European Journal of Scientific Research - EuroJournals

445 Omubo-Pepple V. B, Opara F. E, Ogbonda C and Pekene D.B
where
∂
i Γ
∂t
+ iδ
+
∑
k ′′
k1k2
k3k
4 ( t1
, t 2 , t3
, t 4 ; τ 1,
τ 2 ) = ε k Γ k1k2
k3k
4 ( t1
, t 2 , t3
, t 4 ; τ 1 τ 2 ) +
( t1
− t 4 ) δk1
k 4 Fk2k
3 ( t 2 , t3
; τ 1,
τ 2 ) − iδ
( t1
− t3
) δ k2k
4 Fk4
( t 2 , t 4 ; τ 1,
τ 2 )
V Ψ T a
+ ( t ) b ( t ) b ( t ) a ( t
+ ) a ( t
+ ) a ( t
+ ) b ( τ ) b ( τ ) Ψ
k1k
′′
N
{ }
k ′′
1
1
+
+
( t , t ; τ , τ ) = Ψ T a&
( t ) a&
& ( t ) b&
& ( τ ) b&
& ( τ )
1
k3
2
k3
3
k4
{ Ψ }
F
k2k
4
2 3 1 2 N & k2
2 k3
3 1 2 N
(8)
Eq. (7) is rewritten as
⎛ ∂
⎜
⎝ ∂t1
⎞
+ iε
⎟ k Γ k k k k ( t1,
t2
, t3
, t4
; τ 1τ
2 ) = δ ( t1
−t
4 ) δk
1 2 3 4
k11k
4
⎠
Fk
k ( t2
, t3;
τ 1τ
2 ) −
2 3
−δ
t −t
δk
F t , t ; τ τ
(9)
( 1 3 ) 1k1k
3 k ( 2k4
2 4 1 2 )
V Ψ T a
+ ( t ) b ( t ) a ( t
+ + ) a ( t ) a ( t
+ ) b(
τ ) b ( τ ) b + ( τ )
{ }
−∑
k ′′
k′
k′
′ N k ′′ 1 1 k2
2 k3
3 k4
4 1 1 2 ΨN
The last term on right of eq. (9) vanishes unless τ2 < t1 < τ1, in which case b(t1) b+ (t1) = 1
Laramore, (1971). Eq. (9) then takes the closed form
⎛ ∂
⎜
⎝ ∂t1
⎞
+ iε
⎟ k Γ k1k2k3k
4
⎠
t1,
t2
, t3
, t4
; τ 1τ
2 = δ t1
−t
−δ
t −t
δk
F t , t ; τ τ
with
−i
( 1 3 ) k1
1 k3
k k ( 2 3 2 4 1 2 )
V ( t , t , t , t ; τ τ )
∑
k′
′
k ′ Γ 1k
k′
′ k2k
4
1
( ) ( ) δk
F ( t , t ; τ τ )
2
3
4
1
2
4
4
k11k
4
We next set up the equation of motion for the function F and obtain
⎛ ∂
⎜
⎝ ∂t1
⎞
+ iε
k ⎟ Fk1k2
⎠
( t1,
t 2;
τ 1 τ 2 ) = δ ( t1
− T − 2)
δ k1k
2
g ( τ 1,
τ 2 ) −
−i
V F t , t ; τ τ
∑
k ′′
k1k′
′
k ′′ k2
( )
+
( , ) = Ψ T b(
) b ( τ ) τ
τ
N
1
2
1
2
{ 1 2 } N
g τ Ψ
(12)
1
2
Finally, the equation of motion for g (τ1τ2) yields
⎛ ∂
⎜
⎝ ∂ 1
⎞
+
+ ε ⎟ ( τ 1,
τ 2 ) = δ ( τ 1,
τ 2 ) − ∑ k′
k ′′ k ′′ k′
( τ 1 τ 1 τ 1τ
2 )
⎠
k1k′
′
F V i
g iE
t
(13)
Eqs. (10), (11) and (13) will determine the response function for our model. These equations
take on a rather simple form if final state interactions are neglected. We then find
Γ k ( 1,
2 , 3,
4;
1,
2 ) ( 1 4 ) ( 2 , 3;
1,
2 )
1k2k3
k t t t t τ τ = G
4
k1k
t −t
F t t tau tai
4
k2k
−
3
−G
t − t F t , t ; τ , τ
(14)
( 1 3 ) ( 2 4 1 2 )
3
k2k
4
( t t ; τ , τ ) G ( t −t
)( τ , τ )
k1k
Fk1k2 2 , 4 1 2 k1k2
1
2
1
2
1
k2k3
2
3
2
1
2
N
+
(7)
(10)
(11)
= (15)
−iEc
( τ 1 , τ 2 )
g ( τ1
, τ 2 ) = θ ( τ1,
τ 2 ) e
(16)
In these equations we have converted the differential equations (10), (11), (13) into the integral
forms, and have put
−iε
k ( t1
− t2
)
1
G k ( ) ( )
1k
t 2 1 −t
2 = θ t1
−t
2 δ k1k
e
(17)
2
where G k is the one-electron Green’s function, while 8 ( τ ) is the core-hole Green’s function. We
1k2
may then interpret Γ k1k2k
3k
as two-particle propagator in the presence of a core-hole. The second term
4