Add Wentzel macro xs calculator and fix a_sq_factor

[amandalund]

• Fix an error in the calculation of a_sq_factor for Wentzel OK&VI
• Fix calculation of inv_mass_cbrt_sq
• Fix a_sq_factor units
• Add a Wentzel total xs calculator (to be used in the Wentzel VI msc model)
• Move some of the common code for the Urban and Wentzel VI tests into a msc test base.

[sethrj]

LGTM, but I'll let @whokion take a look before merging. I'm going to remove the bug label since this is mostly an enhancement and the bug hasn't been seen in non-testing use cases.

[sethrj]

Maybe a Quantity (FmSq?) would help elucidate this factor? The divide vs multiply has bitten me several times...

[amandalund]

I would also be grateful if someone (maybe @whokion?) could take a second look at this factor and where it's used (in geant4 and celeritas) and make sure everything looks ok. I'm still not really sure where it comes from or if I have all the units right...

[whokion]

Mo problem, I will take a second look for this today.

[whokion]

Okay, it looks all good to me. This factor is related tothe screening parameter, $A \equiv \frac{1}{4}(\frac{\hbar}{pR})^2$ (where $p$ is the particle momentum) which is originated from the Wentzel SS differential cross section based on the first Born approximation using a simple potential $V(r) \sim \frac{exp(-r/R)}{r}$ where $R \sim 0.885 Z^{1/3} a_o$ is the screening radius in terms of the Bohr radius $a_o$ and the atomic (charge) number $Z$ (i.e, based on Thomas-Fermi model) and then is also a parameter of the angular distribution of the single scattering - please see eq 23, 26 of the paper by J.M. Fernrindez-Varea et al.. When the SS is combined with the WentzelVI multiple scattering, the maximum angle of the scattering is therefore limited by $1-A$ (i.e., the line that you pointed in Geant4).

[amandalund]

Thanks for the explanation and reference! That makes more sense now. I'll take another look and see if I can understand the differences between the screening parameter and the value that's used in the code to limit the angle.

[amandalund]

Ok I fixed another bug (in the calculation of the effective atomic mass^(-2/3)), but the quantity that we're subtracting here is still about 4 orders of magnitude smaller than the screening parameter. Should they be roughly the same? The screening parameter itself is also very small (like O(1e-10)) which means the maximum angle is going to be limited to near zero instead of the polar angle limit... so I think I'm still missing something.

[whokion]

Yes, I guess that the magnitude, 1-factorA^2/ [(mass number)^{2/3} * p^2] should be close to 1, but not too close to 1 - based on my rough calculation, it is about 0.95/0.98 for electrons of 100/200 MeV (kinetic) energy on Pb (A~200).

[sethrj]

Since the factor's always being set here (it looks like??) maybe just default the CoulombParameters to zero and reduce the duplication here.

[whokion]

LGTM too.

[amandalund]

One more fix... I think a_sq_factor should have units MeV^2. The cos(theta) limit looks more reasonable now.

[sethrj]

Good work!