[MAP] Liouville's theorem and electromagnetic fields

Alexey Burov burov at fnal.gov
Thu Mar 10 19:22:48 EST 2011 

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 <mailto:kirkmcd at Princeton.EDU>
> *Sent:* Thursday, March 10, 2011 6:15 PM
> *To:* Alexey Burov <mailto:burov at fnal.gov>
> *Cc:* map-l at lists.bnl.gov <mailto: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 <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?
>>>
>>>
>>> _______________________________________________
>>> MAP-l mailing list
>>> MAP-l at lists.bnl.gov
>>> https://lists.bnl.gov/mailman/listinfo/map-l
>>
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