Using AT for non-relativistic particles

[lfarv]

I started looking at the possibility to use AT to track non-relativistic particles. It's rather easy to associate a particle to a lattice so that the relativistic β and γ are available where needed. However the tracking has a severe problem. With the choice of cτ and Δp/p as longitudinal variables, it does not work. This choice implies β = 1 and Δp/p = ΔE/E. To make it work, we need to include some information on velocity in the particle coordinates. Looking around, I found two ways of doing this: the "classical" choice ('elegant', for instance), and the MAD-X choice.

1. Classical coordinates: the velocity information is included in the 6^th coordinate which becomes βcτ = Δs instead of cτ. This does not make any difference in AT for all passmethods except cavities because:

• the tracking through these elements does not depend upon the arrival time of the particle (6^th coordinate),
• What is computed there as cτ is indeed already Δs. It's pure geometry. I checked DriftPass and the drift case of ExactHamitonianPass (where there is a bug, by the way): it's indeed Δs.
  Only CavityPass has to be modified, because it computes an energy kick and not a momentum kick (they are identical for relativistic particles).

2. MAD-X coordinates: the velocity information is included in the 5^th coordinate (6^th in MAD-X), which is ΔE/pc = β.Δp/p. This cannot be envisaged since it implies rewriting the whole tracking.

To summarise, what is needed to have AT work for non-relativistic particles is rather easy, on the principle:

• Acknowledge the fact that the 6^th coordinate is Δs, which it already is, in reality,
• Modify CavityPass to introduce a 1/β² factor to convert the ΔE/E energy kick into a Δp/p kick.

This modification is small enough to minimise the risk of breaking anything, however, this is a critical point which has to be carefully checked before implementation. For instance, in usual electron machines, β is not exactly equal to one, so it will induce a small change of the nominal RF or revolution frequency. Also, while most pass methods are based on drift/kick sequences, which are safe, there may be a few ones to be checked.

@lnadolski, @swhite2401, @simoneliuzzo, @carmignani, any ideas ?

[swhite2401]

Hello @lfarv, this is certainly an interesting development and I would even extend the discussion to arbitrary particles (protons, ions, etc...). I believe I saw the electron mass hard-coded in many places and it would probably be a good thing to clean this up as a first step.
Introducing general at variables to define the particle type (starting with electrons only) that all function refer to would be a first step in this direction. This would also be coherent with the suggestion made in #285 to introduce a Constants.py module to centralized physics constants.

For your specific problem, I would have to check the impedance passmethod, in its present state it is not compatible with non-relativistic particles (approximations were made assuming beta=1) but this is low priority.

What about Rad passmethods? I think there are constants hard coded there as well that assume beta=1, we should probably make sure these are correct in all cases...

There might be others too, so I agree we should be very careful with this kind of change

[lfarv]

Introducing general at variables to define the particle type (starting with electrons only) that all function refer to would be a first step in this direction. This would also be coherent with the suggestion made in #285 to introduce a Constants.py module to centralized physics constants.

In fact, I already have this done in the particle_data branch. The constants.py module is there, there is a Particle object, defined by name, rest mass and charge, which is set as an attribute of the Lattice object. This is the easy part, but it's useless until the tracking works. So I'm now thinking of going the other way: set up the tracking first, still restricted to electrons, and then step by step, introduce the Particle object, move all hard-coded references to it, and finally allow other particles than electrons.

[swhite2401]

Sounds very good! Let me know in case help is needed for testing etc...