Fix hydrogen ionization scattering

[muffinfan]

Modifications were made to correct mistakes in the hydrogen scattering implementation concerning ionization. Affected functions:

• KSIntCalculatorHydrogenIonisation::ExecuteInteraction link
• KSIntCalculatorHydrogenIonisation::CalculateFinalEnergy link

Fixed issues:

1) Sampling from the singly differenctial cross section (SDCS), i.e. the ionization eloss function:

The sampling method originally implemented in KSIntCalculatorHydrogenIonisation::CalculateFinalEnergy to sample from the SDCS was erroneous. Firsty, when all outgoing electrons after ionization are taken into account (outgoing primary and secondary), the histogram of sampled energy losses should extend all the way down to zero. This was not the case in the previous implementation. Furthermore, it seems it was attempted to implement rejection-based sampling with a (inefficient) uniform proposal distribution. However, there are some mistakes and the distribution of sampled energies therfore does not follow the SDCS as intended.

These issues are adressed by re-writing most parts of the function. For the new implementation, several new proposal functions for the rejection sampling were investigated, in order to make the new implementation much more efficient. An ideal proposal function must be as close to the target distribution, and one must ideally be able to create samples via inversion sampling. Since the SDCS is symmetrical around the middle, it suffices to define a proposal for one half. The second half is then autmatically taken into account, by generating the secondary with the energy E_secondary = E_input-E_binding-E_primary. The investigated proposal distributions are:

Out of these proposals, the custom powerlaw is closest and should be most efficient. A test in a standalone python implementation of the methods was used to assess the acceptance efficiency and the target trueness:

Target trueness:

Number of trials for successful sample generation:

Acceptance ratios:
Acceptance ratio (Kassiopeia sampling method) : 00.123 % 1e5 samples in 08:08 minutes
Acceptance ratio (Uniform rejection samping) : 00.121 % 1e5 samples in 08:01 minutes
Acceptance ratio (Exponential rejection samping) : 01.824 % 1e6 samples in 07:06 minutes
Acceptance ratio (Powerlaw rejection samping) : 53.575 % 1e6 samples in 00:19 minutes

To conclude, the custom powerlaw proposal distribution is true to the target and performs orders of magnitude better than the other methods, especially much better than the original implementation. Therefore, the powerlaw was chosen for the new Kassiopeia implementation.

2) Secondary electron angle

In the original implementation of KSIntCalculatorHydrogenIonisation::ExecuteInteraction, the phi and theta angles of the secondary electron from ionization were simply drawn from uniform distributions. However, due to the kinematics (momentum conservation) there should be a correlation between the scattering angles of the outgoing primary and secondary. For low electron energies this may be difficult to describe accurately. But for energies much larger than the binding energy (which is usally the case), one can assume free electron-electron scattering.

Therefore, in the new implementation, the momentum direction (and thus the phi and theta angles) is simply computed by subtracting the momentum of the outgoing primary from the momentum of the incoming electron.

Both changes have been tested and are used by Julanan Songwadhana (SUT) for his KATRIN WGTS magnetic trap and source scattering simulations. No issues with the implementation were found so far.