2 The Interaction of Radiation with Matter
19
E [eV]
20.0
10.0
4.0
2.0
1.0
0.4
0.2
0.1
1
3
10
30
100 300 1000 3000 10000 30000 100000
dσ/dE/dσ
R /dE
Fig. 2.6 Differential cross section dσ/dE, divided by the Rutherford cross section dσ R /dE, for
particles with βγ = 4 in silicon, calculated with two methods. The abscissa is the energy loss E
in a single collision. The Rutherford cross section is represented by the horizontal line at 1.0. The
solid line was obtained [41] with the Bethe-Fano method (Sect. 2.3.2). The cross section calculated
with the FVP method (Sect. 2.3.3) is shown by the dotted line. The functions all extend to E max ∼
16 MeV. The moments are M 0 = 4 collisions/μm and M 1 = 386 eV/ μm (Table 2.2)
is attributed to the scattering off free electrons (close collisions). This term is
thus of the form Cδ
E − q 2 / (2m)
, with the factor C being determined by the
normalisation (2.23),
C =
1
E
E
0
E
Im
−1
ε (E )
dE
.
Combining the two terms, the longitudinal loss function becomes
Im
−1
ε (q, E)
= Im
−1
ε (E)
E −
q 2
2m
+
δ
E −
q 2
2m
E
E
0
E
Im
−1
ε (E )
dE
.
In the transverse term, the largest contribution to the integral comes from the
region E ∼ qc/
√
ε, i.e. from the vicinity of the (real) photon dispersion relation,
and one consequently approximates ε (q, E) by ε (E) throughout.