PENELOPE 2003 - OECD Nuclear Energy Agency

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112 Chapter 3. Electron and positron interactions
1Ε−6
ρ R(E ) / E ((g/cm 2 ) /eV )
1Ε−7
1Ε−8
1Ε−9
el ec t r o n s
p o s i t r o n s
1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9
E
(eV)
Figure 3.15: CSDA ranges for electrons and positrons in aluminium and gold as functions of
the kinetic energy of the particle.
A parameter of practical importance is the so-called CSDA range (or Bethe range),
which is defined as the path length travelled by a particle (in an infinite medium) before
being absorbed and is given by
R(E) =
∫ E
E abs
dE ′
S(E ′ ) , (3.145)
where we have considered that particles are effectively absorbed when they reach the
energy E abs . Notice that the CSDA range gives the average path length, actual (or
Monte Carlo generated) path lengths fluctuate about the mean R(E); the distribution
of ranges has been studied by Lewis (1952). Fig. 3.15 displays CSDA ranges for electrons
and positrons in aluminium and gold, this information is useful e.g. in estimating the
maximum penetration depth of a beam and for range rejection (a variance-reduction
method). Compare fig. 3.15 with figs. 3.10 and 3.14 (right plots only) to get a feeling of
how differences in stopping power between electrons and positrons are reflected on the
CSDA ranges of these particles.
3.3.3 Angular distribution of emitted photons
The direction of the emitted bremsstrahlung photon is defined by the polar angle θ (see
fig. 3.1) and the azimuthal angle φ. For isotropic media, with randomly oriented atoms
or molecules, the bremsstrahlung DCS is independent of φ and can be expressed as
d 2 σ br
dW d(cos θ) = dσ br
Z2 1
p(Z, E, κ; cos θ) =
dW β 2 W
χ(Z, E, κ) p(Z, E, κ; cos θ), (3.146)