European Journal of Scientific Research - EuroJournals

Equation of Motion in Appearance Potential Spectra of Simple Metals 446
on the right of equation (14) then corresponds to the inclusion of exchange of the two conduction
electrons. If this exchange is neglected, we obtain
Γ k1k2k
3k
4
−iEct
() t = θ () t e Gk1k
4 ( t)
Gk
k ( t)
2 3
(18)
Correspondingly, the response function becomes
t =θ t
2 −iEct
M e
*
M k k K G t G t
(19)
G
we obtain
() () ( ) ( ) ( )
S ∑ 1 2 k1k4
k2k
3
k1k2
> kF
k3
, k4
> kF
Taking the matrix elements to be constant, we get
t =θ t
2 −iEct
M e G t G t
(20)
() () ( ) ( )
S ∑ k1k4
k2k3
k1k2
> kF
k3
, k4
> kF
If we substitute
S
k1k4
() t θ () t
() t = () t
k1k
−ε
k t
11
= δ e
(21)
2 −iE
−i(
k + k t )
ct
ε1
ε
1 2
θ M e e
(22)
∑
k k > k
1 2
F
The k -summation is performed by means of the transformation
∑ ∫ ( ) ( )
→
1
f k N ε f ε dε
Ω k
(23)
where Ω is the normalization volume which is taken to be unity and N ( ε ) is the density of states at
energy ε . The use of (23) then leads to
S
2 −iEct
i(
ε1
+ ε 2 ) t
() t = θ () t M e dε
dε
N ( ε ) N ( ε ) e
∫
ε1
> ε F
∫
1
ε1
> ε F
2
1
Transition Probability and X-ray Yield
We take real part of the Fourier transform of (24) to obtain transition probability per unit time P (E):
P
=
2
i(
E −Ec
−ε1
− ε 2 ) t
( E)
M R e dε
dε
N ( ε ) N ( ε ) d tθ
( t)
e
M
2
∫
∫
dε
1
∫
1
ε1
> ε F ε1
> ε F
∫ 2 1 2 ∫
∞
−∞
i( E − Ec
−ε
1 − ε 2 + iδ
) t
( ε ) N ( ε ) lim Re d t e
∫dε2N12∫ ∫
δ → 0
( ε ) ( ε ) πδ ( − −ε
−ε
) E E N
2
= ε1 ε 2 1 2
c 1 2
N d d M
2
( ε ) N ( E − E −ε
)
∞
0
= π M ∫ dε1
N
c
(25)
ε > ε f
The limits of the ε -integration are ε = 0 at the Fermi level and c E E − = ε which is the
difference between the energy of the incident electron E and the core-level binding energy E c
The X-ray yield is proportional to P ( E)
and so we obtain
Y
where ( E)
E Ec
( E)
α dε
N ( ε ) N ( E − E −ε
)
∫ −
0
c
Y represents the X-ray yield
The Appearance Potential Spectrum consists of the derivation of the X-ray yield with respect to
the incident electron energy. So from (26) we get
( ) α ∫ ε ( ε ) ( − −ε
)
− d E Ec
A E d N N E Ec
(27)
dE 0
2
(24)
(26)