уравнения фаддеева и метод гиперсферических гармоник в

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2012. ’. 43. ‚›. 3
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2012. ’. 43. ‚›. 3
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¤²Ö § ¤ Ψ ¨´Ë¨´¨É´μ£μ ¤¢¨¦¥´¨Ö ¢¸¥Ì É·¥Ì ¸¨²Ó´μ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì Î ¸É¨Í ¨ μ¸ÊÐ¥¸É¢²¥´ ³μ¤¨Ë¨± Í¨Ö Ê· ¢´¥´¨° ” ¤¤¥¥¢ ¤²Ö ¸¨¸É¥³Ò É·¥Ì ¤·μ´μ¢ ¸ ±Ê²μ´μ¢¸±¨³ ¢§ ¨³μ¤¥°¸É¢¨¥³ ¢
±μ´É¨´Êʳ¥. ·¥¤¸É ¢²¥´ μ¡§μ· ´ ²μ£¨Î´ÒÌ ³¥Éμ¤μ¢ ¨¸¸²¥¤μ¢ ´¨Ö É·¥ÌÎ ¸É¨Î´ÒÌ ¸¨¸É¥³.
The method of solving Faddeev's equations in conˇguration space for three-nucleon system in
continuum using expansions in hyperspherical basis is developed. The wave functions of N d-system,
phases, and cross sections of N d-scattering at subthreshold energies are calculated. Within the
framework of this method, one-dimensional integral equations for the problem of inˇnite motion of
all three strongly interacting particles are also formulated and the modiˇcation of Faddeev's equations
for the system of three hadrons with Coulomb interaction in continuum is made. The review of
similar methods of investigation of three-particle systems is presented.
PACS: 03.65.Nk; 21.45.-v; 31.15.xj
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fλ (ρ)Φiλ (ρ, Ω).
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Í¥²ÓÕ μÌ¢ ɨÉÓ ¢¥¸Ó μ¡Ï¨·´¥°Ï¨° ¸¶¥±É· · ¡μÉ ¶μ ¶·¨³¥´¥´¨Õ Ê· ¢´¥´¨°
” ¤¤¥¥¢ ¨ ³¥Éμ¤ £¨¶¥·¸Ë¥·¨Î¥¸±¨Ì £ ·³μ´¨± ± § ¤ Î¥ É·¥Ì É¥², ÎÉμ, ¥¸É¥¸É¢¥´´μ, ¶·μ¸Éμ ´¥¢μ§³μ¦´μ ¢ · ³± Ì μ¤´μ° ¸É ÉÓ¨. ¶¨¸ ÉÓ ¸μμÉ¢¥É¸É¢ÊÕШ¥ Ô±¸¶¥·¨³¥´ÉÒ É ±¦¥ ´¥ ¡Ò²μ ¸ ³μÍ¥²ÓÕ. ŒÒ ¶μ¶ÒÉ ²¨¸Ó μ¡Ñ¥¤¨´¨ÉÓ
³ É¥³ ɨΥ¸±ÊÕ ±μ··¥±É´μ¸ÉÓ Ê· ¢´¥´¨° ” ¤¤¥¥¢ (¶Ê¸ÉÓ ¨ ³μ¤¨Ë¨Í¨·μ¢ ´´ÒÌ) ¸ μÉ´μ¸¨É¥²Ó´μ° ¶·μ¸ÉμÉμ° ³¥Éμ¤ £¨¶¥·£ ·³μ´¨±, ¨¸¶μ²Ó§ÊÖ ¶·¨ ÔÉμ³
¨¤¥Õ μ · §¤¥²¥´¨¨ ±μ´Ë¨£Ê· Í¨μ´´μ£μ ¶·μ¸É· ´¸É¢ ´ ®¢´ÊÉ·¥´´ÕÕ¯ ¨
®¢´¥Ï´ÕÕ¯ μ¡² ¸É¨, ¨ ¨³¥´´μ ¢ § ¤ Î¥ ¸ ´¥¶·¥·Ò¢´Ò³ ¸¶¥±É·μ³ (É ± ± ±
¤²Ö ¸¢Ö§ ´´μ£μ ¸μ¸ÉμÖ´¨Ö “” ¨ Œƒƒ ´¥¶²μÌμ · ¡μÉ ÕÉ ± ¦¤Ò° ¢ μɤ¥²Ó´μ¸É¨) μ¶·μ¡μ¢ ÉÓ ÔÉμÉ ¶μ¤Ìμ¤ ¶·¨ μ¡· ¡μɱ¥ Ô±¸¶¥·¨³¥´É ²Ó´ÒÌ ¤ ´´ÒÌ
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 579
¸ ¶μ³μÐÓÕ ¶·μ¸ÉÒÌ ¤¢Ê̴ʱ²μ´´ÒÌ ¶μÉ¥´Í¨ ²μ¢, ±μÉμ·Ò¥ ¤ ¢´μ ¶·¨³¥´ÖÕÉ¸Ö ¢ ± Î¥¸É¢¥ É¥¸Éμ¢ÒÌ ¢ ¶μ¤μ¡´ÒÌ § ¤ Î Ì.
1. …„“Š–ˆŸ “‚…ˆ‰ ”„„……‚
Š ‘ˆ‘’…Œ… “‚…ˆ‰ „‹Ÿ ”“Š–ˆ‰ „‰ ……Œ…‰
1.1. ”μ·³ ²¨§³. ˆ¸Ì줨³ ¨§ ¨§¢¥¸É´ÒÌ Ê· ¢´¥´¨° ” ¤¤¥¥¢ [1, 4], § ¶¨¸ ´´ÒÌ ¤²Ö ¸¨¸É¥³Ò É·¥Ì ¸¨²Ó´μ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì ¡¥¸¸¶¨´μ¢ÒÌ Î ¸É¨Í ¸
줨´ ±μ¢Ò³¨ ³ ¸¸ ³¨ m, ¢ ±μÉμ·μ° Î ¸É¨Í 1 · ¸¸¥¨¢ ¥É¸Ö ´ Î ¸É¨Í Ì 2 ¨
3, ´ Ìμ¤ÖÐ¨Ì¸Ö ¢ ¸¢Ö§ ´´μ³ ¸μ¸ÉμÖ´¨¨:
Ψ(1) = Φ + G0 (Z)T23 (Z)(Ψ(2) + Ψ(3) ),
Ψ(2) = G0 (Z)T31 (Z)(Ψ(3) + Ψ(1) ),
(3)
Ψ(3) = G0 (Z)T12 (Z)(Ψ(1) + Ψ(2) ),
£¤¥ Ψ = Ψ(1) + Ψ(2) + Ψ(3) ¥¸ÉÓ ¶μ²´ Ö ¢μ²´μ¢ Ö ËÊ´±Í¨Ö ¸¨¸É¥³Ò; Φ Å
¸¨³¶ÉμɨΥ¸± Ö ¢μ²´μ¢ Ö ËÊ´±Í¨Ö, 춨¸Ò¢ ÕÐ Ö ¨´Ë¨´¨É´μ¥ ¤¢¨¦¥´¨¥ 1-°
Î ¸É¨ÍÒ μÉ´μ¸¨É¥²Ó´μ ¸¢Ö§ ´´μ° ¶ ·Ò (23); G0 (Z) = (Z −H0 )−1 Å μ¶¥· Éμ·
ƒ·¨´ ; Z = E ± i0; E Å ¶μ²´ Ö Ô´¥·£¨Ö ¸¨¸É¥³Ò 1 + (23); H0 Å μ¶¥· Éμ·
±¨´¥É¨Î¥¸±μ° Ô´¥·£¨¨; Tij Å ¤¢ÊÌÎ ¸É¨Î´Ò¥ 춥· Éμ·Ò ¶¥·¥Ìμ¤ , ¸¢Ö§ ´´Ò¥
¸ ¶ ·´Ò³¨ ¶μÉ¥´Í¨ ² ³¨ Vij (ij = 12, 23, 31) ¶·¨ ¶μ³μШ Ê· ¢´¥´¨°
Tij (Z) = Vij + Vij G0 (Z)Tij (Z).
(4)
ˆ¸¶μ²Ó§ÊÖ (4) ¢ (3), ¸²μ¦¨³ ¶μ²ÊΨ¢Ï¨¥¸Ö Ê· ¢´¥´¨Ö ¨ ¶μ²ÊΨ³
Ψ = (1 − G0 (Z)V23 )Φ + G0 (Z)U Ψ,
U = V12 + V23 + V31 .
(5)
Éμ Ê· ¢´¥´¨¥ Ô±¢¨¢ ²¥´É´μ Ê· ¢´¥´¨Õ ˜·¥¤¨´£¥· ¸ £· ´¨Î´Ò³ ʸ²μ¢¨¥³, 춨¸Ò¢ ÕШ³ ¸¨³¶ÉμɨΥ¸±¨ ¸¢μ¡μ¤´μ¥ ¤¢¨¦¥´¨¥ Î ¸É¨ÍÒ 1 μÉ´μ¸¨É¥²Ó´μ Í¥´É· ³ ¸¸ ¸¢Ö§ ´´μ° ¶ ·Ò (23). „¥°¸É¢¨É¥²Ó´μ, ¥¸²¨ ¶μ¤¥°¸É¢μ¢ ÉÓ
¨ ÊÎ¥¸ÉÓ, ÎÉμ ¤²Ö ¤ ´´μ° ¸¨³¶Éμ´ Ê· ¢´¥´¨¥ (5) ¸²¥¢ 춥· Éμ·μ³ G−1
0
ɨ±¨ (H0 + V23 − E)Φ = 0, ¶·¨Ì줨³ ± Ê· ¢´¥´¨Õ ˜·¥¤¨´£¥· ¤²Ö Ψ. ’ ±¨³ μ¡· §μ³, Ψ Ê¤μ¢²¥É¢μ·Ö¥É Ê· ¢´¥´¨Õ ˜·¥¤¨´£¥· , ¨, ¸²¥¤μ¢ É¥²Ó´μ, (5)
´¥ ¨³¥¥É ´¥Ë¨§¨Î¥¸±¨Ì ·¥Ï¥´¨°. Š·μ³¥ Éμ£μ, ¥¸²¨ ¢Ò¶μ²´¨ÉÓ ¶μ¤·Ö¤ ´¥¸±μ²Ó±μ ¨É¥· ͨ° ¤²Ö Ê· ¢´¥´¨Ö (5), Éμ ³μ¦´μ Ê¡¥¤¨ÉÓ¸Ö, ÎÉμ ´ j-° ¨É¥· ͨ¨ ·Ö¤ ¤²Ö Ψ(j) ´¥ ¡Ê¤¥É ¸μ¤¥·¦ ÉÓ ¸² £ ¥³ÒÌ ¸ ´¥±μ³¶ ±É´Ò³ Ö¤·μ³,
§ ¨¸±²ÕÎ¥´¨¥³ (G0 V23 )j+1 Φ, ÔÉμ ¸² £ ¥³μ¥, ¢ ¸¢μÕ μÎ¥·¥¤Ó, ¨¸Î¥§ ¥É ´ (j + 1)-° ¨É¥· ͨ¨ ¨ É. ¤. ’ ±¨³ μ¡· §μ³, ¢¥¸Ó ¡¥¸±μ´¥Î´Ò° ·Ö¤ ¤²Ö Ψ ¡Ê¤¥É
¸μ¤¥·¦ ÉÓ Éμ²Ó±μ ¸² £ ¥³Ò¥ ¸ ±μ³¶ ±É´Ò³¨ Ö¤· ³¨. ‘²¥¤μ¢ É¥²Ó´μ, Ê· ¢´¥´¨¥ (5) ¨³¥¥É ¥¤¨´¸É¢¥´´μ¥ ·¥Ï¥´¨¥ (¤²Ö ´¥£μ ¢Ò¶μ²´Ö¥É¸Ö ²ÓÉ¥·´ ɨ¢ ”·¥¤£μ²Ó³ Ä” ¤¤¥¥¢ [4]).
580 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
¥·¥¶¨Ï¥³ Ê· ¢´¥´¨¥ (5) ¸²¥¤ÊÕШ³ μ¡· §μ³:
Ψ − Φ = G0 (Z)U (Ψ − Φ) + G0 (Z)(V12 + V31 )Φ,
(6)
¨ · §²μ¦¨³ · §´μ¸ÉÓ Ψ − Φ ¢ ·Ö¤ ¶μ K-£ ·³μ´¨± ³ (¸³. ¶·¨²μ¦¥´¨¥ ):
ψKn (ρ)uKn (Ω).
(7)
Ψ−Φ=
Kn
‚ ¢Òϥʶμ³Ö´ÊÉÒÌ · ¡μÉ Ì ¶μ ¨´É¥·¶μ²ÖÍ¨μ´´μ³Ê ¶μ¤Ìμ¤Ê [9, 44, 45],
É ±¦¥ ¢ [40, 66] ¡Ò²μ ¶μ± § ´μ, ÎÉμ ¨´É¥·¶μ²ÖÍ¨μ´´μ¥ ¢Ò· ¦¥´¨¥ Ψ =
Ψint + Ψext ¤²Ö ¢μ²´μ¢μ° ËÊ´±Í¨¨ Ö¢²Ö¥É¸Ö μÎ¥´Ó Ìμ·μϨ³ ¶·¨¡²¨¦¥´¨¥³.
·¨ ÔÉμ³ · §²μ¦¥´¨¥ Ψint (ÎÉμ ¢ ´ Ï¥³ ¶μ¤Ì줥 μÉ¢¥Î ¥É · §´μ¸É¨ Ψ − Φ)
¶μ £¨¶¥·¸Ë¥·¨Î¥¸±¨³ ËÊ´±Í¨Ö³ μ¡´ ·Ê¦¨¢ ¥É ¡Ò¸É·ÊÕ ¸Ì줨³μ¸ÉÓ ¸μμÉ¢¥É¸É¢ÊÕÐ¥£μ ·Ö¤ (¢ μɲ¨Î¨¥ μÉ ËÊ´±Í¨¨ Ψ) Å μ¡¸ÉμÖÉ¥²Ó´μ³Ê ¤μ± § É¥²Ó¸É¢Ê ÔÉμ£μ ¶μ¸¢ÖÐ¥´ Í¥²Ò° · §¤¥² ¢ [40], £¤¥, ¢ Î ¸É´μ¸É¨, ¶μ± § ´μ, ± ±
´ °É¨ ¸±μ·μ¸ÉÓ Ê¡Ò¢ ´¨Ö Ψint ¸ ·μ¸Éμ³ K. ɳ¥É¨³, ÎÉμ ¤²Ö ʸ±μ·¥´¨Ö
¸Ì줨³μ¸É¨ · §²μ¦¥´¨° ¶μ ¶μ²´μ³Ê ¡ §¨¸Ê K-£ ·³μ´¨± Ë·μ¸ ¨ ‘³μ·μ¤¨´¸±¨° [67Ä71], § É¥³ ” ¡· [72,73] ¶·¥¤² £ ²¨ ¨¸¶μ²Ó§μ¢ ÉÓ Ê¸¥Î¥´´Ò¥ ¡ §¨¸Ò
K-£ ·³μ´¨±, ´ §Ò¢ Ö ¨Ì, ¸μμÉ¢¥É¸É¢¥´´μ, ´ ¡μ· ³¨ £² ¢´ÒÌ ¨ ¶μÉ¥´Í¨ ²Ó´ÒÌ
£¨¶¥·£ ·³μ´¨±.
μ¤¸É ¢¨¢ · §²μ¦¥´¨¥ (7) ¢ (6) ¨ ¨¸¶μ²Ó§ÊÖ Ê¸²μ¢¨Ö μ·Éμ´μ·³¨·μ¢±¨ ¨
¶μ²´μÉÒ ¤²Ö ¸μ¡¸É¢¥´´ÒÌ ËÊ´±Í¨° · ¤¨ ²Ó´μ° Î ¸É¨ 춥· Éμ· H0 , É ±¦¥
ʸ²μ¢¨¥ μ·Éμ´μ·³¨·μ¢±¨ ¤²Ö K-£ ·³μ´¨±, ¶μ²ÊΨ³ ¸¨¸É¥³Ê μ¤´μ³¥·´ÒÌ ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨° ¤²Ö ±μÔË˨ͨ¥´Éμ¢ · §²μ¦¥´¨Ö (¸³. ¶·¨²μ¦¥´¨¥ )
nn
ψKn (ρ) = fKn (ρ) + λ
RKK
(8)
(ρ, ρ̄) ψK n (ρ̄),
K n
£¤¥
fKn (ρ) =
dρ̄ QK (ρ, ρ̄) WKn (ρ̄),
nn
nn
dρ̄ QK (ρ, ρ̄) UKK
RKK
(ρ, ρ̄) =
(ρ̄)
(9)
(10)
Å ¨´É¥£· ²Ó´Ò° 춥· Éμ·. Šμ´¸É ´É λ ¢ (8) ¸μ¤¥·¦¨É ¢ ¸¥¡¥ Ψ¸²μ¢Ò¥
±μÔË˨ͨ¥´ÉÒ ¨ ´Ê±²μ´´ÊÕ ³ ¸¸Ê m. ”Ê´±Í¨¨ Q, W ¨ U , ¸ÉμÖШ¥ ¶μ¤
§´ ±μ³ ¨´É¥£· ² ¢ (9), (10), ¨³¥ÕÉ ¸²¥¤ÊÕШ° ¢¨¤:
QK (ρ, ρ̄) = −(ρ̄ 3 /ρ2 ) I2 (ρ ξK (ρ))K2 (ρ̄ ξK (ρ))Θ(ρ̄ − ρ)+
+ I2 (ρ̄ ξK (ρ))K2 (ρ ξK (ρ))Θ(ρ − ρ̄) , (11)
£¤¥
ξK (ρ) =
K(K + 4) 2mE
− 2 ,
ρ2
E = Ei − ,
(12)
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 581
WKn (ρ̄) =
dΩ u∗Kn (Ω)(V12 (ρ̄, Ω) + V31 (ρ̄, Ω)) Φ(ρ̄, Ω),
nn
UKK
(ρ̄)
=
dΩ u∗Kn (Ω)U (ρ̄, Ω) uK n (Ω).
(13)
(14)
‚¥²¨Î¨´Ò I2 , K2 ¢ (11) Ö¢²ÖÕÉ¸Ö ³μ¤¨Ë¨Í¨·μ¢ ´´Ò³¨ ËÊ´±Í¨Ö³¨ ¥¸¸¥²Ö,
Θ Å ËÊ´±Í¨Ö •¥¢¨¸ °¤ , Ei ¨ ¢ (12) Å Ô´¥·£¨Ö ¶ ¤ ÕÐ¥° Î ¸É¨ÍÒ ¨
Ô´¥·£¨Ö ¸¢Ö§ ´´μ£μ ¸μ¸ÉμÖ´¨Ö ¢ ¸¨¸É¥³¥ É·¥Ì Î ¸É¨Í ( > 0) ¸μμÉ¢¥É¸É¢¥´´μ.
“· ¢´¥´¨Ö (8) ¨³¥ÕÉ ¸ ³Ò° μ¡Ð¨° ¢¨¤ ¨ ³μ£ÊÉ ¡ÒÉÓ ·¥Ï¥´Ò μ¡Òδҳ¨
Ψ¸²¥´´Ò³¨ ³¥Éμ¤ ³¨ ¤²Ö ¶·μ¨§¢μ²Ó´μ£μ ´ ¡μ· K-£ ·³μ´¨±. ‘²¥¤Ê¥É μɳ¥É¨ÉÓ, ÎÉμ ¢ ¶μ²ÊÎ¥´´ÒÌ ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨ÖÌ Ê¦¥ § ²μ¦¥´Ò ´¥μ¡Ì줨³Ò¥
¸¨³¶Éμɨ± ¨ £· ´¨Î´Ò¥ ʸ²μ¢¨Ö, ¢ Éμ ¢·¥³Ö ± ± ¢ ¸²ÊÎ ¥ ¨¸¶μ²Ó§μ¢ ´¨Ö “”
¢ ¤¨ËË¥·¥´Í¨ ²Ó´μ° Ëμ·³¥ [8] £· ´¨Î´Ò¥ ʸ²μ¢¨Ö ´¥μ¡Ì줨³μ ´ ±² ¤Ò¢ ÉÓ
¤μ¶μ²´¨É¥²Ó´μ.
·¨´Í¨¶ ¶μ¸É·μ¥´¨Ö Ê· ¢´¥´¨° (8) ²¥£±μ μ¡μ¡Ð¨ÉÓ É ±¦¥ ¨ ´ ¸²ÊÎ ° § ¢¨¸¨³μ¸É¨ ¢μ²´μ¢μ° ËÊ´±Í¨¨ Ψ μÉ ¸¶¨´ ¨ ¨§μ¸¶¨´ ´Ê±²μ´μ¢. ‚ ÔÉμ³ ¸²ÊÎ ¥
· §´μ¸ÉÓ Ψ − Φ · ¸±² ¤Ò¢ ¥É¸Ö ¶μ ¡ §¨¸´Ò³ ´É¨¸¨³³¥É·¨Î´Ò³ ¸μ¸ÉμÖ´¨Ö³
ψKn (ρ)ΓKn (Ω; σ, τ ).
(15)
Ψ−Φ=
Kn
[g]
‘μ¸ÉμÖ´¨Ö ΓKn ¸É·μÖÉ¸Ö ¨§ £¨¶¥·¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨° uKn (Ω) ¸ μ¶·¥¤¥²¥´´Ò³ ɨ¶μ³ ¶¥·¥¸É ´μ¢μÎ´μ° ¸¨³³¥É·¨¨ [g]: [a], [s], [ ], [ ] (¸μμÉ¢¥É¸É¢¥´´μ ´É¨¸¨³³¥É·¨Î´Ò³, ¸¨³³¥É·¨Î´Ò³ ¨ ¸³¥Ï ´´μ° ¸¨³³¥É·¨¨) ¨ ¸¶¨´[ḡ]
¨§μ¸¶¨´μ¢ÒÌ ËÊ´±Í¨° ξn (σ, τ ) ¸ ¸μ¶·Ö¦¥´´μ° ¸¨³³¥É·¨¥° [ḡ]: [s], [a], [ ], [ ].
’ ±, ¤²Ö ¸¨¸É¥³Ò É·¥Ì ´Ê±²μ´μ¢ ¸ S = 1/2, T = 1/2 ΓKn ¨³¥¥É ¢¨¤ [74]
[a]
[ ]
[s]
[ ]
ΓKn = uKn ξ [s] − uKn ξ [a] + uKn ξ [ ] − uKn ξ [ ] .
(16)
…¸²¨ ¦¥, ´ ¶·¨³¥·, ¸¨¸É¥³ ´ Ìμ¤¨É¸Ö ¢ ¸μ¸ÉμÖ´¨¨ ¸ S = 3/2, T = 1/2
(±¢ ·É¥É´μ¥ N d-· ¸¸¥Ö´¨¥), Éμ
[ ]
[ ]
ΓKn = uKn ξ [ ] − uKn ξ [ ] .
(17)
Œ¥É줨± ¶μ²ÊÎ¥´¨Ö Ê· ¢´¥´¨° ¤²Ö · ¤¨ ²Ó´ÒÌ ±μ³¶μ´¥´É ψKn (ρ) ´ ²μ£¨Î´ ¨§²μ¦¥´´μ° ¢ÒÏ¥, ´μ ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ ¤μ¶μ²´¨É¥²Ó´μ£μ ¸μμÉ´μÏ¥´¨Ö μ·Éμ£μ´ ²Ó´μ¸É¨ ¤²Ö ξn (σ, τ ):
†
ξ [g] ξ [g ] = δgg .
(18)
στ
·¨ ÔÉμ³, ±μ´¥Î´μ, ´ ¤μ ¥Ð¥ ÊÎ¥¸ÉÓ, ÎÉμ ¶ ·´Ò° ¶μÉ¥´Í¨ ² ¡Ê¤¥É ¸μ¤¥·¦ ÉÓ
³ É·¨ÍÒ Ê²¨ ¨ ¸¶¨´-¨§μ¸¶¨´μ¢Ò¥ ¶·μ¥±Í¨μ´´Ò¥ 춥· Éμ·Ò.
582 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
1.2. ¥¶·¥·Ò¢´Ò° ¸¶¥±É·: N d-· ¸¸¥Ö´¨¥. ¥ μ£· ´¨Î¨¢ Ö μ¡Ð´μ¸É¨,
· ¸¸³μÉ·¨³ ¢´ Î ²¥ ¸²ÊÎ ° nd-· ¸¸¥Ö´¨Ö [75Ä79]. ʤ¥³ É ±¦¥ ¸Î¨É ÉÓ, ÎÉμ
´Ê±²μ´ · ¸¸¥¨¢ ¥É¸Ö ¤¥°É·μ´μ³ ¶·¥¨³ÊÐ¥¸É¢¥´´μ ¢ ±¢ ·É¥É´μ³ ¸μ¸ÉμÖ´¨¨.
‚ · §²μ¦¥´¨¨ (7), (15) μ£· ´¨Î¨³¸Ö ¸² £ ¥³Ò³¨ ¸ K = 0, 1, 2, É ± ÎÉμ ¸¨ LM
(Ω) (x, y Å
¸É¥³ (8) ¡Ê¤¥É ¸μ¤¥·¦ ÉÓ 27 K-£ ·³μ´¨± uKn (Ω) ≡ uKx y
±μμ·¤¨´ ÉÒ Ÿ±μ¡¨, ¸³. ¶·¨²μ¦¥´¨¥ A) [76], É. ¥. 27 ¸¢Ö§ ´´ÒÌ ¨´É¥£· ²Ó´ÒÌ
Ê· ¢´¥´¨° ¤²Ö ´¥¨§¢¥¸É´ÒÌ ËÊ´±Í¨° ψKn (¶μ Ψ¸²Ê ´ ¡μ·μ¢ Kn [2]). ´É¨¸¨³³¥É·¨Î´Ò¥ ¸μ¸ÉμÖ´¨Ö ΓKn , ¸μμÉ¢¥É¸É¢ÊÕШ¥ S = 3/2, T = 1/2, ¨³¥ÕÉ
¢¨¤ (17).
„²Ö § ¤ Î · ¸¸¥Ö´¨Ö ¸¨¸É¥³ (8) ¢ Í¥²μ³ ¡Ê¤¥É ´¥μ¤´μ·μ¤´μ°, É ± ± ±
Φ = 0. ’¥³ ´¥ ³¥´¥¥ ´¥±μÉμ·Ò¥ Ê· ¢´¥´¨Ö, ¢Ìμ¤ÖШ¥ ¢ (8), μ± ¦ÊÉ¸Ö μ¤´μ·μ¤´Ò³¨: WKn = 0 ¨§-§ ¸¢μ°¸É¢ ËÊ´±Í¨° uKn . ‚ ¸¨²Ê μ¸´μ¢´μ° É¥μ·¥³Ò ”·¥¤£μ²Ó³ [80,81] Ôɨ Ê· ¢´¥´¨Ö ¨³¥ÕÉ Éμ²Ó±μ É·¨¢¨ ²Ó´Ò¥ ·¥Ï¥´¨Ö, ¶μ¸±μ²Ó±Ê
¤²Ö ´¥¶·¥·Ò¢´μ£μ ¸¶¥±É· ´¥¢μ§³μ¦´μ μ¡¥¸¶¥Î¨ÉÓ ¢Ò·μ¦¤¥´´μ¸ÉÓ ³ É·¨Í
¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì ¨´É¥£· ²Ó´ÒÌ μ¶¥· Éμ·μ¢ (ÎÉμ ¶μ¤É¢¥·¦¤ ¥É¸Ö É ±¦¥ ¨ ´¥¶μ¸·¥¤¸É¢¥´´Ò³¨ · ¸Î¥É ³¨). ¸É ¢Ï¨¥¸Ö ´¥μ¤´μ·μ¤´Ò¥ Ê· ¢´¥´¨Ö ¸μμÉ¢¥É, u0110
, u0000
, u2020
¨ u0220
. ‹¥£±μ ¢¨¤¥ÉÓ, ÎÉμ ´ ¡μ·Ò
¸É¢ÊÕÉ ËÊ´±Í¨Ö³ u0000
0
1
2
2
2
¨Ì ±¢ ´Éμ¢ÒÌ Î¨¸¥² {
x
y LM } ¶·¥¤¸É ¢²ÖÕÉ ¸μ¡μ° ¢μ§³μ¦´Ò¥ ¸μ¸ÉμÖ´¨Ö
¶·¨ · ¸¸¥Ö´¨¨ ¢ ¸¨¸É¥³¥ É·¥Ì Î ¸É¨Í: x = 0, 2 ¸μμÉ¢¥É¸É¢ÊÕÉ ¸¢Ö§ ´´μ³Ê
¸μ¸ÉμÖ´¨Õ ¶μ¤¸¨¸É¥³Ò (23) Å ¤¥°É·μ´ , y = 0, 1, 2 Å μ·¡¨É ²Ó´Ò° ³μ³¥´É
Î ¸É¨ÍÒ 1 μÉ´μ¸¨É¥²Ó´μ Í¥´É· ³ ¸¸ ¶ ·Ò (23), L = 0, 2 Å ¶μ²´Ò° μ·¡¨É ²Ó´Ò° ³μ³¥´É ¸ ´Ê²¥¢μ° ¶·μ¥±Í¨¥° M (§´ Î¥´¨Ö M = 0 ¶·¨¢μ¤ÖÉ ± WKn = 0
¨ ¤¥² ÕÉ ¸μμÉ¢¥É¸É¢ÊÕШ¥ Ê· ¢´¥´¨Ö (8) μ¤´μ·μ¤´Ò³¨).
¸±² ¤Ò¢ Ö · §´μ¸ÉÓ Ψ − Φ ¶μ ¡ §¨¸´Ò³ ¸μ¸ÉμÖ´¨Ö³ (17) ¨ ¢ÒΨ¸²ÖÖ ³ É·¨Î´Ò¥ Ô²¥³¥´ÉÒ μÉ £¨¶¥·¸Ë¥·¨Î¥¸±¨Ì ¨ ¸¶¨´-¨§μ¸¶¨´μ¢ÒÌ ËÊ´±Í¨°, ¶μ²ÊΨ³ ¶ÖÉÓ Ê· ¢´¥´¨° ɨ¶ (8) ¤²Ö ËÊ´±Í¨° ψ00000 , ψ10110 , ψ20000 , ψ22020 ¨ ψ20220 .
‚¨¤ Ê· ¢´¥´¨° (9)Ä(14) ¶·¨ ÔÉμ³ μ¸É ´¥É¸Ö ´¥¨§³¥´´Ò³, ´μ Vij = V (3,1) (ri,j )
¢ (13), (14).
“· ¢´¥´¨Ö (8) ¡Ò²¨ ·¥Ï¥´Ò Ψ¸²¥´´μ [78,79], ¢ ± Î¥¸É¢¥ ¢μ²´μ¢μ° ËÊ´±Í¨¨ ¤¥°É·μ´ ϕd (x), ¢Ìμ¤ÖÐ¥° ¢ Φ ≡ Φ(x, y) = ϕd (x) exp (iky) (k Å ¨³¶Ê²Ó¸ ´¥°É·μ´ ), ¨¸¶μ²Ó§μ¢ ² ¸Ó ¢μ²´μ¢ Ö ËÊ´±Í¨Ö •Õ²ÓÉ¥´ ϕd (x) ≡ ϕd (x) =
αβ(α + β) exp (−αx) − exp (−βx)
,
2π(β − α)2
x
(19)
√
£¤¥ α = md , d = 2,226 ŒÔ‚, β = 5,94α.
·¨¸. 1 ¶·¨¢¥¤¥´Ò · ¤¨ ²Ó´Ò¥ § ¢¨¸¨³μ¸É¨ ËÊ´±Í¨° ψKn , · ¸¸Î¨É ´´Ò¥ ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ ¶μÉ¥´Í¨ ²μ¢ Œ ²Ë²¨Ä’¨μ´ (Œ’) [82]
VMT I (r) = (1438,72 e−3,11r − 513,968 e−1,55r )/r,
VMT III (r) = (1438,72 e−3,11r − 626,885 e−1,55r )/r,
(20)
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 583
¨¸. 1. ¤¨ ²Ó´Ò¥ § ¢¨¸¨³μ¸É¨ ËÊ´±Í¨° ψKn , · ¸¸Î¨É ´´Ò¥ ¤²Ö ¶μÉ¥´Í¨ ²μ¢
Œ ²Ë²¨Ä’¨μ´ ( ), ‚μ²±μ¢ (¡) ¨ °±¥³ °¥· Ä• ±¥´¡·μ°Ì (EH) (¢) ¶·¨ En =
1,5 ŒÔ‚ (ÏÉ·¨Ìμ¢Ò¥ ±·¨¢Ò¥), 2,45 ŒÔ‚ (¶Ê´±É¨·´Ò¥) ¨ 3,27 ŒÔ‚ (¸¶²μÏ´Ò¥)
‚μ²±μ¢ [83, 84]
Vt (r) = 144,86 e−(r/0,82) − 83,34 e−(r/1,6) ,
Vs (r) = 0,63Vt (r)
2
2
(21)
584 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
¨ °±¥³ °¥· Ä• ±¥´¡·μ°Ì (EH) [85]
Vt (r) = 600 e−5,5r − 70 e−0,5r − 27,6 e−0,38r ,
2
2
2
Vs (r) = 880 e−5,4r − 70 e−0,64r − 21 e−0,48r
2
2
2
(22)
¤²Ö Ô´¥·£¨° ¶ ¤ ÕÐ¨Ì ´¥°É·μ´μ¢ En = 1,5, 2,45 ¨ 3,27 ŒÔ‚ (¢ ² ¡μ· Éμ·´μ°
¸¨¸É¥³¥).
ˆ§ ´ ²¨§ ¶μ¢¥¤¥´¨Ö ±·¨¢ÒÌ, ¶·¥¤¸É ¢²¥´´ÒÌ ´ ÔÉμ³ ·¨¸Ê´±¥, ¸²¥¤Ê¥É,
ÎÉμ, ¢μ-¶¥·¢ÒÌ, ¢¥²¨Î¨´Ò ψKn ¸² ¡μ § ¢¨¸ÖÉ μÉ Ô´¥·£¨¨ ´¥°É·μ´ ¢ · ¸¸³μÉ·¥´´μ³ ¤¨ ¶ §μ´¥ ¥¥ §´ Î¥´¨°. ‚μ-¢Éμ·ÒÌ, ³ ±¸¨³ ²Ó´Ò¥ §´ Î¥´¨Ö ψKn ¤²Ö
μ¸´μ¢´μ° K-£ ·³μ´¨±¨ (K = 0) ´ ¶μ·Ö¤μ± ¨ ¡μ²¥¥ ¶·¥¢ÒÏ ÕÉ ¶μ ¡¸μ²ÕÉ´μ° ¢¥²¨Î¨´¥ ³ ±¸¨³ ²Ó´Ò¥ §´ Î¥´¨Ö ËÊ´±Í¨° ψKn ¤²Ö K = 1 ¨ K = 2.
” §Ò nd-· ¸¸¥Ö´¨Ö ³μ£ÊÉ ¡ÒÉÓ μ¶·¥¤¥²¥´Ò ¸²¥¤ÊÕШ³ μ¡· §μ³. ·¥¤¸É ¢¨³ ¢μ²´μ¢ÊÕ ËÊ´±Í¨Õ § ¤ Ψ ¢ ¢¨¤¥ (2), £¤¥
ψKn (ρ)uKn (Ω)
(23)
Ψint =
Kn
¥¸ÉÓ ·¥Ï¥´¨¥ Ê· ¢´¥´¨° ” ¤¤¥¥¢ (3) ¢ μ¡² ¸É¨ ¢§ ¨³μ¤¥°¸É¢¨Ö, Ψext Å ·¥Ï¥´¨¥ Ê· ¢´¥´¨Ö ˜·¥¤¨´£¥· ¢ ¸¨³¶ÉμɨΥ¸±μ° μ¡² ¸É¨ ¤²Ö ¤¢ÊÌÎ ¸É¨Î´μ°
§ ¤ Ψ · ¸¸¥Ö´¨Ö ´¥°É·μ´ ´ ÉμÎ¥Î´μ³ ¤¥°É·μ´¥ ¸ ¶μÉ¥´Í¨ ²μ³ V12 + V31 .
§²μ¦¨³ É¥¶¥·Ó Ψext ¢ ·Ö¤ ¶μ ¸Ë¥·¨Î¥¸±¨³ ËÊ´±Í¨Ö³ [86]:
Ψext (y) =
ψ,k (y)Y∗m (k/k)Y m (y/y),
(24)
m
£¤¥ ψ,k Å ¶ ·Í¨ ²Ó´ Ö ±μ³¶μ´¥´É , 춨¸Ò¢ ÕÐ Ö ¸μ¸ÉμÖ´¨¥ ¸ μ·¡¨É ²Ó´Ò³
³μ³¥´Éμ³ :
(25)
ψ,k (y) j (ky) + (−1)+1 tg δ n (ky).
‡¤¥¸Ó j (kr), n (kr) Å ¸Ë¥·¨Î¥¸±¨¥ ËÊ´±Í¨¨ ¥¸¸¥²Ö ¨ ¥°³ ´ ; δ Å Ë § · ¸¸¥Ö´¨Ö, ±μÉμ· Ö ³μ¦¥É ¡ÒÉÓ μ¶·¥¤¥²¥´ ¨§ ʸ²μ¢¨° ¸Ï¨¢ ´¨Ö ·¥Ï¥´¨°
¢ ¸¨³¶ÉμɨΥ¸±μ° μ¡² ¸É¨ ψ,k ¸ ¸μμÉ¢¥É¸É¢ÊÕШ³¨ ·¥Ï¥´¨Ö³¨ ¢ μ¡² ¸É¨
¢§ ¨³μ¤¥°¸É¢¨Ö ψKn (É ¡².1). ‚ ¶μ¸²¥¤´¥° ±μ²μ´±¥ É ¡².1 ¶·¨¢¥¤¥´Ò É ±¦¥
’ ¡²¨Í 1. Š¢ ·É¥É´Ò¥ Ë §Ò nd-· ¸¸¥Ö´¨Ö (¢ £· ¤Ê¸ Ì)
0
1
2
MT
Ä67,4
23,4
Ä3,27
δ
‚μ²±μ¢
Ä68,1
21,8
Ä3,19
EH
Ä65,9
22,9
Ä3,73
0
1
2
Ä74,2
28,1
Ä4,31
Ä74,8
26,7
Ä4,19
Ä72,8
27,8
Ä4,76
En , ŒÔ‚
2,45
3,27
„ ´´Ò¥
¤·Ê£¨Ì ¢Éμ·μ¢
Ä66,7 [87, 88]
22,2 [89]
Ä3,41 [35]
Ä73,6 [34, 88]
27,3 [90]
Ä4,66 [90]
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 585
¨¸. 2. „¨ËË¥·¥´Í¨ ²Ó´Ò¥ ¸¥Î¥´¨Ö nd-· ¸¸¥Ö´¨Ö, · ¸¸Î¨É ´´Ò¥ ¤²Ö ¶μÉ¥´Í¨ ²μ¢
Œ ²Ë²¨Ä’¨μ´ (¸¶²μÏ´ Ö ±·¨¢ Ö), ‚μ²±μ¢ (ÏÉ·¨Ìμ¢ Ö) ¨ °±¥³ °¥· Ä• ±¥´¡·μ°Ì (¶Ê´±É¨·´ Ö) ¶·¨ En = 2,45 ŒÔ‚. ±¸¶¥·¨³¥´É ²Ó´Ò¥ ¤ ´´Ò¥ ¢§ÖÉÒ ¨§ [91] (•)
¨ [92] ()
¨¸. 3. ’μ ¦¥, ÎÉμ ¨ ´ ·¨¸. 2, ´μ ¤²Ö En = 3,27 ŒÔ‚
¢¥²¨Î¨´Ò Ë §μ¢ÒÌ ¸¤¢¨£μ¢, ¶μ²ÊÎ¥´´ÒÌ ¤·Ê£¨³¨ ³¥Éμ¤ ³¨ ¤²Ö ´ ²μ£¨Î´μ°
§ ¤ Ψ.
·¨¸. 2, 3 ¶·¥¤¸É ¢²¥´Ò Ê£²μ¢Ò¥ § ¢¨¸¨³μ¸É¨ ¸¥Î¥´¨° nd-· ¸¸¥Ö´¨Ö,
· ¸¸Î¨É ´´Ò¥ ¶μ Ëμ·³Ê²¥ [86]
σ(θ) =
2 (4)
|A (θ)|2 ,
3
A(4) (θ) =
2
1
(2
+ 1) exp iδ sin δ P (cos θ). (26)
k
=0
586 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
ˆ§ ´ ²¨§ ¶μ¢¥¤¥´¨Ö ±·¨¢ÒÌ, ¶·¥¤¸É ¢²¥´´ÒÌ ´ ·¨¸. 2, 3, ¸²¥¤Ê¥É, ÎÉμ
´ ¨²ÊÎÏ¥¥ ¸μ£² ¸¨¥ ¸ Ô±¸¶¥·¨³¥´É ²Ó´Ò³¨ ¤ ´´Ò³¨ ¤μ¸É¨£ ¥É¸Ö ¤²Ö ¶μÉ¥´Í¨ ² Œ ²Ë²¨Ä’¨μ´ (20).
„²Ö Éμ£μ ÎÉμ¡Ò · ¸¸Î¨É ÉÓ ËÊ´±Í¨¨ ψKn ¸ ÊÎ¥Éμ³ ±Ê²μ´μ¢¸±μ£μ ¢§ ¨³μ¤¥°¸É¢¨Ö (Š‚) ³¥¦¤Ê ¶·μÉμ´ ³¨, ¢¢¥¤¥³ Ô±· ´¨·μ¢ ´´Ò° ±Ê²μ´μ¢¸±¨°
¶μÉ¥´Í¨ ²
α e−r/R
,
(27)
VCe (r) =
r
£¤¥ α Å ¶μ¸ÉμÖ´´ Ö Éμ´±μ° ¸É·Ê±ÉÊ·Ò; R Å · ¤¨Ê¸ Ô±· ´¨·μ¢ ´¨Ö. É ¶·μÍ¥¤Ê· Ö¢²Ö¥É¸Ö ¸É ´¤ ·É´μ° ¨ ¤μ¸É Éμδμ Ϩ·μ±μ ¶·¨³¥´Ö¥É¸Ö ¢ ¶μ¤μ¡´ÒÌ
§ ¤ Î Ì. ¡μ§´ Ψ³ § ·Ö¦¥´´Ò¥ Î ¸É¨ÍÒ ¨´¤¥±¸ ³¨ 1, 2 ¨ ¢¢¥¤¥³ ¶μÉ¥´Í¨ ² (27) ¢ ¢Ò· ¦¥´¨Ö (13), (14): V12 → V12 + VCe . ‚ÒΨ¸²ÖÖ ¤ ²¥¥ ËÊ´±Í¨¨
ψKn ¸ · §´Ò³¨ R, ³μ¦´μ É ±¨³ μ¡· §μ³ ¨¸¸²¥¤μ¢ ÉÓ ¸Ì줨³μ¸ÉÓ ¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì ·¥Ï¥´¨° (8).
·¨¸. 4 ¢ ± Î¥¸É¢¥ ¶·¨³¥· ¶·¥¤¸É ¢²¥´μ ¸¥³¥°¸É¢μ · ¸¸Î¨É ´´ÒÌ § ¢¨¸¨³μ¸É¥° ψ00000 (ρ) ¸ · ¤¨Ê¸ ³¨ Ô±· ´¨·μ¢ ´¨Ö R = 1, 2, 10, 20, 100 ˳ ¤²Ö
¶μÉ¥´Í¨ ² (20) ¨ Ô´¥·£¨¨ ¶·μÉμ´μ¢ Ep = 1,5 ŒÔ‚ (¢ ² ¡μ· Éμ·´μ° ¸¨¸É¥³¥).
• · ±É¥· ¸Ì줨³μ¸É¨ μ¸É ²Ó´ÒÌ ËÊ´±Í¨° (ψ10110 , ψ20000 , ψ22020 , ψ20220 )
¶μ¤μ¡¥´ ¶·¨¢¥¤¥´´μ³Ê ´ ·¨¸. 4: ·¥Ï¥´¨Ö ¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì Ê· ¢´¥´¨° (8)
¨¸. 4. ¸¸Î¨É ´´Ò¥ ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ ¶μÉ¥´Í¨ ² M ²Ë²¨Ä’¨μ´ § ¢¨¸¨³μ¸É¨
ψ00000 (ρ) ¤²Ö Ô´¥·£¨¨ ¶ ¤ ÕÐ¨Ì ¶·μÉμ´μ¢ Ep = 1,5 ŒÔ‚: ¡¥§ Ê봃 Š‚ (ÏÉ·¨Ì¶Ê´±É¨·´ Ö ±·¨¢ Ö) ¨ ¸ ¥£μ ÊÎ¥Éμ³ (¸¶²μÏ´ Ö ±·¨¢ Ö, · ¤¨Ê¸ Ô±· ´¨·μ¢ ´¨Ö R = 100 ˳).
¢¸É ¢±¥ ¢ Ê¢¥²¨Î¥´´μ³ ³ ¸ÏÉ ¡¥ ¶μ± § ´ μ±·¥¸É´μ¸ÉÓ ³¨´¨³Ê³ · ¸¸Î¨É ´´ÒÌ
ËÊ´±Í¨° ¸ · §´Ò³¨ R: 1 Å 1 ˳; 2 Å 2 ˳; 3 Å 10 ˳; 4 Å 20 ˳; 5 Å 100 ˳
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 587
¸Ìμ¤ÖÉ¸Ö ± ·¥Ï¥´¨Õ ¸ R = 100 ˳ (´ ·¨¸Ê´±¥ ¢¨¤´μ, ÎÉμ ·¥Ï¥´¨Ö ¸ R =
20 ˳ ¨ R = 100 ˳ ¶μÎɨ ¸μ¢¶ ¤ ÕÉ, Å É ±μ¥ ¶μ¢¥¤¥´¨¥ Ì · ±É¥·´μ É ±¦¥
¨ ¤²Ö ·¥Ï¥´¨° ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ ¶μÉ¥´Í¨ ²μ¢ (21) ¨ (22)).
’ ±¨³ μ¡· §μ³, ³μ¦´μ ¸Î¨É ÉÓ, ÎÉμ ¨¸¶μ²Ó§μ¢ ´¨¥ ¢ · ¸Î¥É Ì ¶μÉ¥´Í¨ ² (27) ¸ R = 100 ˳ ¸μμÉ¢¥É¸É¢Ê¥É ÊÎ¥ÉÊ Š‚ ¤²Ö ¤ ´´μ° § ¤ Ψ · ¸¸¥Ö´¨Ö.
ˆ§ ¨¸¸²¥¤μ¢ ´¨Ö ¸Ì줨³μ¸É¨ ·¥Ï¥´¨° ψKn ¤²Ö Ô´¥·£¨° ¶·μÉμ´μ¢ Ep = 2,45
¨ 3,27 ŒÔ‚ É ±¦¥ ¸²¥¤Ê¥É, ÎÉμ ·¥Ï¥´¨Ö ¸Ìμ¤ÖÉ¸Ö ± ψKn ¸ R = 100 ˳.
‡ ³¥É¨³, ÎÉμ ¢ [93], £¤¥ ·¥Ï ²¨¸Ó Ê· ¢´¥´¨Ö ” ¤¤¥¥¢ ¢ ¨³¶Ê²Ó¸´μ³ ¶·¥¤¸É ¢²¥´¨¨ ¤²Ö ´ ²μ£¨Î´μ° § ¤ Ψ pd-· ¸¸¥Ö´¨Ö, ¡Ò²μ Ê¸É ´μ¢²¥´μ, ÎÉμ ¶·¨
¨§³¥´¥´¨¨ · ¤¨Ê¸ Ô±· ´¨·μ¢ ´¨Ö ¢ ¶·¥¤¥² Ì μÉ 100 ¤μ 2000 ˳ · ¸¸Î¨É ´´Ò¥ ¢¥²¨Î¨´Ò Ë § · ¸¸¥Ö´¨Ö μɲ¨Î ÕÉ¸Ö ´¥ ¡μ²¥¥ Î¥³ ´ 1 %. ‚ [94] ¶·¨
¢ÒΨ¸²¥´¨¨ ¸¥Î¥´¨° pd-· ¸¸¥Ö´¨Ö (Ep = 10 ŒÔ‚) ¸ ¨¸¶μ²Ó§μ¢ ´¨¥³ Ê· ¢´¥´¨Ö ‹¨¶¶³ ´ Ę¢¨´£¥· ¨ ¸¥¶ · ¡¥²Ó´μ£μ ¶ ·¨¦¸±μ£μ ¶μÉ¥´Í¨ ² PEST 1Ä6
· ¤¨Ê¸ Ô±· ´¨·μ¢ ´¨Ö É ±¦¥ ¸μ¸É ¢²Ö² 100 ˳, Ê¢¥²¨Î¨¢ Ö¸Ó ¤μ 300 ˳ ¶·¨
· ¸Î¥É¥ ¶μ²Ö·¨§ Í¨μ´´ÒÌ Ì · ±É¥·¨¸É¨±.
„²Ö ¢ÒΨ¸²¥´¨Ö Ë § pd-· ¸¸¥Ö´¨Ö ¶·¨³¥´¨³ ¶·μÍ¥¤Ê·Ê, 춨¸ ´´ÊÕ ¢ÒÏ¥.
·¥¤¸É ¢¨³ ¢μ²´μ¢ÊÕ ËÊ´±Í¨Õ § ¤ Ψ ¢ ¢¨¤¥ (2) ¨ · §²μ¦¨³ Ψext ¢ ·Ö¤ ¶μ
¸Ë¥·¨Î¥¸±¨³ ËÊ´±Í¨Ö³ (24), ¶·¨Î¥³
ψ,k (y) F (ky) + (−1)+1 tg δ G (ky),
(28)
£¤¥ F , G Å ·¥£Ê²Ö·´ Ö ¨ ´¥·¥£Ê²Ö·´ Ö ±Ê²μ´μ¢¸±¨¥ ËÊ´±Í¨¨ [86] ¸μμÉ¢¥É¸É¢¥´´μ. ‚ É ¡². 2 ¶·¨¢¥¤¥´Ò ±¢ ·É¥É´Ò¥ Ë §Ò pd-· ¸¸¥Ö´¨Ö, ´ °¤¥´´Ò¥ ¨§
ʸ²μ¢¨Ö ¸Ï¨¢ ´¨Ö ·¥Ï¥´¨° Ψext c Ψint , ¶·¥¤¸É ¢²¥´´μ£μ ¢ ¢¨¤¥ ·Ö¤ (23).
’ ¡²¨Í 2. Š¢ ·É¥É´Ò¥ Ë §Ò pd-· ¸¸¥Ö´¨Ö (¢ £· ¤Ê¸ Ì)
Ep , ŒÔ‚
1,5
2
0
1
2
0
1
2
MT
Ä47,8
16,1
Ä1,28
Ä53,5
19,8
Ä2,22
δ
‚μ²±μ¢
Ä48,6
14,8
Ä1,21
Ä54,1
18,7
Ä2,11
EH
Ä45,7
15,5
Ä1,67
Ä51,6
19,7
Ä2,45
„ ´´Ò¥ ¨§ [35]
Ä46,0
15,2
Ä1,59
Ä52,8
19,1
Ä2,37
·¨¸. 5, 6 ¶·¥¤¸É ¢²¥´Ò ¤¨ËË¥·¥´Í¨ ²Ó´Ò¥ ¸¥Î¥´¨Ö · ¸¸¥Ö´¨Ö ¶·μÉμ´μ¢ ¤¥°É·μ´ ³¨, · ¸¸Î¨É ´´Ò¥ ¶μ Ëμ·³Ê²¥ [86]
2
σ(θ) = |AR (θ) + A(θ)|2 ,
(29)
3
£¤¥ θ Å Ê£μ² · ¸¸¥Ö´¨Ö; AR Å ·¥§¥·Ëμ·¤μ¢¸± Ö ³¶²¨Éʤ :
AR (θ) = −
η Γ(1 + iη) exp (−2i ln (sin (θ/2)))
,
2k Γ(1 − iη) sin2 (θ/2)
(30)
588 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
¨¸. 5. „¨ËË¥·¥´Í¨ ²Ó´Ò¥ ¸¥Î¥´¨Ö pd-· ¸¸¥Ö´¨Ö, · ¸¸Î¨É ´´Ò¥ ¤²Ö ¶μÉ¥´Í¨ ²μ¢
Œ ²Ë²¨Ä’¨μ´ (¸¶²μÏ´ Ö ±·¨¢ Ö), ‚μ²±μ¢ (ÏÉ·¨Ìμ¢ Ö) ¨ °±¥³ °¥· Ä• ±¥´¡·μ°Ì (¶Ê´±É¨·´ Ö) ¶·¨ Ep = 1,5 ŒÔ‚. ±¸¶¥·¨³¥´É ²Ó´Ò¥ ¤ ´´Ò¥ (Éμα¨) ¢§ÖÉÒ ¨§ [95]
¨¸. 6. ’μ ¦¥, ÎÉμ ¨ ´ ·¨¸. 5, ´μ ¤²Ö Ep = 2 ŒÔ‚. ±¸¶¥·¨³¥´É ²Ó´Ò¥ ¤ ´´Ò¥
(Éμα¨) ¢§ÖÉÒ ¨§ [96]
η = 2αm/(3k) Å ¶ · ³¥É· ‡μ³³¥·Ë¥²Ó¤ ,
A(θ) =
2
1
(2
+ 1) exp (i(2ω + δ )) sin δ P (cos θ),
k
=0
ω = arg (Γ(
+ 1 + iη)) Å ±Ê²μ´μ¢¸± Ö Ë § .
(31)
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 589
• · ±É¥· ¶μ¢¥¤¥´¨Ö ±·¨¢ÒÌ, ¶·¥¤¸É ¢²¥´´ÒÌ ´ ·¨¸. 5, 6, ¶μ§¢μ²Ö¥É ¸¤¥² ÉÓ ¢Ò¢μ¤, ÎÉμ ´ ¨²ÊÎÏ¥¥ ¸μ£² ¸¨¥ ¸ Ô±¸¶¥·¨³¥´É ³¨, É ± ¦¥ ± ± ¨ ¤²Ö
nd-· ¸¸¥Ö´¨Ö, ¤μ¸É¨£ ¥É¸Ö ¤²Ö ¶μÉ¥´Í¨ ² M ²Ë²¨Ä’¨μ´ . ¥¡μ²ÓÏ Ö ¶μ²μα · ¸¸Î¨É ´´ÒÌ ±·¨¢ÒÌ ¢ μ±·¥¸É´μ¸É¨ θ 30−40◦ Å ·¥§Ê²ÓÉ É ¨¸¶μ²Ó§μ¢ ´¨Ö ¶·¨¡²¨¦¥´¨Ö (29): ¥¸²¨ ³Ò ¢ÒΨ¸²Ö¥³ ¸¥Î¥´¨Ö ¶μ ÔÉμ° Ëμ·³Ê²¥,
Éμ ¶·¥´¥¡·¥£ ¥³ ´¥±μÉμ·μ° ´¥¡μ²ÓÏμ° μ¡² ¸ÉÓÕ Ê£²μ¢, ¢ ±μÉμ·μ°, ¸É·μ£μ
£μ¢μ·Ö, ¶μ²´ Ö ³¶²¨Éʤ ´¥ · ¢´ ¶·μ¸Éμ ¸Ê³³¥ AR ¨ A.
‘μ£² ¸¨¥ ¸ Ô±¸¶¥·¨³¥´É ³¨ ¶μ N d-· ¸¸¥Ö´¨Õ ³μ¦´μ ¡Ò²μ ¡Ò, ´ ´ Ï
¢§£²Ö¤, ʲÊÎϨÉÓ, ¥¸²¨ ¤μ¶μ²´¨É¥²Ó´μ ÊÎ¥¸ÉÓ ¤Ê¡²¥É´ÊÕ ±μ³¶μ´¥´ÉÊ ³¶²¨ÉÊ¤Ò A(2) . Š ± ¶μ± § ´μ ¢ [90], Ê봃 A(2) ¶·¨¢μ¤¨É ± ´¥±μÉμ·μ³Ê Ê¢¥²¨Î¥´¨Õ
§´ Î¥´¨° σ(θ) ¢ μ¡² ¸É¨ ³ ²ÒÌ ¨ ¡μ²ÓÏ¨Ì ¢¥²¨Î¨´ Ê£²μ¢ · ¸¸¥Ö´¨Ö.
’ ±¨³ μ¡· §μ³, ¶·¥¤²μ¦¥´´Ò° ´ ³¨ ³¥Éμ¤ · ¸Î¥É ¸¥Î¥´¨° ´Ê±²μ´-¤¥°É·μ´´μ£μ · ¸¸¥Ö´¨Ö ¸ ¶·μ¸ÉÒ³¨ ³μ¤¥²Ö³¨ N N -¢§ ¨³μ¤¥°¸É¢¨Ö ¶μ§¢μ²Ö¥É ¤μ¸É¨ÎÓ Ê¤μ¢²¥É¢μ·¨É¥²Ó´μ£μ ¸μ£² ¸¨Ö ¸ ¸μμÉ¢¥É¸É¢ÊÕШ³¨ Ô±¸¶¥·¨³¥´É ²Ó´Ò³¨ ¸¥Î¥´¨Ö³¨ ¶·¨ ¤μ¶μ·μ£μ¢ÒÌ §´ Î¥´¨ÖÌ Ô´¥·£¨¨ ¶ ¤ ÕÐ¥° Î ¸É¨ÍÒ.
·¥¤¸É ¢¨¢ ¶μ²´ÊÕ ¢μ²´μ¢ÊÕ ËÊ´±Í¨Õ ¸¨¸É¥³Ò Ψ ± ± ¸Ê³³Ê ¥¥ ¸¨³¶ÉμɨΥ¸±μ° Î ¸É¨ Φ ¨ Î ¸É¨, ±μÉμ· Ö μ¶¨¸Ò¢ ¥É É·¥Ì´Ê±²μ´´ÊÕ ¸¨¸É¥³Ê ¢ μ¡² ¸É¨ ¢§ ¨³μ¤¥°¸É¢¨Ö, ³Ò ¸¢¥²¨ § ¤ ÎÊ μ ´ Ì즤¥´¨¨ Ψ ± ¸¨¸É¥³¥ μ¤´μ³¥·´ÒÌ
¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨°, ·¥Ï¥´¨¥ ±μÉμ·μ° ´¥ É·¥¡Ê¥É §´ Ψɥ²Ó´ÒÌ § É· É
³ Ϩ´´μ£μ ¢·¥³¥´¨. μÔÉμ³Ê ± ´¥¸μ³´¥´´Ò³ ¤μ¸Éμ¨´¸É¢ ³ ³¥Éμ¤ ¸²¥¤Ê¥É
μÉ´¥¸É¨ ¥£μ μÉ´μ¸¨É¥²Ó´ÊÕ ¶·μ¸ÉμÉÊ ¨ ¡Ò¸É·μ¤¥°¸É¢¨¥ ¶μ ¸· ¢´¥´¨Õ ¸ É· ¤¨Í¨μ´´Ò³¨ ³¥Éμ¤ ³¨, μ¸´μ¢ ´´Ò³¨ ´ ´¥¶μ¸·¥¤¸É¢¥´´μ³ Ψ¸²¥´´μ³ ·¥Ï¥´¨¨
¤¢Ê³¥·´ÒÌ ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨° ¢ ¨³¶Ê²Ó¸´μ³ ¶·¥¤¸É ¢²¥´¨¨. ‘ ¤·Ê£μ°
¸Éμ·μ´Ò, ³¥Éμ¤ Ö¢²Ö¥É¸Ö ¶·¨¡²¨¦¥´´Ò³, É ± ± ± ¤²Ö ¶· ±É¨Î¥¸±¨Ì Í¥²¥°
³ Ϩ´´μ£μ ¸Î¥É ·Ö¤ (7) ´¥μ¡Ì줨³μ μ¡μ·¢ ÉÓ: ±μ²¨Î¥¸É¢μ ¨´É¥£· ²Ó´ÒÌ
Ê· ¢´¥´¨° ¢ ´¥³ ´ · ¸É ¥É ² ¢¨´μμ¡· §´μ Å ¥¸²¨ ¤²Ö Kmax = 2 ¸¨¸É¥³ (8)
¸μ¤¥·¦¨É 27 Ê· ¢´¥´¨°, Éμ ¤²Ö Kmax = 4 ʦ¥ 182. ˆ, ´¥¸³μÉ·Ö ´ Éμ, ÎÉμ
¡μ²ÓϨ´¸É¢μ ÔÉ¨Ì Ê· ¢´¥´¨° ¨³¥¥É É·¨¢¨ ²Ó´μ¥ ·¥Ï¥´¨¥, ´¥μ¡Ì줨³μ¸ÉÓ μɸ¥ÖÉÓ ¨Ì ¢¥¤¥É ± 祧³¥·´μ³Ê Ê¢¥²¨Î¥´¨Õ μ¡Ñ¥³ ¢Ò±² ¤μ± ®¢·ÊδÊÕ¯ (¨, · §Ê³¥¥É¸Ö, ¶·μ£· ³³´μ£μ ±μ¤ ). ˆ ¢¸¥ ¦¥ ¶·¥¤²μ¦¥´´Ò° ³¥Éμ¤ ¶·¥¤¸É ¢²Ö¥É¸Ö
´ ³ ¶¥·¸¶¥±É¨¢´Ò³, ¶μ¸±μ²Ó±Ê ¶μ§¢μ²Ö¥É ¨§ÊÎ ÉÓ ¸μ¸ÉμÖ´¨Ö ´¥¶·¥·Ò¢´μ£μ
¸¶¥±É· ¢ ²Õ¡ÒÌ É·¥ÌÎ ¸É¨Î´ÒÌ ¸¨¸É¥³ Ì (´ ¶·¨³¥·, Λ + d, α + d ¨ É. ¶.),
¥¸²¨ ¨§¢¥¸É´Ò ¨Ì ¶ ·´Ò¥ ¶μÉ¥´Í¨ ²Ò ¢§ ¨³μ¤¥°¸É¢¨Ö. …£μ ³μ¦´μ É ±¦¥
μ¡μ¡Ð¨ÉÓ ¨ ´ ¸¨¸É¥³Ò ¸ Ψ¸²μ³ Î ¸É¨Í, ¡μ²ÓϨ³ É·¥Ì [48].
1.3. ‘¨¸É¥³ É·¥Ì ´¥¸¢Ö§ ´´ÒÌ ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì Î ¸É¨Í. Š· ɱμ
μ¸É ´μ¢¨³¸Ö ´ ¢μ¶·μ¸¥ ¨´Ë¨´¨É´μ£μ ¤¢¨¦¥´¨Ö ¢¸¥Ì É·¥Ì Î ¸É¨Í. ‘μμÉ¢¥É¸É¢ÊÕШ¥ Ê· ¢´¥´¨Ö ” ¤¤¥¥¢ ¨³¥ÕÉ ¢¨¤ [1, 97]
(1)
(2)
(3)
(2)
(3)
(1)
(3)
Ψ123
(1)
G0 (Z)T12 (Z)(Ψ123
Ψ123 = Φ1(23) − Φ123 + G0 (Z)T23 (Z)(Ψ123 + Ψ123 ),
Ψ123 = Φ2(31) − Φ123 + G0 (Z)T31 (Z)(Ψ123 + Ψ123 ),
= Φ3(12) − Φ123 +
+
(2)
Ψ123 ),
(32)
590 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
(1)
(2)
(3)
£¤¥ Ψ123 = Φ123 + Ψ123 + Ψ123 + Ψ123 Å ¶μ²´ Ö ¢μ²´μ¢ Ö ËÊ´±Í¨Ö ¸¨¸É¥³Ò.
‡¤¥¸Ó Φ123 = exp (ip1 ρ1 +ik23 r23 ) Å ¸¨³¶ÉμɨΥ¸± Ö ¢μ²´μ¢ Ö ËÊ´±Í¨Ö ¨´Ë¨´¨É´μ£μ ¤¢¨¦¥´¨Ö É·¥Ì Î ¸É¨Í, μ´ Ö¢²Ö¥É¸Ö ¨´¢ ·¨ ´É´μ° μÉ´μ¸¨É¥²Ó´μ
§ ³¥´Ò ±μμ·¤¨´ É Ÿ±μ¡¨, Φi(jk) μɲ¨Î ÕÉ¸Ö μÉ Φ123 ÊÎ¥Éμ³ ¢§ ¨³μ¤¥°¸É¢¨Ö
³¥¦¤Ê Î ¸É¨Í ³¨ j ¨ k, É. ¥. Φi(jk) = exp (ipi ρi )ϕkjk (rjk ), £¤¥ ϕkjk (rjk ) Å
·¥Ï¥´¨¥ Ê· ¢´¥´¨Ö (−Δjk /(2μjk ) + Vjk − Ejk )ϕkjk (rjk ) = 0 ¤²Ö ¶μ²μ¦¨2
É¥²Ó´ÒÌ §´ Î¥´¨° Ô´¥·£¨¨ μÉ´μ¸¨É¥²Ó´μ£μ ¤¢¨¦¥´¨Ö Ejk = kjk
/(2μjk ) > 0,
±μÉμ·μ¥ ´ ¡¥¸±μ´¥Î´μ¸É¨ ¨³¥¥É ¢¨¤ ¸Ê³³Ò ¶²μ¸±μ° ¢μ²´Ò ¨ · ¸Ìμ¤ÖÐ¥°¸Ö
(¨²¨ ¸Ìμ¤ÖÐ¥°¸Ö) ¸Ë¥·¨Î¥¸±μ° ¢μ²´Ò. ·¨ ÔÉμ³ · §´μ¸ÉÓ Φi(jk) − Φ123 , ± ±
´¥É·Ê¤´μ Ê¡¥¤¨ÉÓ¸Ö, Ö¢²Ö¥É¸Ö · ¸Ìμ¤ÖÐ¥°¸Ö (¸Ìμ¤ÖÐ¥°¸Ö) ¢μ²´μ° ¤²Ö ¡μ²ÓÏ¨Ì §´ Î¥´¨° μÉ´μ¸¨É¥²Ó´μ° ±μμ·¤¨´ ÉÒ rjk .
ˆ¸¶μ²Ó§ÊÖ ¶·μÍ¥¤Ê·Ê, ¨§²μ¦¥´´ÊÕ ¢ ¶. 1.1, ¢ ¶¥·¢μ³ ¶·¨¡²¨¦¥´¨¨ · §²μ¦¥´¨Ö ¶μ £¨¶¥·¸Ë¥·¨Î¥¸±¨³ £ ·³μ´¨± ³ ¶μ²ÊΨ³ μ¤´μ³¥·´μ¥ ¨´É¥£· ²Ó´μ¥
Ê· ¢´¥´¨¥ μÉ´μ¸¨É¥²Ó´μ ψ123 (ρ) [98]:
ψ123 (ρ) = πmρ
−2
∞
dρ ρ
3
ψ123 (ρ )U + π 3/2 (Φ2(31) + Φ3(12) )V23 +
0
+ (Φ3(12) + Φ1(23) )V31 + (Φ1(23) + Φ2(31) )V12 − 2Φ123 U
P± (k0 , ρ, ρ ),
(33)
£¤¥ ψ123 (ρ) Å ±μÔË˨ͨ¥´ÉÒ · §²μ¦¥´¨Ö ¶μ K-£ ·³μ´¨± ³ (· ¤¨ ²Ó´Ò¥
ËÊ´±Í¨¨), U μ¶·¥¤¥²Ö¥É¸Ö Ê· ¢´¥´¨¥³ (5) ¨ ¢¢¥¤¥´μ μ¡μ§´ Î¥´¨¥
dΩ
A≡
A.
(34)
π3
Š·μ³¥ Éμ£μ,
P± (k0 , ρ, ρ ) =
(1,2)
(1,2)
= ∓i J2 (k0 ρ)H2 (k0 ρ )Θ(ρ − ρ) + J2 (k0 ρ )H2 (k0 ρ)Θ(ρ − ρ ) , (35)
√
(1,2)
£¤¥ k0 = 2Em, H2
Å ËÊ´±Í¨¨ • ´±¥²Ö. ·¨¸ÊɸɢÊÕÐ¥¥ ¢ (33) ¨´É¥£·¨·μ¢ ´¨¥ ¶μ Ê£² ³ (34) ³μ¦´μ Î ¸É¨Î´μ ¢Ò¶μ²´¨ÉÓ ´ ²¨É¨Î¥¸±¨:
16
Vij =
π
π/2
dθ sin2 θ cos2 θ Vij (αij ρ cos θ),
(36)
0
16
Φ123 Vij =
π αij βk pk ρ2
π/2
dθ sin θ cos θ Vij (αij ρ cos θ)×
0
× sin (αij kij ρ cos θ) sin (βk pk ρ sin θ),
(37)
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 591
Φi(jk) Vij, ki
1
= 3
π
π/2
π
2
2
dθ sin θ cos θ dθx sin θx ϕkjk (rjk )×
0
π
×
2π
dθy sin θy
0
0
2π
dφx
0
dφy exp (iβi pi ρ sin θ cos θpy )Vij, ki (|rij, ki |),
(38)
0
£¤¥
|rij, ki | =
m
m
mk,j M
M
k,j
2
2
=ρ
cos θ +
sin θ ∓ 2
sin θ cos θ cos θxy ,
mj + mk mj,k
mi
mi mj,k
cos θxy = cos θx cos θy + sin θx sin θy cos (φx − φy ),
cos θpy = cos θp cos θy + sin θp sin θy cos φy ,
m(mi + mj )
mM
αij =
,
, βi =
mi mj
mi (mj + mk )
mi Å ³ ¸¸ i-° Î ¸É¨ÍÒ, M = m1 + m2 + m3 .
2. Œ„ˆ”ˆŠ–ˆŸ “‚…ˆ‰ ”„„……‚
ˆ ‹ˆ—ˆˆ Š“‹‚‘Šƒ ‚‡ˆŒ„…‰‘’‚ˆŸ
‚ ¶. 1.2 ³Ò · ¸¸³μÉ·¥²¨ ¢ ·¨ ´É ¶·¨¡²¨¦¥´´μ£μ Ê봃 Š‚: ¢¢¥¤¥´¨¥
Ô±· ´¨·μ¢ ´´μ£μ ¶μÉ¥´Í¨ ² (27) ¢ ¶μ²ÊÎ¥´´Ò¥ · ´¥¥ Ê· ¢´¥´¨Ö (8)Ä(14) ¤ ²μ
´ ³ ¢μ§³μ¦´μ¸ÉÓ ·¥Ï¨ÉÓ § ¤ ÎÊ μ · ¸¸¥Ö´¨¨ ¶·μÉμ´μ¢ ¤¥°É·μ´ ³¨. ‡ ³¥É¨³,
ÎÉμ ¥¸²¨ ¶μÉ¥´Í¨ ² ŠÊ²μ´ ¢¢μ¤¨ÉÓ ´¥¶μ¸·¥¤¸É¢¥´´μ ¢ ¨¸Ìμ¤´ÊÕ ¸¨¸É¥³Ê (3),
Éμ “” μ± §Ò¢ ÕÉ¸Ö ´¥· §·¥Ï¨³Ò³¨ ¢¢¨¤Ê ¢μ§´¨± ÕÐ¨Ì ¸¨´£Ê²Ö·´μ¸É¥° ¢
Ö¤· Ì ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨°. μ¶Òɱ¨ ³μ¤¨Ë¨Í¨·μ¢ ÉÓ “” ¸ Í¥²ÓÕ Ê봃 Š‚ μ¸ÊÐ¥¸É¢²Ö²¨¸Ó ¢ ·Ö¤¥ · ¡μÉ (¸³., ´ ¶·¨³¥·, [99Ä104]). ¤¨´ ¨§ ²£μ·¨É³μ¢ Ê봃 Š‚ ¢Ò£²Ö¤¥² ¸²¥¤ÊÕШ³ μ¡· §μ³. ˆ§ Ê· ¢´¥´¨° ɨ¶ ” ¤¤¥¥¢ ,
¸μ¤¥·¦ Ð¨Ì Š‚, ¢Ò¤¥²Ö² ¸Ó £² ¢´ Ö ±Ê²μ´μ¢¸± Ö μ¸μ¡¥´´μ¸ÉÓ, ¨ ¢μ§´¨± ÕШ° ¶·¨ ÔÉμ³ ± ± μ¸É Éμ± ÔËË¥±É¨¢´Ò° Ö¤¥·´μ-±Ê²μ´μ¢¸±¨° ¶μÉ¥´Í¨ ² ÌμÉÖ
¨ ´¥ ¡Ò² É ±¨³ ¦¥ ±μ·μÉ±μ¤¥°¸É¢ÊÕШ³, ± ± ¨¸Ìμ¤´Ò° ¶μÉ¥´Í¨ ² Ψ¸Éμ
Ö¤¥·´μ£μ ¢§ ¨³μ¤¥°¸É¢¨Ö, ´μ Ê¡Ò¢ ² ¸ Ê¢¥²¨Î¥´¨¥³ · ¸¸ÉμÖ´¨Ö ³¥¦¤Ê ´Ê±²μ´ ³¨ £μ· §¤μ ¡Ò¸É·¥¥ ¶μÉ¥´Í¨ ² ŠÊ²μ´ .
‚ ´ ¸ÉμÖÐ¥³ · §¤¥²¥ ³Ò É ±¦¥ ¢Ò¤¥²¨³ ¸´ Î ² ¨§ “” £² ¢´ÊÕ ±Ê²μ´μ¢¸±ÊÕ Î ¸ÉÓ ¨, · §²μ¦¨¢ ¸²μ¦´Ò° Ö¤¥·´μ-±Ê²μ´μ¢¸±¨° μ¸É Éμ± ¢μ²´μ¢μ°
592 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
ËÊ´±Í¨¨ ¢ ·Ö¤ ¶μ ¶·μ¨§¢¥¤¥´¨Ö³ ¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨°, ¸Ëμ·³Ê²¨·Ê¥³ Ê· ¢´¥´¨Ö ¤²Ö ´¥¨§¢¥¸É´ÒÌ ±μÔË˨ͨ¥´Éμ¢ · §²μ¦¥´¨Ö (· ¤¨ ²Ó´ÒÌ ËÊ´±Í¨°)
¤¢ÊÌ (¨§ Ï¥¸É¨) μÉ´μ¸¨É¥²Ó´ÒÌ ¶¥·¥³¥´´ÒÌ, ¨³¥ÕÐ¨Ì · §³¥·´μ¸ÉÓ ¤²¨´Ò.
Š·μ³¥ · §¢¨Éμ° μ¡Ð¥° É¥μ·¨¨ ³Ò É ±¦¥ ¶μ¤·μ¡´μ μ¸É ´μ¢¨³¸Ö ´ £² ¢´μ³
¶·¨¡²¨¦¥´¨¨ ¤²Ö ´Ê²¥¢ÒÌ §´ Î¥´¨° ¤¢ÊÌ μÉ´μ¸¨É¥²Ó´ÒÌ μ·¡¨É ²Ó´ÒÌ ³μ³¥´Éμ¢ 1 ¨ 2 . Šμ´¥Î´μ, ¤²Ö ¡μ²¥¥ Éμδμ£μ ·¥§Ê²ÓÉ É ´¥μ¡Ì줨³μ ¡Ò²μ
¡Ò μ¸É ¢¨ÉÓ ¢ · §²μ¦¥´¨ÖÌ ¡μ²ÓÏ¥¥ Ψ¸²μ ¸² £ ¥³ÒÌ. ’ ± ÎÉμ ·¥§Ê²ÓÉ É ¢
¶·¨¡²¨¦¥´¨¨ 1 = 2 = 0 ¶·¨ ÊΥɥ Š‚ ¡Ê¤¥É ¢ μ¸´μ¢´μ³ ²¨ÏÓ μÍ¥´μδҳ,
± Î¥¸É¢¥´´Ò³. ’μδμ¸ÉÓ · ¸Î¥Éμ¢ ¢¸¥ ¦¥ ³μ¦´μ ¶μ¢Ò¸¨ÉÓ, ´μ É죤 § ¤ Î ¸É ´μ¢¨É¸Ö ¡μ²¥¥ £·μ³μ§¤±μ°.
μ²ÊÎ¥´´ Ö ¢ [102] ¶¥·¥Ëμ·³Ê²¨·μ¢ ´´ Ö ¸¨¸É¥³ Ê· ¢´¥´¨° ¸ ÊÎ¥Éμ³
Š‚ ¸μ¤¥·¦¨É ÔËË¥±É¨¢´Ò¥ (Ö¤¥·´μ-±Ê²μ´μ¢¸±¨¥) ¶μÉ¥´Í¨ ²Ò ʦ¥ ¸ ±μ´¥Î´Ò³ · ¤¨Ê¸μ³ ¢§ ¨³μ¤¥°¸É¢¨Ö, ¶μÔÉμ³Ê ¡Ê¤¥³ ¨¸Ì줨ÉÓ ¨§ ·¥§Ê²ÓÉ Éμ¢ ÔɨÌ
· ¡μÉ ¨ ¶μ²Ó§μ¢ ÉÓ¸Ö ¢ μ¸´μ¢´μ³ ¨Ì μ¡μ§´ Î¥´¨Ö³¨.
μ²´ÊÕ ¢μ²´μ¢ÊÕ ËÊ´±Í¨Õ Ψ ¸¨¸É¥³Ò É·¥Ì Î ¸É¨Í ¤²Ö ¸²ÊÎ Ö · ¸¸¥Ö´¨Ö
§ ·Ö¦¥´´μ° Î ¸É¨ÍÒ 1 ´ ¸¢Ö§ ´´μ° ¸¨¸É¥³¥ ¤¢ÊÌ ¤·Ê£¨Ì Î ¸É¨Í, § ·Ö¦¥´´μ° 2
¨ ´¥°É· ²Ó´μ° 3, ³μ¦´μ § ¶¨¸ ÉÓ ¢ ¢¨¤¥ [102]
C
)(Ψ(12) + Ψ(23) + Ψ(31) ),
Ψ = (1 + G0 T12
(39)
£¤¥ É·¨ ±μ³¶μ´¥´ÉÒ Ψ(ij) (§¤¥¸Ó ¨ ¤ ²¥¥: i, j = 1, 2, 3; i = j) Ê¤μ¢²¥É¢μ·ÖÕÉ
¸¨¸É¥³¥ ¸¢Ö§ ´´ÒÌ Ê· ¢´¥´¨°
N
C
(1 + G0 T12
)(Ψ(12) + Ψ(31) ),
Ψ(23) = Φ(23) + G0 T̃23
N
C
Ψ(31) = G0 T̃31
(1 + G0 T12
)(Ψ(12) + Ψ(23) ),
Ψ
(12)
=
N
G0 T̃12
(1
+
C
G0 T12
)(Ψ(23)
+Ψ
(31)
(40)
),
£¤¥ ®¸¢μ¡μ¤´Ò°¯ β¥´ Φ(23) μ¶·¥¤¥²Ö¥É¸Ö ´¥§ ¢¨¸¨³Ò³ ¨´É¥£· ²Ó´Ò³ Ê· ¢´¥´¨¥³ ¤²Ö ¸¨¸É¥³Ò É¥Ì ¦¥ É·¥Ì Î ¸É¨Í ¸ ®Ê·¥§ ´´Ò³¯ ¶μÉ¥´Í¨ ²μ³ ¢§ N
≡ V23 ³¥¦¤Ê
¨³μ¤¥°¸É¢¨Ö, Ö¢²ÖÕШ³¸Ö ¸Ê³³μ° Ö¤¥·´μ£μ ¶μÉ¥´Í¨ ² V23
C
Î ¸É¨Í ³¨ 2 ¨ 3 ¨ ±Ê²μ´μ¢¸±μ£μ ¶μÉ¥´Í¨ ² V12 ³¥¦¤Ê Î ¸É¨Í ³¨ 1 ¨ 2:
N
C (23)
Φ(23) = Φ23 + G0 T23
G0 T12
Φ .
(41)
‡¤¥¸Ó Φ23 Å ®´ Î ²Ó´ Ö¯ ¢μ²´μ¢ Ö ËÊ´±Í¨Ö, ¶·¥¤¸É ¢²Ö¥³ Ö ¢ ¢¨¤¥ ¶·μ¨§¢¥¤¥´¨Ö ¢μ²´μ¢μ° ËÊ´±Í¨¨ ϕ(1, 23) μÉ´μ¸¨É¥²Ó´μ£μ ¤¢¨¦¥´¨Ö Î ¸É¨ÍÒ 1
¨ ¸¨¸É¥³Ò (23) (¶²μ¸± Ö ¢μ²´ ) ¨ ψ23 (23) Å ¢´ÊÉ·¥´´¥° ¢μ²´μ¢μ° ËÊ´±Í¨¨
¸¨¸É¥³Ò Î ¸É¨Í 2 ¨ 3 ¢ ¸¢Ö§ ´´μ³ ¸μ¸ÉμÖ´¨¨ (23 Å ¸μμÉ¢¥É¸É¢ÊÕÐ Ö Ô´¥·£¨Ö
¸¢Ö§¨ ¸¨¸É¥³Ò ¤¢ÊÌ Î ¸É¨Í):
Φ23 = ϕ(1, 23)ψ23 (23).
(42)
¶μ³´¨³ μ¶·¥¤¥²¥´¨Ö 춥· Éμ·μ¢ [102], ¢Ìμ¤ÖÐ¨Ì ¢ Ê· ¢´¥´¨Ö (40),
C
C
(41): T12
≡ T12
(Z) Å ¤¢ÊÌÎ ¸É¨Î´Ò° ±Ê²μ´μ¢¸±¨° 춥· Éμ· ¶¥·¥Ìμ¤ , ʤμC
C
C
= V12
(1 + G0 T12
); T̃ijN ≡
¢²¥É¢μ·ÖÕШ° 춥· Éμ·´μ³Ê Ê· ¢´¥´¨Õ T12
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 593
T̃ijN (Z) Å ¤¢ÊÌÎ ¸É¨Î´Ò¥ Ö¤¥·´μ-±Ê²μ´μ¢¸±¨¥ 춥· Éμ·Ò, Ê¤μ¢²¥É¢μ·ÖÕШ¥
N
C
C
C
Ê· ¢´¥´¨Õ T̃ijN = Vij (1 + GC
12 T̃ij ); G12 ≡ G12 (Z) = G0 (1 + T12 G0 ) ≡
C
(1 + G0 T12 )G0 Å ±Ê²μ´μ¢¸± Ö ËÊ´±Í¨Ö (춥· Éμ· ƒ·¨´ ) ¤²Ö ¸¨¸É¥³Ò É·¥Ì
Î ¸É¨Í, ¤¢¥ ¨§ ±μÉμ·ÒÌ, 1 ¨ 2, § ·Ö¦¥´Ò. „²Ö ¤ ²Ó´¥°Ï¨Ì ¶·¥μ¡· §μ¢ ´¨°
¨ μ¡Ð¥£μ ¨¸¸²¥¤μ¢ ´¨Ö ´ ³ ¥Ð¥ ¶μ´ ¤μ¡ÖÉ¸Ö ¡μ²¥¥ ¶·μ¸ÉÒ¥, Î¥³ T̃ijN , ¤¢ÊÌÎ ¸É¨Î´Ò¥ 춥· Éμ·Ò ¶¥·¥Ìμ¤ TijN ≡ TijN (Z), Ê¤μ¢²¥É¢μ·ÖÕШ¥ Ê· ¢´¥´¨Ö³
TijN = Vij (1 + G0 TijN ) ¨ ¸¢Ö§ ´´Ò¥ Éμ²Ó±μ ¸ Ö¤¥·´Ò³¨ ¶μÉ¥´Í¨ ² ³¨ ¢§ ¨³μ¤¥°¸É¢¨Ö Vij . ‘ Ôɨ³¨ 춥· Éμ· ³¨ Ê· ¢´¥´¨Ö ¤²Ö 춥· Éμ·μ¢ T̃ijN ³μ¦´μ
N
C
§ ¶¨¸ ÉÓ ¥Ð¥ ¨ ¢ ¢¨¤¥ T̃ijN = TijN [1 + (GC
12 − G0 )T̃ij ]. ·¨ V12 → 0 ¡Ê¤¥³,
C
C
N
N
(23)
μÎ¥¢¨¤´μ, ¨³¥ÉÓ T12 → 0, G12 → G0 , T̃ij → Tij , Φ
→ Φ23 .
2.1. ¥·¥Ëμ·³Ê²¨·μ¢± Ê· ¢´¥´¨° ¤¢¨¦¥´¨Ö. ‘¨¸É¥³ Ê· ¢´¥´¨° (39),
(40) ¤²Ö ¶μ²´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨ Ψ É·¥ÌÎ ¸É¨Î´μ° ¸¨¸É¥³Ò ¸ ¤¢Ê³Ö § ·Ö¦¥´´Ò³¨ Î ¸É¨Í ³¨ Ö¢²Ö¥É¸Ö ¢¥¸Ó³ ¸²μ¦´μ° ¨ ¤²Ö μ¡Ð¥£μ ¨¸¸²¥¤μ¢ ´¨Ö,
¨ ¤²Ö Ψ¸²¥´´ÒÌ · ¸Î¥Éμ¢ (· ¢´μ ± ± ¨ Ê· ¢´¥´¨¥ (41), ·¥Ï¥´¨¥ ±μÉμ·μ£μ
´¥μ¡Ì줨³μ ¤²Ö ´ Ì즤¥´¨Ö ËÊ´±Í¨¨ Ψ ¨ ¶·¥¤¸É ¢²Ö¥É ¸μ¡μ° μɤ¥²Ó´ÊÕ § ¤ ÎÊ). μÔÉμ³Ê ¢´ Î ²¥ ³Ò Ôɨ Ê· ¢´¥´¨Ö ¶·¥μ¡· §Ê¥³ ± ¡μ²¥¥ Ê¤μ¡´μ³Ê ¤²Ö
Ψ¸²¥´´μ£μ ·¥Ï¥´¨Ö ¢¨¤Ê, ʶ·μÐ Ö, ¶·¥¦¤¥ ¢¸¥£μ, ¢Ìμ¤ÖШ¥ ¢ ´¨Ì ¸²μ¦´Ò¥
춥· Éμ·Ò ¨ ¨§¡ ¢²ÖÖ¸Ó μÉ Î ¸É¨ É ±¨Ì 춥· Éμ·μ¢ [105].
N
μ¤¸É ¢¨¢, ´ ¶·¨³¥·, ¢μ ¢Éμ·μ¥ Ê· ¢´¥´¨¥ (40) ¢³¥¸Éμ 춥· Éμ· T̃31
C N
C
¸Ê³³Ê V31 + V31 G12 T̃31 ¨ ¢μ¸¶μ²Ó§μ¢ ¢Ï¨¸Ó μ¶·¥¤¥²¥´¨¥³ G12 , ¶μ²ÊΨ³
C
Ψ(31) = G0 V31 (1 + G0 T12
)(Ψ(12) + Ψ(23) )+
C
N
C
+ G0 V31 (1 + G0 T12
) G0 T̃31
(1 + G0 T12
)(Ψ(12) + Ψ(23) ) . (43)
‚Ò· ¦¥´¨¥ ¢ ±¢ ¤· É´ÒÌ ¸±μ¡± Ì ¥¸ÉÓ ´¥ ÎÉμ ¨´μ¥, ± ± ±μ³¶μ´¥´É Ψ(31) ,
¶μÔÉμ³Ê c ÊÎ¥Éμ³ (39) Ê· ¢´¥´¨¥ (43) §´ Ψɥ²Ó´μ ʶ·μ¸É¨É¸Ö:
Ψ(31) = G0 V31 Ψ,
(44)
£¤¥ ¢ ¶· ¢μ° Î ¸É¨ ʦ¥ ¨³¥¥³ ¨¸±μ³ÊÕ ¢μ²´μ¢ÊÕ ËÊ´±Í¨Õ Ψ. ´ ²μ£¨ÎN
N
´Ò³ μ¡· §μ³ ¨§¡ ¢¨³¸Ö μÉ μ¶¥· Éμ·μ¢ T̃12
¨ T̃23
¢ μ¸É ²Ó´ÒÌ Ê· ¢´¥´¨ÖÌ
¸¨¸É¥³Ò (40):
(45)
Ψ(12) = G0 V12 Ψ,
(23)
(23)
C
(23)
.
(46)
Ψ
−Φ
= G0 V23 Ψ − (1 + G0 T12 )Φ
줥°¸É¢Ê¥³ É¥¶¥·Ó ´ ²¥¢Ò¥ ¨ ¶· ¢Ò¥ Î ¸É¨ Ê· ¢´¥´¨° (44)Ä(46) ±Ê²μ´μ¢C
). ‘²μ¦¨¢ ¶μ²ÊΨ¢Ï¨¥¸Ö Ê· ¢´¥´¨Ö, ¸ ÊÎ¥Éμ³ (39)
¸±¨³ 춥· Éμ·μ³ (1+G0 T12
C
¨ μ¶·¥¤¥²¥´¨Ö G12 ¶μ²ÊΨ³ μ¤´μ Ê· ¢´¥´¨¥ ¤²Ö ¶μ²´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨ Ψ:
C
C
(23)
Ψ − (1 + G0 T12
)Φ(23) = GC
+
12 U Ψ − (1 + G0 T12 )Φ
C
(23)
+ GC
. (47)
12 (V31 + V12 )(1 + G0 T12 )Φ
594 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
”Ê´±Í¨Ö Φ(23) , ¢Ìμ¤ÖÐ Ö ¢ (47), Ö¢²Ö¥É¸Ö, ± ± ¨ Ψ, ´¥¨§¢¥¸É´μ°, ¨ ¤ ²¥¥
³Ò ¶μ± ¦¥³, ± ± ¥¥ ³μ¦´μ ´ °É¨, ¨¸¸²¥¤ÊÖ Ê· ¢´¥´¨¥ (41). μ ¸´ Î ² ¶·¥¤²μ¦¨³ ³¥Éμ¤ ´ Ì즤¥´¨Ö Ψ ¨§ Ê· ¢´¥´¨Ö (47), ¸Î¨É Ö Φ(23) ¨§¢¥¸É´μ°,
É. ¥. ʦ¥ ´ °¤¥´´μ° ¨§ (41).
Î¥¢¨¤´μ, ¢ (47) ¢³¥¸Éμ Φ(23) Ê¤μ¡´¥¥ ¢¢¥¸É¨ ËÊ´±Í¨Õ
C
)Φ(23) .
χ(23) = (1 + G0 T12
(48)
É ´μ¢ Ö ËÊ´±Í¨Ö ¨³¥¥É ¶·μ¸Éμ° Ë¨§¨Î¥¸±¨° ¸³Ò¸². …¸²¨ ¶¥·¥¶¨¸ ÉÓ Ê· ¢´¥´¨¥ (47) ± ±
(23)
(23)
) + GC
,
Ψ − χ(23) = GC
12 U (Ψ − χ
12 (V31 + V12 )χ
(49)
Éμ ¢¨¤´μ, ÎÉμ ¶·¨ ®μɱ²ÕÎ¥´¨¨¯ Ö¤¥·´ÒÌ ¢§ ¨³μ¤¥°¸É¢¨° Î ¸É¨ÍÒ 1 ¸ ¤¢Ê³Ö
μ¸É ²Ó´Ò³¨ (É. ¥. ¶·¨ V12 = V31 = 0) ·¥Ï¥´¨¥³ ¶μ²ÊÎ ÕÐ¥£μ¸Ö Ê· ¢´¥´¨Ö
(23)
)
Ψ − χ(23) = GC
12 V23 (Ψ − χ
(50)
¡Ê¤¥É ËÊ´±Í¨Ö Ψ = χ(23) . Éμ ¢¨¤´μ É ±¦¥ ¨ ¨§ ¨¸Ìμ¤´ÒÌ Ê· ¢´¥´¨° (39)
¨ (40), É ± ± ± ¶·¨ ÔÉμ³
Ψ(31) = Ψ(12) = 0,
Ψ(23) = Φ(23)
C
¨ Ψ = (1 + G0 T12
)Ψ(23) = χ(23) .
’ ±¨³ ¦¥ μ¡· §μ³ ¶·¥μ¡· §Ê¥³ Ê· ¢´¥´¨¥ (41), ¨§¡ ¢¨¢Ï¨¸Ó μ¤´μ¢·¥N
³¥´´μ μÉ ¸²μ¦´μ£μ 춥· Éμ· T23
. ‡ ³¥É¨³ ¥Ð¥, ÎÉμ Ê· ¢´¥´¨¥ (41) ¢ μɲ¨Î¨¥ μÉ (47) ´¥ § ¢¨¸¨É μÉ Ö¤¥·´ÒÌ ¶μÉ¥´Í¨ ²μ¢ V12 ¨ V31 , É ± ± ± μÉ ´¨Ì ´¥
§ ¢¨¸ÖÉ ¨§¢¥¸É´ Ö ËÊ´±Í¨Ö Φ23 ¨ ¢¸¥ 춥· Éμ·Ò, ¢Ìμ¤ÖШ¥ ¢ (41), ± ± ¨ 춥(23)
¨ χ(23) É ±¦¥ ´¥ § ¢¨¸ÖÉ μÉ ¶μÉ¥´Í¨ ²μ¢
· Éμ· GC
12 . μÔÉμ³Ê ËÊ´±Í¨¨ Φ
V12 ¨ V31 ¨ ´¥ ³¥´ÖÕɸÖ, ¥¸²¨ ¶μ¸²¥¤´¨¥ § ´Ê²¨ÉÓ.
N
ˆ¸¶μ²Ó§ÊÖ Ê· ¢´¥´¨¥ ¤²Ö 춥· Éμ· T23
, § ¶¨Ï¥³ Ê· ¢´¥´¨¥ (41) ¢ ¢¨¤¥
(23)
C (23)
N
C (23)
Φ
(51)
= Φ23 + G0 V23 G0 T12 Φ
+ G0 V23 G0 T23
G0 T12
Φ
¨ § ³¥É¨³, ÎÉμ ¢Ò· ¦¥´¨¥ ¢ ±¢ ¤· É´ÒÌ ¸±μ¡± Ì ¸μ£² ¸´μ (41) ¥¸ÉÓ · §´μ¸ÉÓ
Φ(23) − Φ23 . „¥°¸É¢ÊÖ ¸²¥¢ ´ ¶μ²ÊÎ¥´´μ¥ Ê· ¢´¥´¨¥
C
Φ(23) = (1 − G0 V23 )Φ23 + G0 V23 (1 + G0 T12
)Φ(23)
C
1 + G0 T12
,
춥· Éμ·μ³
Í¨Õ ( ´ ²μ£ Φ23 )
¨¸¶μ²Ó§ÊÖ (48), ¢¢μ¤Ö 춥· Éμ·
GC
12
C
χC = (1 + G0 T12
)(1 − G0 V23 )Φ23 ,
¤²Ö ËÊ´±Í¨¨ χ
(23)
(52)
¨ μ¶·¥¤¥²ÖÖ ËÊ´±(53)
μ±μ´Î É¥²Ó´μ ¶μ²ÊΨ³ Ê· ¢´¥´¨¥
(23)
χ(23) = χC + GC
.
12 V23 χ
(54)
‚ ÔÉμ³ Ê· ¢´¥´¨¨ ¢¸¥, ±·μ³¥ ËÊ´±Í¨¨ χ(23) , ¸Î¨É ¥É¸Ö ¨§¢¥¸É´Ò³.
„ ²¥¥ ¡Ê¤¥³ ¨¸Ì줨ÉÓ ¨§ ¶·¥μ¡· §μ¢ ´´ÒÌ Ê· ¢´¥´¨° (49) ¨ (54), ´μ ¤²Ö
μ¡μ¸´μ¢ ´¨Ö ´¥±μÉμ·ÒÌ ÊÉ¢¥·¦¤¥´¨° ´ ³ É ±¦¥ ¶μ´ ¤μ¡ÖÉ¸Ö ¨ ¶¥·¢μ´ Î ²Ó´Ò¥ Ê· ¢´¥´¨Ö (39)Ä(41).
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 595
2.2. §²μ¦¥´¨¥ ¢ ¡Ò¸É·μ¸Ìμ¤ÖШ¥¸Ö ·Ö¤Ò. ˆ§ Ê· ¢´¥´¨° (39), (40),
¸μ¤¥·¦ Ð¨Ì ±·μ³¥ ±μ·μÉ±μ¤¥°¸É¢ÊÕÐ¨Ì Ö¤¥·´ÒÌ ¶μÉ¥´Í¨ ²μ¢ Vij É ±¦¥ ¨
C
¤ ²Ó´μ¤¥°¸É¢ÊÕШ° ±Ê²μ´μ¢¸±¨° ¶μÉ¥´Í¨ ² V12
, É¥³ ´¥ ³¥´¥¥ ¸²¥¤Ê¥É, ÎÉμ
μɤ¥²Ó´Ò¥ Î ¸É¨ ¶μ²´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨, ¨³¥´´μ Ψ(23) − Φ(23) , Ψ(31) ,
C
)Φ(23) = Ψ − χ(23) , ³μ¦´μ · §²μ¦¨ÉÓ ¢ ¡ÒΨ(12) ¨ · §´μ¸ÉÓ Ψ − (1 + G0 T12
¸É·μ¸Ìμ¤ÖШ¥¸Ö ·Ö¤Ò ¶μ K-£ ·³μ´¨± ³. Éμ£μ, μ¤´ ±μ, ´¥²Ó§Ö ¸¤¥² ÉÓ ¤²Ö
C
· §´μ¸É¨ Φ(23) − Φ23 ¢¢¨¤Ê ¤ ²Ó´μ¤¥°¸É¢¨Ö ¶μÉ¥´Í¨ ² V12
, ÎÉμ ´¥¶μ¸·¥¤¸É¢¥´´μ ¢¨¤´μ ¨§ (41). μÔÉμ³Ê μ¸É ´μ¢¨³¸Ö ¶μ± ´ · §²μ¦¥´¨¨ · §´μ¸É¨
Ψ − χ(23) .
¥¸³μÉ·Ö ´ ¢μ§³μ¦´μ¸ÉÓ · §²μ¦¥´¨Ö Ψ − χ(23) ¢ ·Ö¤ ¶μ £¨¶¥·¸Ë¥·¨Î¥¸±μ³Ê ¡ §¨¸Ê (¶μ¤μ¡´μ Éμ³Ê, ± ± ÔÉμ ¡Ò²μ ¸¤¥² ´μ ¢ ¶·¥¤Ò¤ÊÐ¥³ · §¤¥²¥),
¢ ´ ¸ÉμÖÐ¥³ ¸²ÊÎ ¥ ÔÉμ ¢Ò¶μ²´¨ÉÓ ±· °´¥ ´¥Ê¤μ¡´μ ¨§-§ ´ ²¨Î¨Ö ¢ ¶μ²C
. μÔÉμ³Ê · Í¨μ´ ²Ó´Ò³ §¤¥¸Ó ¡Ê¤¥É ¢Ò´μ³ £ ³¨²ÓÉμ´¨ ´¥ ¶μÉ¥´Í¨ ² V12
¡μ· ¸²¥¤ÊÕÐ¨Ì ¶¥·¥³¥´´ÒÌ: ¤¢ÊÌ · §³¥·´ÒÌ ¶¥·¥³¥´´ÒÌ ¤²¨´Ò r = |r| ¨
ρ = |ρ| ¨ ¤¢ÊÌ ¶ · ¸μμÉ¢¥É¸É¢ÊÕÐ¨Ì Ê£²μ¢ Ω1 ≡ (θ1 , ϕ1 ) ¨ Ω2 ≡ (θ2 , ϕ2 ),
μ¶·¥¤¥²ÖÕÐ¨Ì μ·¨¥´É ͨ¨ ¤¢ÊÌ É·¥Ì³¥·´ÒÌ ¢¥±Éμ·μ¢ r = r1 − r2 ¨ ρ =
r3 − (m1 r1 + m1 r2 )/(m1 + m2 ), £¤¥ rj ¨ mj Å · ¤¨Ê¸-¢¥±Éμ· ¨ ³ ¸¸ j-°
Î ¸É¨ÍÒ. Í¨μ´ ²Ó´μ¸ÉÓ É ±μ£μ ¢Ò¡μ· ¶¥·¥³¥´´ÒÌ μ¡ÑÖ¸´Ö¥É¸Ö É¥³, ÎÉμ
¢ ÔÉμ³ ¸²ÊÎ ¥ ±Ê²μ´μ¢¸±¨° ¶μÉ¥´Í¨ ² ¡Ê¤¥É § ¢¨¸¥ÉÓ Éμ²Ó±μ μÉ μ¤´μ° ¶¥C
C
≡ V12
(r). §²μ¦¨³ É¥¶¥·Ó · §´μ¸ÉÓ Ψ − χ(23) ¢ ·Ö¤ ¶μ
·¥³¥´´μ°: V12
¶·μ¨§¢¥¤¥´¨Ö³ ¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨°:
ψ1 m1 ,2 m2 (r, ρ)Y1 m1 (Ω1 )Y2 m2 (Ω2 ),
(55)
Ψ − χ(23) =
1 m1 2 m2
±μÉμ·Ò°, ± ± ¨ ·Ö¤ ¶μ K-£ ·³μ´¨± ³, ¡Ê¤¥É ¡Ò¸É·μ¸Ìμ¤ÖШ³¸Ö, ¶μ¸±μ²Ó±Ê
¶·¨ r → ∞ ¨ ρ → ∞ · §´μ¸ÉÓ Ψ − χ(23) ¨ ¢¸¥ · ¤¨ ²Ó´Ò¥ ËÊ´±Í¨¨
ψ1 m1 ,2 m2 (r, ρ) ¡Ò¸É·μ ¸É·¥³ÖÉ¸Ö ± ´Ê²Õ.
μ¤¸É ¢¨³ · §²μ¦¥´¨¥ (55) ¢ (49) ¨ ʳ´μ¦¨³ ¶μ²ÊΨ¢Ï¥¥¸Ö Ê· ¢´¥´¨¥
´ Y∗ m (Ω1 )Y∗ m (Ω2 ), ¨´É¥£·¨·ÊÖ ¥£μ ¤ ²¥¥ ¶μ Ê£²μ¢Ò³ ¶¥·¥³¥´´Ò³ Ω1
1
1
2
2
¨ Ω2 , ¶μ²ÊΨ³ ¡¥¸±μ´¥Î´ÊÕ ¸¨¸É¥³Ê ¸¢Ö§ ´´ÒÌ Ê· ¢´¥´¨° ¤²Ö · ¤¨ ²Ó´ÒÌ
ËÊ´±Í¨° ψ1 m1 ,2 m2 (r, ρ). ·¨¢¥¤¥³ §¤¥¸Ó Éμ²Ó±μ ·¥§Ê²ÓÉ É ¤²Ö 1 = m1 = 0
¨ 2 = m2 = 0, ±μÉμ·Ò³ ¡Ê¤¥³ ¶μ²Ó§μ¢ ÉÓ¸Ö ´¨¦¥, ¢¢μ¤Ö ¤²Ö ±· ɱμ¸É¨
μ¡μ§´ Î¥´¨¥ ψ00,00 (r, ρ) = ψ(r, ρ):
ψ(r, ρ) =
1
ψ1 m1 ,2 m2 (r, ρ)Y1 m1 (Ω1 )Y2 m2 (Ω2 )+
dΩ1 dΩ2 GC
=
12 U
4π
1 m1 2 m2
1
(23)
dΩ1 dΩ2 GC
+
. (56)
12 (V31 + V12 )χ
4π
¥¨§¢¥¸É´ Ö ËÊ´±Í¨Ö χ(23) ¢ (56) ¤μ ¸¨Ì ¶μ· ¸Î¨É ² ¸Ó § ¤ ´´μ°. „²Ö ¥¥
´ Ì즤¥´¨Ö ¢¢¥¤¥³ É¥¶¥·Ó ¸¨³¶ÉμɨΥ¸±ÊÕ ËÊ´±Í¨Õ χ23 , ±μÉμ·ÊÕ ³μ¦´μ
596 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
¶μ²ÊΨÉÓ ¨§ χ(23) ¶·¨ ʸ²μ¢¨¨ r → ∞ ¨ ±μÉμ· Ö ¡Ê¤¥É · ¢´ ¶·μ¨§¢¥¤¥´¨Õ
¨§¢¥¸É´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨ ¸¢Ö§ ´´μ£μ ¸μ¸ÉμÖ´¨Ö Î ¸É¨Í 2 ¨ 3 ψ23 (23) ¨
ËÊ´±Í¨¨ ϕC (1, 23) μÉ´μ¸¨É¥²Ó´μ£μ ¤¢¨¦¥´¨Ö ¸¢Ö§ ´´μ° ¸¨¸É¥³Ò (23) ¨ § ·Ö¦¥´´μ° Î ¸É¨ÍÒ 1. ”Ê´±Í¨Ö ϕC (1, 23) μ¶·¥¤¥²Ö¥É¸Ö ¸μ£² ¸´μ [106] ¸²¥¤ÊÕШ³ μ¡· §μ³:
(57)
ϕC (1, 23) = ϕ(1, 23) exp iη12 ln(k12 r − k12 r) , k12 = |k12 |,
£¤¥ η12 Å ¶ · ³¥É· ‡μ³³¥·Ë¥²Ó¤ ; k12 Å μÉ´μ¸¨É¥²Ó´Ò° ¨³¶Ê²Ó¸ Î ¸É¨Í 1
¨ 2 ´ ¡¥¸±μ´¥Î´μ¸É¨. ”Ê´±Í¨Ö (57) ¶¥·¥Ìμ¤¨É ¢ ¶²μ¸±ÊÕ ¢μ²´Ê ϕ(1, 23)
C
→ 0. ’ ±¨³ μ¡· §μ³, ËÊ´±Í¨Ö χ23 = ϕC (1, 23) ψ23 (23) ¸Î¨É ¥É¸Ö
¶·¨ V12
¨§¢¥¸É´μ°, ± ± ¨ Φ23 = ϕ(1, 23) ψ23 (23).
‚ Éμ ¢·¥³Ö ± ± · §´μ¸ÉÓ Φ(23) − Φ23 ´¥ · §² £ ¥É¸Ö ¢ ¡Ò¸É·μ¸Ìμ¤ÖШ°¸Ö
·Ö¤ ¶μ ¶μ²´μ³Ê ´ ¡μ·Ê Ê£²μ¢ÒÌ ËÊ´±Í¨°, · §´μ¸ÉÓ χ(23) − χ23 ³μ¦´μ · §²μ¦¨ÉÓ, ± ± ¨ Ψ − χ(23) :
χ(23) − χ23 =
χ1 m1 ,2 m2 (r, ρ)Y1 m1 (Ω1 )Y2 m2 (Ω2 ).
(58)
1 m1 2 m2
—Éμ¡Ò ¶μ²ÊΨÉÓ Ê· ¢´¥´¨Ö ¤²Ö · ¤¨ ²Ó´ÒÌ ËÊ´±Í¨° χ1 m1 ,2 m2 (r, ρ), ³μ¦´μ
¶μ¸Éʶ¨ÉÓ É ±¨³ ¦¥ μ¡· §μ³, ± ± ¨ ¶·¨ ¸μ¸É ¢²¥´¨¨ Ê· ¢´¥´¨° ¤²Ö ËÊ´±Í¨°
ψ1 m1 ,2 m2 (r, ρ), ¨¸¶μ²Ó§ÊÖ ´ ²μ£ Ê· ¢´¥´¨Ö (49) ¤²Ö ËÊ´±Í¨¨ χ(23) :
(23)
C
χ(23) − χ23 = GC
− χ23 ) + GC
12 V23 (χ
12 V23 χ23 + χ − χ23 ,
(59)
ÎÉμ ´¥¶μ¸·¥¤¸É¢¥´´μ ¸²¥¤Ê¥É ¨§ (54).
2.3. “· ¢´¥´¨Ö ¤²Ö · ¤¨ ²Ó´ÒÌ ËÊ´±Í¨° ¢ ¸²ÊÎ ¥ 1 = 2 = 0. ¥ μ£· ´¨Î¨¢ Ö μ¡Ð´μ¸É¨, · ¸¸³μÉ·¨³ ¢ ± Î¥¸É¢¥ ¶·¨³¥· ¶·μ¸É¥°Ï¥¥ Ê· ¢´¥´¨¥
¤²Ö · ¤¨ ²Ó´ÒÌ ËÊ´±Í¨° (56), ¢Ìμ¤ÖÐ¥¥ ¢ ¡¥¸±μ´¥Î´ÊÕ ¸¨¸É¥³Ê ¸¢Ö§ ´´ÒÌ
¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨°.
·¥¦¤¥ ¢¸¥£μ · ¸¸³μÉ·¨³ ɨ¶¨Î´Ò° μ¡Ð¨° Î¥ÉÒ·¥Ì±· É´Ò° ¨´É¥£· ² ¶μ
Ê£²μ¢Ò³ ¶¥·¥³¥´´Ò³, ¶·¨¸ÊɸɢÊÕШ° ¢ ¨¸¸²¥¤Ê¥³ÒÌ Ê· ¢´¥´¨ÖÌ:
J(r, ρ) = dΩ1 dΩ2 GC
(60)
12 f (r, ρ, Ω1 , Ω2 ),
£¤¥ f (r, ρ, Ω1 , Ω2 ) Å ¶μ± ¶·μ¨§¢μ²Ó´ Ö ËÊ´±Í¨Ö, GC
12 Å ±Ê²μ´μ¢¸± Ö ËÊ´±Í¨Ö ƒ·¨´ , ¢Ìμ¤ÖÐ Ö ¢ (56):
GC
12
≡
−1
2
2
C
= E + i0 +
Δr +
Δρ − V12
=
2μ12
2μ3(12)
−1
2 ρ2
2 r2
C
= E + i0 − Tr −
− Tρ −
− V12
, (61)
2μ12 r2
2μ3(12) ρ2
C −1
(E+i0−H0 −V12
)
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 597
∂
2 ∂
r
,
Tr = −
2μ12 r2 ∂r
∂r
1
∂
2 ∂
Tρ = −
ρ
,
2μ3(12) ρ2 ∂ρ
∂ρ
1
(62)
£¤¥ μ12 = m1 m2 /m12 ¨ μ3(12) = m3 m12 /(m3 + m12 ) Å ¶·¨¢¥¤¥´´Ò¥ ³ ¸¸Ò,
m12 = m1 + m2 . §²μ¦¨³ É¥¶¥·Ó f (r, ρ, Ω1 , Ω2 ) ¢ ·Ö¤ ¶μ ¶·μ¨§¢¥¤¥´¨Ö³
¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨°
f (r, ρ, Ω1 , Ω2 ) =
f1 m1 ,2 m2 (r, ρ)Y1 m1 (Ω1 )Y2 m2 (Ω2 )
(63)
1 m1 2 m2
¨ ¶μ¤¸É ¢¨³ ¸μμÉ´μÏ¥´¨Ö (61)Ä(63) ¢ (60):
J(r, ρ) = 4πGC
12 (r, ρ)f00,00 (r, ρ),
(64)
£¤¥ GC
12 (r, ρ) Å ®Ê±μ·μÎ¥´´ Ö¯ ±Ê²μ´μ¢¸± Ö ËÊ´±Í¨Ö ƒ·¨´ C −1
GC
,
12 (r, ρ) = (E + i0 − Tr − Tρ − V12 )
ËÊ´±Í¨Ö f00,00 (r, ρ) μ¶·¥¤¥²Ö¥É¸Ö ¸μ£² ¸´μ (63) ¢Ò· ¦¥´¨¥³
1
f00,00 (r, ρ) =
dΩ1 dΩ2 f (r, ρ, Ω1 , Ω2 ).
4π
(65)
(66)
μ¸±μ²Ó±Ê ¢ ¡¥¸±μ´¥Î´Ò° ¡Ò¸É·μ¸Ìμ¤ÖШ°¸Ö ·Ö¤ ¢ (56) μ¸´μ¢´μ° ¢±² ¤
¢´μ¸¨É ¸² £ ¥³μ¥ ¸ 1 = m1 = 0 ¨ 2 = m2 = 0, ³Ò ¨ · ¸¸³μÉ·¨³ ÔÉμÉ ¶·μ¸É¥°Ï¨° ¸²ÊÎ °. ˆ¸¶μ²Ó§ÊÖ ¸μμÉ´μÏ¥´¨Ö (64)Ä(66), ¨§ (56) ¶μ²ÊΨ³ ²¨´¥°´μ¥
´¥μ¤´μ·μ¤´μ¥ ¨´É¥£· ²Ó´μ¥ Ê· ¢´¥´¨¥ ¤²Ö ψ(r, ρ):
ψ(r, ρ) =
1
C
G
(r,
ρ)
ψ(r,
ρ)
dΩ
dΩ2 U +
1
(4π)2 12
1 C
G (r, ρ) dΩ1 dΩ2 (V31 + V12 )χ(23) . (67)
+
4π 12
´ ²μ£¨Î´Ò³ μ¡· §μ³ ³μ¦´μ ¶μ²ÊΨÉÓ, ¨¸¶μ²Ó§ÊÖ (58) ¨ (59), ¨´É¥£· ²Ó´μ¥ Ê· ¢´¥´¨¥ ¤²Ö · ¤¨ ²Ó´μ° ËÊ´±Í¨¨ χ(r, ρ) = χ00,00 (r, ρ):
1
C
G
(r,
ρ)
χ(r,
ρ)
dΩ
dΩ2 V23 +
1
(4π)2 12
1 C
−1
G12 (r, ρ) dΩ1 dΩ2 V23 χ23 + (GC
(χC − χ23 ) . (68)
+
12 )
4π
χ(r, ρ) =
μ¸±μ²Ó±Ê Ê· ¢´¥´¨¥ (68) ¸μ¤¥·¦¨É Éμ²Ó±μ μ¤´Ê ´¥¨§¢¥¸É´ÊÕ ËÊ´±Í¨Õ
χ(r, ρ), Í¥²¥¸μμ¡· §´μ ¸´ Î ² ·¥Ï¨ÉÓ ¨³¥´´μ ÔÉμ Ê· ¢´¥´¨¥, ±μÉμ·μ¥ § ¶¨Ï¥³ ¢ ±· É±μ° Ëμ·³¥:
χ(r, ρ) = GC
12 (r, ρ)χ(r, ρ)V 23 + X(r, ρ),
(69)
598 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
£¤¥
V 23
1
≡ V 23 (r, ρ) =
(4π)2
dΩ1
dΩ2 V23 (|r2 − r3 |)
¨ X(r, ρ) (¢Éμ·μ¥ ¸² £ ¥³μ¥ ¢ ¶· ¢μ° Î ¸É¨ (68)) Å ¨§¢¥¸É´Ò¥ ËÊ´±Í¨¨.
C
(r) ¨ Tρ [106]
‚¢¥¤¥³ ¸μ¡¸É¢¥´´Ò¥ ËÊ´±Í¨¨ 춥· Éμ·μ¢ Tr + V12
1/2
πη12
2
πη12
ϕk12 (r) =
k12 exp ik12 r −
F (1 + iη12 , 2, −2ik12 r),
π
2
sh (πη12 )
(70)
2 sin (k3(12) ρ)
,
ϕk3(12) (ρ) =
π
ρ
£¤¥ k3(12) Å ¢μ²´μ¢μ¥ Ψ¸²μ μÉ´μ¸¨É¥²Ó´μ£μ ¤¢¨¦¥´¨Ö Î ¸É¨ÍÒ 3 ¨ ¸¨¸É¥³Ò (12). ˆ¸¶μ²Ó§ÊÖ μ·Éμ´μ·³¨·μ¢±Ê ¨ ¶μ²´μÉÊ ÔÉ¨Ì ËÊ´±Í¨°, ¨´É¥£· ²Ó´μ¥ Ê· ¢´¥´¨¥ (69) ¤²Ö χ(r, ρ) ¸μ£² ¸´μ (64)Ä(66) ³μ¦´μ ¶¥·¥¶¨¸ ÉÓ ¢ ¢¨¤¥
∞
χ(r, ρ) − X(r, ρ) =
2
∞
dr r
0
dρ ρ 2 χ(r , ρ )V 23 (r , ρ )×
0
∞
×
0
dq ϕ∗q (r ) ϕq (r)
∞
dκ
0
ϕ∗κ (ρ ) ϕκ (ρ)
2 2
2 . (71)
q
2 κ
Z−
+
2μ12
2μ3(12)
ˆ´É¥£· ² ¶μ dκ ¢ (71) ³μ¦´μ ¶·¥¤¸É ¢¨ÉÓ ¢ Ëμ·³¥ ¨´É¥£· ² ¸ ¡¥¸±μ´¥Î´Ò³¨ ¶·¥¤¥² ³¨ ¨, ¨¸¶μ²Ó§ÊÖ É¥μ·¥³Ê μ ¢ÒÎ¥É Ì ¨ ËÊ´±Í¨¨ •¥¢¨¸ °¤ ,
¢ÒΨ¸²¨ÉÓ ¥£μ ¢ Ö¢´μ³ ¢¨¤¥:
2 1 1
π ρρ 2
∞
dκ
−∞
R± (q) = −
sin (κρ ) sin (κρ)
2 2
2 =
q
2 κ
E + i0 −
+
2μ12
2μ3(12)
2 1 =
R+ (q)Θ(bq ) + R− (q)Θ(−bq ) ,
π ρρ
πμ3(12) exp ±i bq ρ sin bq ρ Θ(ρ − ρ )+
2
bq
+ exp ±i bq ρ sin
bq ρ Θ(ρ − ρ) ,
2μ3(12)
2 q 2
bq =
E
−
.
2
2μ12
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 599
μ¸²¥ ´ Ì즤¥´¨Ö ËÊ´±Í¨¨ χ(r, ρ) ¨ ¶μ¤¸É ´μ¢±¨ ¥¥ ¢ (67) ³μ¦´μ ´ ²μ£¨Î´Ò³ μ¡· §μ³ ¶·¥μ¡· §μ¢ ÉÓ ¨ ¨´É¥£· ²Ó´μ¥ Ê· ¢´¥´¨¥ (67) ¤²Ö ËÊ´±Í¨¨ ψ(r, ρ).
2.4. ‚Ò¢μ¤Ò. μ¤¢¥¤¥³ ¨É죨 ¨§²μ¦¥´´μ£μ ¢ ÔÉμ³ · §¤¥²¥.
1. μ²ÊÎ¥´´ Ö · ´¥¥ ¸¨¸É¥³ ¸¢Ö§ ´´ÒÌ ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨° ¤²Ö
μɤ¥²Ó´ÒÌ Î ¸É¥° ¶μ²´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨ Ψ ¸¨¸É¥³Ò É·¥Ì ¸¨²Ó´μ¢§ ¨³μ¤¥°¸É¢ÊÕÐ¨Ì Î ¸É¨Í, ¤¢¥ ¨§ ±μÉμ·ÒÌ § ·Ö¦¥´Ò ¨ Éμ²Ó±μ μ¤´ § ·Ö¦¥´´ Ö
Î ¸É¨Í ¸¢Ö§ ´ ¸ ´¥°É· ²Ó´μ°, ¸¢¥¤¥´ ± μ¤´μ³Ê ¨´É¥£· ²Ó´μ³Ê Ê· ¢´¥´¨Õ
¤²Ö Ψ.
2. ¨¡μ²¥¥ ¸²μ¦´Ò¥ Ë· £³¥´ÉÒ ¶μ²´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨, § ¢¨¸ÖШ¥
μ¤´μ¢·¥³¥´´μ μÉ Ö¤¥·´μ£μ ¨ ±Ê²μ´μ¢¸±μ£μ ¢§ ¨³μ¤¥°¸É¢¨°, ¶·¥¤¸É ¢²¥´Ò ¢
¢¨¤¥ ¡Ò¸É·μ¸Ìμ¤ÖÐ¨Ì¸Ö ·Ö¤μ¢ ¶μ ¶·μ¨§¢¥¤¥´¨Ö³ ¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨°, ¤²Ö · ¤¨ ²Ó´ÒÌ ËÊ´±Í¨°, § ¢¨¸ÖÐ¨Ì μÉ ¤¢ÊÌ · §³¥·´ÒÌ ¶¥·¥³¥´´ÒÌ, ¸μ¸É ¢²¥´ ¡¥¸±μ´¥Î´ Ö ¸¨¸É¥³ ¸¢Ö§ ´´ÒÌ ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨° ¨ ʱ § ´ ³¥Éμ¤
Ψ¸²¥´´μ£μ ·¥Ï¥´¨Ö μ¡·¥§ ´´μ° ¸¨¸É¥³Ò.
3. ¶·¨³¥·¥ μ¸´μ¢´μ£μ ¶·¨¡²¨¦¥´¨Ö ¸ ´Ê²¥¢Ò³¨ §´ Î¥´¨Ö³¨ μÉ´μ¸¨É¥²Ó´ÒÌ ³μ³¥´Éμ¢, ±μÉμ·μ¥ ¸¢μ¤¨É¸Ö ± ¤¢Ê³ ¸¢Ö§ ´´Ò³ ¨´É¥£· ²Ó´Ò³ Ê· ¢´¥´¨Ö³ ¤²Ö ¤¢ÊÌ · ¤¨ ²Ó´ÒÌ ËÊ´±Í¨°, Ê· ¢´¥´¨Ö ¤¢¨¦¥´¨Ö § ¶¨¸ ´Ò ¢ Ëμ·³¥,
Ê¤μ¡´μ° ¤²Ö ¨Ì Ψ¸²¥´´μ£μ ·¥Ï¥´¨Ö.
‡Š‹—…ˆ…
¸´μ¢´ Ö ¨¤¥Ö ¶·¥¤²μ¦¥´´μ£μ ¶μ¤Ìμ¤ Å ¢Ò¤¥²¥´¨¥ ¢ ³μ¤¨Ë¨Í¨·μ¢ ´´ÒÌ Ê· ¢´¥´¨ÖÌ ” ¤¤¥¥¢ ¤²Ö ´¥¶·¥·Ò¢´μ£μ ¸¶¥±É· Ô´¥·£¨° Ë· £³¥´Éμ¢ ¨¸±μ³μ° ¶μ²´μ° ¢μ²´μ¢μ° ËÊ´±Í¨¨ É·¥ÌÎ ¸É¨Î´μ° ¸¨¸É¥³Ò, ±μÉμ·Ò¥ ³μ¦´μ
¡Ò²μ ¡Ò · §²μ¦¨ÉÓ ¢ ¡Ò¸É·μ¸Ìμ¤ÖШ¥¸Ö ·Ö¤Ò, É. ¥. Ë ±É¨Î¥¸±¨ Å ·¥ ²¨§ ꬅ ¨¤¥¨ μ · §¤¥²¥´¨¨ ±μ´Ë¨£Ê· Í¨μ´´μ£μ ¶·μ¸É· ´¸É¢ ´ ®¢´ÊÉ·¥´´ÕÕ¯
¨ ®¢´¥Ï´ÕÕ¯ μ¡² ¸É¨. ˆ ÌμÉÖ ·¥ ²¨§ Í¨Õ ÔÉμ° ¨¤¥¨ ´Ê¦´μ · ¸¸³ É·¨¢ ÉÓ
± ± ³μ¤¥²Ó (¨¡μ μ´ ´¥ ¢μ§´¨± ¥É ± ± ·¥§Ê²ÓÉ É ·¥Ï¥´¨Ö ³´μ£μÎ ¸É¨Î´μ£μ
Ê· ¢´¥´¨Ö ˜·¥¤¨´£¥· ), É¥³ ´¥ ³¥´¥¥ μ´ μ± §Ò¢ ¥É¸Ö ¢¥¸Ó³ ÔËË¥±É¨¢´μ°
¨ ¶·μ¤Ê±É¨¢´μ°. Œ¥Éμ¤ £¨¶¥·¸Ë¥·¨Î¥¸±¨Ì £ ·³μ´¨± ± ± ¤μ¸É ÉμÎ´μ ³μдҰ ¶¶ · É μɲ¨Î´μ ¸μ봃 ¥É¸Ö ¸ ·¥Ï ¥³μ° § ¤ Î¥°. •μÉÖ, · §Ê³¥¥É¸Ö, ¶·¨
ÔÉμ³ ´¥ ¨¸±²ÕÎ ¥É¸Ö ¢μ§³μ¦´μ¸ÉÓ ¨¸¶μ²Ó§μ¢ ´¨Ö ¨ ¤·Ê£¨Ì ¡ §¨¸μ¢: ³´μ£μ¥
μ¶·¥¤¥²Ö¥É¸Ö · ¸¸³ É·¨¢ ¥³μ° ±μ´±·¥É´μ° ˨§¨Î¥¸±μ° ¸¨¸É¥³μ°, ÎÉμ ¶·μ¤¥³μ´¸É·¨·μ¢ ´μ, ¢ Î ¸É´μ¸É¨, ¶·¨ ¢Ò¢μ¤¥ Ê· ¢´¥´¨° ¤²Ö ¸¨¸É¥³Ò É·¥Ì Î ¸É¨Í,
¤¢¥ ¨§ ±μÉμ·ÒÌ § ·Ö¦¥´Ò, É. ¥. ¶·¨ ·¥Ï¥´¨¨ É ±μ° ´¥É·¨¢¨ ²Ó´μ° § ¤ Ψ, ± ±
Ê봃 ±Ê²μ´μ¢¸±μ£μ ¢§ ¨³μ¤¥°¸É¢¨Ö ¢ ±μ´É¨´Êʳ¥ É·¥Ì´Ê±²μ´´μ° ¸¨¸É¥³Ò ´ μ¸´μ¢¥ Ê· ¢´¥´¨° ” ¤¤¥¥¢ .
‡ ³¥É¨³, ÎÉμ ¶·¥¤² £ ¥³Ò° ¶μ¤Ìμ¤ ¶μ³¨³μ ¶·μÎ¥£μ ¤ ¥É ´¥ Éμ²Ó±μ ¤μ¶μ²´¨É¥²Ó´Ò¥ μ¡μ¸´μ¢ ´¨Ö ¨´É¥·¶μ²ÖÍ¨μ´´μ° ³μ¤¥²¨ Ö¤· , ʶμ³Ö´ÊÉμ° ¢μ
¢¢¥¤¥´¨¨ ¨ ¶·¨³¥´Ö¥³μ° ¶·¨ ¨¸¸²¥¤μ¢ ´¨¨ ¶·μÍ¥¸¸μ¢ ¢ ´¥¶·¥·Ò¢´μ³ ¸¶¥±É·¥
600 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
¤²Ö É·¥ÌÎ ¸É¨Î´ÒÌ Ö¤¥·´ÒÌ ¸¨¸É¥³, ´μ ¨ ¶μ§¢μ²Ö¥É ÔËË¥±É¨¢´μ · ¸¸Î¨É ÉÓ
ÊÉμδ¥´¨Ö ¤²Ö ÔÉμ° ³μ¤¥²¨. ¥³ ²μ¢ ¦´Ò³ Ë ±Éμ·μ³ ¤²Ö ¶μ§¨É¨¢´μ° μÍ¥´±¨
¨¸¶μ²Ó§Ê¥³μ£μ ¢ ´ ¸ÉμÖÐ¥° · ¡μÉ¥ ¶μ¤Ìμ¤ Ö¢²Ö¥É¸Ö μÉ´μ¸¨É¥²Ó´ Ö ¶·μ¸ÉμÉ ¶μ²ÊÎ¥´´ÒÌ Ê· ¢´¥´¨°, ¨Ì μ¤´μ³¥·´μ¸ÉÓ ¨ μ¤´μ§´ δμ¸ÉÓ. Œμ¦´μ ´ ¤¥ÖÉÓ¸Ö,
ÎÉμ ¶·¥¤² £ ¥³Ò° ¶μ¤Ìμ¤ ¸É ´¥É ¥Ð¥ μ¤´¨³ Ï £μ³ ± ¨¸¸²¥¤μ¢ ´¨Õ ¡μ²¥¥
¸²μ¦´ÒÌ Ö¤¥·´ÒÌ ¸¨¸É¥³.
ˆ, ´ ±μ´¥Í, ¢Éμ·Ò ¸Î¨É ÕÉ ¸¢μ¨³ ¤μ²£μ³ ¸ ¶·¨§´ É¥²Ó´μ¸ÉÓÕ ¢¸¶μ³´¨ÉÓ
¨³¥´ . ƒ. ‘¨É¥´±μ ¨ ‚. Š. ’ ·É ±μ¢¸±μ£μ, ¸ÉμÖ¢Ï¨Ì Ê ¨¸Éμ±μ¢ ÔÉμ° · ¡μÉÒ,
¢Ò· §¨ÉÓ ¡² £μ¤ ·´μ¸ÉÓ ‚. ”. • ·Î¥´±μ ¨ ‚. …. ŠÊ§Ó³¨Î¥¢Ê § μ¡¸Ê¦¤¥´¨¥ ·Ö¤ ¢μ¶·μ¸μ¢, ± ¸ ÕÐ¨Ì¸Ö μ¡μ¡Ð¥´¨° Ê· ¢´¥´¨° ” ¤¤¥¥¢ ¶·¨ ´ ²¨Î¨¨ ±Ê²μ´μ¢¸±μ£μ ¶μ²Ö, É ±¦¥ ˆ. ‚. ‘¨³¥´μ£Ê § ¢¥¸Ó³ ¶²μ¤μÉ¢μ·´Ò¥ ±μ´¸Ê²ÓÉ Í¨¨.
ˆ‹†…ˆ… ¡Ð¨¥ ¸μμÉ´μÏ¥´¨Ö ¤²Ö É·¥ÌÎ ¸É¨Î´ÒÌ K-£ ·³μ´¨± ¨³¥ÕÉ ¢¨¤ [7]
l l m m
l l LM
(Ω) =
lx ly mx my |LM uKx y x y (Ω),
uKx y
mx my
l l mx my
uKx y
l l
(Ω) = NKx y (cos θ)lx (sin θ)ly ×
× Puly +1/2, lx +1/2 (cos 2θ)Ylx mx (x̂)Yly my (ŷ),
(A.1)
£¤¥
l l
NKx y
=
2u!(K + 2)(u + lx + ly + 1)
,
Γ(u + lx + 3/2)Γ(u + ly + 3/2)
u=
K − lx − ly
,
2
u u+α u+β
(z − 1)u−s (z + 1)s
s
u
−
s
s=0
p
Γ(p + 1)
; lx , ly Å μ·¡¨É ²Ó´Ò¥ ³μ³¥´ÉÒ
Å ¶μ²¨´μ³ Ÿ±μ¡¨,
=
q
q!Γ(p − q + 1)
¸¢Ö§ ´´μ° ¶ ·Ò (23) ¨ Î ¸É¨ÍÒ 1 μÉ´μ¸¨É¥²Ó´μ ¶ ·Ò (23), ¸μμÉ¢¥É¸É¢ÊÕШ¥
±μμ·¤¨´ É ³ Ÿ±μ¡¨ x ¨ y.
l l LM
(Ω): K Å ¢¥²¨Î¨´ £¨‚ μ¡μ§´ Î¥´¨¨ K-£ ·³μ´¨± uKn (Ω) ≡ uKx y
¶¥·³μ³¥´É , n Å ³Ê²Óɨ¨´¤¥±¸, ¢±²ÕÎ ÕШ° ¢ ¸¥¡Ö ¢¥²¨Î¨´Ò μ·¡¨É ²Ó´ÒÌ
³μ³¥´Éμ¢ lx , ly , É ±¦¥ ¶μ²´Ò° μ·¡¨É ²Ó´Ò° ³μ³¥´É L μÉ´μ¸¨É¥²Ó´μ£μ ¤¢¨¦¥´¨Ö ¢¸¥Ì É·¥Ì Î ¸É¨Í ¨ ¥£μ ¶·μ¥±Í¨Õ M , Ω = {Θ, θx , φx , θy , φy } Å ¸μ¢μ±Ê¶´μ¸ÉÓ ¶Öɨ Ê£²μ¢ ¢ Ï¥¸É¨³¥·´μ³
¶·μ¸É· ´¸É¢¥, μ¶·¥¤¥²ÖÕШ¥ μ·¨¥´É ͨÕ
Ï¥¸É¨³¥·´μ£μ ¢¥±Éμ· ρ = x2 + y2 .
Puα, β = 2−u
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 601
ˆ‹†…ˆ… ¶¥· Éμ· ±¨´¥É¨Î¥¸±μ° Ô´¥·£¨¨ H0 ¢ £¨¶¥·¸Ë¥·¨Î¥¸±μ³ ¡ §¨¸¥ (ρ, Ω)
¨³¥¥É ¢¨¤
2
2 ∂ 5 ∂
ρ
.
(.1)
H0 = T 0 −
ΔΩ , T0 = −
2
2mρ
2mρ 5 ∂ρ ∂ρ
‘μ¡¸É¢¥´´Ò³¨ ËÊ´±Í¨Ö³¨ 춥· Éμ· ΔΩ Ö¢²ÖÕÉ¸Ö K-£ ·³μ´¨±¨ (A.1)
ΔΩ uKn (Ω) = −K(K + 4)uKn (Ω).
”Ê´±Í¨¨ uKn (Ω) μ·Éμ£μ´ ²Ó´Ò ¨ ¨Ì ´μ·³¨·ÊÕÉ μ¡Òδҳ μ¡· §μ³:
dΩ u∗Kn (Ω) uK n (Ω) = δKK δnn .
(.2)
(.3)
‘μ¡¸É¢¥´´Ò¥ ËÊ´±Í¨¨ ωq (ρ) 춥· Éμ· T0 ¢ (.1) μ¶·¥¤¥²ÖÕÉ¸Ö Ê· ¢´¥´¨¥³ [107]
√
q
2 q 2
ωq (ρ), ωq (ρ) = 2 J2 (qρ).
(.4)
T0 ωq (ρ) =
2m
ρ
´¨ Ê¤μ¢²¥É¢μ·ÖÕÉ Ê¸²μ¢¨Ö³ μ·Éμ´μ·³¨·μ¢±¨ [108]
∞
dρ ρ 5 ωq∗ (ρ) ωq (ρ) = δ(q − q )
(.5)
1
δ(ρ − ρ ).
ρ5
(.6)
0
¨ ¶μ²´μÉÒ
∞
dq ωq∗ (ρ) ωq (ρ ) =
0
μ¤¸É ¢¨¢ (7) ¢ (6), ʳ´μ¦¨³ ²¥¢ÊÕ ¨ ¶· ¢ÊÕ Î ¸É¨ ¶μ²ÊÎ¥´´μ£μ Ê· ¢´¥´¨Ö ´ u∗K n ¨ ¶·μ¨´É¥£·¨·Ê¥³ ¶μ ¶¥·¥³¥´´Ò³ Ω. ‚ ·¥§Ê²ÓÉ É¥ ¸ ÊÎ¥Éμ³ (.3)
¶μ²ÊΨ³ ¸¨¸É¥³Ê ¨´É¥£· ²Ó´ÒÌ Ê· ¢´¥´¨° ¤²Ö ψK n (ρ) [109]:
ψK n (ρ) =
dΩ u∗K n (Ω)G0 (Z)f (ρ, Ω)+
+ dΩ u∗K n (Ω)G0 (Z)(V12 + V31 )Φ, (.7)
f (ρ, Ω) = U
ψKn (ρ)uKn (Ω),
Kn
£¤¥ U Å ¸Ê³³ É·¥Ì ¶ ·´ÒÌ ¶μÉ¥´Í¨ ²μ¢ (5).
(.8)
602 Š‚‹œ—“Š ‚. ˆ., Š‡‹‚‘Šˆ‰ ˆ. ‚.
μ¤¸É ¢¨³ É¥¶¥·Ó ¢ Ê· ¢´¥´¨¥ (.7) 춥· Éμ· ƒ·¨´ G0 (Z) = Z − T0 +
−1
2
Δ
Ω
2mρ2
(.9)
¨ · §²μ¦¨³ f (ρ, Ω) ¢ (.8) ¶μ K-£ ·³μ´¨± ³:
f (ρ, Ω) =
fK n (ρ)uK n (Ω),
(.10)
dΩ u∗K n (Ω)f (ρ, Ω).
(.11)
K n
fK n (ρ) =
’죤 ¸ ÊÎ¥Éμ³ (.2), (.3) ¸¨¸É¥³ Ê· ¢´¥´¨° (.7) ¶·¥μ¡· §Ê¥É¸Ö ± ¢¨¤Ê
ψK n (ρ) = Z − T0 −
F (ρ) =
−1
2
K
(K
+
4)
F (ρ),
2mρ2
dΩ u∗K n (Ω) f (ρ, Ω) + (V12 + V31 )Φ .
(.12)
(.13)
§²μ¦¨³ É¥¶¥·Ó ËÊ´±Í¨Õ F (ρ) ¶μ ¶μ²´μ° ¸¨¸É¥³¥ ËÊ´±Í¨° 춥· Éμ· T0 (.4):
∞
(.14)
F (ρ) = dq ωq (ρ)Fq ,
0
∞
Fq =
dρ̄ ρ̄ 5 F (ρ̄)ωq∗ (ρ̄).
(.15)
0
μ¤¸É ¢¨³ (.14) ¢ (.12), (.13) ¨ ¢μ¸¶μ²Ó§Ê¥³¸Ö ¸μμÉ´μÏ¥´¨Ö³¨ (.4)Ä(.6)
¨ (.15). μ¸²¥ ÔÉ¨Ì μ¶¥· ͨ° ¸¨¸É¥³ Ê· ¢´¥´¨° (.12) ¶·¨μ¡·¥É ¥É μ±μ´Î É¥²Ó´Ò° ¢¨¤
ψK n (ρ) =
πm
2 ρ2
∞
dρ̄ ρ̄ 3 P± (ρ, ρ̄)
0
dΩ u∗K n (Ω)×
ψKn (ρ̄)uKn (Ω) + (V12 + V31 )Φ(ρ̄, Ω) ,
× U
(.16)
Kn
2
P± (ρ, ρ̄) = −
π
∞
dq q
0
J2 (qρ)J2 (q ρ̄)
2 ∓ i0 ,
q 2 − kK
(.17)
“‚…ˆŸ ”„„……‚ ˆ Œ…’„ ƒˆ…‘”…ˆ—…‘Šˆ• ƒŒˆŠ 603
2
2
2
2
2
2
£¤¥ kK
(E Å ¶μ²´ Ö Ô´¥·£¨Ö
≡ kK (ρ) = k0 − K (K + 4)/ρ , k0 = 2mE/
¸¨¸É¥³Ò). ’ ±¨³ μ¡· §μ³, ¡² £μ¤ ·Ö · §²μ¦¥´¨Õ f (ρ, Ω) ¢ ·Ö¤ ¶μ £¨¶¥·¸Ë¥·¨Î¥¸±¨³ £ ·³μ´¨± ³, É ±¦¥ · §²μ¦¥´¨Õ F (ρ) ¢ ·Ö¤ ¶μ ¸μ¡¸É¢¥´´Ò³
ËÊ´±Í¨Ö³ ωq (ρ) 춥· Éμ· T0 ¢ (.7) Î ¸É¨Î´μ, ´μ ¤μ¸É ÉμÎ´μ ¸ÊÐ¥¸É¢¥´´μ,
¢Ò¶μ²´¥´μ ¤¥°¸É¢¨¥ ·¥§μ²Ó¢¥´ÉÒ G0 (Z). „ ²¥¥ ¨´É¥£· ² ɨ¶ (.17) ³μ¦¥É
¡ÒÉÓ ¢ÒΨ¸²¥´ ´ ²¨É¨Î¥¸±¨ [110]:
Jν (ax)Jν (bx)
dx x
=
x2 + c2
⎧
Iν (bc)Kν (ac),
⎪
⎪
⎪
⎨ I (ac)K (bc),
ν
ν
=
⎪
(−bc)K
I
ν
ν (−ac),
⎪
⎪
⎩
Iν (−ac)Kν (−bc),
0 < b < a,
0 < a < b,
0 < b < a,
0 < a < b,
Re c > 0,
Re c > 0,
Re c < 0,
Re c < 0,
Re ν
Re ν
Re ν
Re ν
> −1,
> −1,
> −1,
> −1.
(.18)
‚¢μ¤Ö ËÊ´±Í¨¨ (11)Ä(14), ¸ ÊÎ¥Éμ³ (.17), (.18) ¸¨¸É¥³Ê Ê· ¢´¥´¨° (.16)
³μ¦´μ É¥¶¥·Ó § ¶¨¸ ÉÓ ¢ ±μ³¶ ±É´μ³ ¢¨¤¥ (8).
‘ˆ‘Š ‹ˆ’…’“›
1. ” ¤¤¥¥¢ ‹. „. ’¥μ·¨Ö · ¸¸¥Ö´¨Ö ¤²Ö ¸¨¸É¥³Ò É·¥Ì Î ¸É¨Í // †’”. 1960. T. 39,
¢Ò¶. 4. ‘. 1459Ä1467.
2. ‘¨³μ´μ¢ . . ‡ ¤ Î É·¥Ì É¥². μ²´ Ö ¸¨¸É¥³ Ê£²μ¢ÒÌ ËÊ´±Í¨° // Ÿ”. 1966.
’. 3, ¢Ò¶. 4. ‘. 630Ä638.
3. ¤ ²Ö´ . Œ., ‘¨³μ´μ¢ . . ‡ ¤ Î É·¥Ì É¥². “· ¢´¥´¨¥ ¤²Ö ¶ ·Í¨ ²Ó´ÒÌ
¢μ²´ // Ÿ”. 1966. ’. 3, ¢Ò¶. 5. ‘. 1032Ä1047.
4. Œ¥·±Ê·Ó¥¢ ‘. ., ” ¤¤¥¥¢ ‹. „. Š¢ ´Éμ¢ Ö É¥μ·¨Ö · ¸¸¥Ö´¨Ö ¤²Ö ¸¨¸É¥³ ´¥¸±μ²Ó±¨Ì Î ¸É¨Í. Œ.: ʱ , 1985. 398 ¸.
5. Avery J. Hyperspherical Harmonics, Application in Quantum Theory. Dordrecht;
Boston; London: Kluwer Acad. Publ., 1989. 257 p.
6. „¦¨¡Êɨ . ˆ., ˜¨É¨±μ¢ Š. ‚. Œ¥Éμ¤ £¨¶¥·¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨° ¢ Éμ³´μ° ¨
Ö¤¥·´μ° ˨§¨±¥. Œ.: ´¥·£μ É쳨§¤ É, 1993. 272 ¸.
7. „¦¨¡Êɨ . ˆ., Š·Ê¶¥´´¨±μ¢ . . Œ¥Éμ¤ £¨¶¥·¸Ë¥·¨Î¥¸±¨Ì ËÊ´±Í¨° ¢ ±¢ ´Éμ¢μ° ³¥Ì ´¨±¥ ´¥¸±μ²Ó±¨Ì É¥². ’¡¨²¨¸¨: Œ¥Í´¨¥·¥¡ , 1984. 181 ¸.
8. ˜³¨¤ ., –¨£¥²Ó³ ´ •. ·μ¡²¥³ É·¥Ì É¥² ¢ ±¢ ´Éμ¢μ° ³¥Ì ´¨±¥: ¥·. ¸ ´£².
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9. §Ó . ˆ. Œμ¤¥²Ó Ê· ¢´¥´¨° Ö¤¥·´μ° ˨§¨±¨. ·¥¶·¨´É ˆ´-É É¥μ·. ˨§¨±¨ “‘‘ ˆ’”-71-79. Š¨¥¢, 1971. 38 ¸.;
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