European Journal of Scientific Research - EuroJournals

Equation of Motion in Appearance Potential Spectra of Simple Metals 444
excitation of the core hole leads to the emission of characteristic X-rays. The method is also convenient
for surface composition studies.
The threshold for the excitation of a core electron is signaled by a sudden change in the
bremsstrahllung background. The excitation edge is detected in the derivative of the photocurrent while
the edge shape is related to the self-convolution of the empty density of states above the Fermi level,
Park and Houston, (1992). Thus APS provides direct information on the binding energies of the core
levels. Levels obtained by Park and Houston compare favourably with X-ray values given by Nearden
and Burr (1967).
In this paper we derive the expression for the excitation probability of a core hole using the
equation of motion method. Assuming constant matrix element and neglecting final state interactions,
we find that the transition rate above the thresholds is given by the self-convolution of the empty
density of states.
The Response Functions
The Coulomb interaction responsible for the core hole excitation is given by
= M K
+
a
+
a a
∑
( ) b
H1 + +
k k
k;
k1
k2
k1k2
k1
k2
k
(1)
b
where a , are the creation operators of a conduction electron of energy εk and a core electron,
respectively, while ak annihilates an incident electron of energy εk. M k k ( K ) is the matrix element
1 2
describing the scattering process. The dynamics of the system is described by the Hamiltonian.
H
+
ε k a a
+
+ E b b + V
+
a a
+
bb
(2)
∑ k k c ∑
′ ′ ′
=
k k , k
k′
k ′′
k′
k ′′
where Ec (< 0) is the energy of the core level. The third term on the right of equation (2) represents
final state interactions between the core hole and the conduction electrons. We take the view that the
core hole presents a sudden potential to the system and that this potential is subsequently canceled
when the core hole is filled with the concomitant emission of X-rays. Following Nozieres and de
Dominicis, (1969), we assume that the core hole does not possess any dynamical degree of freedom,
except to appear and disappear. The transition rate is proportional to the real part of the Fourier
S t − t′
, where
transform of the response function ( )
S ( t −t′ ) = ΨN
T { H ( t)
H ( t′
1 ) } ΨN
1 (3)
Ψ N is the initial ground state wave-function of the N-particle system with the core hole filled.
We treat the incident electron K as an independent particle and the response function is then
obtained as
S t −t
′ =
*
M K M K x
( ) ( ) ( )
∑
k1
k
k k
k
1
k
2
k k
3 4
x
2
3 4
+
+ +
ΨN
T{
a k () t ak
() t b ( t ) b ( t′
) ak
( t′
) a&
& k ( t′
) } Ψ
1 2
3
4 N
(4)
where the operators are in the Heisenberg representation for the full Hamiltonian of the solid.
Next we generate the equation of motion of the function Γ k1k2k3k
Laramore, (1971), where
4
+ +
+
Γ k1k2k
3k
( t )
4 1 , t2
, t3
, t4
; τ 1,
τ 2 = ΨN
T{ a k ( t1
) a k ( t2
) ak
( t3
) at
b + ( τ 1 ) b ( τ 2 ) } Ψ
1
2
3
4
N
Thus the equation of motion
(5)
∂
i Γ = [ H,
Γ]
∂t
leads to the result
(6)