEHVF>K>dgHVUGU DLbebgDB»"b½§R#hEGbedeK>N YGD"YNpj*N)RVYF>deEcÚVbgD
Y R#U ]
αdip
αquad
bgD]7YGHVUGU SD]a`hfDLK>K/R#U*KoTLbqR#deF<DLhfDLUKrDLUfSLw*dgYGDLU TD}bgDrŒŽR#deKj*N½ NGU DE)R#F>K>dgD}YNfj*N)RVYF>deEÚVbgDuD]>K£F<DTHVU ]>K>F>NGdeK<D
THVhhfDpNGUXYdeEÚVbgD`n£C‘R%­ P`NGF<D2œ ÷zn _QBdebebeN ]>K>F<DŠbqR!THVUwQDLF>PQDLU TD%YGD
bgHVF<]/jN D
R#NGP`hfDLU*K<D¬wQDLF<]

αdip
N
NGU D!wVR#bgDLNGF¬EGF<HVEcHVF>K>dgHVUGU DLbebgD‰»
nvÉ£R#FˆDtsDLhEGbgD`a]<d–b½¾HVUžTHVU ]>dgYGiLF<D¤NGU«• Ns‹hfH*YNGbgS!E)R#F¬NGU
αquad
EGNGF¬jN)RVYF>deEcÚVbgD¤YGD
Y˜½§R#hEGbedeK>N YGDªY R#U ]pNGU«bgHVKŠYGD¤YGHVUGU SD]¬THVU*K<DLU)R#UK
SLwQSLU DLhfDLU*K<]abqR
10%
105
hfSLK>l H*YGDmYGDˆF<DTHVU ]>K>F>N TLK>dgHVU‹YNYdeEcÚVbgDmR#NËEGF<DLhdgDLF'HVF<YF<DmdeU YdgjN DLF/R‰DLUËhfH¼QDLUGU D%NGU YdeEcÚVbgDmYGD
[À‚Wˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛoe&)fˆf)Ü;P’&‘Û;#0܏õ ò#
Ägc¿
˜Y ½§R#hEGbedeK>N YGD`n‡rHVhEGK<D-K<DLU*N%[email protected]]/]/R#U TD-YGDxF<DTHVU ]>K>F>N TLK>dgHVU!YNm]>deK<Dx]>N YmYGDxb½ š [ ]<DLF>w`R#K<HVdeF<D
aDLUXNGK>debedg]/R#U*Kfb½¾SjN)R#K>dgHVUôœ ÷zn§Ñ ~ \HVUXEDLNGKfD]>K>dehfDLFbqR¤]<deP`UGd“­)TR#K>dew*deK<SˆYGD
É7dgDLF>F<D¬MuNGPQDLF
Kα ≈ 0.33
TDLK>K<DfhfD]>NGF<DŠ»
n<DÞb4D]>K"YGHVU TkU STD]<]/R#deF<D'YGDfPQSLU SLF/R#bedg]<DLFBU HVK>F<DkhfSLK>l HzYGDkDLUż!deU TLbeN)R#U*KBbqR
≈ 4.2 σ
EH`]<]<de[GdebedeK<Skj*N DfbqRˆYdg]<K>F>de[GNGK>dgHVUXR#UGP`NGbqR#deF<D'EcH`]<]<iYGDfNGU DŠTHVhEcH`]/R#UK<DŠjN)RVYF>deEcHVbqR#deF<D`n‡rDTLd4EcDLF>hfDLK
YGDAF<DLEHVN ]<]/DLFu»@b½¾HVF<YF<D
b½¾HVF>deP`deU D-Y˜½ NGU D-SLwQDLU*K>N DLbebgDBTHVU*K/R#hdeU)R#K>dgHVU!YNmYdeEcÚVbgD`n
4Y
l=3
ü2£üFE
¸
žuIL
IVtQ_I>J+„î6}«
Õ R#U ]-bgDfTRV]-HVȉbqR¬Ydg]>K>F>de[GNGK>dgHVUXR#UGP`NGbqR#deF<DkYGD]-F/R¼QHVU ]"TH`]>hdgj*N D]"D]<K"]>NGEGEcH`]<SDfyLK>F<DhfHzYNGbgSD
x
E)R#FNGUmYdeEÚVbgD"DLKNGUmjN)RVYF>deEcÚVbgD`aGbgD–• NsˆY R#U ]NGU D-YdeF<DTLK>dgHVU
EcDLNGK}]½¾STLF>deF<D-]<HVN ]}bqRBŒ’HVF>hfDf”
u
Φ(u) =
X
X
Φ0
Φ0
(1 + αD · u +
Qij ui uj ) =
(1 + αD · u +
λi (qi · u)2 ),
4π
4π
i,j
HVÈ
œ ÷zn „_Q
i,j
F<DLEGF<S]<DLUK<[email protected]'THVhEcH`]/R#UK<[email protected]]<HVK>F<HVEcD`ac»kbqRVj*N DLbebgD\deb‘ŒR#NGK-RIJHVNGK<DLFANGU¤YdeEÚ[email protected]˜½§R#hEGbedeK>N YGD
œ
–Y R#U ]"bqRpYdeF<DTLK>dgHVU
a˜DLKxNGU‰jN)RVYF>deEcÚVbgDYGSTLF>deK-E)R#F-NGUªK<DLU ]/DLNGF-YGDf]<DTHVU YªHVF<YF<D`a
α 0≤α≤1
D
]>¼*hfSLK>F>dgj*N DDLK4YGDK>F/RVTD}UNGbebgD`aQYGHVU*K4bgD
] ðxTHVhEcH`]/R#U*K<D]deU YGSLEcDLU Y R#UK<D]r]<HVUKF<DLbedgSD]rR#NsfTH*Dt¥kTLdgDLU*K<]
YNÆYGSLwQDLbgHVEGEDLhfDLU*K\]<NGF-bgD]-l)R#F>hfHVUGdgj*N D]@]>EGl SLF>dgjN D]nC‘R¬hkR#K>F>dgTD
THVF>F<D]>EcHVU Y R#UK<DŠD]>K
l =2
Qij
F<SDLbebgD}DLK]>¼zhfSLK>F>dgjN D`a`DLK7EcDLNGK4YGHVU TyLK>F<D}YdqR#PQHVU)R#bedg]<SDO]>NGF7NGU Dv[)RV]<DHVF>K>l HVPQHVU)R#bgD`Tn OHVN ]YGS]<deP`U DLF<HVU ]
E)R#F
DLK
bgD]oK>F<HVdg]ow`R#bgDLNGF<]oEGF<HVEGF<D]rDLKowQDTLK<DLNGF<]rEGF<HVEGF<D]Aœ’HVF>K>l HVU HVF>hfS]t4YGD
n*C½§R#hEGbedeK>N YGDxYN
λi qi
Qij
jN)RVYF>deEcÚVbgD D]>KoYGSt­ UGdgD–YGD}hkR#UGdgiLF<DA]>dehdebqR#deF<DA»-TDLbebgD–YN'YdeEcÚVbgD`azTHVhhfDOSLK/R#U*KrNGUŠTHVUK>F/RV]>K<DADLU*K>F<D
β
bgDAhkRszdehBNGh DLKbgDxhdeUGdehBNGh YN¬• Nsª”
Φ0 /4π
β≡
Φmax − Φmin
λ+ − λ −
=
Φmax + Φmin
2 + λ + + λ−
œ ÷zn „ 

HVÈ D]<K;dgTLdGbgDr• NsDLUb½§R#[ ]<DLU TDYGDYdeEcÚVbgD`a
THVF>F<D]<EHVU Yf»–bqR–EGbeN ]£P`F/R#U YGDw`R#bgDLNGF7EGF<HVEGF<DEcH`]>deK>dewQD
Φ
λ+
RV]<]<HzTLdgSDBR#N!wQDTLK<DLNGF
DLK
bqRkEGbeN ]OEDLK>deK<D\w`R#bgDLNGFOEGF<HVEGF<DRV]<]<HzTLdgSDB»
nC‘D]–TLdeU jˆE)R#F/R#hfiLK>F<D]
q+
λ−
q−
deU YGSLEDLU Y R#U*K<]OEcDLF>hfDLK>K/R#UKxYGDBYGSTLF>deF<DBK<HVK/R#bgDLhfDLU*KAbgD"j*N)RVYF>deEÚVbgD\]<HVUK–YGHVU TBYGDLNs%R#hEGbedeK>N YGD]”;œ
β
DLK
ADLK-K>F<HVdg]BR#UGP`bgD]k”4œ a
AYGSt­ UGdg]/]/R#UKBb½¾HVF>dgDLUK/R#K>dgHVUZYNªwQDTLK<DLNGF
DLK
YGSt­ UGdg]<]/R#U*K"bqR
λ+
θ+ ϕ+
q+
ϕ−
YdeF<DTLK>dgHVUBYGD
Y R#U ]bgDEGbqR#U-EDLF>EcDLU YdgTLNGbqR#deF<Do»
nCD7K>F<HVdg]>dgiLhfD4wQDTLK<DLNGF‘EGF<HVEGF<D
D]<KHVF>K>l HVPQHVU)R#b
q−
q+
q0
R#NspYGDLNsˆR#NGK>F<D]aGDLKu]/RBw`R#bgDLNGF}EGF<HVEGF<D"RV]<]/H*TLdgSD`a awQSLF>d“­)[email protected] YdeK>dgHVU
œK>F/RVTD
λ0
λ− + λ + + λ 0 = 0
YGD
U*NGbebgDtn
Qij
ArdgDLUfjN DrU HVN ]£K>F/R#deK<DLF<HVU ]bgDvTRV]Y˜½ NGUfj*N)RVYF>deEcÚVbgDvPQSLU SLF/R#ba`YGHVUKbgD]£w`R#bgDLNGF<]7EGF<HVEGF<D]]<HVU*K7K<HVNGK<D]
Yd“—cSLF<DLU*K<D]aQbgHVF<]<j*N DYGDLNswVR#bgDLNGF<]4EGF<HVEGF<D]4]<HVU*K4SLPR#bgD]a`bgDujN)RVYF>deEcÚVbgDEcH`]<]<iYGDvNGU Du]<¼*hfSLK>F>dgD}RsdqR#bgD`a
Ägc)ú
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
DLEGF<S]<DLU*K/R#K>dgHVUˆYGDubqR\]>NGFJŒRVTDOYGSt­ UGdgD–E)R#FvNGUpjN)RVYF>deEcÚVbgD¶æ1änêFÙ*܏ßfœŽ»BPR#N T<l DoDLK Ú#Þæ!êFÙ*܏ß
 tnV2sÕxa R#U ]‘bgD4EGF<DLhdgDLF7TRV]abqRuwVR#bgDLNGF;EGF<HVEGF<DvRV]<]<HzTLdgSDoR#N\wQDTLK<DLNGF;YGSt­ UGdg]<]ÎR#UK£b½§RsDoYGDo]<¼*hfSLK>F>dgD
Žœ »}YF<HVdeK<`D
D]>KEGbeN ]EcDLK>deK<Dkœ’DLU¬w`R#bgDLNGFOR#[ ]<HVbeN Dvj*N DxbgD]}YGDLNsˆR#NGK>F<D]n
,.-F/.1÷zn
F<SYNGdg]/R#UKpR#deU ]>d}bgD¬U HVhB[GF<DmYGD¬E)R#F/R#hfiLK>F<D]'deU YGSLEcDLU Y R#UK<]p»¤K>F<HVdg]m”£NGU D!R#hEGbedeK>N YGDŜ @DLKkYGDLNs
β
R#UGP`bgD]-œ a
4YGDt­ UGdg]/]/R#UKbqR\YdeF<DTLK>dgHVUˆYGD–b½§RsDAYGDA]>¼zhfSLK>F>dgD`nMžK>deK>F<DxY˜½¾DtsDLhEGbgDAbqR-­ P`NGF<Dfœ ÷zn 
θ+ ϕ+
`
F<DLEGF<S]<DLUK<DBbqR\ŒŽHVF>hfD"Y˜½ NGU!jN)RVYF>deEcÚVbgD"]>¼zhfSLK>F>dgj*N D-YdeKæ1äêúÙ*܏ßpœŽ»PR#N T<l D}bgHVF<]<jN D"[email protected]`R#bgDLNGFuEGF<HVEGF<D
THVF>F<D]>EcHVU Y R#UK\»Šb½§RsGDfYGDf]<¼*hfSLK>F>dgDf»pNGU DfwVR#bgDLNGFxEGbeN ]-ŒŽR#de[GbgDfj*N [email protected]]-YGDLNs‰R#NGK>F<D]aDLK{Ú_Þ'æ!êFÙ*܏ߤœŽ»
YF<HVdeK<DvY R#U ]bgD-TRV]uTHVUK>F/R#deF<D`n
ü2£üû
¸
žb
Q%Qÿ4ÿ Qÿ%[email protected]$+g „ IV€Kÿ IVNXFÿS ‘ÿJ Qÿ~JVJIVKÿ
C RfhfSLK>l H*[email protected]¡zMOɍEDLNGK–yLK>F<D-PQSLU SLF/R#bedg]<[email protected]#N%TRV]–Y˜½ NGU DBR#UGdg]/HVK>F<HVEGdgDBYdeEcHVbqR#deF<DBDLK–jN)RVYF>deEcHVbqR#deF<D
‘
jN DLbgTHVU j*N D-EGF<S]/DLUK<D"Y R#U ]uNGUmbgHVK–YGD-YGHVUGU SD]n‡rHVhhfD"EGF<STSYGDLhhfDLUKacU HVN ]OYGSTLF>deF<HVU ][email protected]<HzTSt
YNGF<DODLU¬THVhhfDLU WR#U*KE)R#FrbgD–TRV]rbgDuEGbeN ]v]>dehEGbgDx»"K>F/R#deK<DLFx”bgHVF<]<j*N DObgD–YGSLK<DTLK<DLNGFTHVU ]>dgYGSLF<S–EcH`]<]<iYGD
NGU DxTHVNGwQDLF>K>NGF<D-K<HVK/R#bgD"hkR#dg]E)RV]vŒŽHVF<TSLhfDLUKuNGUGd“ŒŽHVF>hfD-YNmTLdgDLbn
C DEGF>deU TLdeEcDfYGDbqR¬hfSLK>l H*YGDF<D]<K<[email protected]”R­ U‰Y˜½¾D]<K>dehfDLF"bgD]xU DLNŒE)R#F/R#hfiLK>F<D]BU STD]<]/R#deF<D]
EHVNGF}TR#F/RVTLK<SLF>dg]/DLF}bgDAhfHVU HVEÚVbgD`a bgDxYdeEÚVbgDxDLK}bgDxj*N)RVYF>deEcÚVbgD`aU HVN ]}Ydg]>EcH`]<HVU ]}YGD]hfHVhfDLU*K<]uY˜½¾HVF<YF<D
ÔSLF<HGaQNGUkDLKrYGDLNskYGDbqR-Ydg]<K>F>de[GNGK>dgHVUpR#UGP`NGbqR#deF<D
a*YGHVUK4b½¾DtsEGF<D]<]>dgHVUŠD]>K4YGHVUGU SDOE)R#F–œ ÷zn „_Qtn‡rD]
Φ(u)
hfHVhfDLUK<]O]½¾STLF>dewQDLUK–YGHVU T-]<HVN ]bqRBŒŽHVF>hfDf”
I0 =
Z
Φ(u)dΩ
,
I=
Z
uΦ(u)dΩ
et
Iij =
Z
1
Φ(u)(ui uj − δij )dΩ,
3
œ ÷zn „ÑQ
[À‚Wˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛoe&)fˆf)Ü;P’&‘Û;#0܏õ ò#
Ägc
EHVNGF–bgD]<j*N DLbg]ObqRŠ]<HVhhkR#K>dgHVUª]½¾Dt—˜DTLK>N [email protected]]>NGF–K<HVNGK<DBbqR']>EGl iLF<D`ncCD\F<S]>NGbeK/R#KxYGD\b½ deUK<SLP`F/R#K>dgHVUªEcDLF>hfDLK
YGDAF<DLbedgDLFuTD]w`R#bgDLNGF<]OR#NspE)R#F/R#hfiLK>F<D]Oj*N DAb½¾HVUmT<l DLF<T/l DB»\F<DTHVU ]>K>F>NGdeF<D'”
1
I = Φ0 × αD
3
I0 = Φ 0
,
et
Iij =
œ ÷zn „ ~ 
2
Φ0 × Qij
15
É HVNGF<]<NGdewVR#U*KobgDvF/R#dg]<HVUGU DLhfDLU*Krj*NGd U HVN ]4R-THVU YNGdeKr»ASLK/R#[GbedeF4bqRxhfSLK>l H*YGD`aQTD]4hfHVhfDLUK<]oEDLNGwQDLU*K
yLK>F<DxdgYGDLUK>d“­)S]xR#Nsˆ]<HVhhfD]uYdg]/TLF<iLK<D]Oj*NGdbgDLNGFOTHVF>F<D]<EHVU YGDLU*Kn|7UmU HVK/R#UKATHVhhfD-EGF<STSYGDLhhfDLUK
b½¾DtsEH`]<deK>dgHVUÆYGD'b½¾DtszEcSLF>dgDLU TDpTHVU ]>dgYGSLF<SDŠY R#U ]BbqR%YdeF<DTLK>dgHVU
YN
dgiLhfD'SLwQSLU DLhfDLU*Ka£bgD]
E(uk )
uk
k
]<HVhhfD]}Ydg]<TLF<iLK<D] a DLK
THVF>F<D]>EHVU Y R#U*K–» a DLK
]½¾STLF>dewQDLUKB”
S0 S
S0 =
X
k
Sij
1
E(uk )
I0 I
,
S=
X uk
E(uk )
et
Iij
Sij =
k
œ ÷zn „`„*
X (ui )k (uj )k − 1 δij
3
E(uk )
k
VH È
D]>K}bgDx]<¼*hB[HVbgD-YGDuAF<HVU DT Ç DLFaSLPR#b» j*N)R#U Y
DLKuSLPR#b» ]>deU HVUn‡rD]U DLNŒj*N)R#UK>deK<S]
δij
1
i=j
0
EDLNGwQDLU*KvyLK>F<DxYdeF<DTLK<DLhfDLU*KuTR#bgTLNGbgSD]}»BE)R#F>K>deFYGD]YGHVUGU SD]aYGi]vbgHVF<]rj*N D–b½¾DtszEcH`]>deK>dgHVU¬YNpYGSLK<DTLK<DLNGF
D]>KxTHVUGUN D`n‘CRpYGSLF>dewVR#K>dgHVU‰YGD]-E)R#F/R#hfiLK>F<D]"YN2YdeEcÚVbgDDLK-YN2jN)RVYF>deEcÚVbgDfYGSTHVNGbgDfYdeF<DTLK<DLhfDLU*KBYGD
b½ dgYGDLUK>d“­)TR#K>dgHVU¤YGDxTD]u]<HVhhfD]uRwQDTxb½¾DtsEGF<D]<]>dgHVUŜ ÷zn „`„*vYGD]}hfHVhfDLUK<]\”
αD = 3
S
S0
et
Qij =
œ ÷zn „$ðQ
15 Sij
2 S0
EcDLNGKfR#bgHVF<]fyLK>F<D¬YdqR#PQHVU)R#bedg]<SDpEcHVNGFfHV[GK<DLUGdeFbgD]@wVR#bgDLNGF<]fEGF<HVEGF<D][email protected]<DLNGF<]
C R!hkR#K>F>dgTD
‘
Qij
EGF<HVEGF<D]oYNŠjN)RVYF>deEcÚVbgD`n‡rHVhhfDubgD}hfHVU*K>F<DOb½¾Sj*N)R#K>dgHVU!œ ÷zn „$ðQta*b½¾STLF>deK>NGF<D–YNkwQDTLK<DLNGFvYdeEÚVbgDuF<DTHVU ]J
K>F>NGdeKoD]>KdgYGDLU*K>dgjN DAR#NkTRV]rY˜½ NGUf• NshfHzYNGbgS}E)R#F4NGUkEGNGFoYdeEcÚVbgD`nz‡rHVhhfDOR#UGU HVU TSOEGF<STSYGDLhhfDLUKa
bqRfF<DTHVU ]>K>F>N TLK>dgHVU2YN¤j*N)RVYF>deEÚVbgD\U D"ŒŽR#deK–deUK<DLF>wQDLUGdeF-j*N DBYGD]A]<HVhhfD]–bedgSD]xR#NsmhfHVhfDLU*K<]-Y˜½¾HVF<YF<D
YGDLNsca DLKOD]>K}E)R#FOTHVU ]/SjN DLU*KudeU YGSLEcDLU Y R#UK<D\[email protected]<DTHVU ]>K>F>N TLK>dgHVU¤YNmYdeEcÚVbgD`bn –HVN ]}wQDLF>F<HVU ]uE)R#FubqR
]>NGdeK<DvbgD]THVU ]<Sj*N DLU TD]DLUfK<DLF>hfD}YGDvEGNGdg]/]/R#U TDYGDF<DTHVU ]>K>F>N TLK>dgHVUŠjN½ dehEGbedgj*N D}TDLK>K<DvdeU YGSLEcDLU Y R#U TD
YGD]hfH*YGD]n
ü2£ü‚
¸
þ•Y‹KÿÅÿ€]^ IV Y‹K†ÿrIÿ%r+ÿS ‘ÿJ ÿr+I> J†ÿ%KK†ÿ
C RkPQSLU SLF/R#bedg]/R#K>dgHVU‰R#N!TRV]–Y˜½ NGU [email protected]>K>NGF<D\E)R#F>K>dgDLbebgD\YGD\bqRfEGF<H*TSYNGF<[email protected]<DTHVU ]>K>F>N TLK>dgHVU2]>NGdeK
‘
bgDˆhfyLhfD%EGF>deU TLdeED¤j*N DˆEGF<STSYGDLhhfDLUK2”4bgD]ŠhfHVhfDLUK<] a R#U Y
]<HVUKpTR#bgTLNGbgS]pDLU bedehdeK/R#U*K
I0 I
Iij
b½ deUK<SLP`F/R#K>dgHVUŠ»ObqR–EcHVF>K>dgHVUfYGDTLdgDLbGHVÈb½¾DtszEcH`]>deK>dgHVUfD]>K£U HVUUNGbebgD`aTHVUK<DLUN DODLUK>F<DbqR–YGSTLbedeU)R#dg]<HVU
δmin
DLK
œ’DLU‰THzHVF<YGHVUGU SD]"Sj*N)R#K<HVF>dqR#bgD]Νta‘THVF>F<D]<EHVU Y R#U*[email protected]»ŠbqR¬THzHVF<YGHVUGU SD]\]>EGl SLF>dgjN D THVUK<DLUN D
δmax
θ
ÄgcW
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
LD UK>F<D
DLK
nzC½¾STLF>deK>NGF<DœbgSLPQiLF<DLhfDLUK}Yd“—˜SLF<DLUK<DAYNpTRV]oTHVNGwQDLF>K>NGF<D–K<HVK/R#bgDoYGD]ohfHVhfDLUK<]D]>K
θmin θmax
YGHVU T-YGHVUGU SD-]<HVN ]Œ’HVF>hfDxTHVU YGDLU ]<SD"E)R#F-”
[ I0 ; I ; Iij ] =
Z
θmax
dθ sin θ
θmin
Z
2π
dϕ [ Φ(u) ; uΦ(u) ; ui uj Φ(u) ]
0
œ ÷zn „ 
Ø
ÕxR#U ]bgDvTRV]7HVÈ@bqR–THVNGwQDLF>K>NGF<DYNTLdgDLbD]<K;E)R#F>K>dgDLbebgD`aVbqROF<DLbqR#K>dgHVUfDLU*K>F<DvbgD]hfHVhfDLU*K<]DLK£bgD]£E)R#F/R#hfiLK>F<D]
YN!YdeEcÚVbgD\DLK–YN¤jN)RVYF>deEcÚVbgDBU D"EcDLNGKOEGbeN ]A]½¾STLF>deF<D\YGDBhkR#UGdgiLF<D\SLbgSLPR#U*K<Dp”)bgD\]>¼]>K<iLhfD"Y˜½¾Sj*N)R#K>dgHVU ]
D]>KoTHVNGEGbgSADLKr]<HVU'deU*wQDLF<]>dgHVUpU STD]<]>deK<DONGU'EcDLNkEGbeN ]vY˜½§R#bePQiL[GF<DAj*N½§R#NGE)R#F/Rw`R#UKnGCD]vTHVhEcH`]/R#UK<D]YN
wQDTLK<DLNGFOYdeEcÚVbgD"DLK–YGD-bqRhkR#K>F>dgTD\YGSTLF>dew`R#UK–bgD-jN)RVYF>deEcÚVbgD\]<DLF<HVUKOU HVK<S]uE)R#FObqRf]>NGdeK<DB]<HVN ]}bqRŒ’HVF>hfD
]>NGdew`R#UK<Df”

Dx




αD = 
D
y


Dz
et

Qxx Qxy Qxz

Q=
 Qxy
Qxz
Qyy
Qyz


Qyz 

Qzz
œ ÷zn „÷`
Y R#U ]£[email protected]<StŒ’SLF<DLU*K>dgDLb)TR#F>K<S]>dgDLUkRwQDTvb½§RsGD HVF>dgDLUK<SwQDLF<]£bgDvÉ;ÚVbgDzOHVF<YTSLbgD]<K<D`n`CDvTR#bgTLNGbGU STD]<]/R#deF<D}»
z
bqR}YGSLF>dew`R#K>dgHVU\YGD]‘F<DLbqR#K>dgHVU ]£DLUK>F<DobgD]‘E)R#F/R#hfiLK>F<D]£YGDb½§R#UGdg]/HVK>F<HVEGdgDrDLKbgD]]/HVhhfD]]>NGF‘bgD]SLwQSLU DLhfDLU*K<]
SLK/R#UKOŒŽRV]<K>dgYdgDLNscaTDLbeNGd“†TLd;D]>KOF<DLEHVF>K<SBDLU!R#UGU DtsGD`nCRF<S]<HVbeNGK>dgHVU¤YN%]>¼]>K<iLhfDBY˜½¾SjN)R#K>dgHVU ]ATHVU YNGdeK
»@DtszEGF>dehfDLFubgD]}THVhEH`]ÎR#UK<D]–YN¬wQDTLK<DLNGFOYdeEcÚVbgDx]<HVN ]bqR\ŒŽHVF>hfD”
Di =
1
1 + S̃S̃zz
(Didip + ξi S̃iz )
(i = x, y, z)
œ ÷zn „¦Q
0
VH ÈBbgD]7j*N)R#UK>deK<S] dip ]<HVU*K;bgD];w`R#bgDLNGF<]7YGD]£THVhEH`]/R#U*K<D][email protected]ÚVbgDrF<DTHVU ]>K>F>NGdeK<D]4R#N\EGF<DLhdgDLFHVF<YF<D`a
Di
bgD]}w`R#F>dqR#[GbgD]}U HVK<SD] ]<HVU*KOYGD]uTHVhB[GdeU)R#dg]<HVU ]–bedeU SR#deF<DBYGD]O]<HVhhfD]uYdg]/TLF<iLK<D]O]>NGFubgD]uSLwQSLU DLhfDLUK<]
S̃
YGSt­ UGdgD]vY R#U ]obgDOTRV]vYGD}bqRBTHVNGwQDLF>K>NGF<DOK<HVK/R#bgDADLKxœ a 7YGD]vTHVU ]>K/R#U*K<D]rYGSLEcDLU Y R#UK}YGDubqRxŒŽHVF>hfD–YGDubqR
ξi
THVNGwQDLF>K>NGF<[email protected]¤TLdgDLbnCD]xYd“—˜SLF<DLUK<]AK<DLF>hfD]ATHVF>F<DTLK>d“ŒŽ]x]½§R#EGEGbedgjN)R#U*K-»fb½¾DtsEGF<D]<]>dgHVUªYGD]–E)R#F/R#hfiLK>F<D]
YNŠYdeEcÚVbgDAR#NkEGF<DLhdgDLFHVF<YF<DO]/HVUKx”QNGU'ŒŽRVTLK<DLNGFrh"NGbeK>deEGbedgTR#K>d“Œ£P`bgHV[)R#baTHVF>F<D]>EcHVU Y R#UK}»-bqR"THVF>F<DTLK>dgHVU
YNˆ]<DTHVU YˆHVF<YF<D-YGD-b½¾D]>K>dehkR#K>dgHVUmYNˆhfHVU HVEcÚVbgDf”
Φ0 = Φdip
0 (1 + DLKNGUˆK<DLF>hfD
ξi S̃iz
S̃zz
)
S̃0
dehEGbedgjN)R#U*KOYGD]}]/HVhhfD]}Y˜½¾HVF<YF<D"YGDLNs˜n
œ ÷zn „^Q
[À‚Wˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛoe&)fˆf)Ü;P’&‘Û;#0܏õ ò#
ÄgcV[
CD]}THVhEcH`]/R#UK<D]–YNˆj*N)RVYF>deEÚVbgD-]/HVUKuYGHVUGU SD]uE)R#F"”
Qxx =
Qzz =
Qxy =
Qxz =
Qyz =
S̃xx
S̃zz
4
[
+
]
Φ0 ∆31 − ∆53 12(γ2 − γ1 )
S̃zz
2
3Φ0 (γ1 − γ2 )
,
Qyy = −Qxx − Qzz
S̃xy
8
Φ0 ∆31 − ∆53
2
S̃x
2
Φ0 (∆42 − ∆31 ∆53 )
2
S̃y
2
Φ0 (∆42 − ∆31 ∆53 )
œ ÷zn:ð`_Q
œ ÷zn:ð 

œ ÷zn:ð`ÑQ
œ ÷zn:ð ~ 
œ ÷zn:ð#„*
œ ÷zn:ðJðQ
VH ÈfbgD]w`R#F>dqR#[GbgD] DLK
]<HVU*KoYGD]oTHVU ]>K/R#UK<D]rYGSLEDLU Y R#U*KrYGD}bqR–Œ’HVF>hfDuYGD}bqRAEcHVF>K>dgHVU'YGD}TLdgDLbHV[ ]<DLFJ
γi ∆ij
wQSD`nCD]AE)R#F/R#hfiLK>F<D]AF<DTHVU ]<K>F>NGdeK<]xYN!YdeEcÚ[email protected]–YN¤jN)RVYF>deEcÚVbgDBŒ’HVU*KxYGHVU TBdeU*K<DLF>wQDLUGdeF-YGD]A]<HVhhfD]
Ydg]<TLF<iLK<D]}YNˆEGF<DLhdgDLFuDLKOYNˆ]<DTHVU YˆHVF<YF<D]a DtsEGbedgTLdeK/R#UKAR#deU ]>d˜bgD-THVNGEGbqR#PQDBYdeEÚVbgDt†j*N)RVYF>deEÚVbgD`n
ü2£ü¨§
¸
ÿ ÿ Qÿ%+O_g žb+_IV
C ½¾SLK>N YGD–YGDbqRxEGF<STLdg]>dgHVUŠYGD}bqRAF<DTHVU ]<K>F>N TLK>dgHVUpY˜½ NGUŠYdeEÚVbgDuDLKoY˜½ NGU'j*N)RVYF>deEcÚVbgDORwQDTObqRAhfSLK>l HzYGD

¡zMOÉ2ROSLK<SvDt—cDTLK>N SDYGDrbqROhfyLhfDoŒŽRVWHVUj*N DoEGF<STSYGDLhhfDLU*Ku”ÎIL½§R#dG]>dehBNGbgSrNGU\P`F/R#U YDLU ]/DLh"[GbgDYGDrbgHVK<]
YGDuYGHVUGU SD]THVF>F<D]>EcHVU Y R#UK}»-Yd“—˜SLF<DLUK<D]vw`R#bgDLNGF<]vYGDuYdeEcÚVbgDODLKvYGDuj*N)RVYF>deEcÚ[email protected]œU HVUp]>¼zhfSLK>F>dgjN DtazDLK
IL½§R#d˜SLK>N YdgSxbgD]Ydg]>K>F>de[GNGK>dgHVU ]uYGD]E)R#F/R#hfiLK>F<D]}F<DTHVU ]>K>F>NGdeK<]}E)R#FF/R#EGEHVF>K<]uR#NsŠE)R#F/R#hfiLK>F<D]uY˜½¾DLU*K>F<SD`n
C‘[email protected]<HzTSYNGF<DBY˜½§R#U)R#be¼]<D-DLKObgD]}F<S]>NGbeK/R#K<]OSLK/R#UK–K>F<i]u]>dehdebqR#deF<D]–R#N%TRV]OYGDxbqRF<DTHVU ]>K>F>N TLK>dgHVUªR#N
EGF<DLhdgDLFvHVF<YF<D`azU HVN ]DtsEH`]/DLF<HVU ]4EGbeN ]vF/R#EGdgYGDLhfDLU*KbgD]oF<S]<NGbeK/R#K<]]>NGdewVR#U*K<]nzCD–bgDTLK<DLNGFvSLPR#F<S–EcHVNGF>F/R
YGDK<HVNGK<DŒRVWHVU']<D}F<StŒ’SLF<DLFv»AbqRxYGD]<TLF>deEGK>dgHVUpYGD}bqRAEGF<STLdg]<dgHVU'YGD}F<DTHVU ]<K>F>N TLK>dgHVUŠYNŠYdeEÚVbgDuR#N'EGF<DLhdgDLF
HVF<YF<D`n
C D]'Ydg]<K>F>de[GNGK>dgHVU ]ŠYGD]kE)R#F/R#hfiLK>F<D]ŠF<DTHVU ]J
K>F>NGdeK<]a4R#hEGbedeK>N YGD]pDLKkYdeF<DTLK>dgHVU ]a4EGF<S]<DLUK<DLU*KpTHVhhfD!Rw`R#UK'NGUTHVhEcHVF>K<DLhfDLUKŠPR#N ]<]>dgDLUTDLU*K>F<S
R#NGK<HVNGFrYGDObqRxwzF/R#dgDuw`R#bgDLNGFoYNŠE)R#F/R#hfiLK>F<D`nCDO[GdqR#dg]oDLKrb½¾STR#F>KoKÞ¼*EcDuYGD–TD]rYdg]>K>F>de[GNGK>dgHVU¬]>NGdewQDLUKvSLPR
√ F<D]<EDTLK>dewQDLhfDLU*KnGCDATHVU TDLEGK}YGDOEGNGdg]/]/R#U TDOYGD–F<DTHVU ]>K>F>N TLK>dgHVU
bgDLhfDLUKNGU DObgHVdcDLU
DLKvDLU
1/N
1/ N
D]>KxK<HVNVIJHVNGF<]"wVR#bqR#[GbgDkDLKxEH`]<]/[email protected]]xhfyLhfD]-EGF<HVEGF>dgSLK<S]\jN DY R#U ]-bgDfTRV]-YGDbqRŠF<DTHVU ]>K>F>N TLK>dgHVUXR#N
EGF<DLhdgDLFoHVF<YF<DuDtsGDLEGK<Suj*N D}]/RxwVR#bgDLNGFrD]>KEGbeN ]7ŒR#de[GbgD`n|U'Dt—˜DLKa`EHVNGFoNGU DŒŽHVU TLK>dgHVU'Y˜½¾DtsEH`]<deK>dgHVUkTHVFJ
ï *ÃQfV&‘Ã`>fôÛ;&ôÛ£0Ü$õ« & *ÃQf<ÛÙ)Û;
Ägcc
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
F<D]>EcHVU Y R#UKR#N\]<deK<D4]>N Y\YGD4b½ š [ ]/DLF>wVR#K<HVdeF<DvÉ7dgDLF>F<DvMONGPQDLFabqRuEGNGdg]<]/R#U TDrYGD4F<DTHVU ]<K>F>N TLK>dgHVUhfH¼QDLUGU D ø
YGDxb½§R#hEGbedeK>N YGD\YNmYdeEÚVbgDBR#N%]<DTHVU YˆHVF<YF<D"wVR#NGK
a»fTHVhE)R#F<DLF–RwQDT"bqRw`R#bgDLNGFuj*N D
(Kα )2nd ≈ 0.14
U HVN ]RwQHVU ]oHV[GK<DLUN DuR#NEGF<DLhdgDLF4HVF<YF<Dx”
nQC‘RxEGF<STLdg]>dgHVUfYGDvF<DTHVU ]>K>F>N TLK>dgHVU'R#UGP`NGbqR#deF<D
(Kα )1st ≈ 0.33
D]>KBR—˜DTLK<SDkYGDbqRˆhfyLhfDfEcDLF>K<DfYGDfEGNGdg]/]/R#U TD`n‘CDfEGF>d“sÅ»pE)R¼QDLF\EHVNGFBHV[GK<DLUGdeF"NGU DkD]<K>dehkR#K>dgHVUÆYN
YdeEÚVbgDpj*NGdU Dp]<HVdeKE)RV]ˆ÷tEHVbebeN SDQ÷ŠE)R#Fb½¾HVF<YF<D¬]<NGESLF>dgDLNGFfD]>KYGHVU T¬NGU DpYGSLP`F/RVY R#K>dgHVU‹YNÆEcHVNGwQHVdeF
YGDfF<DTHVU ]>K>F>N TLK>dgHVU‚
n ”±K>deK>F<DkY˜½¾DtsGDLhEGbgD`aEHVNGFBF<DTHVU ]>K>F>NGdeF<DkNGUÆYdeEÚVbgDŠ»¬b½¾HVF<YF<D'YGDLNsÅRwQDT'bgDk]>deK<D
]>N Y!YGDBb½ š [ ]/DLF>wVR#K<HVdeF<[email protected]É7dgDLF>F<DMONGPQDLFadebŒR#NGKADLUwzdeF<HVU
ŒŽHVdg]OEGbeN ]xY˜½¾SLwQSLU DLhfDLU*K<]xEcHVNGFxRwQHVdeFANGU D
5.6
YGSLK<DTLK>dgHVUpR¼R#UKbqR-hfyLhfDO]>deP`UGd“­)TR#U TDAj*N DuTDLbebgDAR#N'EGF<DLhdgDLFHVF<YF<D`gn D†bD]>K4deU*K<SLF<D]<]/R#U*KvYGDOU HVK<DLFvjN D
Y R#U ]vbgDATRV]oE)R#F>K>dgTLNGbedgDLFOY˜½ NGU DuŒŽHVU TLK>dgHVUˆY˜½¾DtsEH`]>deK>dgHVU¬THVF>F<D]>EcHVU Y R#UK–R#Np]<deK<D–]<N YŠYGD–b½ š [ ]<DLF>w`R#K<HVdeF<D
É7dgDLF>F<DfMONGPQDLFabqRpEGNGdg]/]/R#U TDfYGDF<DTHVU ]>K>F>N TLK>dgHVUÆYN‰j*N)RVYF>deEÚVbgD'D]>KxbgSLPQiLF<[email protected]]>NGEcSLF>dgDLNGF<D'»pTDLbebgD
YNˆYdeEÚVbgD`n
‡rD\F<S]>NGbeK/R#KxTHVU­ F>hfDbgD\P`F/R#U YªRw`R#UK/R#PQDfj*N D\EH`]<]/iYGD"NGUªYGSLK<DTLK<DLNGF"HV[ ]<DLF>w`R#UKxbqR'K<HVK/R#bedeK<SYN
TLdgDLb˜EHVNGFb½¾SLK>N YGD"YGD]}R#UGdg]<HVK>F<HVEGdgD]O»\P`F/R#U YGDxST/l DLbebgDk”Y R#U ]}TDxTRV]bgD]vhfHzYGD]}SLK/R#UKudeU YGSLEDLU Y R#U*K<]a
bqRfEGNGdg]/]/R#U TDBYGD\F<DTHVU ]>K>F>N TLK>dgHVUªD]>K–dgYGDLUK>dgj*N Dj*N DLb;j*N DB]/HVdeKOb½¾HVF<YF<DR#N j*N DLbbqRkF<DTHVU ]>K>F>N TLK>dgHVU2D]>K
Dt—cDTLK>N SD`mn –HVN ]–wQDLF>F<HVU ]AE)R#FObqRŠ]>NGdeK<DBb½ deU• N DLU TDjN DBEcDLNGKxRwQHVdeFAbqRŒ’F/RVTLK>dgHVUªYN¤TLdgDLb£THVNGwQDLF>K<D]>NGF
bqR\EGNGdg]<]/R#U TD-YGDAF<DTHVU ]<K>F>N TLK>dgHVUn
C D]Ydg]>K>F>de[GNGK>dgHVU ]uF<DTHVU ]>K>F>NGdeK<D]}YGD]vE)R#F/R#hfiLK>F<D]OYNpj*N)RVYF>d“
EÚVbgD\]>NGdewQDLU*K–NGUªTHVhEcHVF>K<DLhfDLUK-PR#N ]/]>dgDLU%dgYGDLU*K>dgjN Df»'TDLbeNGd7YNªYdeEÚVbgDp”bqRfhfH¼QDLUGU DfTHVUwQDLF>PQDDLU
√ n
wQDLF<]bqR\wzF/R#dgDxw`R#bgDLNGFuYN¬E)R#F/R#hfiLK>F<D`aRwQDT-NGUˆSTR#F>K}KÞ¼*EcD-YdehdeU*N)R#UKODLU
1/N
1/ N
C‘R­ P`NGF<DH÷zn ÑdebebeN ]<K>F<D-b½¾SLwQHVbeNGK>dgHVU¤YGDxbqRfYdg]>EcDLF<]>dgHVUmYGD
DLK
œ’HVÈ YGS]>deP`U D"b½ NGU D

(δβ/β)
(δλ/λ)
λ
YGD]uK>F<HVdg]Ow`R#bgDLNGF<]uEGF<HVEGF<D]ΝORwQDT DLK F<D]>EcDTLK>dewQDLhfDLUKncC‘R'Ydg]>EcDLF<]>dgHVU%YGD\TD]OYdg]<K>F>de[GNGK>dgHVU ]OEcDLNGK
β
λ
YGHVU T-yLK>F<DAE)R#F/R#hfSLK>F>dg]<SDBTHVhhfD-EGF<[email protected]”
ï *ÃQfV&‘ÃQò>fSÛ£&s’&‘Û;#0Ü$õ
[σ(δx/x)]−1 = Kx × x ×
√
N,
(x = β
ou λ)
œ ÷zn:ð 
Ø
C DmTHVU TDLEGKpYGDˆEGNGdg]/]/R#U TDmYGDmF<DTHVU ]>K>F>N TLK>dgHVUF<D]<K<DmYGHVU T%wVR#bqR#[GbgDmEHVNGFŠYGSTLF>deF<Dmb½§R#EGK>deK>N YGD¤»2YGSt
K<DLF>hdeU [email protected]]@E)R#F/R#hfiLK>F<D]fYNÆj*N)RVYF>deEcÚVbgD`n7C½§R#hEGbedeK>N YGD
D]>[email protected])R#F/R#hfiLK>F<DpbgDŠEGbeN ]@dehEcHVF>K/R#UKk»
β
YGSLK<DLF>hdeU DLFa debTR#F/RVTLK<SLF>dg]<DBDLUˆDt—cDLKbgD"THVUK>F/RV]>K<DBDLUK>F<DxbgDAhkRsdeh"NGh¢DLK}bgDAhdeUGdeh"NGh¢YN¬• Nscn –HVN ]
SLK>N YdgDLF<HVU ]oYGHVU TuEGbeN ]E)R#F>K>dgTLNGbedgiLF<DLhfDLU*KrbqRxEGNGdg]<]/R#U TDuYGD}F<DTHVU ]>K>F>N TLK>dgHVUŠYGD}b½§R#hEGbedeK>N YGDuEGbeNGK<ÚVKrjN D
TDLbebgDYGD]AwVR#bgDLNGF<]-EGF<HVEGF<D]n˜ÕxR#U ]AbgDTRV]xY˜½ NGU [email protected]Œ’HVU TLK>dgHVU2Y˜½¾DtsEcH`]>deK>dgHVU2THVF>F<D]>EcHVU Y R#UKB»'NGU2]>deK<DYGD
ùJïì<îLßJà#àO9Jß;ëŽéâ˜áìtéáß¯ë‘ä§ßJë‘ìÎâê§ßJàá†ã<áê›ìÎà#ëûé3ûê§ñ2þtä§ß7ñ`ìÎëŽëŽêŠƒä§ßJë
[À‚Wˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛoe&)fˆf)Ü;P’&‘Û;#0܏õ ò#
reconstruction accuracy of the
quadripole eigenvalues and amplitude
2.13
!
100
dispersion of (#x/x) $ N
1/2
Ägcv
1.85
"
(full sky detector)
10
0,01
0,1
x = ! or " (quadripole eigenvalue or amplitude)
,6-0/.1÷zn Ñ É7F<STLdg]>dgHVU¬YGDubqR-F<DTHVU ]>K>F>N TLK>dgHVU¬YGDOb½§R#hEGbedeK>N YGDADLKrYGD]rw`R#bgDLNGF<]4EGF<HVEGF<D]vYNpjN)RVYF>deEcÚVbgD`n
CD]}RI¯N  ]<K<DL2 hfDLUK<]}K>F/RVTS]DLU¬EcHVdeUK>debebgS]uhfHVUK>F<DLU*KujN DAbqR\Ydg]>EcDLF<]>dgHVUˆYGDAb½§R#hEGbedeK>N YGDxF<DTHVU ]<K>F>NGdeK<Dkœ
β
HVN rw`R#F>dgDxYGDAŒRVWHVU¬deUwQDLF<]<DLhfDLU*KOEGF<HVEcHVF>K>dgHVUGU DLbebgDB»@TDLK>K<DxhfyLhfD\R#hEGbedeK>N YGD`n
λ
b½ š [ ]<DLF>wVR#K<HVdeF<DxɏdgDLF>F<DxMuNGPQDLFabqRBEGNGdg]<]/R#U TDxYGDAF<DTHVU ]<K>F>N TLK>dgHVUˆhfH¼QDLUGU D-YGD–b½§R#hEGbedeK>N YGD-Y˜½ NGUˆj*N)R
YF>deEÚVbgDrw`R#NGK
DLK
]>d*b½¾HVURI>HVNGK<DobgDv]>deK<DoU HVF<Y˜nQ‡rHVhhfDY R#U ];bgDrTRV]£YNYdeEcÚVbgD`abqR
Kβ ' 0.23 Kβ ' 0.52
EGNGdg]<]/R#U TDxYGD–F<DTHVU ]>K>F>N TLK>dgHVU
YGD]YGDLNs¬]>deK<DAYGDAb½ š ÉM D]>KvK>F<i]vEGF<HzT<l D-YGD–bqR\w`R#bgDLNGF
Kβ
Kβ ' 0.54
HV[GK<DLUN Dfœ’DLUp]>deh"NGbqR#K>dgHVUrY R#U ]vbgD–TRV]vY˜½ NGU DxDtszEcH`]>deK>dgHVUŠK<HVK/R#bgD–DLKvNGUGd“ŒŽHVF>hfD`nC½ š [ ]<DLF>w`R#K<HVdeF<DxÉ7dgDLF>F<D
MuNGPQDLF\THVhEGbgDLK\EcHVNGF>F/RˆYGHVU TŠYGSLK<DTLK<DLFfRwQDT'bqRˆhfyLhfDŠ]>deP`UGd“­)TR#U TD'NGUÆj*N)RVYF>deEcÚVbgDŠRwQDT'TLdeU j!ŒŽHVdg]
hfHVdeU ]rY˜½¾SLwQSLU DLhfDLU*K<]aGDLKvYGHVU TODLUpDLUwzdeF<HVU¬Yd“sfŒ’HVdg]ohfHVdeU ]vYGDuK<DLhE ]vj*N½§RwQDT–bgD–]>deK<DO]<N Yk]<DLNGbgDLhfDLU*Kn
C‘ROEGNGdg]<]/R#U TDvYGDrF<DTHVU ]>K>F>N TLK>dgHVUkD]>K;NGU DjN)R#U*K>deK<S
jNGd`EcDLF>hfDLK‘YGD7hfD]>NGF<DLF‘bqRvTR#E)RVTLdeK<S4Y˜½ NGU DDtsESLF>dgDLU TD4YGHVUGU SDo»vD]>K>dehfDLFbqRwVR#bgDLNGFY˜½ NGU DrR#UGdg]<HVK>F<HVEGdgD`n
‡rDLK>K<DOEGNGdg]/]/R#U TDAYGDOF<DTHVU ]<K>F>N TLK>dgHVUˆwVR#F>dgDA]>NGdew`R#UKvbgD–K†¼zEcD–Y˜½§R#UGdg]/HVK>F<HVEGdgDfœ’YdeEÚVbgDAHVN¬jN)RVYF>deEcÚVbgDrDLK
bqR'Œ’HVF>[email protected]'ŒŽHVU TLK>dgHVU2Y˜½¾DtszEcH`]>deK>dgHVUnMu­ UªY˜½¾SLK>N YdgDLFBTDLK>K<DYGDLF>UGdgiLF<DfYGSLEcDLU Y R#U TD`aI½§R#dTR#bgTLNGbgSbqR
EGNGdg]<]/R#U TDvhfH¼QDLUGU DOYGDvF<DTHVU ]>K>F>N TLK>dgHVU'YGD}b½§R#hEGbedeK>N YGD}YNkYdeEcÚVbgD`a
a`DLKYNkj*N)RVYF>deEÚVbgD`a
aVEcHVNGF
Kα
Kβ
YGD]uŒŽHVU TLK>dgHVU ]-Y˜½¾DtsEH`]<deK>dgHVU ]AE)R#F>K/R#UK-Y˜½ NGU D\ŒR#de[GbgD\THVNGwQDLF>K>NGF<DxI¯N ]<j*N½§»'NGU [email protected]>K>NGF<DK<HVK/R#bgD`ncC‘R
Œ’HVF>hfD'YGD'TD]BŒŽHVU TLK>dgHVU ]\D]<K"bqRmEGbeN ]B]>dehEGbgD'EcH`]<]>de[GbgDm”b½¾DtsEH`]>deK>dgHVUÆw`R#NGK EcHVNGF\bgD]BYGSTLbedeU)R#dg]<HVU ]
1
R#EGE)R#F>K<DLU)R#UKr»
L
D
K
<
]
e
d
U
V
H
U
`
n

C
¾
½
L
S
Q
w
V
H
e
b
G
N
>
K
g
d
V
H
Š
U
G
Y
v
D
q
b
x
R
G
E
G
N
g
d
<
]
/
]
#
R
U
T
}
D
G
Y
D
<
F
D
THVU ]<K>F>N TLK>dgHVU'YNfYdeEcÚVbgD`a
[−90◦ , δmax ] 0
R#NËEGF<DLhdgDLF'DLKpR#N ]<DTHVU Y HVF<YF<D`arR#deU ]>duj*N DˆTDLbebgDmYNjN)RVYF>deEcÚVbgDmDLUXŒŽHVU TLK>dgHVUYGDmbqR!Œ’F/RVTLK>dgHVUYN
õ
f‰7&‘gf‘ÃQHÛ;fõò'†#GÃ`fyÛ£&!Ã*õÃ&‘
Ägv
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
 &ÛV0Ü$õˆ‘Üi‰ÁQVÀ’&‘
= â[email protected]$â€ë^‘± ³@¢+§œ²O³@«¬šœ¯3šœ˜¡“§·¡†©$™J¢À§e«É³*ª†šœ·O˜$›¡ª†©$n¸
¡¢À@˜y™± © ˜yÈO©³@™Jª†¢¬¨g×@«Àš}›Ø_¯"§·¡ª†¢ÉÈO©$š@ÁV³µØO³@˜O¡"™Jš°© ¦N´Å³Å¸
«Àš°© ªU›H¨ ªO¨ ª†šœ›z§œ²O³@«¬š¶›°½ ‰ O©J¡š¶›H«¬š¶›}¨$ªO¨ ª¢À§·¡†§œ›}™Jšx«À³
ª†šœ·O˜$›¡ª†©$n¡†¢¬O˜B›@˜¡V¢À™Jšœ˜¡¢ÉÈO©$šœ›œÁŚ·¦ ·šœ¨J¡§«É³.™J§·¡†š°ª†¯3¢¬¸
˜$³¿¡¢À@˜j™Jš”«0± ³@¯3¨$«¬¢¬¡©$™ š“È© ¢›œ± ³µ´Oઆš’°¡ª†š3«À§°²Oઆš°¯"š°˜¡
ž ¢É³¿¢É›§œš@½ Í ³+ª†³@¢À›@˜|šœ˜š¶› ¡L«À³6›© ¢¬´Å³@˜O¡†šëÅ·O˜$›¢À™J§œªO˜$›«Àš
°³O›^™± ©$˜uÈ©$³O™Jª†¢¬¨g×@«Àš6³µØ³¿˜¡
Á
λ3 > λ1 = λ2 = −λ3 /2
³µ´@š¶©$˜ š|³¿¯"¨ «À¢¤¡†©$™Jš|§°²³¿«Àš6¹
3λ ½
β = λ −λ
¯3¢À˜ ¢À+λ
¯”³¿«Àš = 4+뚜›¡
Í ± šœ›¡¢À¯”³Å¡†¢¬O˜e™Jš«É³z´Å³@«¬šœ© ª+¨$ªO¨ ªš2+λ
·šœ¨šœ˜$™ ³@˜O¡Zž ¢É³¿¢É›§œšN°³@ªe¯o’°¯"šN›¢B«Àšœ›Z™J¢É› ¡†ª¢Àž λ©J−¡†¢¬O˜$›
+
+
−
−
3
3
™Jš
š·¡
›@˜¡u·š°˜¡†ª§œšœ›“›© ª
š·¡
Á
λ1«Àšu¯3¢Àλ˜ 2¢¬¸
«É³x´µ(λ
³@«¬1šœ)© rec
ª”™Jš (λ2š¶)›rec
¡zOžJ¡šœ˜_© šuš°˜y¨ ª†š°˜$³@˜O¡”
¯©$¯ š°˜¡†ªš λ− š·¡
½ Í ± ³¿¯"¨ «À¢¬¡©$™Jšš¶› ¡™ @˜$
™ ³@˜$›.·š"°³@›6(뛐Ø_1›)¡rec
§œ¯”³Å¡(λ
¢ÉÈ© 2 š°)rec
¯"šœ˜O¡|›@©$›¸0š¶› ¡†¢¬¯"§œš@½g¾>O© ª
ª†§œ›@©$™Jª†šV·š°¡¡š<¢¬˜™J§·¡†š°ª†¯3¢À˜$³¿¡¢À@˜Á·¢À«¿£³@©J¡³¿Ÿ @©J¡†š°ª«0± Ç_ظ
¨g¿¡Ç$Μ›š}æ¹XŸ ©$›¡¢¬¥$š°ªe; é
š·¡6ªš¶·O˜$› ¡†ª©$¢¬ª†šX© ˜ ¢¬¸
È© š°¯"šœ˜O¡
¨g@©$ª3š¶› ¡†¢¬λ¯"1šœ=
ª3«0λ± ³¿2¯"¨ «À¢¤¡†©$™Jšu™J©!È©$³O™Jª†¢¤¸
¨g×@«Àš@½%Ó<šž λ¢É3³¿¢É›6™J¢¬¯"¢À˜_© š«À@ªU›È© šB«¬š¶›´Å³@«¬šœ© ª†›¨ ª†@¨ ª†šœ›
š°¡
›O˜¡)›©JÖz›³@¯3šœ˜¡^™J¢¬Æ§°ª†š°˜¡š¶›¨g@© ªÈ© š6«¬šœ© ª
λ
™J¢É1› ¡†ª¢Àž λ©J2¡†¢¬O˜$›ª†šœ°@˜$›¡ª†© ¢¬¡šœ›)˜ š|›š|ªš¶·O© ´_ªšœ˜O¡^¨ «À©$›œ½
TLdgDLb;THVNGwQDLF>K<D\D]>K–EGF<S]<DLUK<[email protected]]>NGFObqR­ P`NGF<Dpœ ÷zn ~ tncM}­ U¤YGD-EGF<STLdg]/DLF–bgD]–THVUwQDLU*K>dgHVU ]xNGK>debedg]<SD]aU HVN ]

R
YGSt­ UGdg]<]<HVU ][email protected]ŒF/RVTLK>dgHVUmYNˆTLdgDLbTHVNGwQDLF>K<D"THVhhfDf”
n
1
A=
4π
dΩE(u)
‡ [email protected]%EcDLNGKA]½ ¼mR#K>K<DLU YF<[email protected]>NGdeK>dewQDLhfDLU*Ka˜EGbeN ]–bgDBYGSLK<DTLK<DLNGFxHV[ ]<DLF>wQD\NGU DBP`F/R#U YGD"Œ’F/RVTLK>dgHVU
r
YNkTLdgDLb)DLKEGbeN ]bqRAEGNGdg]<]/R#U TD}YGD}F<DTHVU ]>K>F>N TLK>dgHVUŠD]<K7P`F/R#U YGD`nÉ;HVNGFoTHVhEGF<DLU YF<D–TDF<S]>NGbeK/R#KO÷RwQDT}bgD]
]>NGFxbqRpYdg]>K>F>de[GNGK>dgHVUÅYGD]
hkR#deU ]÷Va˜THVU ]>dgYGSLF<HVU ]"[email protected]<HV[GbgiLhfDfY˜½§RI¯N ]<K<DLhfDLUKBY˜½ NGU DŒ’HVU TLK>dgHVU2DLU
cos δ
YGSTLbedeU)R#dg]<HVU ]}hfD]<NGF<SD]}E)R#FNGU D-DtsESLF>dgDLU TD"YGHVUGU SD`%n DÞb˜E)R#F/R#deKuSLw*dgYGDLU*KOj*N DAEGbeN ][email protected]>K>dgHVUmYNˆTLdgDLb
HV[ ]<DLF>wQSD]<DLF/R–P`F/R#U YGD`a`EGbeN ]b½§RI¯N ]>K<DLhfDLU*K4]<DLF/RATHVU*K>F/R#deUKaTDj*NGdGTHVF>F<D]>EcHVU YfY R#U ]7bgDTRVYF<DYGDvU HVK>F<D
hfSLK>l H*YGDf»'NGU [email protected] ]xP`F/R#U [email protected]]/]/R#U [email protected]<DTHVU ]>K>F>N TLK>dgHVUn¡*d7b½¾HVU2]<HVNGl)R#deK<DDLU¤EGbeN ]"RIJN ]>K<DLFANGU D
Œ’HVU TLK>dgHVUpDLU
EcHVNGFrD]>K>dehfDLFrNGUpjN)RVYF>deEcÚVbgD`aTDLbqRBRI>HVNGK<DOYGD]vYGDLP`F<S]vYGDObede[DLF>K<S–]>NGEGbgSLhfDLUK/R#deF<D]
cos2 δ
DLKb½§RI¯N ]<K<DLhfDLUKuD]>K}DLU THVF<DAhfHVdeU ]uTHVUK>F/R#deUKAj*N DOEGF<STSYGDLhhfDLU*Ka»\hfHVdeU ]uY˜½¾HV[ ]<DLF>wQDLF}K<HVNGKbgDxTLdgDLbn
‡rDLKoDtsGDLhEGbgDudebebeN ]<K>F/R#K>d“ŒEDLF>hfDLKvYGDuTHVhEGF<DLU YF<DAj*N)R#bedeK/R#K>dewQDLhfDLUKvbgD]4F<S]<NGbeK/R#K<]oYGDubqRA­ P`NGF<[email protected]œ ÷zn ~ }”

bqRBEGNGdg]<]/R#U TDxYGDAF<DTHVU ]>K>F>N TLK>dgHVUmYN¬YdeEcÚVbgD-R#Nˆ]<DTHVU Y¬HVF<YF<DxD]>KYGDAb½¾HVF<YF<DxYGDxTDLbebgDAYNˆjN)RVYF>deEcÚVbgD`a
[email protected]#NGP`hfDLU*K<DmRwQDTŠbqR%EcHVF>K>dgHVUZYGDŠTLdgDLbTHVNGwQDLF>K<DpEHVNGF\F<D¯I>HVdeU YF<DpbqRˆw`R#bgDLNGF
YNZTRV]YGD
Kα ' 0.58
THVNGwQDLF>K>NGF<D-K<HVK/R#bgD`n
š UËEcDLNGKkSLPR#bgDLhfDLUKŠF<DLhkR#F<j*N DLF'j*N D¬bgD]kTHVF>F<DTLK>dgHVU ]ŠYN]<DTHVU Y HVF<YF<DmYGD¬b½¾STLF>deK>NGF<D!YGD]fE)R
F/R#hfiLK>F<D]'YN‹YdeEÚVbgDmF<DTHVU ]>K>F>NGdeK'Œ’HVU*KkdeUK<DLF>wQDLUGdeFpbgD]khfyLhfD]']<HVhhfD]ŠYdg]<TLF<iLK<D]
j*N DˆEcHVNGFkbgD
S̃ij
jN)RVYF>deEcÚVbgD`a˜DLKxdeUK>F<HzYNGdg]<DLU*KBR#deU ]>dYGD]–• N TLK>N)R#K>dgHVU ]"]<dehdebqR#deF<D]-Y R#U ]xbqRŠF<DTHVU ]>K>F>N TLK>dgHVU‰YGD
DLK n
α
β
CHVF<]<j*N DŠbqR%ŒF/RVTLK>dgHVUËYGD¬TLdgDLbuR#NGP`hfDLU*K<D`a7bgD]fTHVF>F<DTLK>dgHVU ]fYNË]<DTHVU YXHVF<YF<DpK<DLU YGDLU*KfwQDLF<] a£DLKbgD]
0
EGNGdg]<]/R#U TD]}YGD-F<DTHVU ]>K>F>N TLK>dgHVU ]}F<D¯I>HVdeP`U DLUK–TDLbebgD]uYNˆTRV]uYGDxTHVNGwQDLF>K>NGF<D"K<HVK/R#bgD`n
CHVF<]<j*N D%bqRÅTHVNGwQDLF>K>NGF<DªYN«TLdgDLbxD]>KŠE)R#F>K>dgDLbebgD`aobqRÅTR#F/RVTLK<SLF>dg]/R#K>dgHVUÌYGD]¬R#UGdg]<HVK>F<HVEGdgD]ˆ»‰P`F/R#U YGD
[À‚Wˆ‡ŠÁgf‘Á* õ’f &ÃQGsÛoe&)fˆf)Ü;P’&‘Û;#0܏õ ò#
ÄgvÄ
0,6
0,5
Reconstruction power
Dipole only
0,4
quadripole
amplitude
0,3
0,2
0,1
Dipole amplitude
(with quadrupole)
0
20
30
40
50
60
70
80
90
100
Observed sky fraction in %
,6-0/.1÷zn ~ |7wQHVbeNGK>dgHVU¬YGDubqR-EGNGdg]<]/R#U TD–YGD}F<DTHVU ]>K>F>N TLK>dgHVU¬YGDOb½§R#hEGbedeK>N YGDAYNŠYdeEcÚVbgDuDLKvYN'j*N)RVYF>d“
EÚVbgDODLUf ŒŽHVU 2 TLK>dgHVUŠYGDubqRxŒ’F/RVTLK>dgHVUŠYNŠTLdgDLbTHVNGwQDLF>K<D–E)R#F4bgD–YGSLK<DTLK<DLNGFnzÉHVNGFvbgDuYdeEcÚVbgD`a*bqRxEGNGdg]<]/R#U TD–YGD
bqR-F<DTHVU ]>K>F>N TLK>dgHVU¬D]>KvTR#bgTLNGbgSDxR#N'EGF<DLhdgDLFDLKR#NŠ]/DTHVU Y'HVF<YF<D`nC‘RBTHVNGF>[DADLUŠEHVdeU*K>debebgS]rdeU YdgjN D–bqR
K<DLU Y R#U TDxPQSLU SLF/R#bgD"YGDAb½¾SLwQHVbeNGK>dgHVU!YGDxbqR\EGNGdg]<]/R#U TD-YGD-F<DTHVU ]>K>F>N TLK>dgHVUn
ST<l DLbebgDfRwQDT\U HVK>F<D\hfSLK>l HzYGDBU STD]/]>deK<[email protected]˜½¾Dt—cDTLK>N DLFxNGU¤THVhEGF<HVhdg]f”bqRkF<DTHVU ]<K>F>N TLK>dgHVU2Y˜½ NGU!hfHzYGD
Y˜½¾HVF<YF<D'EGbeN ]@SLbgDLwQSpEcDLF>hfDLK\Y˜½¾HV[GK<DLUGdeFNGU DpD]>K>dehkR#K>dgHVUZU HVUZ[GdqR#dg]<SDpYGD]\hfH*YGD]Y˜½¾HVF<YF<DŠdeUŒŽSLF>dgDLNGFa
hkR#dg][email protected] D\EGNGdg]<]/R#U TD\deUŒ’SLF>dgDLNGF<D`ncC‘R'F<S]<HVbeNGK>dgHVU‰R#U)R#be¼zK>dgj*N D\YNª]>¼]>K<iLhfD\Y˜½¾SjN)R#K>dgHVU ]AbedqR#U*KxbgD]
hfHVhfDLUK<]-R#NsmE)R#F/R#hfiLK>F<D]xYGDBb½§R#UGdg]<HVK>F<[email protected]*dgDLU*K-R#bePQSL[GF>dgjN DLhfDLU*K-bgHVNGF<YGD"EcHVNGF–bgD]OhfHzYGD]–EGbeN ]
SLbgDLwQS]4j*N DvbgD}jN)RVYF>deEcÚVbgD`^n –HVK>F<DhfSLK>l HzYGDU½¾D]>K4YGHVU T}E)RV]4YdeF<DTLK<DLhfDLUKrPQSLU SLF/R#bedg]/R#[GbgDO]<HVN ]bqR–ŒŽHVF>hfD
EGF<S]<DLUK<SDOdgTLd$n DÞb]<DLF/R#deKoEH`]/]>de[GbgDuY˜½¾HV[GK<DLUGdeFrbgDu]>¼]>K<iLhfDOY˜½¾SjN)R#K>dgHVU ]rDLKoYGDOb½ deUwQDLF<]<DLFrUNGhfSLF>dgj*N DLhfDLUK
EHVNGF'F<DTHVU ]>K>F>NGdeF<D!bgD]kE)R#F/R#hfiLK>F<D]¬YGD%b½§R#UGdg]<HVK>F<HVEGdgD¤DLK'SLK<DLU YF<DªR#deU ]>dOU HVK>F<D¤R#U)R#be¼]<D!R#NsHVF<YF<D]
]>NGESLF>dgDLNGF<]@»!Ña;RwQDT'TDLEcDLU Y R#[email protected]'EGF<HV[GbgiLhfDŠYGD'EcDLF>K<D'YGDkEGNGdg]<]/R#U TD'hfDLUK>dgHVUGU SpEGF<STSYGDLhhfDLUK
EHVNGFNGU D-THVNGwQDLF>K>NGF<D"E)R#F>K>dgDLbebgD`n
C‘R'F<DTHVU ]>K>F>N TLK>dgHVUÅR#UGP`NGbqR#deF<DEGF<S]/DLUK<[email protected]\hfyLhfDTHVhEHVF>K<DLhfDLU*K"j*N)R#UK"»'bqRŠYGSLEDLU Y R#U TDkRwQDT
bqR¬THVNGwQDLF>K>NGF<DkYNÆTLdgDLbnMÒK>deK>F<DfY˜½¾DtsDLhEGbgD`a‘bqRp­ P`NGF<b
D ÷zn „%debebeN ]>K>F<Db½¾SLwQHVbeNGK>dgHVUZYGDbqRpEGNGdg]<]ÎR#U TDkYGD

F<DTHVU ]>K>F>N TLK>dgHVU
YGD–bqR\YGSTLbedeU)R#dg]<HVUˆYNŠYdeEcÚVbgDADLUŠŒ’HVU TLK>dgHVUˆYGD–bqR-ŒF/RVTLK>dgHVUˆYGD–TLdgDLbTHVNGwQDLF>K<D`n
Kθ = K δ
C‘RAw`R#bgDLNGFYGD
R#Nf]<DTHVU YkHVF<YF<D}THVUwQDLF>PQDuwQDLF<]4]/R–w`R#bgDLNGFoR#NEGF<DLhdgDLFoHVF<YF<D}bgHVF<]<jN D}bqRATHVNGwQDLF>K>NGF<D
Kθ
YGSLE)RV]<]<DAbgD]
YNmTLdgDLbHV[ ]/DLF>wQSD`n
90%
ÉHVNGFrTHVU TLbeNGF<D}TDLK>K<DOR#U)R#be¼]<DYGDb½¾SLwQHVbeNGK>dgHVUŠYGDbqR–EGNGdg]<]/R#U TDuYGDF<DTHVU ]>K>F>N TLK>dgHVUŠRwQDT}bqROŒ’F/RVTLK>dgHVU
YN2TLdgDLba˜U HVN ]BR#K>K>deF<HVU ]"b½§R#K>K<DLUK>dgHVUÅYN‰bgDTLK<DLNGF"]>[email protected]ŒR#deK-jN [email protected] HVN ]\RwzdgHVU ]xK>F<HVNGwQSfNGU [email protected]]/]/R#U TD
ÄgvMv
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
0,6
-1
Reconstruction power (in rad )
0,5
0,4
Dipole only
0,3
0,2
0,1
Dipole direction
(with quadrupole)
0
20
30
40
50
60
70
80
90
100
Observed sky fraction in %
,.-F/.1÷zn „
|7wQHVbeNGK>dgHVU¬YGD}bqR-EGNGdg]<]/R#U TDuYGDuF<DTHVU ]>K>F>N TLK>dgHVU¬YGD}bqR"YGSTLbedeU)R#dg]<HVU¬YN'YdeEcÚVbgDuDLUfŒŽHVU TLK>dgHVU

2
YGD4bqRuŒ’F/RVTLK>[email protected]>K<DvE)R#F£bgDvYGSLK<DTLK<DLNGFn`C‘R}EGNGdg]/]/R#U TDvYGDobqROF<DTHVU ]>K>F>N TLK>dgHVUfD]<K;TR#bgTLNGbgSD}R#N
EGF<DLhdgDLFrDLKR#Np]<DTHVU Y'HVF<YF<D`nC‘RBTHVNGF>[cDODLUŠEHVdeU*K>debebgS]vdeU YdgjN DObqR"K<DLU Y R#U TDOPQSLU SLF/R#bgDAYGDOb½¾SLwQHVbeNGK>dgHVU
[email protected]]<]/R#U TDxYGD-F<DTHVU ]>K>F>N TLK>dgHVU!R#UGP`NGbqR#deF<D`n
#R N!YGDLNsdgiLhfDHVF<YF<D\EHVNGF–NGU DBŒ’HVU TLK>dgHVUªY˜½¾DtsEH`]>deK>dgHVU¤]>dehdebqR#deF<DR#Nª]>deK<D\]>N Y%YGD\b½ š ÉMBa
jNGdTHVNGw*F<DODLUwzdeF<HVU
YNŠTLdgDLbn*|U'THVhE)R#F/R#UK]<D}F<S]>NGbeK/R#KvRwQDT}bqRx­ P`NGF<D\œ ÷zn ~ tabqRxw`R#bgDLNGFoYGD

33%
Kα
EHVNGFoNGU DuDtsEH`]<deK>dgHVUkEGbqR#K<DAR¼R#UKvbqR-hfyLhfD}Œ’F/RVTLK>dgHVUŠYGDOTLdgDLbTHVNGwQDLF>K<DAD]>KrYGDu]<DLNGbgDLhfDLU*K
n
Kα ' 0.08
M ŒF/RVTLK>dgHVU¤YN!TLdgDLb;THVNGwQDLF>K<[email protected]#bgD`a)bqRŒ’HVU TLK>dgHVU¤Y˜½¾DtszEcH`]>deK>dgHVU¤YN%]>deK<DB]>N YmYGDBb½ š ÉMƒD]>KOYGHVU TBEGbeN ]
Dt¥kTRVTD`nQ|7UfDt—˜DLKaV[GdgDLUfjN½¾SLK/R#U*KhfHVdeU ]4]/DLU ]>de[GbgDY R#U ]7TDLF>K/R#deU D]4YdeF<DTLK>dgHVU ]aDLbebgDHV[ ]<DLF>wQDNGU DvF<SLP`dgHVU
EGbeN ]vSLK<DLU YN D–DLUpYGSTLbedeU)R#dg]<HVUª”*EGF/R#K>dgjN DLhfDLU*K ◦ YGDOEGbeN ]aTDOj*NGdEDLF>hfDLKvYGDuhdgDLNsŠTHVUK>F/R#deU YF<DxbqR
45
hfH*YNGbqR#K>dgHVU%YN¥ Nsp]<NGFNGU DxP`F/R#U YGD-ST<l DLbebgD`n
‡rDF<S]>NGbeK/R#KBhfHVUK>F<DŠjN DfbqRpŒ’F/RVTLK>dgHVU‰YNÅTLdgDLboU½¾D]>K-E)RV]-bgD']<DLNGbE)R#F/R#hfiLK>F<DŠjNGdoTHVU YdeK>dgHVUGU DkbqR
wVR#[email protected]'EGNGdg]<]ÎR#U TDfYGD\F<DTHVU ]>K>F>N TLK>dgHVUabqR'ŒŽHVF>[email protected]@bqR'Œ’HVU TLK>dgHVUÅDtszEcH`]>deK>dgHVUªYGSLK<DLF>hdeU D
b½¾Dt¥kTRVTLdeK<S"YGDAF<DTHVU ]>K>F>N TLK>dgHVUn
Kα ' 0.14
–,³
³
“«µC?:/¸¹:AB)º ? :[email protected]}1
³\5ôó<º ¸¹¶‚?²@x³B5
C RvEGNGdg]/]/R#U TD7YGD7F<DTHVU ]>K>F>N TLK>dgHVU\Y˜½ NGU D4R#UGdg]<HVK>F<HVEGdgDYdeEcHVbqR#deF<D4DLKj*N)RVYF>deEHVbqR#deF<DoR#NGP`hfDLUK<DobgHVF<]<jN D
‘
[email protected]>K>NGF<D-YNˆTLdgDLb‘]½¾SLK<DLU Y˜ahkR#dg]}EHVNGFNGU D-DtsEcSLF>dgDLU TDxYGHVUGU [email protected]>K>NGF<D-hkRsdehkR#bgD"RVTTD]J
]>de[GbgDmD]<Kk­GsGSD`+
n DÞbOD]<KkU SR#UGhfHVdeU ]pEcH`]<]>de[GbgDmY˜½§R#NGP`hfDLU*K<DLF¬bqR2EGNGdg]/]/R#U TD%YGD%F<DTHVU ]>K>F>N TLK>dgHVUžY˜½ NGU D
[À0[2qܑ4i‰>fÛ;%õ•i2Á@h‘VÛ;
Ägv¿
DtszEcSLF>dgDLU TD"R¼R#U*KuNGU DxTHVNGwQDLF>K>NGF<D"E)R#F>K>dgDLbebgD-U HVUˆNGUGd“ŒŽHVF>hfD`a YNˆKÞ¼*EcDxMuNGPQDLFA¡*N Y˜n
C RphfSLK>l HzYGD'¡zM–Éaj*N DU HVN ]"RwQHVU ]"YGSTLF>deK<D'YGD\ŒŽRVWHVU‰YGSLK/R#debebgSD'Y R#U ]-TDfT/l)R#EGdeK>F<D`aEGF<S]<DLU*K<DfbgD
‘
YGStŒŽR#NGK4]<NGdewVR#U*KO”QY R#U ]obgD]4]<HVhhfD]oYdg]<TLF<iLK<D] deUK<DLF>wQDLU)R#U*KvY R#U ]obqRxYGSLK<DLF>hdeU)R#K>dgHVU¬YGD]E)R#F/R#hfiLK>F<D]
S̃
F<DTHVU ]>K>F>NGdeK<]a b½¾DtsEcH`]>deK>dgHVUI>HVN DxbgD-F<ÚVbgD-Y˜½ NGU%EHVdgYG]O]>K/R#K>dg]>K>dgj*N D"R—˜DTLK<SBR#NsˆSLwQSLU DLhfDLUK<]n˜‡rDTLdEcH`]<D
EGF<HV[GbgiLhfDªY R#U ]pbgD]pF<SLP`dgHVU ]ˆYN¨TLdgDLbxHVȞb½¾DtszEcH`]>deK>dgHVUžD]>KŠŒŽR#de[GbgDªTR#FpEcDLNžY˜½¾SLwQSLU DLhfDLUK<]%RwQDTªNGU
Œ’HVF>KAEcHVdgYG]x]>K/R#K>dg]>K>dgj*N DBwQHVU*K"R#bgHVF<]-THVUK>F>de[GN DLF\R#Ns!]<HVhhfD]aR#NGP`hfDLU*K/R#[email protected]#deU ]>d£bgD]–• N TLK>N)R#K>dgHVU ]"DLK
YGSLP`F/RVY R#UKubqR\EGNGdg]/]/R#U TD-YGDxF<DTHVU ]>K>F>N TLK>dgHVUn
M ­ UYGDTHVU*K<HVNGF>U DLF4TDvEGF<HV[GbgiLhfDvdeUGl SLF<DLUKv»ubqROhfSLK>l HzYGDu¡zMOÉ;aLI½§R#dGYGSLwQDLbgHVEGESvNGUhfH¼QDLUf]>dehEGbgD
}
EHVNGFuYdehdeU*N DLFAbgD]• N TLK>N)R#K>dgHVU ]” YdehdeU*N DLF–bqRfF<SLP`dgHVU%Y˜½¾HV[ ]<DLF>w`R#K>dgHVU •CD-EGF>deU TLdeEcDBD]<KuYGHVU TBDtsK>F<yt
hfDLhfDLUKx]>dehEGbgDŠ” debŒŽR#NGKObedehdeK<DLF–bqRfÔHVU DBY˜½¾HV[ ]/DLF>wVR#K>dgHVUªYN¤TLdgDLb;YGD"hkR#UGdgiLF<[email protected]»'TD"j*N DBb½¾DtszEcH`]>deK>dgHVU
YGDLhfDLNGF<DfK<HVNVI>HVNGF<]"]>NGEcSLF>dgDLNGF<D'»pNGU DkTDLF>K/R#deU Dkw`R#bgDLNGFaj*N½ deb7ŒŽR#NGKBYGSLK<DLF>hdeU DLFnC‘R¬ŒF/RVTLK>dgHVUÅYGDkTLdgDLb
THVNGwQDLF>K<DkU DfYGHVdeK\TDLEDLU Y R#U*[email protected])RV]"yLK>F<DfK>F<HVEÅF<SYNGdeK<D`a‘EcHVNGFBTHVU ]/DLF>wQDLF"bgDfhkRsdeh"NGh Y˜½¾SLwQSLU DLhfDLUK<]
YGSLK<DTLK<S]nC‘R¤w`R#bgDLNGFfbedehdeK<D%YGDˆTHVNGEGNGF<DˆYGD¬b½¾DtsEH`]<deK>dgHVU YGHVdeKkYGHVU TmyLK>F<DmHVEGK>dehdg]<SD¬EcHVNGFkT<l)RVj*N D
DtszEcSLF>dgDLU TD`a YGDAhkR#UGdgiLF<DB»@HV[GK<DLUGdeFONGU DxYGSLK<DTLK>dgHVUmbqR\EGbeN ]ODt¥kTRVTDxEH`]/]>de[GbgD`n
C RO]>deP`UGd“­)TR#K>dew*deK<SvYGDobqRhfD]>NGF<DrYGDob½§R#hEGbedeK>N YGDYN\YdeEcÚVbgDoHVN\YN\j*N)RVYF>deEcÚVbgDrD]<K‘EGF<HVEcHVF>K>dgHVUGU DLbebgD
‘
√
»AbqRAEGNGdg]<]/R#U TDOYGDF<DTHVU ]>K>F>N TLK>dgHVUpDLKo»xbqRxF/RVTLdeU DuTR#F>F<SDuYNkU HVh"[GF<D–Y˜½¾SLwQSLU DLhfDLUK<]A”
nσ = K x x N
œ
HVN tn|7U THVU ]>dgYGSLF/R#U*K'j*N DpbgD¬U HVh"[GF<D%Y˜½¾SLwQSLU DLhfDLUK<]ŠYGSLK<DTLK<S]kD]>KEGF<HVEcHVF>K>dgHVUGU DLb–»ªbqR
x = α
β
ŒF/RVTLK>dgHVUXYNXTLdgDLbrTHVNGwQDLF>K<D`a£bqR!YGSLK<DLF>hdeU)R#K>dgHVU YGD'bqR%wVR#bgDLNGFHVEGK>dehkR#bgD¬YGDŠTHVNGEGNGF<D¬YGDŠb½¾DtszEcH`]>deK>dgHVU
p
aF<DLwzdgDLUK–»@K>F<HVNGwQDLF}[email protected]`R#bgDLNGF
jNGdhkRsdehdg]<DAbgDxEGF<HzYNGdeK"”
n
Eopt
Eopt
Kx (Eopt ) ×
A(Eopt )
>K deK>F<DBY˜½¾DtsGDLhEGbgD`a*IL½§R#d;YGSLK<DLF>hdeU S\bgD]Ow`R#bgDLNGF<]–HVEGK>dehkR#bgD]xYGDBTHVNGEGNGF<D dip EcHVNGFObgD\YdeEÚVbgDpœŽR#N
Eopt
EGF<DLhdgDLFODLKuR#Nm]<DTHVU Y%HVF<YF<DvDLK quad EHVNGF}bgD"jN)RVYF>deEcÚVbgD`aGEcHVNGF}bqR\ŒŽHVU TLK>dgHVU%Y˜½¾DtsEH`]>deK>dgHVU%YNm]>deK<D
Eopt
q
]>N Y2YGDb½ š ÉMBn;C½¾SLwQHVbeNGK>dgHVUZYGDbqR¬w`R#F>dqR#[GbgD
A(Eopt ) DLU2ŒŽHVU TLK>dgHVUÅYGDbqRˆwVR#bgDLNGFBYGDfbqR
Kx (Eopt ) ×
Apao
THVNGEGNGF<Dk]>NGFxb½¾DtsEH`]<deK>dgHVUÅD]>KxF<DLEGF<S]<DLU*K<SD']<NGFxbqRŠ­ P`NGF<Z
D ÷zn ðnCDhkRsdeh"NGh YGDkTDLK>K<DŒŽHVU TLK>dgHVU‰]<D

K>F<HVNGwQD\Y R#U ]–bqRkÔHVU D

EcHVNGFObgD\YdeEcÚVbgDBDLKAbgD\jN)RVYF>deEcÚVbgD`aU HVN ]–T/l HVdg]>deF<HVU ]
Eopt ' 0.08 Eopt ' 0.15
YGHVU TOE)R#FvbqR\]>NGdeK<D dip
EHVNGFD]>K>dehfDLFvbgDOPR#deUˆYGDuEGNGdg]/]/R#U TDAHV[GK<DLUNm»Bb½§R#dgYGDAYGD–TDLK>K<D
quad
Eopt ' Eopt
' 0.1
HVEGK>dehdg]/R#K>dgHVUn
”
É HVNGF\U HVK>F<DŠDtsDLhEGbgD`a£DLU‰F<SYNGdg]ÎR#UKYGD ◦ DLUÆYGSTLbedeU)R#dg]<HVUÆbqRmÔHVU D'Y˜½¾HV[ ]<DLF>w`R#K>dgHVUZYNÆ]>deK<D']>N Y
3
YGDBb½ š ÉM\a˜bqR'ŒF/RVTLK>dgHVU2YN2TLdgDLbTHVNGwQDLF>K<DkD]>KxYdehdeUN SDfYGD]<DLNGbgDLhfDLU*K
[email protected]#deU2DLU¤EGNGdg]<]/R#U TD
0.5%
YGDxF<DTHVU ]>K>F>N TLK>dgHVU¤D]>K}E)R#FuTHVU*K>F<D"U HVU%U SLP`bedePQDR#[GbgD'” YGD-b½¾HVF<YF<D"YGD
»
Y˜½§R#NGP`hfDLUK/R#K>dgHVU •cCD
40 50%
K/R#[GbgDR#NfTLd“†YGD]<]<HVN ]£deU Ydgj*N DobgD]£w`R#bgDLNGF<]7YGDrbqROEGNGdg]<]/R#U TDYGDrF<DTHVU ]>K>F>N TLK>dgHVUkYGDrb½§R#hEGbedeK>N YGD}HV[GK<DLU*N D]
EHVNGF}Yd“—˜SLF<DLUK<D]uŒ’HVU TLK>dgHVU ]uY˜½¾DtszEcH`]>deK>dgHVU‰”
Ägv)ú
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
,.-F/.1÷zn ð
|7wQHVbeNGK>dgHVU GY DŠbqR%EGNGdg]<]/R#U TD¬YGDŠF<DTHVU ]>K>F>N TLK>dgHVUô÷LDt—cDTLK>dewQDQ÷Va7YGSt­ UGdgDpE)R#Fb½¾DtsEGF<D]<]>dgHVU
A(Eopt ) DLUŌ’HVU TLK>dgHVUXYGD'bqR%wVR#[email protected]ŠbqRmTHVNGEGNGF<Dp]>NGF\b½¾DtszEcH`]>deK>dgHVUZEcHVNGFBbqRmF<DTHVU ]J
Kx (Eopt ) ×
K>F>N TLK>dgHVU%YN!YdeEÚVAbgpao
D\R#NmEGF<DLhdgDLFODLK–]<DTHVU YmHVF<YF<DBDLKuEcHVNGF}bgD"j*N)RVYF>deEcÚVbgD`n)C‘RŒ’HVU TLK>dgHVU!Y˜½¾DtszEcH`]>deK>dgHVU
THVF>F<D]>EcHVU YmR#Nˆ]>deK<Dx]>N YˆYGDAb½ š [ ]/DLF>wVR#K<HVdeF<DBÉ7dgDLF>F<D-MONGPQDLFn

2 q
M NGPQDLF"”)¡*deK<D\¡*N Y
u
MONGPQDLF-”¡*deK<D"¡*N Y˜a š EGK>dehdg]/S
MuNGPQDLF"”)¡*deK<D]–¡*N YˆDL€
K –HVF<Y
Kα 1er
_n ~`~
_n „

_:n ð`¦
HVF<YF<D
Kα 2me
_n „

_n§Ñ

_:n ð`¦
HVF<YF<D
Kβ
_n§Ñ ~
_n ~ Ñ
_:n ð`Ñ
C R'EGF<STLdg]>dgHVUªYGD\bqR'F<DTHVU ]>K>F>N TLK>dgHVU‰R#UGP`NGbqR#deF<DD]>KxSLPR#bgDLhfDLUK"R#hfSLbedgHVF<SD¬”E)R#FxDtsDLhEGbgD`abqRŠYdg]J
‘
EDLF<]<dgHVUŠYGD–bqRBYGSTLbedeU)R#dg]>dgHVUˆYN¬YdeEÚVbgDOF<DTHVU ]>K>F>NGdeK<DAE)RV]<]<D–YGD ◦ »
EHVNGFvNGU DOF<DTHVU ]>K>F>N TLK>dgHVU%R#N
8
2.1◦
EGF<DLhdgDLFuHVF<YF<D"RwQDT
] •
5 SLwQSLU DLhfDLU*K<G
1.6 10
[À‚c•mo*põÞg #%Ü£
–,•”
³
Ägv
p5gB–?5£· óOõ:\B[@ )? ·[email protected]µ\:/5

M N%THVNGF<]OYGD]O]<DTLK>dgHVU ]}EGF<STSYGDLU*K<D]abgDxEGF>deU TLdeED\YGDxb½¾SLK>N YGD\YGD]OR#UGdg]<HVK>F<HVEGdgD][email protected]>l HzYGD
u
¡zMOÉËR\SLK<SxYGSTLF>deKYGDAhkR#UGdgiLF<DxYGSLK/R#debebgSD`aGDLKbgD]vEGF>deU TLdeE)R#bgD]}TR#F/RVTLK<SLF>dg]>K>dgj*N D]}THVU TDLF>U)R#UKub½¾Dt¥kTRVTLdeK<S
YGD'b½¾D]<K>dehkR#K>dgHVU YGD]@E)R#F/R#hfiLK>F<D]kYNËYdeEcÚVbgDpDLKfYNZj*N)RVYF>deEcÚVbgD¬HVUKkSLK<SpDtsEH`]<SD]n;|U R#EGEGbedgjN)R#U*K
bqRŠhfSLK>l H*YGD'¡zM–ÉÌ»ŠNGU2bgHVK-YGDYGHVUGU SD]"F<SDLbebgD]aHVU2HV[GK>dgDLU*K"NGU [email protected]`R#bgDLNGF"[email protected]½§R#hEGbedeK>N YGDkYN‰YdeEcÚVbgD
DLK\YNXj*N)RVYF>deEÚVbgDŠYd“—cSLF<DLU*K<D¬YGDŠÔSLF<HGa;[email protected]'hfyLhfDŠ]<dobqR!Ydg]>K>F>de[GNGK>dgHVUËR#UGP`NGbqR#deF<D¬YGD]@YGHVUGU SD]D]>K
dg]<HVK>F<HVED`nC‘Rˆj*N D]>K>dgHVUÆD]>KBYGHVU T'YGDk]/RwQHVdeF\]>d4bqR¬w`R#bgDLNGFBYGD]@R#hEGbedeK>N YGD]BK>F<HVNGwQSD]@D]>KBTHVhE)R#K>de[GbgD
RwQDT\b½ l¼zEHVK>l i]/D\j*N DBbgD-• Ns¤D]>KOdg]/HVK>F<HVED`>n D†b£]½§R#P`deKxYGHVU T\Y˜½ NGU!EGF<HV[GbgiLhfD]<K/R#K>dg]>K>dgjN D\Yd“—˜SLF<DLUK-YGD
TDLbeNGdoj*N DU HVN ]\RwQHVU ]BDts R#hdeU S'EGF<STSYGDLhhfDLUK¬”Y R#U ]\TDLK>K<Dk]<DTLK>dgHVUÅU HVN ]BT/l DLF<T<l HVU ]f»¬Dt—˜DTLK>N DLF
NGUŠK<D]<K}Y˜½ l¼zEHVK>l i]<DAEcHVNGFF<SLEcHVU YF<D-»"[email protected]*N D]>K>dgHVU2”[email protected]]>K>F>de[GNGK>dgHVUmR#UGP`NGbqR#deF<D-D]<KJ†DLbebgDxTHVhE)R#K>de[GbgD
RwQDTONGUk• Nskdg]/HVK>F<HVE
D ϖMObgHVF<]oj*N½§R#NGE)R#F/Rw`R#UKU HVN ]vD]>K>dehdgHVU ]rbgD]oE)R#F/R#hfiLK>F<D]]½§RIJN ]>K/R#UKR#N'hdgDLNs
]>[email protected]]>K>F>de[GNGK>dgHVU%YGD]uYdeF<DTLK>dgHVU ]OY˜½§R#F>F>dewQSD]nenen
É HVNGF¬K<D]<K<DLF¬b½ l¼zEHVK>l i]<DªYGD!b½ dg]<HVK>F<HVEGdgD`a;I½§R#dx]>deh"NGbgS¤NGUžP`F/R#U Y«U HVhB[GF<DªYGD¤bgHVK<]¬YGD¤YGHVUGU SD]
THVF>F<D]>EHVU Y R#U*Kv»xNGUf• Nsdg]<HVK>F<HVEcD`aj*N D7I½§R#d F<DTHVU ]>K>F>NGdeKrEHVNGF4DLUŠDtszK>F/R#deF<DubgD]oR#hEGbedeK>N YGD]vYN'YdeEcÚVbgD
DLKBYNZjN)RVYF>deEcÚVbgD`ë
n ”±E)R#F>K>[email protected]ŠTD]BYdg]>K>F>de[GNGK>dgHVU ]@Y˜½§R#hEGbedeK>N YGD]@F<DTHVU ]>K>F>NGdeK<D]adebrD]>K\EH`]<]<de[GbgDkYGD
TR#bgTLNGbgDLF-bqRpEGF<HV[)R#[GdebedeK<S
HVN
Y˜½¾HV[GK<DLUGdeF"NGU DkR#hEGbedeK>N YGD'YGDYdeEÚVbgDfHVNÅYGD
P (αrec ≥ α)
P (βrec ≥ β)
jN)RVYF>deEcÚVbgD4F<D]<EDTLK>dewQDLhfDLU*Ka#j*NGd*]/HVdeK]>NGEcSLF>dgDLNGF<DoHVN\SLPR#bgDr»}NGU D4w`R#bgDLNGFYGHVUGU SD`nVC‘RuTHVUGU)R#dg]<]/R#U TDvYGD
TD]˜ŒŽHVU TLK>dgHVU ]EcDLF>hfDLKYGDoTR#bgTLNGbgDLFbgD4UGdewQDR#N\YGDoTHVU­R#U TDrRwQDT4bgDj*N DLbw`R#bedgYGDLF£HVNBDtsGTLbeNGF<Db½ l¼zEcHVK>l i]<D
YGDAb½ dg]<HVK>F<HVEGdgD`n
z¡ HVdeK
bqRuEGF<HV[)R#[GdebedeK<SvYGDF<DTHVU ]>K>F>NGdeF<Drb½§R#hEGbedeK>N YGD
Žœ R#NBEGF<[email protected]#N\YGDLNsdgiLhfD
p(αrec ) dαrec
αrec
HVF<YF<D}»
EGF<i]a)RwQDT"NGUˆ• Nspdg]/HVK>F<HVED-DLU!DLUK>F<SD`nC‘[email protected]<HV[)R#[GdebedeK<S
Y˜½¾HV[GK<DLUGdeFuNGU D
dαrec
P (αrec ≥ α)
wVR#bgDLNGF}]>NGEcSLF>dgDLNGF<D-HVNmSLPR#bgDB» D]>K}YGHVU T"YGHVUGU SDxE)R#F}[email protected]]<HVhhfDf”
α
P (αrec ≥ α) =
Z
∞
p(αrec )dαrec
œ ÷zn:ð ÷`
α
T HVhhfDOdebebeN ]<K>F<SO]>NGFrbqR-­ P`NGF<[email protected]œ ÷zn tn ¡*db½¾HVU¬T<l HVdg]>deKvNGUŠUGdewQDR#NpYGDATHVU­R#U TDAYGD
azb½ l*¼*EcHVK>l i]<DAYGD
3σ
Ø
b½ dg]<HVK>F<HVEGdgD-YGD]YdeF<DTLK>dgHVU ]uY˜½§R#F>F>dewQSD"YGD]F/R¼QHVU ]OTH`]>hdgjN D]}EHVNGF>F/[email protected]>F<DxSLK/R#[GbedgD-]<d˜b½¾HVU¬hfD]>NGF<DANGU D
R#hEGbedeK>N YGD
K<DLbebgD"jN D
]<HVdeK}deUŒŽSLF>dgDLNGF<D"»
n
αrec
P (αrec ≥ α)
0.27%
ù ½§R#doTR#bgTLNGbgSkUNGhfSLF>dgj*N [email protected]]-EGF<HV[)R#[GdebedeK<S]
œŽR#N2EGF<DLhdgDLF\DLKB]<DTHVU Y‰HVF<YF<D-DLK
P (αrec ≥ α)
EHVNGF\Yd“—cSLF<DLU*K<D]\Œ’HVU TLK>dgHVU ]@Y˜½¾DtsEH`]>deK>dgHVU ]@[email protected]“—cSLF<DLU*K<D]\K/R#debebgD]\YNÆ[email protected]'YGHVUGU SD]n
P (βrec ≥ β)
AodgDLU!jN D"bqRf]>dehBNGbqR#K>dgHVU¤UNGhfSLF>dgj*N D\]/HVdeKubqRf]/DLNGbgD-hkR#UGdgiLF<[email protected]˜½¾D]>K>dehfDLF–EGF<STLdg]<SLhfDLUKAbgD]OEGF<HV[)R#[GdebedeK<S]
ÄgvW
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
,.-F/.1÷zn
Ֆdg]>K>F>de[GNGK>dgHVUZYGDfb½§R#hEGbedeK>N YGDpYNÆYdeEcÚVbgDfF<DTHVU ]<K>F>[email protected] R#U ]BbgDŠTRV]BY˜½ NGU DŠYdg]>K>F>de[GNGK>dgHVU
Ø dg2]<HVK>F<HVEcD`n`C‘RAEGF<HV[)R#[GdebedeK<S
#R UGP`NGbqR#deF<D}Z
Y˜½¾HV[GK<DLUGdeFoNGU Dw`R#bgDLNGF4YGDvb½§R#hEGbedeK>N YGD}F<DTHVU ]>K>F>NGdeK<D
P (αrec ≥ α)
YNˆYdeEÚVbgD"]>NGESLF>dgDLNGF<D"HVNˆSLPR#bgD"» D]<K}EGF<HVEHVF>K>dgHVUGU DLbebgD\»\b½§R#deF<Dxl)RVT<l*NGF<SDB]/HVN ][email protected]>[D`n
α
ca debD]>KAEcH`]<]>de[[email protected]#bgTLNGbgDLFBR#U)R#be¼zK>dgjN DLhfDLU*K
Y R#U ]-bgDTRV]AdgYGSR#bY˜½ NGU D
P (αrec ≥ α)
THVNGwQDLF>K>NGF<D-K<HVK/R#bgD-DLKNGUGd“ŒŽHVF>hfD`a EcHVNGFw`R#bedgYGDLF}bgD]F<S]>NGbeK/R#K<]OYGDAbqR]>deh"NGbqR#K>dgHVUn
C‘RBEGF<HV[)R#[GdebedeK<S-YGDOF<DTHVU ]>K>F>NGdeF<DxNGU¬YdeEcÚVbgD–Y˜½§R#hEGbedeK>N YGD
D]>KvYGHVUGU SDAE)R#Frb½¾Sj*N)R#K>dgHVU‰œ ÷zn ~ _Qta
αrec
jN DAU HVN ]}F/R#EGEDLbgHVU ]udgTLd4”
P (Xrec ≥ X)
p (αrec , N ) =
σα−3
r
α2rec
2 2
α exp − 2
π rec
2σα
q nVC‘ROEGF<HV[)R#[GdebedeK<S
HVÈ@b½¾STR#F>K7K†¼zEDvD]>K£YGHVUGU SvE)R#F
]½¾HV[GK>dgDLU*KYGHVU TvDLU\deU*K<SLP`F/R#UK
σα = N3
P (αrec ≥ α)
œ ÷zn ~ _Qv]>NGFK<HVNGK<D]ubgD]wVR#bgDLNGF<]u]>NGESLF>dgDLNGF<D]–»
”
αrec
P (αrec ≥ α) =
Z
œ ÷zn:ð`¦Q
∞
p(αrec )dαrec
α
= 1 − erf
α
√
2 σα
+
r
α2
2 α
exp − 2
π σα
2σα
œ ÷zn:ð`^Q
œ ÷zn Q_ 
Ø
HVȈ[email protected]ŒŽHVU TLK>dgHVU%DLFJŒ<œòs)D]>K}bqRŒ’HVU TLK>dgHVU!Y˜½¾DLF>F<DLNGF–YGHVU*KObgD]uEGF>deU TLdeE)R#bgD]–EGF<HVEGF>dgSLK<S]O]<HVU*K–YGSTLF>deK<D]–Y R#U ]
[À‚c•mo*põÞg #%Ü£
ÄgvV[
,6-0/.1'÷zn ÷ |7wQHVbeNGK>dgHVU2YGD\bqRfEGF<HV[)R#[GdebedeK<S
YGD\F<DTHVU ]>K>F>NGdeF<[email protected] DR#hEGbedeK>N [email protected]ÚVbgD
]>NGESLF>dgDLNG F<D2 HVNÅSLPR#bgDk»
DLU¤ŒŽHVU TLK>dgHVU‰YGD P (α
EHVrec
NGF-≥
Yd“—˜α)SLF<DLUK<D]"ŒŽHVU TLK>dgHVU ]-Y˜½¾DtszEcH`]>deK>dgHVU ”cbgD]<deK<D]>N Y
YGD%b½ š ÉM œ’DLU‹[GbgDLNžEcHVαNGFpbgD er HVF<YF<D`aDLαU«F<HVNGPQD!EHVNGFpbgD me HVF<YF<Dta}R#EGF<i]¬HVEGK>dehdg]/R#K>dgHVU œ’DLU
EHVdeU*K>debebgS]ΝBDLK\bgD]@YGDLNsZ]>deK<D]@YG1D'b½ š ÉM œ’DLUÅwQDLF>KΝtn7CD]@THVNGF>[c2 D]]<HVUKHV[GK<DLU*N D]EHVNGFYGD]\bgHVK<]YGD
YGHVUGU SD]uTHVUK<DLU)R#U*K
5 SLwQSLU DLhfDLUK<]–T<l)RVTLNGUn
1.6 10
¤‚ ÷J÷†nz‡rDF<S]>NGbeK/R#Kr]½§R#EGEGbedgj*N DuDLUkK>l SHVF>dgDO]>K>F>dgTLK<DLhfDLUKR#N'TRV]oYGD}bqRAF<DTHVU ]<K>F>N TLK>dgHVUŠYNŠYdeEÚVbgD–R#NfEGF<Dt
hdgDLFHVF<YF<DoEHVNGF;Y˜½ NGU D4THVNGwQDLF>K>NGF<DrK<HVK/R#bgDoDLKNGUGd“Œ’HVF>hfD`n`‡rDLEcDLU Y R#UKabqRvŒŽHVF>hfD4EGF<SYdeK<DoE)R#F‘b½¾Sj*N)R#K>dgHVU
TLd“†YGD]<]>N ]BF<D]>K<DfNGU D'K>F<i]"[cHVUGU DpR#EGEGF<HsdehkR#K>dgHVUXEHVNGFBhfH*YGSLbedg]<DLF\bgD]BF<S]>NGbeK/R#K<]@HV[GK<DLUN ]\EHVNGFBNGU D
THVNGwQDLF>K>NGF<DmE)R#F>K>dgDLbebgD%DLKkEcHVNGFfbgD]'R#hEGbedeK>N YGD]pYN]/DTHVU YËHVF<YF<D`+
n (HVN ]kbgD]kEGF<SYdgTLK>dgHVU ]p]>NGdewVR#U*K<D]
]<DLF<HVUKOU SR#UGhfHVdeU ]OHV[GK<DLU*N D]–YdeF<DTLK<DLhfDLU*K-»@b½§R#dgYGD"YGD]–]>deh"NGbqR#K>dgHVU ]a)]/R#U ]OT<l DLF<T<l DLF-»@E)R#F/R#hfSLK>F>dg]/DLF
bgD]}Ydg]>K>F>de[GNGK>dgHVU ]uY˜½§R#hEGbedeK>N YGD]OF<DTHVU ]>K>F>NGdeK<D]n
YGD%F<DTHVU ]>K>F>NGdeF<DmNGU D
C‘R2­ P`NGF<DƜ ÷zn ÷`fEGF<S]<DLUK<D!b½¾SLwQHVbeNGK>dgHVUžYGD%bqR2EGF<HV[)R#[GdebedeK<S
P (αrec ≥ α)

R#hEGbedeK>N YGDBYGD\YdeEÚVbgDB]>NGEcSLF>dgDLNGF<DBHVN¤SLPR#bgD\» DLUmŒŽHVU TLK>dgHVU!YGD EcHVNGFOYd“—˜SLF<DLUK<D]OŒ’HVU TLK>dgHVU ]AY˜½¾Dtsz
α
α
EH`]>deK>dgHVU ]noÉ7beN ]ŠbgD]pTHVNGF>[cD]¬YGDmEGF<HV[)R#[GdebedeK<S
T<l*NGK<DLUKˆF/R#EGdgYGDLhfDLU*KˆwQDLF<]ŠÔSLF<HGaoEGbeN ]
P (αrec ≥ α)
b½¾DtsTLbeN ]>dgHVUªYGDBb½ l¼zEHVK>l i]/D\YGD\b½ dg]<HVK>F<HVEGdgD\EcDLNGKxyLK>F<D"F/R#EGdgYGDLhfDLU*K"SLK/R#[GbedgD`ncÉHVNGFxNGU DR#hEGbedeK>N YGDYGD
YdeEÚ[email protected]<DTHVU ]>K>F>NGdeK<D­GsSD`acbqR'ŒŽHVU TLK>dgHVU‰Y˜½¾DtsEH`]>deK>dgHVU2YGDfb½ š [ ]<DLF>wVR#K<HVdeF<DkÉ7dgDLF>F<DfMONGPQDLF-THVhEGbgDLK"D]>K
TDLbebgDkj*NGdEDLF>hfDLKBYGDfF<D¯I>DLK<DLF"bgDEGbeN ]"F/R#[email protected]½ l¼zEHVK>l i]/DkYGDb½ dg]<HVK>F<HVEGdgD`n š U‰EcDLNGK"SLPR#bgDLhfDLU*K
THVU ]>K/R#K<DLFoj*N D}bqR-F<DTHVU ]>K>F>N TLK>dgHVUŠYNŠYdeEÚVbgD–R#NkEGF<DLhdgDLFvHVF<YF<DOD]<KEGbeN ]oDt¥kTRVTD–j*N D}TDLbebgDOYN']<DTHVU Y
HVF<YF<D`aTHVU ]<Sj*N DLU TD\YGD]}• N TLK>N)R#K>dgHVU ]–EGbeN ]OdehEHVF>K/R#U*K<D]-R—cDTLK/R#U*KObqRkF<DTHVU ]>K>F>N TLK>dgHVUªYN%YGDLNsdgiLhfD
HVF<YF<D`n)C½¾HVEGK>dehdg]/R#K>dgHVUªYGD"[email protected]>l H*YGD¡zM–ɍDLUˆbedehdeK/R#U*[email protected]ŒŽHVU TLK>dgHVU%Y˜½¾DtsEH`]>deK>dgHVU¤R#Ns¬wVR#bgDLNGF<]–]>N
Ägvc
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
,.-F/.16÷zn ¦
|7wQHVbeNGK>dgHVU‹YGD¬bqR!EGF<HV[)R#[GdebedeK<S
YGD¬F<DTHVU ]>K>F>NGdeF<DmNGU DmR#hEGbedeK>N YGD
YGD
(Xrec ≥ X)
YdeEÚ[email protected] N2YG2 [email protected]*N)RVYF>deEÚVbgD]>NGEcSLF>dgDLNGF<DHVNªSLPR#PbgDf
»
DLU!Œ’HVU TLK>dgHVU2YGD
EcHVNGFxYd“—cSLF<DLU*K<D]AŒŽHVU TLXK>dgHVU ]
Y˜½¾DtszEcH`]>deK>dgHVUˆ”VbgD]>deK<D]<N YYGDvb½ š ÉM œ’DLUK>F/R#deK7EGbgDLdeUX£DLK7bgD]YGDLNs]>deK<D]7X
YGDb½ š ÉM œ’DLUK>F/R#deK7EHVdeU*K>debebgStn
CD]}THVNGF>[cD]u]<HVU*K}HV[GK<DLUN D]OEHVNGF}YGD]ubgHVK<]}YGD-YGHVUGU SD]OTHVUK<DLU)R#UK
5 SLwQSLU DLhfDLU*K<]OT/l)RVTLNGUn
1.6 10
EHVNGFrbgDO]<deK<DO]>N YpY˜½¾MONGPQDLFvR#EGEcHVF>K<DASLPR#bgDLhfDLUKNGU DOŒ’F/R#U T<l DxR#hfSLbedgHVF/R#K>dgHVUˆYGD–bqR
ESLF>dgDLNGF<D]»
E = 0.1
]<DLU ]>de[GdebedeK<S"R#N¬K<D]<K}YGDxb½ dg]<HVK>F<HVEGdgD`n
C R'­ P`NGF<Dˆœ ÷zn ¦QOdebebeN ]>K>F<[email protected]ŠTHVhE)R#F/R#dg]<HVUÅYGD\]/DLU ]>de[GdebedeK<SDLUK>F<DbqR'F<DTHVU ]>K>F>N TLK>dgHVUÅYN2YdeEÚVbgDDLK
‘

YN2jN)RVYF>deEcÚ[email protected]<D]>K<DLFxb½ dg]<HVK>F<HVEGdgD`nCHVF<]/jN [email protected]ŠK<HVK/R#bedeK<SfYN2TLdgDLbD]<KAHV[ ]/DLF>wQSD`a˜b½§R#hEGbedeK>N YGDkYN
YdeEÚVbgDmD]>KfbqR¤w`R#F>dqR#[GbgDˆj*NGd}EDLF>hfDLK'Y˜½¾Dt—cDTLK>N DLF'bgD¬K<D]>KfbgD¬EGbeN ]kEGNGdg]<]/R#U*K'YGD¬b½ l*¼*EcHVK>l i]<DmYN˕ Ns
dg]<HVK>F<HVEcD`nÕxR#U ]-bgDfTRV]BY˜½ NGU DfTHVNGwQDLF>K>NGF<DkE)R#F>K>dgDLbebgD'THVhhfDkTDLbebgDfYNÆ]>deK<D]>N Y2YGDfb½ š ÉM\abgD'YdeEÚVbgD
R#NZEGF<DLhdgDLFfHVF<YF<DpF<D]>K<DŠbgD'EGbeN ]Dt¥kTRVTD`a£hkR#dg]@b½§R#hEGbedeK>N YGDmYNZj*N)RVYF>deEcÚVbgDpEDLF>hfDLKNGUZK<D]>[email protected] ]
EGNGdg]<]/R#U*KujN½§RwQDTxbgD-YdeEcÚVbgD-R#Nm]<DTHVU YˆHVF<YF<D`n
C RuYGSLhkR#F<T<l DrPQSLU SLF/R#bgDr»}]>NGdew*F<D4EHVNGF;TR#F/RVTLK<SLF>dg]<DLF;NGU D4SLwQDLUK>N DLbebgDR#UGdg]<HVK>F<HVEGdgDrYGDbqRuYdg]>K>F>de[GNGK>dgHVU
‘
R#UGP`NGbqR#deF<[email protected]]AF/R¼QHVU ]xTH`]>hdgj*N D]AD]>KxYGHVU TBbqR']>NGdew`R#UK<Dp”˜R¼R#U*K-T<l HVdg]>d£NGU D\wVR#F>dqR#[[email protected]]<TLF>dehdeU)R#U*K<D
œ’j*NGdEDLNGK-yLK>F<DfE)R#F"DtsDLhEGbgDfb½§R#hEGbedeK>N YGDkYN‰YdeEcÚVbgD`aHVN2TDLbebgDfYN‰j*N)RVYF>deEcÚVbgDta˜deb7ŒR#NGK-D]>K>dehfDLF
X
bqRˆw`R#[email protected]]<DLNGdeb
K<DLbebgD'j*N D
a7]>dob½¾HVUZ]<Dk­GsD'NGUÆUGdewQDR#NZYGDŠTHVU­R#U TD
X3σ
P (αrec ≥ α3σ ) = 0.27%
THVF>F<D]>EcHVU Y R#UKO»
nGC½¾D]>K>dehkR#K>dgHVUmYGD
YGHVdeK}yLK>F<DAF<SR#bedg]<SD-»BE)R#F>K>deF}YGDx]>deh"NGbqR#K>dgHVU ]uUNGhfSLF>dgj*N D]
3σ
X3σ
YGSTLF>dewVR#U*[email protected]¤EH`]<]<de[GbgDBbqR'ŒŽHVU TLK>dgHVU‰Y˜½¾DtsEH`]<deK>dgHVU2YN2YGSLK<DTLK<DLNGFn‘Mu­ U¤Y˜½¾D]<K>dehfDLF-bqRp]>deP`UGd“­)TR
K>dew*deK<S\YN!K<D]>KO]>NGF–bgD]–YGHVUGU SD]xDtszEcSLF>dehfDLUK/R#bgD]abgD]–bgHVK<]–YGDBYGHVUGU SD]x]>dehBNGbgSD]AYGHVdewQDLUKxTHVUK<DLUGdeFxbgD
[À‚c•mo*põÞg #%Ü£
Ägvv
hfyLhfDBU HVhB[GF<DfY˜½¾SLwQSLU DLhfDLUK<]BjN D\bgD]xYGHVUGU SD]nVDÞb£D]>KA[GdgDLU¤]<NGF–EcH`]<]>de[[email protected]@]>dehBNGbgDLFxbqR']/DLU ]>de[GdebedeK<S
jN½§R#NGF/R-bgDuYGSLK<DTLK<DLNGFrbgHVF<]<jN½ debcR#NGF/R"RVTTLNGhBNGbgSONGU DO]>K/R#K>dg]>K>dgj*N DuYGHVUGU SD–DLU'SLK>N YdqR#U*KrTHVhhfDLU*Krw`R#F>dgD
RwQDT nz™OU DOŒ’HVdg]vj*N D–bqR"w`R#bgDLNGFYGD
D]>KYGSLK<DLF>hdeU SDAEcHVNGFvb½¾DtszEcSLF>dgDLU TDxTHVU ]>dgYGSLF<SD`aGHVU¬F<Dt
X3σ
N
X3σ
THVU ]>K>F>NGdeK
RwQDTObqRxhfSLK>l HzYGDA¡zM–É;a*DLKv]>d
R#bgHVF<]ob½ l¼zEcHVK>l i]<DOYGDOb½ dg]<HVK>F<HVEGdgDOEDLNGKryLK>F<D
Xrec
Xrec ≥ X3σ
F<D¯IJDLK<[email protected]\NGU!UGdewQDR#N¤YGD\THVU­R#U [email protected]
ncÕxR#U ]ATDBTRV]aHVU¤EDLNGKOK<DLUK<DLF-Y˜½¾D]>K>dehfDLFAbqRkhfDLdebebgDLNGF<D
3σ
wVR#bgDLNGFuYGD]}E)R#F/R#hfiLK>F<D]OYGSTLF>dew`R#UKOb½§R#UGdg]<HVK>F<[email protected]]>K>F>de[GNGK>dgHVU!R#UGP`NGbqR#deF<D`n˜¡*dbqR\w`R#bgDLNGFuYGD
Xrec
D]>K-deUŒŽSLF>dgDLNGF<DŠ»
a˜b½ l*¼*EcHVK>l i]<DkYGDkb½ dg]<HVK>F<HVEGdgD'U DEDLNGKBE)RV]ByLK>F<DkF<D¯IJDLK<SD`aDLK"YGHVU T'b½¾D]>K>dehkR#K>dgHVU
X3σ
YGD]E)R#F/R#hfiLK>F<D]uYGDxb½§R#UGdg]/HVK>F<HVEGdgD-EcDLF<Y¬]<HVUˆ]<DLU ]n
ÐßnÔ¹Ù!ÞëÓ~ß
” |7UpK<D]>K/R#U*Krb½ dg]/HVK>F<HVEGdgD-RwQDTAbqR"wVR#bgDLNGFYN¬YdeEÚVbgD–F<DTHVU ]>K>F>NGdeKuR#NpEGF<DLhdgDLFHVF<YF<D`aGHVU
U DAEDLNGKvE)RV]vŒR#deF<DAbqR\Yd“—cSLF<DLU TD-DLU*K>F<DxNGU DxYdg]>K>F>de[GNGK>dgHVU%R#UGP`NGbqR#deF<D-w*F/R#dehfDLU*K}dg]<HVK>F<HVEcDfœNGUGdgjN DLhfDLU*K
bgDohfHVU HVEcÚVbgDtaQDLK7NGU DYdg]>K>F>de[GNGK>dgHVU'THVUK<DLU)R#UKrYGD]7hfHzYGD]Y˜½¾HVF<YF<D]
a`[email protected]ŒŽR#deKYGDvb½¾HVF>K>l HVPQHVU)R#bedeK<S
l>1
YGD]l)R#F>hfHVUGdgj*N D]o]>EGl SLF>dgj*N D]4DLUŠTHVNGwQDLF>K>NGF<DuK<HVK/R#bgD`n¡*d b½¾HVUfŒR#deKb½ l*¼*EcHVK>l i]<Duj*N DbgDOYGSLwQDLbgHVEGEDLhfDLU*K
DLUfl)R#F>hfHVUGdgjN D]o]>EGl SLF>dgj*N D]4YN• Nsf]½§R#F>F<yLK<Du»xb½¾HVF<YF<D aR#bgHVF<]4EHVNGF7K<D]<K<DLFK<HVK/R#bgDLhfDLUKrb½ l¼zEHVK>l i]/D
`
YGDub½ dg]<HVK>F<HVEGdgD`azdebŒŽR#NGKrF<DTHVU ]>K>F>NGdeF<DAU HVUp]<DLNGbgDLhfDLUKbgD–YdeEcÚVbgDuhkR#dg]R#N ]<]>dK<HVN ]rbgD]rhfH*YGD]v]>NGEcSLF>dgDLNGF<]
I¯N ]<j*N½§» EcHVNGFwQHVdeFu]>d˜bgDLNGFuwVR#bgDLNGF}D]>K}]<deP`UGd“­)TR#K>dewQD`n
`
Õ R#U ]bgD4TRV];Y˜½ NGU D4THVNGwQDLF>K>NGF<DrE)R#F>K>dgDLbebgD`abgD]‘hfHzYGD]‘]/HVUK£THVNGEGbgS]aDLK;YGHVU T4]>d*b½¾HVU"K<D]>K<D4b½ dg]<HVK>F<HVEGdgD
x
RwQDTxbgD-YdeEcÚVbgD"R#NˆEGF<DLhdgDLF–HVF<YF<D`a)HVUˆK<D]<K<D-SLPR#bgDLhfDLUK–DLUˆE)R#F>K>[email protected]<S]/DLU TD-YGDxhfHzYGD]u]>NGEcSLF>dgDLNGF<]
jNGd;EDLNGwQDLU*KAEHVbebeN [email protected]ÚVbgD`Ln ArdgDLU¤SLw*dgYGDLhhfDLU*Ka˜bgD]–hfH*YGD]xR¼R#U*KxNGU SLbgDLwQS`acTHVF>F<D]>EHVU Y R#U*K"»
l
YGD]xEDLK>deK<D]"ST<l DLbebgD]BR#UGP`NGbqR#deF<D]a˜U DTHVUK>F>de[GN DLU*KBE)RV]"R#Ns¤hfHzYGD][email protected][)RV] n˜Õ–DTDLK>K<[email protected]#UGdgiLF<D`aDLU
l
THVNGwQDLF>K>NGF<DE)R#F>K>dgDLbebgD`aEHVNGFK<D]<K<DLF]>dQEGbeN ]<dgDLNGF<]hfHzYGD]R#NGK<HVNGFYGD ]/HVUK‘EGF<S]/DLUK<];Y R#U ]bqR}Ydg]<K>F>de[GNGK>dgHVU
l
R#UGP`NGbqR#deF<D`a*deb)ŒŽR#NGKrF<DTHVU ]>K>F>NGdeF<D–bgDOhfH*YGD ]/R#U ]r]<D}EGF<SHzTLNGEDLFvY˜½ NGU¬SLwQDLUK>N DLbc[GdqR#dg]azDLKoDtsEGbgHVF<DLF}R#deU ]>d
l
YGD]PR#hhfD]uYGDAw`R#bgDLNGF<]uYGDAbn
*¡ dcb½¾HVU¬F<DTHVU ]>K>F>NGdeKNGU DAwVR#bgDLNGF}]<deP`UGd“­)TR#K>dewQDxYNˆYdeEÚVbgDxR#N¬EGF<DLhdgDLF}HVF<YF<D`ab½ l¼zEcHVK>l i]<DxYGDAb½ dg]<H#
K>F<[email protected]>F<[email protected] [email protected]#dg]x]ÎR#U ]xyLK>F<D\]\|GFAj*N D\b½§R#UGdg]<HVK>F<HVEGdgDf]<HVdeKxYdeEcHVbqR#deF<D`a˜jN)RVYF>deEcHVbqR#deF<D`aDLK-»
bqRmbedehdeK<D¬HzTLK>NGEcHVbqR#deF<D
nDÞbvD]>KfYGHVU TpEH`]/]>de[GbgDpYGD'F<DTHVU ]<K>F>NGdeF<DˆDLU ]>NGdeK<DpbgDpYdeEÚVbgDˆR#NX]<DTHVU Y
l = 3
HVF<YF<D'R#deU ]>d4j*N DfbgDkjN)RVYF>deEcÚVbgDfEcHVNGF"K<D]>K<DLF\]>d4TDLK>K<DkEGF<SLK<DLU YN DŠR#UGdg]<HVK>F<HVEGdgDkEGF<HwzdgDLUKY˜½ NGUÅYdeEcÚVbgD
HVNˆY˜½ NGU¬hfH*YGD
nGÏ!R#bel DLNGF<DLN ]/DLhfDLUKa bqR\EGNGdg]/]/R#U TD-YGDAF<DTHVU ]>K>F>N TLK>dgHVUmYdehdeU*N)R#UKAYF/R#hkR#K>dgjN Dt
l>2
hfDLUK"»'T<l)RVj*N DfRIJHVNGKxY˜½ NGUªHVF<YF<D]>NGEcSLF>dgDLNGFadeb£U½¾D]>K–E)RV]AEcH`]<]>de[GbgD\YGDBK<D]<K<DLFA]<deh"NGbeK/R#U SLhfDLUKBK<HVN ]
bgD]vhfH*YGD]DLUˆTHVNGwQDLF>K>NGF<DxE)R#F>K>dgDLbebgD`n ”ÌK>deK>F<DxY˜½ debebeN ]>K>F/R#K>dgHVUaGEcHVNGFNGU D–Œ’HVU TLK>dgHVUˆY˜½¾DtsEcH`]>deK>dgHVU¬]>dehEGbgD
THVF>F<D]>EHVU Y R#U*[email protected]#NÅ]>deK<Df]>N Y2YGDkb½ š ÉM œŽRwQDTkHVEGK>dehdg]/R#K>dgHVUZYGDbqRˆhfSLK>l H*YGDp¡zMOÉoxDLK
N = 1.6 105
SLwQSLU DLhfDLUK<]a)[email protected]#bgDLNGFubedehdeK<DBR#NˆEGF<DLhdgDLF–HVF<YF<D-D]>KuYGD
R#bgHVF<]uj*N DAEcHVNGF}bgD-]<DTHVU YˆHVF<YF<D
α3σ ≈ 2%
v
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
HVUmR¬”
n
α3σ ≈ 5%
{;deU)R#bgDLhfDLUKa`bgDvbgDTLK<DLNGF4R#NGF/R–F<DLhkR#F<j*N Sj*N DvK<HVN ]£bgD]£F<S]>NGbeK/R#K<]HV[GK<DLUN ]oY R#U ]7TDLK>K<DvE)R#F>K>dgDTHVU ]/R
TLF<SD"»@b½¾SLK>N YGDBYGD]OR#UGdg]<HVK>F<HVEGdgD]A»@P`F/R#U YGD"ST/l DLbebgD"hfHVU*K>F<DLUK–bqRU STD]<]<deK<S-Y˜½¾HV[ ]<DLF>wQDLFubqRK<HVK/R#bedeK<SBYN
TLdgDLKb •ÉHVNGFdeU ]>dg]<K<DLFv]<NGFrTDOŒR#deKadebŒR#NGKRI>HVNGK<DLFjN D–Y R#U ]vbgDATRV]vY˜½ NGU DxTHVNGwQDLF>K>NGF<DAE)R#F>K>dgDLbebgD`adebD]>K}»
EGF>dgHVF>dcdehEH`]<]<de[GbgDOY˜½¾DtsGTLbeNGF<D–bqR"EcH`]<]>de[GdebedeK<S–j*N DuK<HVN ]vbgD]rhfHzYGD]A÷LTHVU ]<EGdeF<DLUK÷–EcHVNGFvYGHVUGU DLFvNGU DAYdg]J
K>F>de[GNGK>dgHVU¤R#UGP`NGbqR#deF<DBEGbqR#K<DIJN ]>K<D-Y R#U ]ObqRF<SLP`dgHVU¤HV[ ]<DLF>wQSDBDtszEcSLF>dehfDLUK/R#bgDLhfDLU*Knenen‘‡rDLK>K<D"EcH`]<]>de[GdebedeK<S`a
hfyLhfDf]>d4DLbebgDfE)R#F/OR gK-ŒR#FJŒ’DLbeN rD Ë`aD]>KBY˜½§R#NGK/R#UK"hfHVdeU ]BEGF<HV[)R#[GbgDkj*N [email protected]>K>dgHVUÅYN‰TLdgDLboD]>K-P`F/R#U YGD`n
D]>K<D-»\E)R#K>dgDLU*K<DLF–j*N DLbgj*N D]uR#UGU SD]}EcHVNGF}EcHVNGwQHVdeFuDtszEGbgHVdeK<DLFObgD]}YGHVUGU SD]uYGD-b½ š [ ]<DLF>wVR#K<HVdeF<DBÉ7dgDLF>F<D
a
MuNGPQDLF}THVhEGbgDLK[a OHVF<Y¬DLKA¡*N è
Y •
–,•–
É@u¸¢µ\º}·º}:[email protected]}1´º{}579ÃóOõº}1\ºXó4#èB)5±³\5ôò º5#"ó/57:·ö}²
³
ü2—£üÞý
¸
þªÿ 4ÿ KI
Q # ÿ
C R–hfSLK>l HzYGDK>F/RVYdeK>dgHVUGU DLbebgDLhfDLUKvDLhEGbgH¼QSDOY R#U ]7bqRATHVhh"NGU)R#NGK<S–YGD]7F/R¼QHVU ]TH`]>hdgj*N D]YGDLEGNGdg]
‘
YGD]BYGSTSLUGUGdgD]@[email protected]>N YdgDLF\bgD]R#UGdg]<HVK>F<HVEGdgD]»mP`F/R#U YGDpST<l DLbebgDpYGDŠbqRmYdg]>K>F>de[GNGK>dgHVUËR#UGP`NGbqR#deF<DpYGD]
E)R#F>K>dgTLNGbgD]vHV[ ]<DLF>wQSD]vD]>Kob½§R#U)R#be¼]<D–YdeK<DOYGD R¼*bgDLdeP`lpDLU¬RV]<TDLU ]>dgHVUpYF<HVdeK<D`nG‡rDLK>K<DOhfSLK>l HzYGD`a*Y R#U ]v]/R
a
wQDLF<]>dgHVU¤bqR'EGbeN ]-]>dehEGbgD`aEDLF>[email protected]#F/RVTLK<SLF>dg]<DLF"NGUGdgjN DLhfDLU*K-bqRŠYdg]>K>F>de[GNGK>dgHVU2DLU‰RV]<TDLU ]>dgHVUªYF<HVdeK<D
hfD]>NGF<SD¬E)R#FNGU DˆDtsEcSLF>dgDLU TDˆYGHVUK'bqR%Œ’HVU TLK>dgHVU Y˜½¾DtsEcH`]>deK>dgHVUËU D¬YGSLEcDLU YËjN DmYGDpbqRªYGSTLbedeU)R#dg]<HVUn
C½¾D]>K>dehkR#K>dgHVUÆYGDb½§R#UGdg]<HVK>F<HVEGdgDŠD]>K-YGHVUGU SDfE)R#F-NGU DŠR#hEGbedeK>N YGD
DLK-NGU D'YdeF<DTLK>dgHVU
YGSt­ UGdgD]
r1h
ψ1h
E)R#F-”
p
œ ÷zn 
b
Ø
r1h = a2 + b2 et ψ1h = arctan ,
a
RwQDT
a=
N
2 X
cos ϕi
N
i=1
et
b=
N
2 X
sin ϕi ,
N
i=1
œ ÷zn QÑ 
Ø
C‘RBhfSLK>l H*YGD–F<DLwzdgDLUKu»BD]>K>dehfDLFvbgD–EGF<DLhdgDLF}HVF<YF<D–YN¬YGSLwQDLbgHVEGEcDLhfDLUK}YGDA{ HVNGF>dgDLF}YGD–bqR\Ydg]>K>F>de[GNGK>dgHVU
DLU M\a
D]>K¬YGHVU T¤THVhh"NGU SLhfDLUK¤R#EGEDLbebgS2R#hEGbedeK>N YGDªYGD!EGF<DLhdgiLF<D¤l)R#F>hfHVUGdgj*N D`a}DLK
]/R
r1h
ψ1h
a
EGl)RV]<D`nC‘D]oTR#F/RVTLK<SLF>dg]>K>dgj*N D]YGDuTDLK>K<DOhfSLK>l HzYGDuHVUKvSLK<S–DLUkE)R#F>K>dgD–YGSTLF>deK<D]oE)R#FrC˜deU ]>bgDL¼%œ ^ ÷_ðQ4Y R#U ]

¤‚ ÷_ð†n[–HVN ]ONGK>debedg]<DLF<HVU ]AU SR#UGhfHVdeU ]ANGU D\R#EGEGF<HzT<l [email protected]“—˜SLF<DLUK<[email protected]*NGdEcDLF>hfDLK>K>F/RŠYGD-ŒR#deF<DBR#EGE)R#F/ROgK>F<[email protected]
hfSLK>l H*YGDˆ¡zM–ɃTHVhhfDpNGU DpDtszK<DLU ]<dgHVUËDLK\NGU DpPQSLU SLF/R#bedg]ÎR#K>dgHVU YGDŠb½§R#U)R#be¼z]<DpYGD R¼zbgDLdeP`lÆEGF<DLU)R#U*K
a
DLU¬THVhEGK<DANGU D–YdehfDLU ]<dgHVU%R#UGP`NGbqR#deF<Dx]>NGEGEGbgSLhfDLUK/R#deF<D`nM}­ U¬Y˜½§R#bebgSLPQDLFb½¾DtsEH`]/S`a*U HVN ]vF<DLEcHVF>K<HVU ]}DLU
Ì
Ýìtïïß;ä§ß ûê›ëã/ê¾á£æ} ˜ ê§à#ë’áßJê§àð=ˆ†å£ê§ßJéAßJë’á‘ëŽé]ƒáê§ägðï}ãÎê§ë˜ê§äzà$ü ßJë’á‘ñVãÎë˜ï|9¯çý#ãÎàáTœ ˆ
[À‚vö"i~ÜG# f QÃ%õÞef õwÜNÛ; ï ^ܑõ*(h
v
Ä
#R UGU DtsD–¦n§Ñn bqR"YGSLF>dewVR#K>dgHVUpYGD]EGF<HVEGF>dgSLK<S]v]>K/R#K>dg]>K>dgj*N D]rYGD}b½§R#hEGbedeK>N YGDADLKoYGDubqRxEGl)RV]<D`azDLUkNGK>debedg]/R#U*K

THVhhfDEHVNGF7bqR–hfSLK>l HzYGDO¡zM–É2b½§R#U)R#bgHVP`dgD}DLU*K>F<DbgDEGF<HV[GbgiLhfD}YGD}bqR–hkR#F<T<l DuR#bgSR#K<HVdeF<DODLK7b½§R#hEGbedeK>N YGD
F<DTHVU ]>K>F>NGdeK<D`n|7U¬Dt—cDLKazbgDOwQDTLK<DLNGF
YGD–THVhEcH`]/R#UK<D]-œ a 4EcH`]<]<iYGDObgD]rhfyLhfD]oEGF<HVEGF>dgSLK<S]ujN DObgD
r1h
ab
wQDTLK<DLNGF}EH`]<deK>dgHVUmTHVU ]<STLNGK>d“Œ»
YGSLEGbqRVTDLhfDLU*K<]–R#bgSR#K<HVdeF<D]–YGDAbgHVUGP`N DLNGF
T<l)RVTLNGUn
N
2/N
C‘RˆYdg]<K>F>de[GNGK>dgHVUÆYGD]"wVR#bgDLNGF<]BF<DTHVU ]>K>F>NGdeK<D]@YGD
K<DLU Y˜a;»pP`F/R#U Y
œ
2 tawQDLF<]"NGU D
r1h
N N 4/r1h
q
p
PR#N ]<]>dgDLUGU D-YGDAbqR#F>PQDLNGF
DLK
2
σr =
ü2—£üFE
¸
ƒ¤ÿ%K I> <Ìÿ%<JQÿ
2/N
r1h
ÿ
σψ =
2/N r1h
α
M ­ U'YGD}EcHVNGwQHVdeFoTHVhE)R#F<DLFrbqRAEGNGdg]<]/R#U TDOYGDF<DTHVU ]>K>F>N TLK>dgHVU¬YGDb½§R#U)R#be¼]<DuYGD R¼zbgDLdeP`lŠ»"TDLbebgDuYGD
}
a
bqRAhfSLK>l H*YGDA¡zMOÉ;aQdeb)ŒŽR#NGKoSLK/R#[GbedeF4bgD}bedgDLU'DLUK>F<DOb½§R#hEGbedeK>N YGD
DLK4bqRxwzF/R#dgDuR#hEGbedeK>N YGDOYNŠYdeEÚVbgD`n|7U
r1h
Dt—cDLKaQ]<dGb½¾HVUk]>NGEGEcH`]<DjN D}bqRxYdg]>K>F>de[GNGK>dgHVUŠR#UGP`NGbqR#deF<DOYGD]7F/R¼QHVU ]rTH`]>hdgj*N D]4THVhEcHVF>K<DNGUfhfHVU HVEcÚVbgD
DLKkNGUYdeEcÚVbgD!Y˜½§R#hEGbedeK>N YGD abqR‰wVR#bgDLNGFŠYGD
THVF>F<D]>EcHVU)R#UK<D¤]<DLF/R2Yd“—˜SLF<DLUK<D¤TR#F'b½§R#U)R#be¼z]<D!YGD
α
r1h
R¼zbgDLdeP`l-U½¾D]>K]<DLU ]>de[GbgDj*N½§»vNGU D7hfHzYNGbqR#K>dgHVUBYGD7b½§RV]/TDLU ]>dgHVUBYF<HVdeK<D`aDLKD]>KdeU TR#E)R#[GbgD4YGD4YGSLK<DLF>hdeU DLF
a
bqR\w*F/R#dgDxw`R#bgDLNGF}YNmYdeEcÚVbgD`aGUGd]<HVUˆHVF>dgDLU*K/R#K>dgHVUn
ÕxR#U ]Ab½§R#U)R#be¼z]<D\l)R#F>hfHVUGdgj*N D]>K/R#U Y R#F<Y˜acbgD]AjN)R#U*K>deK<S]-YGD\[)RV]<[email protected]#bgTLNGbgSD]"»'E)R#F>K>deF-YGD]AYGHVUGU SD]
]<HVUK DLK œ’TtŒ–Sj*N)R#K>dgHVt
U ÷zn ÑQtn4¡*NGdewVR#U*Kb½§R#EGEGF<H*T/l DˆYGSLwQDLbgHVEGESD]kY R#U ]@bgDŠTRVYF<D¬YGDŠbqR%hfSLK>l HzYGD
Ø
a
b
¡zMOÉ;a U HVN ]uEHVNGwQHVU ]OF<SLdeUK<DLF>EGF<SLK<DLFATD]u]<HVhhfD]–THVhhfD]ObgD]}wQDLF<]>dgHVU ]OYdg]<TLF<iLK<D]OY˜½ deU*K<SLP`F/R#bgD]A]>NGF}bqR
]>EGl iLF<DxjNGda)]>NGdewVR#U*K}bgD]F<DLbqR#K>dgHVU ]\œ ÷zn vDLK"œ ÷zn ÑQta EcDLNGwQDLUKu]½¾STLF>deF<D-THVhhfDf”
Ø

Z
N
2
2 X
cos ϕi =
Φ(u)ε(u) cos ϕdΩ,
a=
N
N
œ ÷zn ~ 
Ø
Z
N
2
2 X
sin ϕi =
Φ(u)ε(u) sin ϕdΩ,
b=
N
N
œ ÷zn *„ 
Ø
i=1
DLK
HVÈ
i=1
ε(u)
D]>Kb½¾DtszEcH`]>deK>dgHVUˆYNmTLdgDLbY R#U ]bqRYdeF<DTLK>dgHVU
N=
Z
u
DLK
N
D]<K}YGHVUGU SxE)R#FBœ’TtŒ7|jn>÷zn QÑ A”

Φ(u)ε(u)dΩ.
œ ÷zn Qð 
Ø
√
C ½§R#hEGbedeK>N YGDuYGDvEGF<DLhdgiLF<Dl)R#F>hfHVUGdgj*N D`a

a`D]>K7YGHVU T}HV[GK<DLU*N D}E)R#F7deUK<SLP`F/R#K>dgHVUŠYdeF<DTLK<D
r1h = a2 + b2
YGD]–DtsEGF<D]<]>dgHVU ]xTLd“†YGD]/]>N ]a]>NGFAbqRkEcHVF>K>dgHVUª[email protected]£HVȤb½¾DtsEH`]<deK>dgHVU
œK<HVNVI>HVNGF<]x]>NGEGEcH`]<SD"deU YGDt
E(θ)
EDLU Y R#U*K<DkYGDb½§RV]<TDLU ]<dgHVU‰YF<HVdeK<DfE)R#F-bqRˆ]>NGdeK<DAD]>K"Yd“—˜SLF<DLUK<D'YGDÔSLF<HGn(HVNGKBTR#bgTLNGb4Dt—˜DTLK>N S`aEcHVNGF
NGU DxHVF>dgDLUK/R#K>dgHVU¤YNˆYdeEÚVbgD
œ’DLUmTH*HVF<YGHVUGU SD]O]>EGl SLF>dgj*N D]}SjN)R#K<HVF>dqR#bgD]tta bqR\F<DLbqR#K>dgHVU%DLU*K>F<D
DLK
θd
r1h
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
v Mv
α
]½¾STLF>deKB”
r1h =
c3 α sin θd
c1 + c2 α cos θd
œ ÷zn

Ø`Ø
VH È
a
DLK
]/HVUKBYGD]\THVU ]>K/R#U*K<D]"U*NGhfSLF>dgjN D]@j*NGdoYGSLEcDLU GY DLUK\NGUGdgjN DLhfDLU*K\YGDkbqRŠŒ’HVF>hfD'YGDfbqR
c1 c2
c3
Œ’HVU TLK>gd HVU%Y˜½¾DtszEcH`]>deK>dgHVUˆTHVU ]>dgYGSLF<SD`nCDLNGFuDtsEGF<D]<]>dgHVUmD]>K}YGHVUGU SD E)R#F"”
c1 =
c2 =
c3 =
Z
θmax
θmin
Z θmax
Z
θmin
θmax
θmin
E(θ) sin θ dθ
E(θ) sin θ cos θ dθ
œ ÷zn `÷ 
Ø
E(θ) sin2 θ dθ
ÕxR#U ]bgD-TRV]uYNˆ]<deK<D-]>N YˆYGDxb½ š [ ]/DLF>wVR#K<HVdeF<D-ɏdgDLF>F<D"MONGPQDLFaGU HVN ]}K>F<HVNGwQHVU ]\”
a
c1 = 0.77 c2 = −0.32
DLK
ncÉ;HVNGFAbgD]–YGDLNsm]<deK<D]–YGDBb½ š ÉMBaU HVN ]ARwQHVU ]”
a
DLK
acTD]
c3 = 0.60
c1 = 1.4 c2 = 0.028
c3 = 1.1
wVR#bgDLNGF<]vSLK/R#U*KrEGF<HzT<l D]vYGDATDLbebgD]vYNŠTRV]rdgYGSR#b˜YGD–THVNGwQDLF>K>NGF<DOK<HVK/R#bgDADLKrNGUGd“Œ’HVF>[email protected]”
a
DLK
c1 = 2 c 2 = 0
n
c3 = π/2
C½¾Sj*N)R#K>dgHVU¤œ ÷zn hfHVUK>F<DATLbqR#deF<DLhfDLU*Kvj*N DObqRxF<DLbqR#K>dgHVUpDLUK>F<D
DLK YGSLEcDLU YŠYGD}bqRBYGSTLbedeU)R#dg]<HVU
Ø`Ø
r1h α
YN'YdeEcÚVbgD`a
a*jNGdU½¾D]>KI¯N ]<K<DLhfDLUKrE)RV]oF<DTHVU ]<K>F>NGdeK<DOE)R#Frb½§R#U)R#be¼]<DOYGD R¼*bgDLdeP`ln ‡rHVhhfD
δd = π2 − θd
a
b½¾HVUXEHVNGw`R#deK]½ ¼ÆR#K>K<DLU YF<D`ab½§R#hEGbedeK>N YGD
]½§R#UGUNGbgDpbgHVF<]/jN D
HVN
a7TD¬j*NGd
r1h
δd = −π/2
δd = π/2
deU YdgjN Dj*N Drb½§R#U)R#be¼z]/DDLUkRV]<TDLU ]>dgHVUfYF<HVdeK<DD]>K7K<HVK/R#bgDLhfDLU*KdeU)RVY R#EGK<SDEcHVNGFSLK>N YdgDLFNGU D}R#UGdg]<HVK>F<HVEGdgD
Y R#U ]}TD-TRV]abqR\K<HVK/R#bedeK<[email protected]#K>dgHVUm]<DAK>F<HVNGw`R#UKO]<NGFbqRYGSTLbedeU)R#dg]<HVUn
ü2—£üû
¸
ÿ ÿ Qÿ%<OJ$g <
žZ_IV
Ï NGUGdg]"YGDfbqR¬F<DLbqR#K>dgHVUÆDLU*K>F<Dkb½§R#hEGbedeK>N YGDŠYGDfEGF<DLhdgiLF<D'l)R#F>hfHVUGdgjN DŠDLK-bqRˆw*F/R#dgDŠR#hEGbedeK>N YGD'YN
%
YdeEÚVbgD`aVU HVN ]EHVNGwQHVU ]oTR#bgTLNGbgDLFbqR–EGNGdg]<]/R#U TDYGDF<DTHVU ]<K>F>N TLK>dgHVUkK>l SHVF>dgj*N DYGDb½§R#U)R#be¼z]/DYGD R¼zbgDLdeP`l
a
EHVNGFb½§R#hEGbedeK>N YGDBDLK}bqR\EGl)RV]<D`n
|7U¬NGK>debedg]/R#U*[email protected]­ UGdeK>dgHVUZœ ÷zn§Ñ ~ vYGDAbqR\EGNGdg]<]/R#U TD-YGDAF<DTHVU ]<K>F>N TLK>dgHVUÅaGHVU¬EcDLNGK}YGSt­ UGdeF}[email protected]]J
]/R#U TD
RwQDT-bqR\F<DLbqR#K>dgHVUŔ
K1h
√
√
r1h
1
= √ r1h N ≡ K1h α N,
σr
2
œ ÷zn Q¦ 
Ø
1 r1h
c3 sin θd
1
K1h = √
=√
.
α
c
+
2
2 1 c2 α cos θd
œ ÷zn Q^ 
Ø
RwQDT
[À‚vö"i~ÜG# f QÃ%õÞef õwÜNÛ; ï ^ܑõ*(h
a
v
¿
DtszEGF>dehkR#U*KObqRfEGNGdg]<]/R#U TD"YGD"F<DTHVU ]>K>F>N TLK>dgHVU¤YGD-b½§R#hEGbedeK>N YGD
a)THVF>F>dePQSD\YNˆŒR#deKujN D"b½§R#U)R#be¼z]/D"YGD
r1h
R¼zbgDLdeP`lˆU DAF<DTHVU ]>K>F>NGdeKOE)RV]}YdeF<DTLK<DLhfDLUK–b½§R#hEGbedeK>N YGD"YNˆYdeEcÚVbgD`n
M‘[email protected]ŠhfyLhfDYGSLhkR#F<T/l DkY˜½§R#U)R#be¼]<Dj*N DTDLbebgDNGK>debedg]<[email protected]ŠhfSLK>l H*YGDk¡zM–ÉaGIL½§R#d]>dehBNGbgSNGU
P`F/R#U Y-U HVh"[GF<DvYGD4bgHVK<]YGDoYGHVUGU SD];DLKF<DTHVU ]>K>F>NGdeK£b½§R#hEGbedeK>N YGDoDLKbqR}EGl)RV]/D4YGDEGF<DLhdgiLF<Dol)R#F>hfHVUGdgj*N D`a
R­ U‰YGDfTHVhE)R#F<DLF\b½¾SLwQHVbeNGK>dgHVUÅYGDbqRpEGNGdg]/]/R#U TDkYGDF<DTHVU ]<K>F>N TLK>dgHVUÆYGDb½§R#U)R#be¼]<DfYGD R¼zbgDLdeP`lÆRwQDT
a
TDLbebgD-YGDAU HVK>F<DxhfSLK>l HzYGD`n
C R2­ P`NGF<Dy÷zn ^‰THVhE)R#F<D%b½¾SLwQHVbeNGK>dgHVUžYGDmbqR2EGNGdg]<]ÎR#U TD%YGDmF<DTHVU ]>K>F>N TLK>dgHVUžYGDmb½§R#hEGbedeK>N YGD¤YN
‘

YdeEÚVbgDpE)R#FbqR!hfSLK>l HzYGDm¡zMOÉ RwQDTˆTDLbebgD¬YGDpb½§R#U)R#be¼]<Dpl)R#F>hfHVUGdgj*N D`nÕxR#U ]@bgD¬TRV]fYNË]>deK<Dp]>N YXYGD
b½ š ÉM\abqRkEGNGdg]/]/R#U TD
D]>KOEGbeN ]AdehEcHVF>K/R#UK<[email protected]]AYGSTLbedeU)R#dg]<HVU ][email protected]Ú[email protected]>dg]<D]xDLU*K>F<D
K1h
R#U Y
aTR#Fxb½¾D]/]<DLUK>dgDLbYGD\bqR'hfH*YNGbqR#K>dgHVUÅYdeEHVbqR#deF<DD]>KAEGF<S]<DLU*K-Y R#U ]AbqRŠYdg]>K>F>de[GNGK>dgHVU‰DLU
−60◦
+60◦
RV]<TDLU ]>dgHVU2YF<HVdeK<D`nÉ;HVNGF-NGU DTHVNGwQDLF>K>NGF<DfK<HVK/R#bgD`acb½¾SLwQHVbeNGK>dgHVUÆYGD
D]>KxEcDLN2Yd“—cSLF<DLU*K<DfYN2TRV]-YGD
K1h
bqRkTHVNGwQDLF>K>NGF<[email protected])R#F>K>dgDLbebgD¬œ’[email protected]]>K–hkR#deUK<DLU)R#UK-]<¼*hfSLK>F>dgj*N D\E)R#FOF/R#EGEcHVF>K-»
tacDLK–D]<KOK<HVNVI>HVNGF<]
δ = 0◦
deUŒ’SLF>dgDLNGF<D"»\bqREGNGdg]<]/R#U TD-YGDAF<DTHVU ]>K>F>N TLK>dgHVU!YNmYdeEcÚVbgDxTHVhEGbgDLKn
,6-0/.1÷zn ^
‡rHVhE)R#F/R#dg]<HVUpDLUK>F<D–bqRxEGNGdg]<]ÎR#U TDOYGDuF<DTHVU ]>K>F>N TLK>dgHVU¬YGD}b½§R#hEGbedeK>N YGDAYNŠYdeEcÚVbgDuDLKrTDLbebgD

2
YGD
DLUmŒŽHVU TLK>dgHVUªYGD"bqR'YGSTLbedeU)R#dg]<HVUªYNªYdeEÚVbgD`neaY R#U ]–bgDBTRV]AYGDBNGU!HVNªYGDLNs%]>deK<D]–YGDBb½ š ÉM\ncC‘R
EGNGdgr]<]/1hR#U TDxYGDxb½§R#U)R#be¼]<D-YGD R¼zbgDLdeP`la
a¾D]>K}YGHVUGU SD"E)R#Fb½¾SjN)R#K>dgHVUZœ ÷zn ^Qtn
K1h
a
Ø
‡ HVU T DLF>U)R#UKxbqR'EGF<STLdg]>dgHVUªYGD\bqRkYGSLK<DLF>hdeU)R#K>dgHVUÅYGDBbqRkEGl)RV]<D adeb£D]>KOEcH`]<]>de[GbgD\[email protected]#bgTLNGbgDLFANGU D
r
ψ
EGNGdg]<]/R#U T\
D YGD"F<DTHVU ]>K>F>N TLK>dgHVU2Sj*NGdew`R#bgDLUK<D`ac]>NGdew`R#UK–bgD]OhfyLhfD]AYGSt­ UGdeK>dgHVU ]xj*N D"EGF<STSYGDLhhfDLU*KaDLU
v
)ú
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
,.-F/.1V÷zn§Ñ`_ ‡rHVhE)R#F/R#dg]<HVU2DLU*K>F<D"bqRkEGNGdg]<]/R#U TD\YGDBF<DTHVU ]>K>F>N TLK>dgHVU2YGD"bqR'YdeF<DTLK>dgHVUªYN!YdeEcÚVbgD\DLK–YGD
bqREGl)RV]<D\YG2DBbqREGF<DLhdgiLF<D\l)R#F>hfHVUGdgjN D
a)DLU%Œ’HVU TLK>dgHVU¤YGDBbqRfYGSTLbedeU)R#dg]<HVUªYN¤YdeEÚVbgD`neaY R#U ]ubgD\TRV]
a*D]<KoYGHVUGU SD–E)R#F
YGD}NGUpHVN'YGDLNs']<deK<D]oYGDOb½ š ÉM\nzC‘R-EGNGdg]/ψ]/1h
R#U TDOYGDub½§R#U)R#be¼z]/DOYGD R¼*bgDLdeP`la
KRA (1h)
a
b½¾SjN)R#K>dgHVUƜ ÷z¤n ÷V_Qtn
NGK>debedg]/R#[email protected]<DLbqR#K>dgHVU
q
2
σψ = 2/N r1h
DLK}DLUmSTLF>dew`R#UKB”
σψ−1
1 r
√ = √ 1h
2 α
α N
1
c3 sin θd
=√
rad−1 .
c
+
c
α
cos
θ
2 1
2
d
KRA (1h) =
œ ÷zn¤÷V_Q
‡ DLK>K<DDtszEGF<D]/]>dgHVU2D]>KAdgYGDLUK>dgj*N [email protected]ŒŽHVF>hfDLbebgDLhfDLUKB»Šb½¾Sj*N)R#K>dgHVUœ ÷zn Q^ –YGSt­ UGdg]<]/R#U*K
r
ahkR#dg][email protected]<Dt
Ø
K1h
EGF<S]<DLUK<D\NGU [email protected]*N)R#UK>deK<SYd“—˜SLF<DLUK<D`a˜DtsEGF>dehfSD\DLU
acR#bgHVF<]xjN DBqb R'EGNGdg]<]/R#U [email protected]<DTHVU ]>K>F>N TLK>dgHVU
rad−1
YGDAb½§R#hEGbedeK>N YGD"D]>K}]ÎR#U ]}YdehfDLU ]>dgHVUn
RwQDTbqRAYGSTLbedeU)R#dg]<HVU'YNfYdeEÚVbgD}D]<K;debebeN ]<K>F<SD}]>NGF7bqRO­ P`NGF<D÷zn§Ñ`_`RAEcHVNGF7bgD
C ½¾SLwQHVbeNGK>dgHVUŠYGD

KRA (1h)
]>deK<Dv]>N YYGDvb½ š ÉMBaQDLK]<NGF+÷zn§Ñ`_#[fEHVNGF7bgD]7YGDLNs]>deK<D]n`ÕxR#U ]7bgDvTRV]YGDvbqRATHVNGwQDLF>K>NGF<DuE)R#F>K>dgDLbebgD}YNf]>deK<D
]>N Y˜abqRŠF<DTHVU ]>K>F>N TLK>dgHVUÅYGD\b½§RV]<TDLU ]<dgHVU‰YF<HVdeK<DYN2YdeEÚVbgDfRwQDTb½§R#U)R#be¼]<DYGD R¼*bgDLdeP`l2D]>KxK<HVNVI>HVNGF<]
a
EGbeN ]ŠEGF<STLdg]<D`nCHVF<]<j*N D!bqR‰K<HVK/R#bedeK<SªYNžTLdgDLbxD]>K¬HV[ ]<DLF>wQSD`avbqRÅhfSLK>l HzYGDª¡zM–É YGHVUGU DªYGD!hfDLdebebgDLNGF<]
F<S]>NGbeK/R#K<]EcHVNGFuD]>K>dehfDLF}b½§RV]/TDLU ]>dgHVUmYF<HVdeK<D-YNˆYdeEcÚVbgD`n
Õ D4hkR#UGdgiLF<DrPQSLU SLF/R#bgD`aVbqROhfSLK>l HzYGD¡zM–É;aVR#EGEcHVF>K<DrNGU D4deUŒŽHVF>hkR#K>dgHVUf]>NGEGEGbgSLhfDLU*K/R#deF<DE)R#FF/R#EGEcHVF>K
A
»2b½§R#U)R#be¼]<DªYGD R¼*bgDLdeP`lžDLU¨R#U)R#be¼]/R#UK¬bqRÆYdg]>K>F>de[GNGK>dgHVU¨DLU¨RV]<TDLU ]>dgHVUžYF<HVdeK<DªDLK¬DLUžYGSTLbedeU)R#dg]<HVUn
a
‡rDLK>K<D-deUŒŽHVF>hkR#K>dgHVU!]½§RwQiLF<D"yLK>F<DBY˜½ NGU D-P`F/R#U YGDBdehEHVF>K/R#U TD`aTR#Fub½§R#U)R#be¼]<D"Y˜½ NGU DB]<DLNGbgD"THzHVF<YGHVUGU SD
[À‚vö"i~ÜG# f QÃ%õÞef õwÜNÛ; ï ^ܑõ*(h
v
U B
D EcDLF>hfDLKxE)RV]-YGD\F<DLhfHVUK<[email protected]»'b½§R#hEGbedeK>N [email protected]½§R#UGdg]<HVK>F<HVEGdgDf]<HVN ]rI<RVTDLUK<D`aUGd7»p]ÎR'YdeF<DTLK>dgHVUn‘CD]
F<S]>NGbeK/R#K<]}YGDxb½§R#U)R#be¼]<D-YGD R¼zbgDLdeP`lm]<HVUKOYGHVU T-Yd“¥kTLdebgDLhfDLUK–deUK<DLF>EGF<SLK/R#[GbgD]–]<HVN ]}TDLK>K<DAŒŽHVF>hfD`n
a
v
W
ï ÃQ>f‘V&Ã`fÛ;kf)Ü;d²ª(f<Û;mÁÃ_h*õ’õ%Ã%õò•i‰Á[email protected]Û Ú $'
!
}
"
#$%&
™
«
š (
( '›šœt(('›š
¢­¬
¬›®š¡
«
¯.°
.ž+
±›²
ŸP¡¢¤£¥œ¥¢*¦ŸIš¡
ž§¨šÒ(
©Išª¡
šœ
³ ´yµ·¶x¸,´¹º¶»·¼.½y¾e¿rÀrÁ4ÂnÃ
G
ÄŽk¸,´ÆÀ7¼AÇL¹]Âȹ]ÂnÁ4ÂQ»QǺÉbµÆÀrÁy¶b»Àr¿?Ǻɹ]ÂyµÊ½WÉi¿:µÇWÀrÁ4Çy¾e¿ËÂyµnÌ,ͺÁM½WÉe¹]Â,´Q»·¼.·Î7ÎNÂÐÏƶiÂÊÉǺ¶]µÈ½ºÑÇWÉbµÊ¹]´Q»ÆÁy¿rÀ•Âo¾eÁ4´Q»´·Ò
¹]·¸Ê¸,·É!À5ÂyµÆÀ5½y¾j¾eÎw¿?Ïƶb´Â\Ìn¶xÉÎÓÇWÀ$¹xÂu¹jÇWÉiÉ´QÂyµo¸,Âyµy¶AÁQ´QÂyµq¾.½ºÁZÎÔÂoÁQ´yµÂQ½º¶Õ¹]ÂRµ·¶xÁ7Ö½j»Â\¹]ÂoÎQׄØmÙyµ ·Á·Ñ½WÀËǺ¿7Á4Â
Úq¿?ÂÆÁyÁQÂ$Û¶WÍ]·Á Ü]Äu× ½ºÉe½WÎHÝgµÂY·µ·ÀÅÂ7ÞÂ4»Àr¶i´QÂP¾.Ǻ¶AÁfÀrÁ4Ǻ¿2µVÂyµÆÀr¿2¸È½WÀr¿?ǺÉbµf¹º¿ Þ´·ÁQ·É!À•ÂyµG¹]ÂfÎN½Z»Q½ºÁÆÀ•ÂY¹]ÂY»Çº¶xѺ·Á·Àr¶AÁQÂyß
ÂÆÀG¹]·¶Wà¾eÁ4Ç »´Q¹º¶xÁQÂyµ%¹× ÂyµÆÀr¿2¸È½WÀr¿?Ǻɥ¹]ÂÎQ× ´·É·Á?ͺ¿ËÂȹ]ÂyµÈ´·Ñg´·É·¸,·É!À7µÜIá\½ºÉbµ»Æ¼A½jÏƶiÂ,»½gµßVÎÓ½kµy¿NͺÉi¿ âf»Q½ºÉe»yÂ
¹]ÂyµÈ½º¸¾eÎH¿rÀr¶b¹]ÂyµnÁ4ÂQ»QǺÉbµÆÀrÁy¶A¿7À•Â·µÐÂyµ·ÀR»Q½ºÎN»Æ¶xÎÔ´QÂãÃ$ÎÔÂyµnѽºÎÔ·¶xÁyµ%¹º¶ä¹º¿0¾.åºÎÔÂ%ÂÀ¹º¶¥Ïƶb½j¹ºÁ·¿0¾.åºÎÔÂÈÁQÂ4»QÇWÉbµ·ÀrÁ·¶x¿rÀ•Âyµ
µÆǺÉ!Àq»QǺ¸¾.½WÀr¿ËÙ·ÎÔÂyµ½ºÑºÂ4»Î4× ¼]Ýy¾.ÇWÀ7¼bæyµÂ¹]ÂuÎQ× ¿2µÆÇWÀrÁ4Çy¾e¿Ë¹]ÂuÎӽйº¿2µ·ÀrÁ·¿?ÙƶAÀr¿?ǺÉ罺Éxͺ¶AÎÓ½W¿2ÁQ Ü
èiéeê
ëÐì]íºîìxïíºî*ìÕðfñÓòió^ô7õ öi÷ ïgö!øIô7ì]õnò]ùeì]íûú7ì]õÈð^öeó^óüxì]õnðfñÔý%þ*ÿ!ìxï
!"$#&%'!(')+*-,/.102!%')3$#4%65879!%:%';<.=(?>[email protected] .=B:7C58DA#FE'GH*I.>1.=D.!>45A;4D879).=5879!%KJ:#L)3.=%:79)!58DA!B:79#L)J:#L)JM7CDA#L>ON
5879!%')+JP,/.=D8D87C02;L#L)J:#L)QD.1R2!%')3>LS)8"79T6('#L)Q%U,V#L)A5QB S))87CW:*9#T6('#)87'*-,V!%X>L!%:% .=7C5+*9# %'!W:DA#&JP,V;402;4%'#4$#4%65A)
.=585A#4%'JM(YJ'.=%')">[email protected] .!T6('#KJM7CDA#L>45879!%JM(Z>479#4*-GQ[\>L!"B .=DA#4D].L02#L>K*9#X%'!W:DA#^JP,V;402;4%'#4$#4%H5A)$DA;L#4*C*9#4$#4%65
J:;45A#L>45A;L)1_a`b.$cd!%'>45879!%eJP,V#OfMB S)87C5879!% E (α, δ) GP;4gH.=*9#4$#4%65.=B:B #4*C*9;L#h>1.=D85A#hJ:#h>L!(:02#4D858(:DA#SGPJ:!%:%'#i*I.
E
)8(:Djc-.!>L#k#OlP#L>4587C02#FJ:#FJ:;45A#L>45879!%m.L02#L>F*I.!T6('#4*C*9#F.;45A;&!W')#4D802;L#&(:%'#&DA;4gS79!%KJM(X>479#4*a>L#4%H58DA;L#<.=(:5A!(:D J:#L)
>L6!DAJ:!%:%';L#L) (α, δ) G'.=(X>L!(:DA)JP, (:%'#FB ;4D879nJ:#kJP,/.!>LT6(:79)87C5879!%KJ:#kJ:!%:%';L#L)o_:F,V#L)A5+*I.J:;45A#4D8"7C% .=5879!%mJ:#
>L#4585A#hTH( .=%6587C5A;iT6(:7B #4D8$#45<J:#i>1.=*9>4(:*9#4D<*9#%'!W:DA#"JP,V;402;4%'#4$#4%H5A).=585A#4%'JM(pJ'.=%')q>[email protected] .!T6('#"JM7CDA#L>45879!%
)8(:D *I.h)8B:@'r4DA#>L;4*9#L)A5A#S_
ks t %$J:# $#L)8(:DA#4DQ(:%'#k.=%:79)!58DA!B:79#kJP, (:%'#&.="B:*C7C58('J:#&J:#T6('#4*9TH('#L)Ba!(:D+>L#4%651GS7C*:c-.=(:5+>L!%:% .vu958DA#*I.
>1.=D85A#J:#>L!(:02#4D858(:DA#h.L02#L>*I.h$w4$#B:DA;L>479)879!%U_ s 587C58DA#JP,V#OfM#4"B:*9#SG)87U!%m7C%658DA6JM(:7C5F(:%'##4D8DA#4(:D&)ARn)jN
5A;4].=5879TH('#qJ:#<x2yz)8(:D*-,V#OfnBaS)87C5879!%UG:J:#&].=%:79r4DA#<>[email protected]';4DA#4%H5A#SG:B .=D #Of:#4"B:*9#F#4%K)8(:DjN{#L)A587C].=%H5 E (α, δ)
E
J:#$x2y|J'.=%')<(:%'#$JM7CDA#L>45879!%p(:%'#hDA;4gS79!%}JM(~>479#4*Q#45#4%p)A!(')jN{#L)8587C].=%65 E (α, δ) J:#"*I.X$w4$#hc-.!L!%UG
E
7C*+B .=D.vu95h;40679J:#4%65iT6(U, (:%'#?.=%:79)!58DA!B:79#XJ:#]*-,V!DAJMDA#?J:#Xx2y€)#4D.^.=D8587 t >479#4*C*9#4$#4%H5$>4DA;L#?J'.=%')*9#L)iJ:!%MN
%';L#L)1_F#]c-.!L!%}g2;4%';4D.=*9#SG*-,V#4D8DA#4(:D")8(:Di*I.^>1.=D85A#XJ:#?>L!(:02#4D858(:DA#SG$w4$#‚)87#4*C*9#]%U, 7C%658DA6JM(:7C5"B .!)hJ:#
>L!D8DA;4*I.=5879!%\[igSD.=%'J:#;L>@'#4*C*9#SG'7C"BaS)A#q(:%'#<0S.=*9#4(:D "7C%:7C].=*9##4%MN{J:#L)A)!(')J:#<*I.!T6('#4*C*9#<(:%'#<J:;45A#L>45879!%
JP,/.=%:79)A!58DA!B:79#<%U,V#L)85kB:*C(') t .=W:*9#SG'>1.=D#4*C*9#<Ba!(:D8D.=7C5DA;L)8(:*C5A#4DkJP, (:%'#].=(:0!.=79)A#<#L)A587C].=5879!%ƒJ:#q*I.h>1.=D85A#
J:#<>L!(:02#4D858(:DA#S_
< .=%')i*9#?>1.!)hJM(„DA;L)A#1.=(ZJ:#?)A(:Djc .!>L#XJ:#X*-,‡†&W')A#4D80!.=5A!7CDA#‚ˆ79#4D8DA# s (:g2#4D1G‰*I.ƒJ:;45A#4D8"7C% .=5879!%YJ:#X*I.
cd!%'>45879!%}JP,V#OfnBaS)87C5879!%}#L)A5W #1.=('>L!(:B}B:*C('))87C"B:*9#X.=(Mfp;4%'#4D8gS79#L)!Šp*-,/.!>L>L#4B:5.=%'>L#X#L)85)‹.=58(:DA;L#^ŒU#4%
#Ola#451G:>L#4*C*9#ON{>47;45.=%H5B:(:DA#4$#4%65kg2;L!$;458D879T6('#SG'7C*b)A(M$5J:#>1.=*9>4(:*9#4D*-, 7C%H5A;4gSD.=*9#XŒ
EE (α, δ) =
Z
cos θ Seff [E, θ(α, δ, t), ϕ(α, δ, t)] dt
t
Ž  _Co‘
J'.=%')X*I.!TH('#4*C*9#\*I.})8(:Djc .!>L#\#OlP#L>4587C02#\J:#’J:;45A#L>45879!% S [E, θ(α, δ, t)] #L)85‚)87C"B:*9#4$#4%H5m;4gH.=*9#e[~(:%'#
eff
>L!%')85.=%H5A#S_`Q,V;402!*C(:5879!%5A#4"B !DA#4*C*9#‚J:#?*I.ƒ)A(:Djc .!>L#X#Ola#L>4587C02#‚J:#XJ:;45A#L>45879!%YJ:!7C5"w458DA#?>L!D8DA#L>45A#4$#4%65
#L)8587C$;L#SG6B !(:DB:DA#4%'JMDA#q#4%‚>L!"B:5A#F*-,/.=gSD.=%'JM79)A)#4$#4%H5JM(XDA;L)A#1.=(‚.102#L>F*9#&5A#4"B') .=7C%')87aT6('#k*9#L)5A#4"B')
JP, 7C% .!>4587C067C5A;iJ:#L)F)85.=5879!%')&JM“’.=(MfmB:DA!W:*9r4$#L)F5A#L>[email protected]:%:79T6('#L)1_U#)8(:7C0n7‰#L)85F.vla#L>458(';gSD”!>L#i.=(Mf t >@:79#4DA)
J:#<•– Ž >Oc8_:>[email protected] .=B:7C58DA#E6‘O_
êP—d˜öuòbï÷ ì]õÈð^ìÕíjöþÅùì]ï÷Lþïgì
èié™
ˆ !(:D^J:#L)
;4%'#4D8gS79#L)37C%Mc ;4D879#4(:DA#L)[ E ≈ 3  #-žhGn*I.)A(:Djc .!>L#F#Ola#L>4587C02#qJ:#FJ:;45A#L>45879!%mJ:;4B #4%'JXJ:#F*-,/.=%:gS*9#FŸ1;4%:[email protected] .=*
sat
4
#
5
:
J
q
#
*
V
,
4
;
'
%
4
#
8
D
S
g
9
7
<
#
:
J
F
#
I
*
i
.
2
g
#4D8W #SGM>L#4585A#<J:;4Ba#4%'J'.=%'>L#.L02#L>q*-,/.=%:gS*9# θ J:!7C5J:!%'><w458DA#q>L!%:%H('#[T6('#4*9T6('#L)
θ
B !(:D>L#4%65B:DArL). t %~JP,V;458('JM79#4DiJ:#L).=%:79)A!58DA!B:79#L)J:#ic-.=7CW:*9#].="B:*C7C58('J:#S_‰`U#L))87C(:*I.=5879!%')J:#hg2#4D8Wa#L)
#45XJM( J:;45A#L>45A#4(:DKDA#4B:DA6JM(:79)#4%H5KgS*9!W .=*9#4$#4%H5KW:79#4% *9#L)‚J:!%:%';L#L)K#OfMB ;4D87C$#4%65.=*9#L)‚B !(:DKJ:#L)K.=%:gS*9#L)
Ÿ1;4%:[email protected] .=(Mfp7C%Mcd;4D879#4(:DA)i[‚EHx ◦ GU>L#4Ba#4%'J'.=%H51G‰7C*Q%U,V#L)85B .!)B S))87CW:*9#"JP,V!W:5A#4%:7CDh.!>458('#4*C*9#4$#4%65h(:%p%:7C02#1.=(
J:#>L!% t .=%'>L#)A(M]).=%H5FB !(:D&)A# t #4Dq.=(MfK)A7C(:*I.=5879!%')FB !(:DF#L)8587C$#4DF[$TH('#4*9T6('#L)qy¡B:DArL)k*I.">1.=D85A#iJ:#
>L!(:02#4D858(:DA#&#4%MN{J:#L)A)!(')3J:#k*I.).=58(:D.=5879!%U_n¢&;1.=%:$!7C%')oGH%'!(')£.=*C*9!%')+02!7CDTH(U, 7C* #L)85+BaS)A)87CW:*9#kJP, (:587C*C79)A#4D
*9#L)J:!%:%';L#L)k#OfMB ;4D87C$#4%65.=*9#L)Ba!(:Dk#L)8587C$#4D*I.h>1.=D85A#J:#>L!(:02#4D858(:DA#S_
ˆ*C(')879#4(:DA)"$;[email protected]'nJ:#L)]!%65$;45A;mJ:;402#4*9!B:Ba;L#L)$J'.=%')$>L#‚W:(:51G%'!5.="$#4%65]*I.\$;[email protected]'nJ:#KJM7C5A#m)A#4"7¤N
.=% .=*CR65879T6('# Ž T6('#q%'!(')J:;L)87CgS%'#4DA!%')&B .=D&¥ s B .=D*I.$)8(:7C5A#o‘#45*9#L) $;[email protected]'nJ:#L)kJM7C5A#<J:#]¦4)[email protected](M§$7C%:g ¦!_
`b.q$;[email protected]'nJ:#k)A#4"7¤N¨.=% .=*CRn5879T6('#k#L)8579J:#4%65879TH('#&J'.=%')Q*9# B:D87C%'>47CBa#&.=(]>1.=*9>4(:* J:# E (α, δ) J'.=%')Q*9#>1.!)
E
!Š$*-,/.!>L>L#4B:5.=%'>L#q#L)85+).=58(:DA;L#ŒS*I.>1.=D85A#&J:#>L!(:02#4D858(:DA#&)o,V!W:5879#4%H5£#4%]7C%H5A;4gSD.=%65£*I.<)8(:Djc-.!>L#k#OlP#L>4587C02#kJ:#
J:;45A#L>45879!%h>L!"$#£J'.=%')*-,V;LT6( .=5879!%  _CSG![k*I.&JM7¤la;4DA#4%'>L#£T6('#+*I.FJ:;4Ba#4%'J'.=%'>L##4% θ J:# S [E, θ(α, δ, t)]
eff
#L)85k!W:5A#4%H('#"[hB .=D8587CDqJ:#*I.]JM79)858D87CW:(:5879!%’J:#L)F.=%:gS*9#L)&Ÿ1;4%:[email protected] .=(Mfƒ#OfMB ;4D87C$#4%65.=*9#S_©&%\.vªj(')85A#4$#4%H5qJ:#
*-, @:79)85A!gSD.="$#~J:#L)^.=%:gS*9#L) θ $#L)8(:DA;L)KBa#4D8$#45^JP,V!W:5A#4%:7CDm*I.ZJ:;4Ba#4%'J'.=%'>L#p>L!D8DA#L>45A#}[«B .=D8587CDƒJ:#L)
J:!%:%';L#L)1GaJ'.=%')&*-, @[email protected]'rL)A#"!Š\*I.?B:DA;L)A#4%'>L#hJP, (:%'#h.=%:79)!58DA!B:79#h%'#$nJM7 t #iB .!)q>L#4585A#iJM79)A58D87CW:(:5879!%UG
)A!(') Ba#47C%'#<JP,/.=W')A!D8W #4Dk*-,/.=%:79)A!58DA!B:79#S_
ˆ‰!(:Di02;4D87 t #4Dh)873>L#4585A#[email protected] [email protected]'rL)#]#L)85kª¬(')8587 t ;L#SGb7C*3#L)85B S))87CW:*9#]J:#]g2;4%';4DA#4D(:%«*9!5J:#?J:!%:%';L#L)
>L!%H5A#4% .=%H5 (:%XJM7CB ­!*9#FJ:# 50% JP,/.="B:*C7C58('J:#SGM#45J:#k02!7CD3B .=D(:%‚)87C"B:*9#k>@ .=%:g2#4$#4%H5kJ:#kDA;Ocd;4DA#4%65879#4*a)A7
*I.<JM79)858D87CW:(:5879!%‚J:#L) θ $#L)8(:DA;L#kB .=D+*9#&¥n®#L)85Q$nJM7 t ;L#k!(]%'!%U_6`b. t gS(:DA#  _C1.2‘7C*C*C(')858DA#k*9#DA;L)8(:*C5.=5FŒ
*I.\B:DA;L)A#4%'>L#^JP, (:%JM7CBa­!*9#mJ:#‚c !D85A#^.="B:*C7C58('J:#^%'#K$6JM7 t #^TH('#‚58DArL)]B #4( Ž ∼ 3% ‘i*I.eJM79)858D87CW:(:5879!%
J:#L).=%:gS*9#L)Ÿ1;4%:[email protected] .=(MfaG‰>L#]T6(:7QB #4D8$#45iJP, (:587C*C79)A#4D*I.^JM79)858D87CW:(:5879!%„#OfnBa;4D87C$#4%H5.=*9#S_  %~DA#40S.=%'>[email protected]'#SG‰*I.
JM79)858D87CW:(:5879!%$J:#L)+.=%:gS*9#L)+.=ŸL7C(:[email protected] .=(Mf"#L)A5‰c !D85A#4$#4%H5+$6JM7 t ;L#£B .=D*I.&B:DA;L)A#4%'>L#JM("JM7CBa­!*9#SGS>L!"$#!%
B #4(:5*9#Q02!7CD)8(:D‰*I. t gS(:DA#  _C4W‘O_!©k%'#J:#L))8(:W:587C*C7C5A;L)J:#+*I.k$;[email protected]'6J:#£)A#4"7¤N¨.=% .=*CRn5879T6('##L)A5T6('#Q*9#>[email protected]'!7¤f
J:#L)c !%'>45879!%')+J:#W .!)# B !(:D3*-,/.vª¬(')A5A#4$#4%H53J:#k*I.<JM79)858D87CW:(:5879!%‚J:#L)3.=%:gS*9#L)3Ÿ1;4%:[email protected] .=(Mf$B #4(:5+7C%H58DAnJM(:7CDA#
(:%'#k#4D8DA#4(:D£)8Rn)A5A;4].=5879TH('#&)8(:D+*I.>1.=D85A#kJ:#F>L!(:02#4D858(:DA#q>L!D8DA;4*9;L#&#4%658DA#k*9#L)JM7¤la;4DA#4%65A#L)DA;4gS79!%')JM(X>479#4*-_
`U#L)&$;