16
H. Bichsel and H. Schindler
The loss function Im (−1/ε (q, E)) and the generalized oscillator strength density
are related by
df (E, q)
dE
= E
2Z
π
¯
hh p
2 Im
−1
ε (q, E)
.
(2.18)
Using this identity, we see that the longitudinal term (first term) in Eq. (2.17) is
equivalent to the non-relativistic quantum mechanical result (2.13). As is the case
with the generalized oscillator strength density, closed-form expressions for the
dielectric loss function Im (−1/ε (q, E)) can only be derived for simple systems
like the ideal Fermi gas [39, 40]. In the following (Sects. 2.3.2 and 2.3.3), we discuss
two specific models of Im (−1/ε (q, E)) (or, equivalently, df (E, q) /dE).
2.3.2 Bethe-Fano Method
The relativistic version of Eq. (2.13) or, in other words, the equivalent of Eq. (2.17)
in oscillator strength parlance, is [1, 41]
d 2 σ
dEdQ
=
2πz 2 (α ¯
hc) 2
mc 2 β 2 Z
⎡
⎢
⎣
|F (E, q)| 2
Q 2
1 +
Q
2mc 2
2 +
|β t · G (E, q)| 2
Q
1 +
Q
2mc 2
− E 2
2mc 2
2
⎤
⎥
⎦
1 +
Q
mc 2
(2.19)
where Q
1 + Q/2mc 2 = q 2 /2m, β t is the component of the velocity perpendicular to the momentum transfer q, and F (E, q) and G (E, q) represent the matrix
elements for longitudinal and transverse excitations.
Depending on the type of target and the range of momentum transfers involved,
we can use Eqs. (2.13), (2.19) or (2.17) as a starting point for evaluating the singledifferential cross section. Following the approach described by Fano [1], we split
dσ/dE in four parts. For small momentum transfers (Q < Q 1 ∼ 1 Ry), we can use
the non-relativistic expression (2.13) for the longitudinal term and approximate the
generalised oscillator strength density by its dipole limit,
dσ (1)
dE
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
E
df (E)
dE
Q 1
Q min
dQ
Q
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
E
df (E)
dE
ln
Q 1 2mc 2 β 2
E 2
.
(2.20)
In terms of the dielectric loss function, one obtains
dσ (1)
dE
=
z 2 α
β 2 π ¯
hcN
Im
−1
ε (E)
ln
Q 1 2mc 2 β 2
E 2
.