Production of secondaries near the volume boundaries

Introduction

It is well known that in a few electromagnetic processes (ionisation and bremsstrahlung) we need to define a low energy cutoff for the production of secondary particles. This feature introduces an arbitrary limit between two models of simulation.
Indeed several physics quantities, like the distribution of the energy released by a charged particle along its trajectory, are independent of that limit. Some others, like the total track length in the shower development or the energy flux across boudaries, may be sensitive to the energy cutoff.
The algorithm described in this note allows:

• to make those quantities less dependent on the choice of the cutoff,
• to gain performance by optimising the number of tracks to be simulated.

Production of delta rays near a volume boundary

When a ionising particle is near a boundary, even the delta rays of very low energy can escape the boundary, reach another part of the detector which could be a sensitive one, and therefore contribute to the response of the detector. In conclusion, in many circumstances it is meaningful to create explicitly those delta rays, even if they have low energy.
With the traditional procedure the only way is to build the and dE/dx tables with the lowest possible tcut. It will generate plenty of delta rays along the full trajectory of the ionising particle, even where this generation is undesirable.
The algorithm we will describe now allows us to keep tcut as high as possible, and to generate low energy delta rays only where they are needed, i.e. near the boundaries. The result can be a drastic improvement of the performance of the simulation, keeping the same quality of physics results as with the lowest cut.
Let us call rcut the range corresponding to default energy threshold tcut. The starting remark is the following: at a given point, if the safety radius is smaller than rcut one has to produce additional delta rays. The energies of these delta rays will correspond to ranges between safety and rcut, since those delta rays can escape a boundary.

1. In fact the delta rays are emitted along the step of the ionising particle. The safety radius along the step is evaluated, as a first implementation, using the spherical safety at beginning and at the end of step: saf1 and saf2. (To be general, it should be evaluated as a cylindrical or toroidal safety along the step).
2. If the minimal safety is smaller than rcut one generates additional delta rays along (a fraction of) the step of the ionising particle.
3. The number of delta rays to be emitted are calculated using an approximate formula based on the differential cross section of the delta ray production.
4. The energies of the delta rays are from the energy corresponding to minimal safety, up to tcut, with a distribution proportional to 1/t**2
5. The energies of the delta rays are substracted from the contribution of the continuous part of the ionising particle, avoiding a double counting of the energy released by the ionising particle along its step.
6. The positions of the delta rays along the step are uniformly distributed along a fraction of the step which is determined by the relative values of saf1, saf2 and rcut. (The reason for uniformly is that, in first approximation, the cross section for delta rays production is constant along the step).

An example

• case 1

  In figure 1 a proton of 500 MeV passes through a block of 5 cm of iron. The production threshold is rcut = 1 mm, which corresponds to tcut = 1.25 MeV in iron. Which such a threshold there are no delta rays produced in iron.
  In fact, in this picture, 100 protons are superimposed, the multiple scattering is off.
  The distribution of the energy deposited in iron is shown in plot 1, with a bigger statistic and multiple scattering included.

• case 2

  In figure 2, protons, with the same energy. The delta rays production threshold is rcut = 10 micron (tcut = 58 keV). The delta rays are emitted along the proton trajectory. The distribution of the vertex position can also be seen in plot 2.
  The distribution of the energy deposited in iron is shown in plot 1.
  The delta rays created at the end of the block of iron can escape the boundary and travel in the gas behind iron. The energy spectrum of those delta rays when leaving iron is in plot 3: it is the energy flux behind the block of iron.

• case 3

  In figure 3, rcut = 1 mm as in case 1, but the algorithm described in this note is applied. The delta rays in iron are not created, except those near the boundaries. The distribution of the vertex position can also be seen in plot 2. The energy deposited in iron is in plot 1.
  The energy flux behind the block of iron is shown in plot 3. It is the same as in case 2.

• conclusion

  The three cases give the distribution of energy deposit in iron. But only the cases 2 and 3 can simulate the energy flux behind the block of iron. Concerning the performance, in this example, case 3 is about 10 time faster than case 2.

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