30
H. Bichsel and H. Schindler
is the total inverse mean free path. After updating the coordinates of the particle,
the collision mechanism to take place is chosen based on the relative frequencies
M
(i)
0 /λ −1 . The energy loss in the collision is then sampled by drawing another
uniform random variate u ∈ [0, 1], and determining the corresponding energy loss
E from the inverse of the cumulative distribution,
E =
−1 (u) .
In general, the new direction after the collision will also have to be sampled from
a suitable distribution. The above procedure is repeated until the particle has left
the absorber. The spectrum f (, x) is found by simulating a large number of
particles and recording the energy loss in a histogram. Advantages offered by the
Monte Carlo approach include its straightforward implementation, the possibility of
including interaction mechanisms other than inelastic scattering (bremsstrahlung,
elastic scattering etc.), and the fact that it does not require approximations to the
shape of dσ/dE to be made.
For thick absorbers, detailed simulations can become unpractical due to the large
number of collisions, and the need to update the inverse mean free path M 0 and the
cumulative distribution (E) following the change in velocity of the particle.
In “mixed” simulation schemes, a distinction is made between “hard” collisions
which are simulated individually, and “soft” collisions (e.g. elastic collisions with a
small angular deflection of the projectile, or emission of low-energy bremsstrahlung
photons) the cumulative effect of which is taken into account after each hard
scattering event. Details on the implementation of mixed Monte Carlo simulations
can be found, for example, in the PENELOPE user guide [74].
2.5.2 Convolutions
For short track segments, one can calculate the distributions f (n) explicitly by
numerical integration and construct f (, x) directly using Eq. (2.35). A computationally more efficient approach is the absorber doubling method [41, 75], which
proceeds as follows. Consider a step x that is small compared to the mean free path
such that n 1 (in practice: n < 0.01 [76]). Expanding Eq. (2.35) in powers of
n and retaining only constant and linear terms gives
f (E, x) ∼ (1 − −n) f
(0) (E) + +nf
(1) (E) .
The straggling function for a distance 2x is then calculated using
f (, 2x) =
0
f ( − E, x) f (E, x) dE.
H. Bichsel and H. Schindler
is the total inverse mean free path. After updating the coordinates of the particle,
the collision mechanism to take place is chosen based on the relative frequencies
M
(i)
0 /λ −1 . The energy loss in the collision is then sampled by drawing another
uniform random variate u ∈ [0, 1], and determining the corresponding energy loss
E from the inverse of the cumulative distribution,
E =
−1 (u) .
In general, the new direction after the collision will also have to be sampled from
a suitable distribution. The above procedure is repeated until the particle has left
the absorber. The spectrum f (, x) is found by simulating a large number of
particles and recording the energy loss in a histogram. Advantages offered by the
Monte Carlo approach include its straightforward implementation, the possibility of
including interaction mechanisms other than inelastic scattering (bremsstrahlung,
elastic scattering etc.), and the fact that it does not require approximations to the
shape of dσ/dE to be made.
For thick absorbers, detailed simulations can become unpractical due to the large
number of collisions, and the need to update the inverse mean free path M 0 and the
cumulative distribution (E) following the change in velocity of the particle.
In “mixed” simulation schemes, a distinction is made between “hard” collisions
which are simulated individually, and “soft” collisions (e.g. elastic collisions with a
small angular deflection of the projectile, or emission of low-energy bremsstrahlung
photons) the cumulative effect of which is taken into account after each hard
scattering event. Details on the implementation of mixed Monte Carlo simulations
can be found, for example, in the PENELOPE user guide [74].
2.5.2 Convolutions
For short track segments, one can calculate the distributions f (n) explicitly by
numerical integration and construct f (, x) directly using Eq. (2.35). A computationally more efficient approach is the absorber doubling method [41, 75], which
proceeds as follows. Consider a step x that is small compared to the mean free path
such that n 1 (in practice: n < 0.01 [76]). Expanding Eq. (2.35) in powers of
n and retaining only constant and linear terms gives
f (E, x) ∼ (1 − −n) f
(0) (E) + +nf
(1) (E) .
The straggling function for a distance 2x is then calculated using
f (, 2x) =
0
f ( − E, x) f (E, x) dE.