[MAP] Liouville's theorem and electromagnetic fields
Kirk T McDonald
kirkmcd at Princeton.EDU
Thu Mar 10 21:22:08 EST 2011
Alex,
"Leaving the vector-potential out of the calculations" changes the partial emittances.
I now don't believe this is true.
Hiding behind words like "symplectic" does not add insight to this issue.
--Kirk
From: Alexey Burov
Sent: Thursday, March 10, 2011 7:22 PM
To: Kirk T McDonald
Cc: map-l at lists.bnl.gov
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
Kirk,
the first question in that business should be: what are sub-emittances for arbitrary coupled beam state? Correct answer is: they are diagonal elements of the sigma-matrix in a basis of its eigenvectors. These two (or 3 in 3D coupled case) values are invariant under any symplectic transformations - that is why they are so important. "Leaving the vector-potential out of the calculations" changes the partial emittances. This non-symplectic procedure is equivalent to neglect of a kick from solenoidal edge fields - which may easily lead to severe errors.
Alexey.
On 3/10/2011 5:27 PM, Kirk T McDonald wrote:
Alexey,
A further comment is the Swann’s method does not show that “subemittances” are invariant in TIME, but it seems to show that they are invariant under leaving the vector potential out of their calculation.
------------
Maybe Lebedev and Bogacz considered an example in which two “subemittances” evolved with time such that one increased and the other decreased, whereby the “total” emittance remained invariant.
This in no way precludes that these subemittances would have the same (time-dependent) values if the vector potential were ignored in their calculation.
--Kirk
From: Kirk T McDonald
Sent: Thursday, March 10, 2011 6:15 PM
To: Alexey Burov
Cc: map-l at lists.bnl.gov
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
Alexey,
I understand your hope to “avoid long discussion”, as the Lededev/Bogacz paper is more or less incomprehensible to me.
It is not clear why parameters epsilon1 and epsilon2 are called “emittances”, since they are not invariants.
And, I don’t know what indices 1 and 2 refer to.
Etc.
If Valery or Alex care to enlighten me, that would be most welcome.
--Kirk
From: Alexey Burov
Sent: Thursday, March 10, 2011 6:02 PM
To: Kirk T McDonald
Cc: map-l at lists.bnl.gov
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
Kirk,
they are not invariant. To avoid long discussion here, please have a look at Lebedev-Bogacz paper:
http://iopscience.iop.org/1748-0221/5/10/P10010/pdf/1748-0221_5_10_P10010.pdf,
the very end of it, pp. 21-23. You see that the 2 emittances of e-beam born at the magnetized cathode, \epsilon_1 and \epsilon_2 may differ by orders of magnitude. This is actual case for e-beam of our e-cooler.
Alexey.
On 3/10/2011 4:42 PM, Kirk T McDonald wrote:
Alexey,
For the subspace (q,p) we have
dq’ dp’ = J dq dp
J = | dq’/dq dq’/dp |
| dp’/dq dp’/dp |
Suppose p = m v + A (in units where e/c = 1)
and we transform
q’ = q
p’ = mv = p – A(q)
Then the Jacobian is
J = | 1 0 |
| –dA/dq 1 | = 1
It looks to me like the partial phase volumes are also invariant under the “transformation” of neglecting the vector potential.
--Kirk
From: Alexey Burov
Sent: Thursday, March 10, 2011 5:33 PM
To: map-l at lists.bnl.gov
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
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
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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