Speaker: Max Topp-Mugglestone
Abstract: The form of magnetic fields required to realise the scaling Vertical excursion Fixed Field Alternating gradient accelerator (VFFA) can imply strong solenoid fringe field components for certain lattice configurations in addition to the skew quadrupole components in the magnet body – both of which in turn imply coupled dynamics that will need special treatment. Currently, all modelling procedures for these machines rely on computationally intensive simulations, and optimisation and lattice design processes require time-consuming parameter scans. Using a method based on the standard Hamiltonian formalism as seen elsewhere in accelerator physics, first steps towards realising and testing a fully analytic model of the VFFA that accounts for the uniquely coupled dynamics of the system are presented. This analytic model would provide a means for rapid lattice design, optimisation, and benchmarking, as well as enabling us to develop an understanding of the effects of various key lattice parameters, such as the VFFA field index m. A rudimentary transfer-matrix based optics code using this model is demonstrated in the context of a muon collider lattice, as well as a number of other sample lattices.
8 replies on “An analytic approach to modelling the VFFA”
Thank you for sharing these results. We understand these results are preliminary. You have a Hamiltonian. Does your integrator (or transfer matrix?) respect symplecticity?
Do you have a strategy (roadmap) for going from coupled linear dynamics to the non-linear version?
Are Lie-algebraic techniques on your horizon?
I don’t understand your coordinate system. Is the path shown in slide 4 assumed to be straight? It should not be. Neither is it circular; with longish ramps at entrance and exit, the path has to be determined first. Only then can you define s, the path length, and rho(s).
Hi Shane, thank you for your question.
Currently, I’m using the Euler method, which is nonsymplectic, to integrate the equations of motion and form transfer matrices for each element (magnet body, magnet entrance, magnet exit). As we decrease the step size of the integration method, each of these matrices tends towards symplecticity.
As yet I don’t have a detailed roadmap for moving to non-linear dynamics — I’d like to have a bit more evidence confirming that the linear model can be used to make useful predictions for the behaviour of the VFFA machine before I start thinking about non-linear dynamics, though it is absolutely the next step once I’ve got a working linear description.
I anticipate Lie-algebraic techniques to have a number of applications in this project — I’ve actually been trying to make use of some already, although I’m still familiarising myself with Lie algebraic techniques in general.
Hi Rick, thank you for your comment. This is a good point to bring up. We’re working with a circular path in the magnet body; for the fringes, at the moment we’re using a path that can be locally defined as circular, with a linearly increasing radius of curvature as we move away from the magnet body.
I agree with your conclusion that edge focusing may be an important contribution, but I think the issue may not be so much the traditional notion of edge focusing (which should be handled by your Maxwellian treatment of the ramp) as with your treatment of the solenoid component. The reason that MAD-X adds the fringes to the solenoid is that they are of the same order as the body contribution, and the solenoid doesn’t behave properly if you don’t include them (without them, a solenoid would leave an off-axis particle with no transverse momentum untouched, whereas with the fringes it acts as a linearly focusing magnet); so I would expect you need to add the equivalent of those solenoid fringes at the ends of your ramps.
The fringes can be computed analytically using Lie Algebraic techniques along with Maxwell’s equations to get the terms with higher longitudinal derivatives (a limit is taken in the length of the fringe); Etienne Forest has a paper on the subject and probably discusses it in his books.
Yes, I think I agree with Scott here.
An interesting parameter of the VFFA fringe is the effective Larmor rotation angle within the solenoidal fringe. So somehow the dynamics of the magnet body are rotated relative to the outside; in Shinji’s neutron source lattice, this angle is vary large (I think about 80 degrees) for one of the magnets. It should be smaller in the muon collider lattice but may be worth watching.
These are some important discussion points – I hoped to raise them in the Q and A session today but I did not get the chance!
To address Stephen’s point first — with the model we’re currently using (i.e. without including something like the solenoid fringes Scott suggests), the principle effect of the solenoid terms is this Larmor rotation. It’s not something I’ve been able to look at in detail with the current version of my model yet, though I did take a quick look when I was experimenting with the MADX solenoids at the behaviour of the lattice as a function of the Larmor angle (by artificially varying the strength of the solenoids I was using to approximate the fringe fields). Perhaps I’d be able to put together a similar investigation using my more updated model… I did notice some fun effects, such as the loss of stability when the Larmor angle through a fringe field becomes 45 degrees (i.e. there’s an effective 90 degree rotation between the coordinate system of one magnet body and the next).
Moving on to Scott’s points, and a few of the things that came up in discussion:
The vector potentials in the Hamiltonian I am currently using come directly from the scaling criterion and Maxwell’s laws, so where f(Z) is well-defined my model should be in accordance with Maxwell’s law.
We have a linear ramp function, which gives rise to a constant solenoid-body-like term in the fringe field, but not any corresponding solenoid fringe fields. This is why I was interested in solenoid-body type dynamics in isolation to begin with!
To include solenoid fringe fields would therefore, as I see it, imply changing this definition of f(Z) shown in slide 4. There may well be justification for doing so — not only for this reason, but also to make the derivative of f(Z) continuous, which in turn would resolve some problems with conversion between kinetic and canonical momenta at the boundaries of fringe regions (and also I think that we’re not strictly Maxwellian at these points because of the discontinuity). That said, I’m not exactly sure that switching to a different (perhaps more physical) f(Z) will by itself incorporate solenoid-fringe dynamics at linear order. In any case, the solenoid fringe effects are absolutely something I should look at (and may very well as you suggest be necessary for a complete depiction), and the Lie Algebraic treatment thereof is something I’d find very interesting to study. I hadn’t come across the Etienne Forest paper yet — thank you very much for that reference.
Thank you both for your comments!
A small point to clarify one sentence:
“That said, I’m not exactly sure that switching to a different (perhaps more physical) f(Z) will by itself incorporate solenoid-fringe dynamics at linear order. ”
I’m not sure that it wouldn’t do that either — merely that it’s close to midnight and I couldn’t quite think it through in my head as I was writing!