[MAP] Liouville's theorem and electromagnetic fields

[Alexey Burov]

One remark to Swann's paper:
His theorem relates to the total emittance, not to the partial ones. 
Partial emittances are sensitive to eA/c term.

A possible way to get rid of eA/c inside solenoidal structures is to 
make a fake 0-length edge of the solenoid at a place where emittances 
are calculated; kicks from the edge solenoidal fields have to be taken 
into account, of course.

Alexey.

On 3/10/2011 4:09 PM, Kirk T McDonald wrote:
> Folks,
> There is a technical question as to how we should be calculating 
> emittance for beams in electromagnetic fields.
> The formal theory of Liouville’s theorem is clear that the invariant 
> volume in phase space is to be calculated with the canonical momentum
> gamma m v + e A / c
> and not the mechanical momentum m v.
> This is awkward in two ways:
> 1.   We don’t always know the vector potential of our fields
> 2.   The vector potential is subject to gauge transformations, so 
> canonical momentum is not gauge invariant.
> The second issue is disconcerting in that it suggests that phase-space 
> volume, and emittance, are not actually invariant  -- with respect to 
> gauge transformations.
> Hence, it is useful to note a very old paper,
> W.F.G. Swann, Phys. Rev. 44, 233 (1933)
> which shows that the phase-space volume for a set of noninteracting 
> particles is the same whether or not the term e A / c is included in 
> the “momentum”.
> This result has the consequence that phase-space volume (and 
> emittance) is actually gauge invariant – although the location of a 
> volume element in space space is gauge dependent.
> ---------------
> This suggests that we could simply calculate emittances based only on 
> the mechanical momentum, and avoid having to worry about the accuracy 
> of our model for the vector potential.
> Of course, our calculations are actually of rms emittance, which is a 
> better representation of the “ideal” emittance if the phase-space 
> volume is more “spherical”, and not elongated/twisted.
> It could be that the shape of the phase-space volume is better for rms 
> emittance calculation if the vector potential, in some favored gauge, 
> is included in the calculation.....
> --Kirk
> PS  I have placed Swann’s paper as DocDB 560
> http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase
> user = ionization pass = mucollider1
> See also the paper by Lemaitre that used Liouville’s theorem for 
> cosmic rays in the Earth’s atmosphere (using mechanical momentum).   
> This may well be the earliest paper about particle beams and 
> Liouville’s theorem.
> PPS  Scott Berg notes that when one evaluates emittance at a fixed 
> plane in space, rather than at a fixed time, it is better to use the 
> “longitudinal” coordinates (E,t) rather than (P_z,z).
> Is there any written reference that explains this “well known” fact?
> How is this prescription affected by electromagnetic fields?
> The vector potential of even a simple rf accelerating cavity has an 
> A_z component (which is zero on axis, but nonzero off it).
> http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf 
> <http://puhep1.princeton.edu/%7Emcdonald/examples/cylindrical.pdf>
> Note that the vector potential is nonzero outside the cavity, even 
> though the E and B fields are zero there!
> Do we know how to include A_z in our longitudinal emittance calculations?
>
>
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