18
H. Bichsel and H. Schindler
Ka o
E=48 Ry
df(E,q)/dE [Ry –1
]
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
2
4
6
8
1 0
1 2
1 4
Fig. 2.5 Generalized oscillator strength density for Si for an energy transfer E = 48 Ry to the
2p-shell electrons [41–44], as function of ka 0 (where k 2 a 2
0 = Q/Ry). Solid line: calculated with
Herman-Skilman potential, dashed line: hydrogenic approximation [45, 46]. The horizontal and
vertical line define the FVP approximation (Sect. 2.3.3)
We will discuss this term in more detail in Sect. 2.3.3. The total single-differential
cross section,
dσ
dE
=
dσ (1)
dE
+
dσ (2)
dE
+
dσ (3)
dE
+
dσ (h)
dE
,
is shown in Fig. 2.6 for particles with βγ = 4 in silicon which, at present, is the
only material for which calculations based on the Bethe-Fano method are available.
2.3.3 Fermi Virtual-Photon (FVP) Method
In the Bethe-Fano algorithm discussed in the previous section, the dielectric
function ε (q, E) was approximated at low momentum transfer by its optical limit
ε (E). In the Fermi virtual-photon (FVP) or Photoabsorption Ionisation (PAI)
model [6, 47, 48], this approximation is extended to the entire domain q 2 <
2mE. Guided by the shape of the hydrogenic GOS, the remaining contribution to
Im (−1/ε (q, E)) required to satisfy the Bethe sum rule
∞
0
E Im
−1
ε (q, E)
dE =
π
2
¯
hh p
2 ∀q,
(2.23)