2 The Interaction of Radiation with Matter
29
E [eV]
n=1
n=2
n=3
n=4
n=5
0.100
0.050
0.020
0.010
0.005
0.002
0.001
0
20
40
60
80
100
s [E] ù f (n)
[a.u]
*n
Fig. 2.10 Distributions f (n) of the energy loss in n collisions for solid silicon. The plasmon peak
at ∼17 eV appears in each spectrum at E ∼ n × 17 eV, and its FWHM is proportional to
√
n. The
structure at ∼2 eV appears at 2 + 17(n − 1) eV, but diminishes with increasing n. For n = 6 (not
shown) the plasmon peak (at 102 eV) merges with the L-shell energy losses at 100 eV, also see
Fig. 2.12
2.5.1 Monte Carlo Method
In a detailed Monte Carlo simulation, the trajectory of a single incident particle is
followed from collision to collision. The required ingredients are the inverse mean
free path M
(i)
0 and the cumulative distribution function,
(i) (E) =
1
M
(i)
0
E
0
N
dσ (i)
dE dE
,
(2.36)
for each interaction process i (electronic collisions, bremsstrahlung, etc.) to be
taken into account in the simulation. The distance x between successive collisions
follows an exponential distribution and is sampled according to
x = −
ln r
λ −1 ,
where r ∈ (0, 1] is a uniformly distributed random number and
λ
−1
=
i
M
(i)
0