2 The Interaction of Radiation with Matter
13
atoms. For an accurate calculation of dσ/dE, the electronic structure of the target
medium therefore needs to be taken into account.
In the non-relativistic first-order Born approximation, the transition of an atom
from its ground state to an excited state |n involving a momentum transfer q is
characterised by the matrix element (inelastic form factor)
F n0 (q) = =n|
Z
j =1
exp
i
¯
h
q · r j
|0
which is independent of the projectile. The differential cross section with respect
to the recoil parameter Q = q 2 / (2m), derived by Bethe in 1930 [16], is given by
[1–3, 16]
dσ n
dQ
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
Q 2 |F n0 (q)|
2
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
f n (q)
QE n
,
where f n (q) denotes the generalised oscillator strength (GOS). In the limit q → 0
it becomes the dipole oscillator strength f n discussed in Sect. 2.2.1. The doubledifferential cross section for transitions to the continuum (i.e. ionisation) is given
by
d
2 σ
dEdQ
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
QE
df (E, q)
dE
,
(2.13)
where df (E, q) /dE is the generalised oscillator strength density. The GOS is
constrained by the Bethe sum rule [2, 16] (a generalisation of the TRK sum rule),
n
f n (q) +
dE
df (E, q)
dE
= Z, ∀q.
(2.14)
Closed-form expressions for the generalised oscillator strength (density) exist
only for very simple systems such as the hydrogen atom (Fig. 2.4). Numerical
calculations are available for a number of atoms and molecules (see e.g. Ref. [37]).
A prominent feature of the generalised oscillator strength density is the so-called
“Bethe ridge”: at high momentum transfers df (E, q) /dE is concentrated along
the free-electron dispersion relation Q = E.
In order to calculate dσ/dE, we need to integrate the double-differential crosssection over Q,
dσ
dE
=
Q max
Q min
dQ
d
2 σ
dEdQ
,
Q min ∼
E 2
2mβ 2 c 2 .
(2.15)