11 Detector Simulation
493
11.3.2 Introduction to the Transport Monte Carlo Method
At each step of a simulation, the Monte Carlo method for particle transport needs:
• the cross sections in the current material of each physical interaction;
• an algorithm to select which interaction occurs next;
• a method to apply the effects of each interaction: to generate new particles, and
change the state of the potential surviving projectile.
The Monte Carlo method [35], general techniques [36] and its application to
particle transport for charged and neutral particle transport [37] are well described
in the literature. We touch on a few of the essential features.
A key ingredient is a source of ‘pseudorandom’ numbers, distributed uniformly
in an interval, usually [0,1). These are obtained from pseudorandom number
generators [38] and must come with guarantees of non-correlation, such as those
provided by the generators based on ergodic theory, MIXMAX [39, 40] and
RANLUX [41, 42], or at least have survived a barrage of empirical tests [43] to
suggest there are no correlations which affect the Monte Carlo estimates.
For a general particle the total interaction cross-section (summed over all
interactions)
σ total = Σ i σ i
is used to sample the step length s, using a random number r from the interval (0,1):
s = − (1/μ) ln r
where the absorption coefficient μ is proportional to the cross-section σ total and
density ρ. Thereafter, the type of interaction that will occur at this step is chosen. The
probability for one particular type of interaction to occur (in one step) is proportional
to its cross section.
In the ideal case, all interactions would be sampled this way. However, in practice
a different approach is needed, as the cross section diverges for the emission of soft
photons and delta rays. A systematic treatment proposed by Berger [44], separates
collisions that alter the state of the particle below a chosen threshold, typically for
the momentum transfer. These are not sampled individually; only their collective
effect is sampled. The collisions above the threshold are simulated individually.
In this approach, the part of the cross section corresponding to an interaction
below this threshold is labeled the continuous part, and it does not contribute to
limiting the step. Its effect is applied separately as an integral over the length of the
step, to the state of a track.
The discrete part of an interaction contributes its cross-section to limiting the step
σ = σ discrete + σ continuous
493
11.3.2 Introduction to the Transport Monte Carlo Method
At each step of a simulation, the Monte Carlo method for particle transport needs:
• the cross sections in the current material of each physical interaction;
• an algorithm to select which interaction occurs next;
• a method to apply the effects of each interaction: to generate new particles, and
change the state of the potential surviving projectile.
The Monte Carlo method [35], general techniques [36] and its application to
particle transport for charged and neutral particle transport [37] are well described
in the literature. We touch on a few of the essential features.
A key ingredient is a source of ‘pseudorandom’ numbers, distributed uniformly
in an interval, usually [0,1). These are obtained from pseudorandom number
generators [38] and must come with guarantees of non-correlation, such as those
provided by the generators based on ergodic theory, MIXMAX [39, 40] and
RANLUX [41, 42], or at least have survived a barrage of empirical tests [43] to
suggest there are no correlations which affect the Monte Carlo estimates.
For a general particle the total interaction cross-section (summed over all
interactions)
σ total = Σ i σ i
is used to sample the step length s, using a random number r from the interval (0,1):
s = − (1/μ) ln r
where the absorption coefficient μ is proportional to the cross-section σ total and
density ρ. Thereafter, the type of interaction that will occur at this step is chosen. The
probability for one particular type of interaction to occur (in one step) is proportional
to its cross section.
In the ideal case, all interactions would be sampled this way. However, in practice
a different approach is needed, as the cross section diverges for the emission of soft
photons and delta rays. A systematic treatment proposed by Berger [44], separates
collisions that alter the state of the particle below a chosen threshold, typically for
the momentum transfer. These are not sampled individually; only their collective
effect is sampled. The collisions above the threshold are simulated individually.
In this approach, the part of the cross section corresponding to an interaction
below this threshold is labeled the continuous part, and it does not contribute to
limiting the step. Its effect is applied separately as an integral over the length of the
step, to the state of a track.
The discrete part of an interaction contributes its cross-section to limiting the step
σ = σ discrete + σ continuous