On a quantum kinetic linked to the Compton effect Modelling and 3D simulation of the satellite charge in plasmic environment Martine Chane-Yook To cite this version: Martine Chane-Yook. On a quantum kinetic linked to the Compton effect - Modelling and 3D simulation of the satellite charge in plasmic environment. Mathematics [math]. Université de Provence Aix-Marseille I, 2004. English. �tel-00008427� HAL Id: tel-00008427 https://tel.archives-ouvertes.fr/tel-00008427 Submitted on 9 Feb 2005 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. 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"c8+- %&= c3 c/ =c/( + + c(% Gc 'cL' %&3!% = ",c g/c L + %&3N % &% *$ > ( ;" % %>/gc/ +4(* +- ( % +-% 3G( % 3hc +4( "c > c<#$ c 3 c7 "?> ",c G/c ( "c %&"6, ",c 3 %&( 0:%9+4( '% ( +4c'H%&3 > `%$'& (*),+ +.-0/213) c -465,+7128:9 b ),+./3+ ;=< c ?c >[email protected] <ED >AFAGcHBJILKM>[email protected]> [email protected] < [email protected] cHNhcHKHB[@CILK:[email protected]@GcHG\B[U >[email protected] cPn]K?I^BgcHO`_acbTWILORbTQ YdU?e ;=< c6M>cHNLOfILG\BgcHGgUJI^BhaEILUi>A_jQAWB i l k m ]n BGc?UCNo>,>B0B < c v2v ; v e ; ; ; D A> FSGc B[IPK QSNhcHU?X]NLQK> BGc?O >A_ QSWB k [email protected] c?U=Q B < c#F [email protected][> < IPK QSNhcHU?e B B < c#[email protected] d >AK cAX3B < c6M>c?NPO @Gc?Uc D _gNhcHU >OI QSN c YdIPB < > OgOc?O [email protected]@c?FSWgNL>[email protected] D jQAGc?G\B[UHX > GgO`B < c M>cHNLO ILUdOgI QSNhc!hNPI" c0W BJQ:OILU B >AGgK c?U7Qed e$#iBJQ #XY < [email protected]%#ILU=B < c c>[email protected] B < @[>AOgIPI e ;=< cOgIPUCB > GBM>cHNLO Qed c?>[email protected] < [email protected]?> B[N^a D Q]OgI^M>cHO_\a%B < c UCQSNo> @dYdIPGgO Xj> < QABdQSWBCbTQ Y d @JQ D B < cUCWgG XTKHQSGgUJIPUCBJILGF Qed UJQSNL>[email protected]'& D >AIPGgN^a < a][email protected] cHG)( D Q+*IPGgF >B>A_jQAWBd\mSm " D-, U YdI^B < Bha.gILK?>AN Oc?GgUCIPBha6> Bc?>[email protected] <0/ U [email protected]_gIPB Qed i IPQSGgU , e ;=< c c?>[email protected] < M>cHNLOd [email protected] D U >AG QA_gUCB[>AK?N c B[Q0B < c#UCQSNL>[email protected] YdIPGgO XB < > B K?QSGMgGc?U B < c7M>cHNLO NPILGc?U=>AGO1gNL>AU D >AU=IPG\B[Q >AG c?NPQSGgFS> BgcHO,K>2*IPBhaAX0"GgQ YdG >AUB < c c>[email protected] B <,D >AFSGcHB[QAU3 < [email protected] e ;=< c0_jQSWGgOT>[email protected] a _6cHBhY c cHG B < c7BhY Q ILU K>ANPNhcHO B < c D >AFSGc B[Q4T> WgUGc e B @JQSWF < N^a6_6c < >5*ec?U=NPI" c >[email protected] cHB QedZNLIPVWgIPO Pc n6 QSUGcHO BJQ6>[email protected] bTQ Y e WB[UCILOc6B < c D >AFSGc B[Q4g>AWgUGc6ILUB < c6_jQ Y U < Q]K7" X Y < [email protected] < c8* cHNLQK?I^BhafQ d B < c6UJQSNL>[email protected] YdIPGgO >[email protected] [email protected]>AUI^B >:[email protected]\> K < c?UB < c D >AFSGc B[QSU; 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< [email protected] < [email protected]=IPU No>AU D >AU3 < [email protected] @JQ D D ILO BJQNPQ Y NL> B[I^B[WOc?UHe$N QAB < Q5*[email protected]>:6 gIPGgF0B < cNo>AU D >AU3 < [email protected]#>AGgO B < [email protected]>AU D > U < c cHBXg>[email protected]@[>AOIo> BJILQAG _6c?N^B[U > GgO @CILGgF [email protected]@GcHG\B2> ;=< c7F c?QAUCB >B[IPQSGT>[email protected] [email protected]_gI^B IPU >ANPUJQ>[email protected] e [email protected]?FSIPQSGgU B[Q0B < c D >AFAGcD B[Q:T>AWgUc >[email protected]?>ANLN c?O _ QSWgGgOg>[email protected] No>a [email protected] ;=< [email protected]=KHWgU3U7> @Gc.U < QYdG QSG B < cEgIPKHB[[email protected] v ePkSe ;=< c-T>[email protected]?Q D IPGgF d @JQ D B < c6UCQSNo> @0YdIPGgO [email protected],> UCQSNL>[email protected]@JW9BJILQAG K?Q D c6ILG\B[Q B < c D >AFAGcHBJQSU3 < [email protected]*Io> B < cKHWgU3U?e ILK B[[email protected] v Pe k [email protected][> BGc?Ud [email protected] D >ANPNPa B < c D c?G\B[[email protected]?FSIPQSGgUHe ILK B[[email protected] v He GU < Q YdU7>:UCQSNL>[email protected] [email protected][IPQSG e > & Y < c?G6UJQSNL>[email protected][email protected] ;=< cc>[email protected] B <D >AFSGc B[QSU; < [email protected] > < QAB K?QSNPNPILUCILQSGNhcHUJU gNo> U D = [email protected]('e > K < *AQSNPW D cY < ILK < OgI D c?gG UCILQAGEIPU.>A_ QSWB#B < cBI cH_ aec hN cHGgFAB < K?QSG\B > ILGgU > NL>[email protected] c.GW D [email protected] dJT>[email protected] B[IPK?N c?UHe v v eHG]e ; ; 3; [email protected] v Le k > @CB < / U D >AFSGc B[QSU; < [email protected] [email protected] v He G QANo>[email protected] [email protected]]B[IPQSG v ; ` $ v e ; ?-i9 c ; ; - +.5,45,+.9 No> U D > cHGgFSIPGc?U WgUc?OR_\a U3T> K cB* c < ILKHNhcHU>[email protected] Y [email protected]?OR_a]ncHGgQSG X >%F\>AU#Y < ILK < ILU c?N c?K B[@CILK?>ANLN^a K < >[email protected] c?O e ;=< c *c [email protected] D cHG\B[U < >2* c U < Q YdG,B < >B B < c gNL>AU D > cHGgFSIPGc?U.> @Gc D [email protected] 6KHIhcHG\B.B < >AG B[@J>AOgI^B[IPQSGT>AN*d WcHN jQ Y [email protected]?O,QSGc?U?Xj_IcHK>AWUGc-cHNo> UCB[IPK @JQ4gWNLUCILQSGUCaUCBGc D U#YcHILF < >fd8c Y < [email protected]?O"[email protected][> D:D cHU?X >AU=Q4jQSUc?O B[Q B < c BJQSGgGcHU=Qed NLIPVWgILO1d WcHNLUHe [email protected]?UJWNPBJUdB < > B B < [email protected]]K7" cHBJU >[email protected] D [email protected] cH:K?I c?G\B >AGgO K>AG K> @[email protected] D [email protected] D > [email protected]>ANPU7IPG\B[QU3T> K cAe ;=< c0QSGgN^a [email protected]_gNhc0ILUB < >B7B < c gNL>AU D >:[email protected][>Bgc?O_a B < c cHGgFSIPGciIPU UCQ < QABB < >B GgQ D > [email protected]>[email protected]?UCILU B I^B[U < c?> Be GgOc c?O X\B < cd >AU BJT>[email protected] B[IPK?N c?U3Qe1d n c?GQSG>[email protected] c?K BgcHOBJQ#IPGMTGgI^Bha >AGO OgQ.GgQAB K?Q D c=_T>AK7" X Y < [email protected]> UB < c UJNPQY T>[email protected]=Bgc?GO B[QKHQ D c_g>AK7" B[Q B < cUJ> BgcHNLNPI^BgcAe IPGgK cB < cgNL>AU D >-c c?KHBGc?O%_\aB < cc?GFSILGc#ILU < QAB?XB < c UJ> BgcHNLNPI^Bgc IPU WUJWT>ANPN^a @c?K?Q5*[email protected] _\a%K [email protected][> D IPK?U D >[email protected]> NLUHe ; 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B! V! ` (+*>% &)(+* >* CE* ,>&IM\20%*>%8/>5)&IR / />YU *?&)(8-D& &)(+*T/_-D&4*>C)C 5)&P*5E/?- /[email protected]+(+*>M\* 243 M - D5)YJ/ P ;hM>5)&P* /]-0/ Ñ. Ó k ` Ó k ××µØ . GD Ó 3Ó4 k × ×c . < GD Ó 3Ó4 : × × Ë ^ 3 4 Ó Ó k ×c Ó P ×× Ó Ó × × GD : . GD k < 354 Ë I ^ = = P = 35 4 Ó Ó : ) c× 6 Ó P ×©× e Ë Ó6 Ë× Ó Ó k c× Ó P ××kÞ d 3 4 P = = }e} F} B{ ! !V! V! +V«/R¡A£E¡á±¶¥+w/¢G¨/¤ ° G¨Û+rR¡R£q¨/?>·E>¨CD+¨C>RfR¥¡ Dkw£q <+¨©^ËGË/²wËG² +rEwE£E¨©G¸¥kR? >©¡á±¶IkE> Ø *Ë L 5 S S b Ñ d )= : = < Ñ Ó k × e G ` k : ××uY . ÑÓ ` Ó C \ 0 ³+R¨© b Ñ E >+( 9¸ Y³´R¡ww/¥+& T+E¥¸f± b Ñ Ó ×µØ d 0 Ñ e 6 d Ë 0 Ñ = e Þ . <o1-34 < 354 Ë<Ë ÈµlÆ$ËGË/² V È ÖÈ Ç o µ Ç ¸Ç ®È 6Æ oÈ VÇ Æ uÆ 9 #J =# á= gc6Ç; ÑØ Ü ^ Ñ Ø Ü ^ ^ w¢<k¨mËGËG²¡Ë ÑØ Ü ^ ¬uR¡Y£R[á±`¥+w/¢G¨I¤ ° G¨R¡R£q¨©G?> Ë <l< Ñ ØÜ GË ËG²¡ËG² È Ç 9 ·;µÇ Ç ´È)ÆÛÇ O Ç á +àEwE£E¨G¶¥+E>©¨©wI<k¡G ° <k+£EwG ^ Ñ E>+/6aER¨m³+E Z[[[ E¡+R¨ ) Ü ` [ Z[[ J = ) = : d Ë P e = : < Ó Ó P × Ó k ×× Ü ` \ Ë/ËG² < × \ = Ó [[[ G¨ [ Ñ [] [] Ü ) < Ó + 0 Ó k ××HÞ Ñ +àEwE£E¨G¶¥+RI >[ᱪ8R£EG¤mH > 0 Ñ. Ó k ` Ó k ××µ!Ø C b Ñ dJ = < Ñ Ó : × e G A b Ñ d )= : = < Ñ Ó k × e G ` 0 0 ³+R¨© Ø P+G )= : d Ë P J= = e = : < Ñ Ó Ó P × Ó k ×× Ü /k¥ ) < Ñ Ó + Ó k ©× × Þ 0 0 +R Ñ. Ó k ` Ó k ××uØ . d <21-0 3Ñ 4 e 6 GD Ó 3Ó4 k × × C GD Ó <Ë 034 Ñ = %× G d 0 Ñ e 6 GD Ó Ó : × × GD Ó Ë 0 Ñ ) = × GD Ó Ë 0 Ñ = %× ) = . <21-354 34 < 354 < 34 Ø ` ³¡ Ø . d <21-0 3Ñ 4 e 6 ID Ó 3Ó4 k × × GD Ó <Ë 0354 Ñ = × G ` C × 6 k Ø d 0 Ñ e ID Ó Ó × GD Ó Ë 0 Ñ ) = × ID Ó Ë 0 Ñ = %× ) = Þ . <21-34 34 < 34 < 34 3 >M>¡+¢n¡ >MD++E¨¡£?/%£RYG¨©¥+w+IC >HA Ø . GD Ó 3Ó4 : × ×qÞ Ó Ë/ËG² Ì × È)GF¤ D?<kHQ¡G? >o ° ² Ë<Ì ÈµlÆ$ËGË/² V È ÖÈ Ç o µ Ç ¸Ç ®È 6Æ oÈ VÇ Æ uÆ 9 #J =# á= gc6Ç; Ó Ó ×× Ñ Ø Ó Ë Ñ × d 0 e 6 GD : B 6 GD 0 ) = ) . <21-354 34 < 34 S F D S D YD F F YD AF F I I D 0 = % = ) ` < 3 4 Ë Ó Ñ × ³+E¨ ) = : = Ñ Ó Ó P × Ó : ×× S Ø P = H H Þ ` d ) 66 Ë e B>.>¡+¢^w+Dc/2/¨ £EA/¨¥+¡TIH> ³¡ª¨©H>MDcE£Eo A ³¨©¡H>·3> Ë Ø <21 d 0 Ñ e 6 GD Ó Ó k × × B 6 × ID Ó 0 Ñ ) = × Ó ) 7 ) . <o1-34 354 < 3 4 S F A F F F D S D YD F D ³+E¨ 6 6S D Z d S 66 ej Z Ë Z jf H H GD Ó 0 Ñ H H = ×% H H Þ Ó ) ×uØ! < 34 +O£HT/+¢/O ° «I/¨©2/kwC> Ø <0 34 Ñ H H ¡ Ó ) × wR/¥?> ) = : = Ñ Ó Ó P × Ó : ×× 5 3 4 Ñ S e Þ Ó ) × Ø Ñ Ë GD d <0 354 ` d ) 66 Ë e 0 & <?> Ø <21 d 0 Ñ e 6 d 34 e GD Ó Ó k × × ID d Ó k ) ×c6 Ó P × e = ` . <o1-34 34 34 Ó6 Ë× ] 0 Ñ ³[ Ë × )` Ø @,B 6 S GD Ó 0 Ñ ) = 7 < 34 ] F D S D D F F YD ) FGF Ñ : P= = Ø @,B 6 S F ) FGF GD <0 34 ) = Ó Ë = P = × e ) Þ F D S D D F YD Ë<% ) ` ËGËG²¡ËG² È Ç A · µ Ç Ç ´È)ÆÛÇ 9 ; O ?A Ó k × × Ë Ó Ó k ×c Ó P ×× Ó ^ 34 < 354 I GD d Ó Ó : ) c× 6 Ó P ×× e Ë P = Ë ^ = 34 Ó6 Ë× Ó ËGËG²~Ì × /k¥ Ó ËGËG² % × wR/¥P+r¨©H>.<k¡R² Ø . GD ?{ "!~! V! V! ? (+*>% k Ç á Ó Ó ×× Ó Ó × Ó P ×× Þ Ó Ë/ËG² % × . GD : < 3 4 = k c = P = 34 P ` Ñ. Ó k ` Ó k ×× . Ñ Ó P × Ø . < GD Ó 35Ó4 P × ×vÞ g ° Y£H<?>o+J³ G¡G?>E²·È)Gf¨I¨©±Ps+àR¡R£E¨GI>HAc³´iT?«/in£Ew¨©£R¡i/+¥/ =±D8R¨©cGw ° G¨uk6¡G^«/R¡A£E¡wC >R²l +R¨©W/¨©oá³´i¥+ _E¨E=Û£?3 >©H > +·£?3 >©·³+E¨ ° <+¨ DcG¡f >o ° ¡fE.¨ >E£EwGÏ8Eá³´RE+O£R¡¨£Ew /+¥P+Of± DcE¨8Gwà AE >©VI+¥ª+ £?3>©O³+E¨r+E¨rE >okT D8G¡= ° ¡fRM¨ >E£E¡G² O * $ & gÊrT?«Grk ° /w¡J³w+¢s¨C>.<+[ [{ ! ! B! / />YU *&V(8-D&[&V(+* /_-D&4*>C)C 5)&P*5E/]-T/[email protected]+(+*>M\*w243WM - D5)YJ/QP ` (+*>% &V(+*W5720% *>%8/>5)&BR 5E/ Ó Ó ©× ×µØ Ë Ó × Ë Ó Ë Ó Ó ×× Ó × Ó Ó Ë× Ë Ó × Ó × Ë Ë Ó Ó ×× Ó × Ó × " k ` k k : GD . . 34 < 1 P k : GD : GD d 6 ) < 3 4 3 4 6 P = : P k e 1 34 d = P = GD k GD d = < 354 354 = P = I P k e d ^ 354 = P = = Ë<n Ó k × × d Ë 34 ×e I Ë ^ P = = Ó k × e 34 Ó k ) c× 6 Ó P × e I Ó6 Ë× ` ;=(+*>% P = = Ó P × x Ó k v× Þ ÈµlÆ$ËGË/² V È ÖÈ Ç o µ Ç ¸Ç ®È 6Æ oÈ VÇ Æ uÆ 9 #J =# á= gc6Ç; }e} 6} Fu1n<{ B{ +OwGP¥+E? >[ᱪI koN> " Ó k ` Ó k ××uØ * L 5 P S b " dJ ) = : = : < " Ó k ×e Ë . C \ +Ow/¶¥kN >¨¡?<kw/ ° <++£q¡G ^ " N>+I «/I+N>©¶³kR 0 [ : Ë J= e = : < k × P ×× ®Ü = ) d Ó Ó Ó ` \ P = [] /k¥ = : = : < k × 0 Ñ Ü I+¥ ) ÚÝÜ k Ú P Þ ) " Ó ` E cW+r£R¡¨£Ewr³¡ªE&¯ <TI0 ¡G ) = : = : < " Ó k ×uØÜ ` /+¥ f± D &< 0 = ) = : d Ë P = e = : < " "Ó !VÓ k × !VÓ P ××uØ ÜkÞ # %$&'( H>')* 8 7D 9: Ø ; Ó ËGËG² n × 0 % D+.>),+-M>-$./')0 1$23$-45' (6 VÓ × VÓ × VÓ ×EÞ VÓ × !VÓ k × ` < P = P k e - P = P ! ` " d = ! = ! 0 ; < " P = ! P = 0 # <2=$->?7( 2'@AB)C?>E> D)C 2E @ ) G ` Ø Ë< ËGË¡Ë È Ç Ç Ç ´È ÆÛÇ 9 #; # # # O Ç ?# ^ Ó × " ØÜ Ø Ü " ^ " Trapped ions Ó × ^ Ø Ü " " ^ " Trapped ions ^ " @< 3ËGË < ØÜ -=$->?7( @'A )C?> Ë< ØÜ ) È #lÆ$ËGË È ÖÈ Ç Ç ¸Ç oÈ ÇVÆ uÆ ®È 6Æ D# 9 # # # ?#= # gc ; # J # = 6Ç # 2( ?>>?7 k N> Ó VÓ ××uØ C BKC 6 D 6 S YD ) @ P " k ` ! k . b " d F b"d Ø )= : = ` VÓ × = : < " ! k e G 0 VÓ × : < k e G ` "! 0 > Ø Ë ÑÚÝÓ Ü VÓ × VÓ ×× Ü ` = : : < P+G ) = d P = e = ! k c ! P ) = : = : < " ! : ` ) 0 ` 0 VÓ × Ü ' ( Ø Ø VÓ × VÓ × Ø : < k e G " b = J d ` 6 "! C BJC 6 @ D S P YD ) F 0 : : < b " d )= " ! k e G : = 0 BA Ø b"d Ø b " Jd VÓ × Ë = : < " ! k e G ` 0 := < k e G ` "! 0 < VÓ × Þ ËGË¡Ë È Ç Ç Ç ´È ÆÛÇ 9 #; # # # Ç ?# O Ø Ë Ñ Ó VÓ × VÓ ×× Ü ` K= ) = : d P e = : < "! k c ! P = 0 < ! k ) " ` 0 P+G ) = : d P J= = e = : < ! : c ! P 0 < ! k < P = P c k e ) " ` d ! ! " = 0 0 < k = Y ) = "! 0 # D 2$. < 3 ,ID '?> >. > ' ( P G VÓ × ÚÜ Ú Ø Ë Ú ^ËGË Ì VÓ × VÓ × VÓ × Ü ÓÑ VÓ × VÓ ×× Ü ` VÓ × ) @< # >3.> <U ' ( # D# 9 # # # ?#= # gc ; # J = # 3>M> @ +>.D $ ' $&3( C>HA % 1 d 0 " e G = GD ! k . <21-34 3 4 ] ' ( " A @,B 6 H H = GD 0 < 34 F S D PZ <K34 G = ! k GD d " e < 3 4 0 # ^ k ! 34 : % d ^ ! k e ` 34 vÞ < . 1 1 - 6 < F 1 Y ` = H H ] ! k : GD ! k 354 < 3 4 AFD?< ?> ') ! k e 34 d ^ Þ $-'D>- B> b " d ) = : = : < " ! k e G ` 0 3>M> @ +>.D $ ' $&3( C>HA <21 . d <21-0 3" 4 e G = GD 3! 4 k @,B 6 S GD 0 " ) = < 34 D PZ j 6 6S D Z P 66 Z e j " H H= e H H ) d Z I H H GD d f j < 30 4 " <21 d 0 e G = GD ! k @,B 6 GD 0 " ) = = ) N . <21-34 34 < 34 S j D PZ > 6 6S DP 6 Z " H= H H Z 6 j G D H H Z = jf < 30 4 : = ! : ! P " )= P d 34 " e GD d 0 e 354 ` ) 6 6 0 Ü Þ #<I> 0 < ` GD ! k d . ^ 354 I GD d ! P) 6 ! k e P = = 354 6 AFD?< ?> ') 2 '2>7A G GD d ! P) 6 ! k e 53 4 6 ?# ,@ B 6 GD 0 " ) = ) ` < 34 S j D PZ A @,B 6 GD d 0 " ) = = J= P e ) = < 34 S j D PZ G " 2< 1 d 0 e G = d 53 4 " e ID ! k . <21-34 34 # O 9 #; # # Þ k e GD ! : ! 34 < 34 = k d ^ = P = I 35! 4 e '2(?> 6S " : " = G ! 6 j D P 6Z d 0 e GD GD 0 ) = S D 6 YD F D ) FAF . <21-34 34 < 34 F DP I GD 0 " = = ) Z ` < 354 3>.>32 @ 2+D$-=( H>> H>MD $ A 0& ' " )= <21 d 0 " e G = GD ! : j 6 S D P 6 Z ) ) ` I D 04 6 S D 6 YD F D ) FAF . <21-34 34 < 3 F DP Z -3 6 6S DP 6 Z " = Z 6 j ) I D H H Z ` 6 < 30 4 CF 6 F D S P YD j) f F : = k P Z = ) " " : : 3 4 !6 ! = P e GD ! GD d ) ) 0 0 " 34 < 34 < 34 ` 6 0 Þ # D# 9 # # # ?#= # gc ; # J = # # d 0 " e G = <21 GD ! : . <21-354 34 > k k 6S " j D P 6Z G D 04 6S 6 ) < 3 F G F F D D Y D F DP 6S Z " j D P 6Z G D 6 ) < 30 4 D S P D 6 D F F YD GF F &?A k Z < " ! k 0 k I #&<?>HA . : I < ^ ! k 354 1 0 " : = k P ! P )! 6 e ) ` 6 = k 'A Þ < P = P k e ! " d = ! ) < 30 4 = P = '2 2< 1-34 P = e GD ! k 6 ! P ) d 34 " d = < 6 0 ! k 34 P = = < P = ^ )= 0 " < 34 GD d 3)C3A2 @ $ ' @ ') <'3 2C> )= CÍ ` k ID ! k : 0 " ) = ) 34 < 34 )= GD ! k 34 34 e " I k 0 d 34 = = P = 3 4 ^ AÞ P d = = ! P ! : e 1 3 4 GD d ! k ! : ) 6 ! P e ^ 34 34 6 GD d ! : ! k ) 6 ! P e 34 34 6 = P = P ! k e d = P = = ! = = P = P ! k e d = P = = ! < Þ ! : 34 = = P = Þ < # ?# &2 $- " k ` ! k . # 9/ # # +ß A' ( ' &> " k ` ! k ! : : GD ! k d . . 34 < 34 1 : GD ! : GD ! P ! k e I )6 d 34 < 34 6 P = P = GD : P ! k e d ! 1 = 34 = < = P = I ID d ! k ) 6 ! P e ^ 354 = P = d = 34 6 ! : e 34 ^ P = = ! k 34 ! P ! : e Þ $ & $e* # ( 2 '@A % N> <?>( &$- <'-&$2C> -3 " N> ' a- # $A D?< ?> )C' @.2 @ < 83D?D-(,,( ?>>?7*' >A2^ 2' B> $-'ADI< 2'?> )C D3D?D-( ( -?>3?7 # .72-'(9 D)C @)CA < 2 [{ ! ! B! / />YU *&V(8-D&[&V(+* /_-D&4*>C)C 5)&P*5E/]-T/[email protected]+(+*>M\*w243WM - D5)YJ/QP ` (+*>% &V(+*W5720% *>%8/>5)&BR 5E/ " : ` ! k . . 1 ! k : GD ! k d ^ 354 < 53 4 # $ &%'! ! k e 354 P = = ID d 3 4 ! k ) 6 ! P e 6 : . 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( 6 ! 6 # )C2E2 @ 3DID &2A 2'?> =<?>3-( ! k E> [email protected]'[email protected] 2 > C>.D-$. ! P D -A GD ) ! k 354 1 & - ! P = = & . 1 ! ! P 34 3)C3A @D0ID ' ?>9')& >.D '$- $ '@' J -9N> >A '=' ( k 0& ' ) >A -$ AFD?< ?> J : P = = G= ) & P = ) = & ) P = = & '2$D -3 E> ''>7+$ ' >3N<& FJ k & A = # ) ?( # # O #'&>0 >H> 3 > $ e* $ & AA B> ' ( GD # # ( ( IA >MD'$ $ @FJ 1 9 F>.D$- . GD ! ) 34 J : ! 3 4 ! k ! k P P = ! (6 ! = 34 6 P = k ! P = P = 3 4 ! ) ! GD ! ! 1 34 34 34 GD ! P = GD ! ( 6 ! P = 34 53 4 6 '3'HA 3<?>.>3- A ( ?> - ' ( ( ' 2 E> D A ' ' ' ( )C'A0 E > ' ( )BU 3>.>M< A-( .7A < D E<=2 @9> D'82' ! ( $-3 3 >3H> 3 > 2*) ?7 2 3D = D 2 @ $-3> # E>N > D 3E<-( +7 D)C2E @ 2A A : "!$#%#&'()!)*+ & -,./10234#&23"5" !768 9:#&'( !)*<;>=?#/ #@59&A39*2#1(&23BC0ED P P1S 0E59&<!GFH 6 #I & !KJ #L*M N # 6 O5QPR# ! ) TS U /VXWYW I tÍ > )C33A - T ! >.< $ QÍ JÍ IÍ ' ( )C @>< -8702 @'% A LD'+ d J T! N> 3>37=AFD $ >3N<& ' E> D?< MD >33A 3>37=AFD $ - E<=< ') J - ! E> >3A '2 ' ( T N> >3.< ( -(3>)C2E > # >'A*3DID .&2A 2'?> B> )C D3D?D 2 @ $ B> ' <?>3-( -3 '3Z2 @ >.D'$0$ '@D ! N> >A ' ' ( T '( # D# ( # # ( # (?# ( # T ! P P1S TS T - ! T ! P MP1S S T J T! ) ! P ! ) ST ! *) ?>3-&< -873A # 3DID 2 @ $-3> * ) P1 S ! P ! A TS P1S ! P ! S J T! T TS T P P S ! P ) S ! T ) ST ! +3DID 2 @ $-3> * ) P1S ! P ! A TS P P S ! P J T ! T ! TS ) T ! P1S ! P TS ) ! TS ! '2$ 4 3D?D @6$ 3>33A, -3 E>1 ' '>7+$ ' 3> N<& ! J T ! # )C'3A +< 8BD?D @%$-3>N> '2 -( *) >?7 D 3-$-E> 73A >E< 2 ') J T ! 2 ) >?7N> ! T ! E > > A ' ' ( T >A ' T A 0 E<' ! [email protected] S ) S T ! P ) ! PMP1S TS ' (9 -3 3D?D-(?> # $- ( 2 E> >N> )-( 2 $ 3>3 - - ' 3DID (92I> $ 3<I> )C T >A ' A ( A-?>32' 2C>.> E>.>3-&<2' ! T @<'H> J,E> [email protected]'2$( >3- $ 3D - E<-CA& Q>3E< ) -&<2SA 2 $ & ! ! ! ( ! & *) & ?7 N> ! % 3D '$ P T ! &T ? E> >$2SA *( *$H< < 9') J > 3H>.D-$. ,'( E<?> 3( - > - > > 3>37=AFD '$0- E<=< # C) '2E2 @ &< A-32$-' <'N< C> 'D '-SA ! * ! P ) T P ' ( & ! # 0 '$ D'+ )C>MD N > ! ! P P T Ion and electron densities with respect to the radius) 2e+06 ’Electron density’ ’Ion density’ 1.8e+06 1.6e+06 1.4e+06 ne,ni 1.2e+06 1e+06 800000 600000 400000 200000 0 0 5 10 15 20 25 r0 @< 3 " * ??> ' ( --$.'?> ( -?>32C> ( $ < #03HD 3H>8 > 2 C( $H< < ( --$.' [email protected] $C< 3<' ( >>H> > 3H>.D-$. * '( N<?> C D'3 $2>3 Q>-2> $ 3<?>3 ' ( % P $ ) '2$9 2 ( ?>>?7 D C>+.> A &A < A )C3 U # D# ( # # ( # (?# ( # Charge density with respect to the radius in the case where there is no trapped ion 0 ’untrapping case’ -0.005 Charge density rho -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 0 5 10 15 r0 20 25 30 [email protected]< * D '$- $ '[email protected] ( @ > > <'E< * *) > ?7A -3 3> 0--$ ( -?>3?7 >3$>72 $-3>C> > ><'E< 2 ) >?7 $ B> ' ) >MD -30$ '@( @8<'->7 # D$H< 3<D 2$. < 3 ,N> >MD'$D$ '@' JFD 2( H>MD $, ( N<I> D ' D' '7+2$-' $C< 3<'D') J3> - 3> 3?>> H>MD $ 9 '( E<?> ( -?>32C>') 2'?>' ( --$ 2$. < 3 3HD 3H>3-8 > 3DID ( [email protected] $C< 3<' ( 2$3 C-( $H< 3< ( -?>32C> > C>.D-$ '( E<?> % '2$ 9 2' ( -?>3?7 D 3H>8 > A &2A < A )C3* ( @ > > <'N< ,D*) >?7A -3> *--$ 3 ( ?>>?7K>3$>7 $-3 3>3H>9> > <'E< *) ?7 $ B> ') R>.D - $ @( @8><7 2$. < 3 N > L>.D$-*$ '@LJD '3-( C>.D-$. 0'( E<?> # A2 2A < AB) J*)C 3D?D-( 2I> E > E ' A A < A ') J*2 QD .<&<?> $ B > ? = $ 3D&-CA >>< 2 3>E< E>.>' 3> 7>3-A 87 = @ 28 '$$-< 8 [email protected]'+ ? ( >32( J $-AFD?<&-( ''>7+$ '>7 .)C ( Ion and electron densities with respect to the radius - Trapping case 2e+06 ’Electron density’ ’Trapped ion density’ 1.8e+06 1.6e+06 1.4e+06 ni,ne 1.2e+06 1e+06 800000 600000 400000 200000 0 0 @< 3 5 10 15 r0 20 25 30 * ??> ' ( --$.'?> ( -?>32C> ( Charge density with respect to the radius - Trapping case 0 ’Charge density - trapping case’ -0.005 rho -0.01 -0.015 -0.02 -0.025 0 5 10 15 r0 @< 3 20 * D'$ $ '@ 25 30 # D# ( # # ( # (?# ( # ) ! 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