¨¸Ó³ ¢ —Ÿ. 2014. ’. 11, º 6(190). ‘. 1118Ä1121 ”ˆ‡ˆŠ ‹…Œ…’›• —‘’ˆ– ˆ ’Œƒ Ÿ„. ’…ˆŸ ‹…Š’ˆ—…‘ŠŸ ˆ Œƒˆ’Ÿ ‹Ÿˆ‡“…Œ‘’ˆ ’—…—„›• —‘’ˆ– ‘ ‘ˆŒ 1/2 . Ÿ. ‘¨²¥´±μ1 ¡Ñ¥¤¨´¥´´Ò° ¨´¸É¨ÉÊÉ Ö¤¥·´ÒÌ ¨¸¸²¥¤μ¢ ´¨°, „Ê¡´ ˆ´¸É¨ÉÊÉ Ö¤¥·´ÒÌ ¶·μ¡²¥³ ¥²μ·Ê¸¸±μ£μ £μ¸Ê¤ ·¸É¢¥´´μ£μ Ê´¨¢¥·¸¨É¥É , Œ¨´¸± ²¥±É·¨Î¥¸± Ö ¨ ³ £´¨É´ Ö ¶μ²Ö·¨§Ê¥³μ¸É¨ ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í ¸μ ¸¶¨´μ³ 1/2, ¨³¥ÕÐ¨Ì ´μ³ ²Ó´Ò° ³ £´¨É´Ò° ³μ³¥´É, · ¸¸Î¨É ´Ò ¶ÊÉ¥³ ¶·¥μ¡· §μ¢ ´¨Ö ¨¸Ìμ¤´μ£μ £ ³¨²ÓÉμ´¨ ´ ± ¶·¥¤¸É ¢²¥´¨Õ ”첤¨Ä‚ ÊÉÌμ°§¥´ . ·μ¨§¢¥¤¥´μ ¸· ¢´¥´¨¥ ¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì ·¥§Ê²ÓÉ Éμ¢ ¤²Ö Î ¸É¨Í ¸μ ¸¶¨´ ³¨ 1/2 ¨ 1. Electric and magnetic polarizabilities of pointlike spin-1/2 particles possessing an anomalous magnetic moment are calculated with the transformation of an initial Hamiltonian to the FoldyÄWouthuysen representation. Comparison of corresponding results for spin-1/2 and spin-1 particles is performed. PACS: 03.65.Pm; 11.10.Ef; 12.20.Ds ‚ ´ ¸ÉμÖÐ¥° · ¡μÉ¥ ³Ò μ¶·¥¤¥²Ö¥³ Ô²¥±É·¨Î¥¸±ÊÕ ¨ ³ £´¨É´ÊÕ ¶μ²Ö·¨§Ê¥³μ¸É¨ ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í ¸μ ¸¶¨´μ³ 1/2, ¨³¥ÕÐ¨Ì ´μ³ ²Ó´Ò° ³ £´¨É´Ò° ³μ³¥´É (ŒŒ), ¶ÊÉ¥³ ¨¸¶μ²Ó§μ¢ ´¨Ö ¶·¥μ¡· §μ¢ ´¨Ö ”첤¨Ä‚ ÊÉÌμ°§¥´ (”‚). “´¨± ²Ó´Ò¥ ¸¢μ°¸É¢ ¶·¥¤¸É ¢²¥´¨Ö ”‚ [1] ¤¥² ÕÉ ¥£μ ¢¥¸Ó³ Ê¤μ¡´Ò³ ¤²Ö ¶¥·¥Ìμ¤ ± ±¢ §¨±² ¸¸¨Î¥¸±μ³Ê ¶·¨¡²¨¦¥´¨Õ ¨ ´ Ì즤¥´¨Ö ±² ¸¸¨Î¥¸±μ£μ ¶·¥¤¥² ·¥²Öɨ¢¨¸É¸±μ° ±¢ ´Éμ¢μ° ³¥Ì ´¨±¨. „ ¦¥ ¤²Ö ·¥²Öɨ¢¨¸É¸±¨Ì Î ¸É¨Í ¢μ ¢´¥Ï´¥³ ¶μ²¥ μ¶¥· Éμ·Ò ¢ ¤ ´´μ³ ¶·¥¤¸É ¢²¥´¨¨ ¶μ²´μ¸ÉÓÕ ´ ²μ£¨Î´Ò ¸μμÉ¢¥É¸É¢ÊÕШ³ μ¶¥· Éμ· ³ ´¥·¥²Öɨ¢¨¸É¸±μ° ±¢ ´Éμ¢μ° ³¥Ì ´¨±¨. ‚ Î ¸É´μ¸É¨, μ¶¥· Éμ·Ò ¶μ²μ¦¥´¨Ö (ÓÕÉμ´ Ä ‚¨£´¥· ) [2] ¨ ¨³¶Ê²Ó¸ · ¢´Ò r ¨ p = −i∇, μ¶¥· Éμ· ¶μ²Ö·¨§ ͨ¨ ¤²Ö Î ¸É¨Í ¸μ ¸¶¨´μ³ 1/2 ¢Ò· ¦ ¥É¸Ö ¤¨· ±μ¢¸±μ° ³ É·¨Í¥° Π. ‚ ¤·Ê£¨Ì ¶·¥¤¸É ¢²¥´¨ÖÌ Ôɨ μ¶¥· Éμ·Ò μ¶·¥¤¥²ÖÕÉ¸Ö §´ Ψɥ²Ó´μ ¡μ²¥¥ £·μ³μ§¤±¨³¨ Ëμ·³Ê² ³¨ (¸³. [1, 3]). 쳨³μ ¶·μ¸Éμ£μ ¨ μ¤´μ§´ δμ£μ ¢¨¤ μ¶¥· Éμ·μ¢, ¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì ±² ¸¸¨Î¥¸±¨³ ´ ¡²Õ¤ ¥³Ò³, ¢ ¦´¥°Ï¨³ ¤μ¸Éμ¨´¸É¢μ³ ¶·¥¤¸É ¢²¥´¨Ö ”‚ Ö¢²Ö¥É¸Ö ¢μ¸¸É ´μ¢²¥´¨¥ ¢¥·μÖÉ´μ¸É´μ° ¨´É¥·¶·¥É ͨ¨ ¢μ²´μ¢μ° ËÊ´±Í¨¨. μ¸±μ²Ó±Ê, ± ± ʱ § ´μ ¢ÒÏ¥, ¨³¥´´μ ¢ ¶·¥¤¸É ¢²¥´¨¨ ”‚ μ¶¥· Éμ· ÓÕÉμ´ Ä‚¨£´¥· , Ì · ±É¥·¨§ÊÕШ° ¶μ²μ¦¥´¨¥ £¥μ³¥É·¨Î¥¸±μ£μ Í¥´É· Î ¸É¨ÍÒ, · ¢¥´ · ¤¨Ê¸-¢¥±Éμ·Ê r, Éμ ±¢ ¤· É ³μ¤Ê²Ö ¢μ²´μ¢μ° ËÊ´±Í¨¨ μ¶·¥¤¥²Ö¥É ¶²μÉ´μ¸ÉÓ ¢¥·μÖÉ´μ¸É¨ ´ Ì즤¥´¨Ö Î ¸É¨ÍÒ ¢ Éμα¥ ¸ ¤ ´´Ò³ · ¤¨Ê¸-¢¥±Éμ·μ³. ɳ¥É¨³, ÎÉμ 1 E-mail: [email protected] ²¥±É·¨Î¥¸± Ö ¨ ³ £´¨É´ Ö ¶μ²Ö·¨§Ê¥³μ¸É¨ ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í ¸μ ¸¶¨´μ³ 1/2 1119 ¢ ¶·¥¤¸É ¢²¥´¨¨ ”‚ £ ³¨²ÓÉμ´¨ ´ ¨ ¢¸¥ μ¶¥· Éμ·Ò ¤¨ £μ´ ²Ó´Ò ¶μ ¤¢Ê³ ¸¶¨´μ· ³ (¡²μ±¤¨ £μ´ ²Ó´Ò). ˆ¸¶μ²Ó§μ¢ ´¨¥ ÔÉμ£μ ¶·¥¤¸É ¢²¥´¨Ö ʸɷ ´Ö¥É ¢μ§³μ¦´μ¸ÉÓ ¶μÖ¢²¥´¨Ö ´¥μ¤´μ§´ δμ¸É¥° ¶·¨ ·¥Ï¥´¨¨ § ¤ Ψ ´ Ì즤¥´¨Ö ±² ¸¸¨Î¥¸±μ£μ ¶·¥¤¥² ·¥²Öɨ¢¨¸É¸±μ° ±¢ ´Éμ¢μ° ³¥Ì ´¨±¨ [1, 4]. Š ´ ¨¡μ²¥¥ ¢ ¦´Ò³ ¶¥· ³¥É· ³, Ì · ±É¥·¨§ÊÕШ³ Î ¸É¨ÍÒ ¨ Ö¤· , ¶·¨´ ¤²¥¦ É ¸± ²Ö·´Ò¥ Ô²¥±É·¨Î¥¸± Ö ¨ ³ £´¨É´ Ö ¶μ²Ö·¨§Ê¥³μ¸É¨. ˆÌ ¢±² ¤ ¢ £ ³¨²ÓÉμ´¨ ´ ¢ ¶·¥¤¸É ¢²¥´¨¨ ”‚ μ¶·¥¤¥²Ö¥É¸Ö ¢Ò· ¦¥´¨¥³ 1 1 ΔHFW = − αs E 2 − βs B 2 . 2 2 (1) ŒÒ · ¸¸³ É·¨¢ ¥³ ¸²ÊÎ ° ¸É Í¨μ´ ·´ÒÌ ¨ μ¤´μ·μ¤´ÒÌ Ô²¥±É·¨Î¥¸±μ£μ (E) ¨ ³ £´¨É´μ£μ (B) ¶μ²¥° ¨ ¨¸¶μ²Ó§Ê¥³ ¸¨¸É¥³Ê ¥¤¨´¨Í = 1, c = 1. …¸²¨ ¢ ±² ¸¸¨Î¥¸±μ° ˨§¨±¥ Î ¸É¨Í ³μ¦¥É ¨³¥ÉÓ ´¥´Ê²¥¢Ò¥ ¶μ²Ö·¨§Ê¥³μ¸É¨ Éμ²Ó±μ ¶·¨ ´ ²¨Î¨¨ Ê ´¥¥ ¢´ÊÉ·¥´´¥° ¸É·Ê±ÉÊ·Ò, Éμ ¢ ±¢ ´Éμ¢μ° ³¥Ì ´¨±¥ μ´¨ ¶μÖ¢²ÖÕÉ¸Ö ¤ ¦¥ Ê ÉμΥδμ¶μ¤μ¡´ÒÌ μ¡Ñ¥±Éμ¢. ¶·¥¤¥²¨ÉÓ Ôɨ ¶ · ³¥É·Ò ¶μ§¢μ²Ö¥É ¶·¥μ¡· §μ¢ ´¨¥ ”‚ ¨¸Ìμ¤´μ£μ Ê· ¢´¥´¨Ö „¨· ± Ä Ê²¨ [5] ¤²Ö Î ¸É¨Í ¸ ŒŒ ¸ ¶μ¸²¥¤ÊÕШ³ ¶¥·¥Ìμ¤μ³ ± ±² ¸¸¨Î¥¸±μ³Ê ¶·¥¤¥²Ê. ¤´ ±μ ¤²Ö ·¥Ï¥´¨Ö ¤ ´´μ° § ¤ Ψ ´¥μ¡Ì줨³μ ¢ÒΨ¸²¥´¨¥ ¸² £ ¥³ÒÌ, ±¢ ¤· ɨδÒÌ ¶μ ¢´¥Ï´¥³Ê ¶μ²Õ. É ¶·μÍ¥¤Ê· É·¥¡Ê¥É μ¶·¥¤¥²¥´´μ° μ¸Éμ·μ¦´μ¸É¨, ¶μ¸±μ²Ó±Ê · §´Ò¥ ³¥Éμ¤Ò ¶·¨¢μ¤ÖÉ ± · §²¨Î´Ò³ ·¥§Ê²ÓÉ É ³ (¸³. μ¡§μ· [6] ¨ ¸¸Ò²±¨ É ³). Š ¶· ¢¨²Ó´Ò³ ·¥§Ê²ÓÉ É ³ ¶·¨¢μ¤¨É ¨¸¶μ²Ó§μ¢ ´¨¥ ³¥Éμ¤ ·¨±¸¥´ [7]. ˆ¸Ìμ¤´Ò° £ ³¨²ÓÉμ´¨ ´ „¨· ± Ä Ê²¨ [5] Ê¤μ¡´μ · §¤¥²¨ÉÓ ´ Υɴҥ ¨ ´¥Î¥É´Ò¥ ¸² £ ¥³Ò¥, ±μ³³Êɨ·ÊÕШ¥ ¨ ´É¨±μ³³Êɨ·ÊÕШ¥ ¸ ¤¨· ±μ¢¸±μ° ³ É·¨Í¥° β ¸μμÉ¢¥É¸É¢¥´´μ: HD = βm + E + O, βE = Eβ, βO = −Oβ. (2) ‡¤¥¸Ó E = eΦ − μ Π · B, O = cα · π + iμ γ · E, (3) £¤¥ μ Å ŒŒ. ŒÒ ¨¸¶μ²Ó§Ê¥³ μ¡Òδҥ μ¡μ§´ Î¥´¨Ö [8] ¤²Ö ¤¨· ±μ¢¸±¨Ì ³ É·¨Í. §²μ¦¥´¨¥ £ ³¨²ÓÉμ´¨ ´ ¢ ¶·¥¤¸É ¢²¥´¨¨ ”‚ ¶μ ¸É¥¶¥´Ö³ 1/m, ¶μ²ÊÎ¥´´μ¥ ³¥Éμ¤μ³ ·¨±¸¥´ , ¶·¨¢¥¤¥´μ ¢ · ¡μÉ Ì [6, 9, 10]. ‚ ´ ²¨§¨·Ê¥³μ³ ¸²ÊÎ ¥ ¸² £ ¥³Ò¥, ¶·μ¶μ·Í¨μ´ ²Ó´Ò¥ Υɢ¥·Éμ° ¨ ¡μ²¥¥ ¢Ò¸μ±¨³ ¸É¥¶¥´Ö³ μ¡· É´μ° ³ ¸¸Ò, ³μ¦´μ ´¥ ÊΨÉÒ¢ ÉÓ. ‚ ÔÉμ³ ¸²ÊÎ ¥ £ ³¨²ÓÉμ´¨ ´ ¢ ¶·¥¤¸É ¢²¥´¨¨ ”‚ μ¶·¥¤¥²Ö¥É¸Ö Ê· ¢´¥´¨¥³ O4 O2 β 1 − [O, [O, F ]] + {O, [[O, F ], F ]} , (4) HFW = β m + +E − 2m 8m3 8m2 16m3 £¤¥ F = E − i∂/∂t. „²Ö · ¸¸³ É·¨¢ ¥³μ° ¸É Í¨μ´ ·´μ° § ¤ Ψ F = E. ¸Î¥É ¶μ Ëμ·³Ê² ³ (3), (4) ¶·¨¢μ¤¨É ± ¸²¥¤ÊÕÐ¥³Ê ¢Ò· ¦¥´¨Õ: HFW π4 π2 1 μ0 (2Σ · [π × E] − ∇ · E) − − + μ =β m+ + eΦ + 2m 8m3 2m 2 μ (μ0 + μ )μ 2 μ20 2 − (μ0 + μ ) Π · B + E B , (5) {Π · π, π · B} + β − β 4m2 2m 2m £¤¥ μ0 = e/(2m) Å ¤¨· ±μ¢¸±¨° ³ £´¨É´Ò° ³μ³¥´É. 1120 ‘¨²¥´±μ . Ÿ. ɳ¥É¨³, ÎÉμ ¶μ¸²¥¤´¥¥ ¸² £ ¥³μ¥ ¢ Ê· ¢´¥´¨¨ (4) ´¥ ¢´μ¸¨É ¢±² ¤ ¢ Ô²¥±É·¨Î¥¸±ÊÕ ¨ ³ £´¨É´ÊÕ ¶μ²Ö·¨§Ê¥³μ¸É¨. ¸Î¥É ¸ ¶μ³μÐÓÕ ³¥Éμ¤ , ¶·¥¤²μ¦¥´´μ£μ ¢ μ·¨£¨´ ²Ó´μ° · ¡μÉ¥ ”첤¨ ¨ ‚ ÊÉÌμ°§¥´ [1], ¨ ¤·Ê£¨Ì ¨É¥· Í¨μ´´ÒÌ ³¥Éμ¤μ¢ (¸³. [10, 11] ¨ ¸¸Ò²±¨ É ³) ¶·¨¢μ¤¨É ± ¨´μ³Ê ¢¨¤Ê ÔÉμ£μ ¸² £ ¥³μ£μ ¨, ± ± ¸²¥¤¸É¢¨¥, ´¥ ¤ ¥É ¶· ¢¨²Ó´μ£μ ¢Ò· ¦¥´¨Ö ¤²Ö Ô²¥±É·¨Î¥¸±μ° ¶μ²Ö·¨§Ê¥³μ¸É¨. ‘· ¢´¥´¨¥ Ê· ¢´¥´¨° (1) ¨ (5) ¶μ± §Ò¢ ¥É, ÎÉμ ¸± ²Ö·´Ò¥ Ô²¥±É·¨Î¥¸± Ö ¨ ³ £´¨É´ Ö ¶μ²Ö·¨§Ê¥³μ¸É¨ ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í, μ¡² ¤ ÕÐ¨Ì ŒŒ, ¨³¥ÕÉ ¢¨¤ αS = − e2 g(g − 2) (μ0 + μ )μ =− , m 16m3 βS = e2 μ20 = . m 4m3 (6) Œ É·¨ÍÊ β ³μ¦´μ μ¶Ê¸É¨ÉÓ, ¶μ¸±μ²Ó±Ê ¢ ¶·¥¤¸É ¢²¥´¨¨ ”‚ ´¨¦´¨° ¸¶¨´μ· · ¢¥´ ´Ê²Õ. ‚ ¦´Ò³ Ö¢²Ö¥É¸Ö ¸· ¢´¥´¨¥ ¶μ²Ö·¨§Ê¥³μ¸É¥° ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í ¸μ ¸¶¨´ ³¨ 1/2 ¨ 1. — ¸É¨ÍÒ ¸μ ¸¶¨´μ³ 1 Ì · ±É¥·¨§ÊÕÉ¸Ö ´¥ Éμ²Ó±μ ¸± ²Ö·´Ò³¨, ´μ ¨ É¥´§μ·´Ò³¨ ¶μ²Ö·¨§Ê¥³μ¸ÉÖ³¨, · ¸¸Î¨É ´´Ò³¨ ¢ [12]. ‘± ²Ö·´Ò¥ ¶μ²Ö·¨§Ê¥³μ¸É¨ É ±¨Ì Î ¸É¨Í μ¶·¥¤¥²ÖÕÉ¸Ö ¢Ò· ¦¥´¨Ö³¨ [12]: αS = − e2 (g − 1)2 , m3 βS = 0. (7) ’ ±¨³ μ¡· §μ³, ¸± ²Ö·´ Ö ³ £´¨É´ Ö ¶μ²Ö·¨§Ê¥³μ¸ÉÓ Î ¸É¨Í ¸μ ¸¶¨´μ³ 1 · ¢´ ´Ê²Õ, ¸± ²Ö·´ Ö Ô²¥±É·¨Î¥¸± Ö ¶μ²Ö·¨§Ê¥³μ¸ÉÓ μɲ¨Î´ μÉ ´Ê²Ö ¤²Ö Î ¸É¨Í ´¥ Éμ²Ó±μ ¸ ´μ³ ²Ó´Ò³, ´μ ¨ ¸ ´μ·³ ²Ó´Ò³ (g = 2) ³ £´¨É´Ò³ ³μ³¥´Éμ³. ɨ ¸¢μ°¸É¢ μɲ¨Î ÕÉ¸Ö μÉ ¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì ¸¢μ°¸É¢ Î ¸É¨Í ¸μ ¸¶¨´μ³ 1/2. ɳ¥É¨³, ÎÉμ ¸± ²Ö·´Ò¥ ¶μ²Ö·¨§Ê¥³μ¸É¨ ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í ¸μ ¸¶¨´μ³ 0 · ¢´Ò ´Ê²Õ (¸³. [13]). ¡μÉ ¶μ¤¤¥·¦ ´ £· ´Éμ³ ¥²μ·Ê¸¸±μ£μ ·¥¸¶Ê¡²¨± ´¸±μ£μ Ëμ´¤ ËÊ´¤ ³¥´É ²Ó´ÒÌ ¨¸¸²¥¤μ¢ ´¨°. ‘ˆ‘Š ‹ˆ’…’“› 1. Foldy L. L., Wouthuysen S. A. On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit // Phys. Rev. 1950. V. 78. P. 29Ä36. 2. Newton T. D., Wigner E. P. Localized States for Elementary Systems // Rev. Mod. Phys. 1949. V. 21. P. 400Ä406. 3. Silenko A. J. FoldyÄWouthuysen Transformation for Relativistic Particles in External Fields // J. Math. Phys. 2003. V. 44. P. 2952Ä2966. 4. Costella J. P., McKellar B. H. J. The FoldyÄWouthuysen Transformation // Am. J. Phys. 1995. V. 63. P. 1119Ä1121. 5. Pauli W. Relativistic Field Theories of Elementary Particles // Rev. Mod. Phys. 1941. V. 13. P. 203Ä 232. 6. de Vries E. FoldyÄWouthuysen Transformations and Related Problems // Fortschr. Phys. 1970. V. 18. P. 149Ä182. 7. Eriksen E. FoldyÄWouthuysen Transformation. Exact Solution with Generalization to the TwoParticle Problem // Phys. Rev. 1958. V. 111. P. 1011Ä1016. 8. ¥·¥¸É¥Í±¨° ‚. ., ‹¨ËÏ¨Í …. Œ., ¨É ¥¢¸±¨° ‹. . Š¢ ´Éμ¢ Ö Ô²¥±É·μ¤¨´ ³¨± . 3-¥ ¨§¤. Œ.: ʱ , 1989. 720 ¸. ²¥±É·¨Î¥¸± Ö ¨ ³ £´¨É´ Ö ¶μ²Ö·¨§Ê¥³μ¸É¨ ÉμΥδμ¶μ¤μ¡´ÒÌ Î ¸É¨Í ¸μ ¸¶¨´μ³ 1/2 1121 9. de Vries E., Jonker J. E. Non-Relativistic Approximations of the Dirac Hamiltonian // Nucl. Phys. B. 1968. V. 6. P. 213Ä225. 10. ‘¨²¥´±μ . Ÿ. ‘· ¢´¨É¥²Ó´Ò° ´ ²¨§ ³¥Éμ¤μ¢ ¶·Ö³μ£μ ¨ ®Ï £ § Ï £μ³¯ ¶·¥μ¡· §μ¢ ´¨Ö ”첤¨Ä‚ ÊÉÌμ°§¥´ // ’Œ”. 2013. T. 176. C. 189Ä204. 11. Neznamov V. P., Silenko A. J. FoldyÄWouthuysen Wave Functions and Conditions of Transformation between Dirac and FoldyÄWouthuysen Representations // J. Math. Phys. 2009. V. 50. P. 122302. 12. Silenko A. J. Quantum-Mechanical Description of Spin-1 Particles with Electric Dipole Moments // Phys. Rev. D. 2013. V. 87. P. 073015. 13. ‘¨²¥´±μ . Ÿ. ¶¥· Éμ· ƒ ³¨²ÓÉμ´ ¨ ±¢ §¨±² ¸¸¨Î¥¸±¨° ¶·¥¤¥² ¤²Ö ¸± ²Ö·´ÒÌ Î ¸É¨Í ¢ Ô²¥±É·μ³ £´¨É´μ³ ¶μ²¥ // ’Œ”. 2008. T. 156. C. 398Ä411. μ²ÊÎ¥´μ 11 ³ ·É 2014 £.
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