stephenbrooks.orgForumMuon1GeneralIn progress: space-charge simulation
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Stephen Brooks
2004-06-30 03:52:47
In Muon1 at present we have charged particles and these are acted on by the fields of magnets and RF cavities.  However the particles themselves also produce electric fields that repel each other and magnetic fields through their motion, but these are quite small in the case of our accelerator.  In high-power accelerators (proton machines, usually), this 'space-charge' repulsion effect is more of a problem.

In the last few days I've got this simulation working, as a prototype of something that could be added to Muon1 in the future.  If you move the mouse around slowly, you move the bunch of positive charge (red plus signs) and it creates electric (blue) and magnetic (orange) fields.  Moving it a little faster makes it move faster than the speed of light in the simulation, so you get 'bow waves' or Cherenkov radiation produced.  This is unphysical for a vacuum, but in other substances, the propagation rate of electromagnetic waves is slowed, so the charge carriers can potentially move faster than them.  Moving the mouse too fast will mess up the simulation, as it won't be able to move the charge from place to place smoothly enough (you'll get large residual 'charges' left behind).
[OCAU] badger
2004-07-01 19:24:04
looks pretty good, although I think you need to do something else with the boundary conditions.

at the moment you will get an eternal oscillation of em waves, I think you should have some sort of damping or loss at the boundaries so you can get to a steady state after a while of the charges being still.
Stephen Brooks
2004-07-02 01:20:27
That is because the easiest boundaries to implement were perfectly conducting metal walls.  They are quite a good model for what happens in an accelerator, as the vacuum tube often has metal walls, though there would be a small amount of loss.  Some RF cavity components have quality factors of Q~10000 meaning that the waves would oscillate 10000 times inside them before being damped.
[OCAU] badger
2004-07-05 18:18:23
The size of an RF cavity would be tuned for the desired wavelength so that one is amplified.  all others are damped.
As for the accelerator, surely it would only have boundaries in some directions (ie where will the particles go?).  I understand that lossy walls are much harder to do though
/thinks back to honours electrodynamics
/shudder
Stephen Brooks
2004-07-07 07:37:40
Yeah, this is the "higher order mode" problem.  For a square cavity such as the one in my simulation, with perfect conducting walls, there are infinitely-many modes at a sequence of increasing frequencies, all with infinite Q. With lossy boundaries, I think a 1st-order friction-like term gets added to the boundary condition, probably representing the loss of power to heat due to the imperfect conductivity of the copper or whatever.  In that case there'd still be higher-order modes, but I think the higher frequency you go, the lower the (now finite) Q would become.  In real RF cavities, there are still higher modes (you can't avoid them), but sometimes extra tubes are added to extract the power of the most problematic ones - essentially it gets pulled out into a transmission line and removed in some sort of load.

In the actual accelerator, the problem becomes harder - essentially I would have to cut off the grid at some point in the unbounded directions.  Fortunately in the forward direction the waves can only propagate at light speed, so I only need to go forward a little bit, but backwards I'd have to decide where to end the grid and how to cut it off.  I already have a basic routine for fitting a rectangular grid around a beam, though, so it should work to a good approximation.
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