2 The Interaction of Radiation with Matter
17
For high momentum transfers (Q > Q 2 ∼ 30 keV), i.e. for close collisions where
the binding energy of the atomic electrons can be neglected, the longitudinal and
transverse matrix elements are strongly peaked at the Bethe ridge Q = E. Using [1]
|F (E, q)|
2
∼
1 + Q/
2mc 2
1 + Q/
mc 2
δ(E − Q),
|β t · G (E, q)|
2
∼ β
2
t
1 + Q/
2mc 2
1 + Q/
mc 2
δ(E − Q)
and
β
2
t =
1
1 + Q/
2mc 2
−
1 − β
2
one obtains (for longitudinal and transverse excitations combined),
dσ (h)
dE
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
Z
E
1 −
E
1 − β 2
2mc 2
.
(2.21)
In the intermediate range, Q 1 < Q < Q 2 , numerical calculations of the generalised
oscillator strength density are used. An example of df (E, q) /dE is shown in
Fig. 2.5. Since the limits Q 1 , Q 2 do not depend on the particle velocity, the integrals
dσ (2)
dE
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
E
Q 2
Q 1
dQ
Q
df (E, q)
dE
need to be evaluated only once for each value of E. The transverse contribution can
be neglected 1 [41].
The last contribution, described in detail in Ref. [1], is due to low-Q transverse
excitations in condensed matter. Setting Im (−1/ε (E, q)) = Im (−1/ε (E)) in the
second term in Eq. (2.17) and integrating over q gives
dσ (3)
dE
=
z 2 α
β 2 πN ¯
hc
×
Im
−1
ε (E)
ln
1
1 − β 2 ε (E)
+
β
2 −
ε 1 (E)
|ε (E)|
2
π
2
− arctan
1 − β 2 ε 1 (E)
β 2 ε 2 (E)
.
(2.22)
1 For particle speeds β < 0.1, this approximation will cause errors, especially for M 0 .