[MAP] Liouville's theorem and electromagnetic fields

[Alexey Burov]

Kirk,

epsilon1 and epsilon2 are called emittances,because they are emittances 
=phase volumes for two eigenvectors of the 4D matrix of beam second 
moments (sigma matrix).They are invariant under any symplectic 
transformations.

You may convince yourself that this result is correct in a following 
numerical way: generate a numerical ensemble of electrons born at 
magnetized cathode, inside a solenoid. Propagate them along that 
solenoid, until they will come out, in a free space. Calculate 
sigma-matrix there, find its eigenvectors. In the basis of its 
eigenvectors, sigma-matrix is diagonal with 2 emittances on its main 
diagonal. See Eqs. (3.26, 3.27) of that paper. Indices 1 and 2 relate to 
2 eigenmodes of the sigma-matrix. Generally, they are not just x and y 
planar modes, because the beam state is x-y coupled.

Alexey.

On 3/10/2011 5:15 PM, Kirk T McDonald wrote:
> 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 <mailto:burov at fnal.gov>
> *Sent:* Thursday, March 10, 2011 6:02 PM
> *To:* Kirk T McDonald <mailto:kirkmcd at Princeton.EDU>
> *Cc:* map-l at lists.bnl.gov <mailto: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 <mailto:burov at fnal.gov>
>> *Sent:* Thursday, March 10, 2011 5:33 PM
>> *To:* map-l at lists.bnl.gov <mailto: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 
>>> <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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>>
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