Grad-B drift velocity

[MCFlowMace]

The Grad-B drift velocity is incorrect:

CRESana/cresana/cyclotronphysics.py

Lines 109 to 112 in 905d542

	def get_v_gradB(E_kin, pitch, B, w, grad):
	v0 = get_relativistic_velocity(E_kin)
	gamma = get_gamma(E_kin)
	return (v0*np.sin(pitch))**2/(2*B*w)*grad*np.sqrt(gamma)

I added the factor sqrt(gamma) because only then I ended up with good agreement to Kassiopeia. Following the equations from literature this factor should not be there. From Jackson one can extract this equation for example:

One can show that this is equivalent to the used value in the code (minus the sqrt(gamma) by using

The version in Jackson is also equivalent to the one in this original publication about the guiding center approximation https://agupubs.onlinelibrary.wiley.com/doi/10.1029/RG001i003p00283
One only has to pay attention to not get confused in the comparison because both Northrop and Jackson use CGS units and therefore they have an additional factor 1/c in their definition of the cyclotron frequency that we usually do not have.

It is odd that using the factor sqrt(gamma) gave good agreement because it should clearly not be there according to literature. I tried to find out what is going on in the Kassiopeias source and the relevant code seems to be here
https://github.com/KATRIN-Experiment/Kassiopeia/blob/c07cbb354be937e26f8a2ce11973639d9f327a64/Kassiopeia/Trajectories/Source/KSTrajTermDrift.cxx#L48
and I assume that the term for the grad-B drift is only the part with tTransMomentum2. The part 2. * tLongMomentum2 has to be the contribution from the curvature drift. If I interpret the variables correctly this results in an equation like v=v_J*2*gamma/(1+gamma) where v_J would be the drift velocity out of Jackson. At the tritium endpoint we have gamma=1.036 and in this case by coincidence it is approximately sqrt(gamma)=2*gamma/(1+gamma)=1.018. This looks like Kassiopeia has indeed this factor wrong and using sqrt(gamma) makes the drift velocity agree to Kassiopeia by coincidence.

To me it seems like Kassiopeia is wrong here and that the factor sqrt(gamma) should be removed on our end. Before opening an issue in Kassiopeia I would like to have someone else look at this and confirm my suspicion.

[renereimann]

The Kassiopeia implementation in https://github.com/KATRIN-Experiment/Kassiopeia/blob/c07cbb354be937e26f8a2ce11973639d9f327a64/Kassiopeia/Trajectories/Source/KSTrajTermDrift.cxx#L48 is in agreement with the Kassiopeia paper [Daniel Furse et al 2017 New J. Phys. 19 053012] (see screenshot below).

I can reproduce the implemented CRESana formular given that equation 12.55 [Section "Particle Drift in Nonuniform, Static Magnetic Fields, p.589, 3rd edition] in Jackson is correct. I verified relativistic cyclotron frequency and cyclotron radius seperately.

[renereimann]

The only reference in Kassiopeia paper is to the original Northrop paper (Screenshot reference list Kassiopeia paper)

The corresponding Northrop paper

discusses only the non-relativistic case

The relativistic case is discussed in a different paper.


https://journals.aps.org/pr/pdf/10.1103/PhysRev.117.215