[MAP] Liouville's theorem and electromagnetic fields

Alexey Burov burov at fnal.gov
Fri Mar 11 09:31:27 EST 2011 

Kirk,

Let me consider an example I already mentioned - e-beam of our e-cooler. 
It is born inside a solenoid and travels a while along the magnetic 
field. If I will use your favorite recipe, that the magnetic field is 
irrelevant for the emittances, I immediately see, that my beam has 2 
identical transverse emittances, equal to
\epsilon_T=(thermal velocity)*(cathode radius) .

OK, now the beam goes out of the solenoid, and gets a kick from its 
transverse edge field. What happened with emittances after that? There 
is no magnetic field any more, but how can I calculate them? The beam 
state is strongly coupled there. Both me and Rob already mentioned here, 
that the emittances are eigen-emittances of the sigma-matrix. If I will 
calculate those, I will see that they are very different from 
\epsilon_T. Emittances are not preserved - something is wrong. Either 
your favorite recipe or the eigen-emittance recipe is incorrect. To find 
out, which recipe is incorrect, I will use an optical scheme, invented 
by Slava Derbenev, and called "Derbenev adapter". This is a triplet of 
skew quads, which transforms our round beam, coming out from the 
solenoid, into an uncoupled beam state. Just 3 skew quads can do that. 
Now, when our beam is uncoupled, moving in a free space, we all know how 
to calculate emittances. And now, the moment of truth is coming, Kirk. 
These 2 uncoupled emittances are the same, as eigen-emittances right 
after the solenoid, before the adapter. The eigen-emittances are 
preserved, but your 'mechanical emittances' are not. That's it.

Alexey.


On 3/10/11 8:48 PM, Kirk T McDonald wrote:
> Folks,
>
> I have added Alex' paper to DocDB 560.  See Appendix A.
>
> It is gratifying to see that the fact that Liouville's theorem holds for
> both mechanical and canonical phase space is "well known to those who know".
>
> The challenge now is to learn how best to use the "freedom" offered to us by
> this apparently nonintuitive result.
>
> --Kirk
>
> --------------------------------------------------
> From: "alex dragt"<dragtnb at comcast.net>
> Sent: Thursday, March 10, 2011 9:23 PM
> To: "Don Summers"<summers at phy.olemiss.edu>
> Cc: "Robert D Ryne"<rdryne at lbl.gov>; "Yuri Alexahin"<alexahin at fnal.gov>;
> "Kirk T McDonald"<kirkmcd at Princeton.EDU>; "Alex Dragt"
> <dragtg5 at comcast.net>; "MAP List"<map-l at lists.bnl.gov>; "Alex Dragt"
> <dragt at physics.umd.edu>
> Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
>
>> Dear all,
>>
>> The fact that Liouville's theorem holds in both mechanical and  canonical
>> phase space is also proved in
>>
>> A. Dragt
>> SOLAR CYCLE MODULATION OF THE RADIATION BELT PROTON FLUX, J.
>> Geophys. Res. 76: 2312-2344 (1971)
>>
>> also done in the context of the Van Allen Radiation, and hence for  motion
>> in the Earth's Magnetic Field.
>>
>> But we are interested in more than Liouville's theorem.  Also note  that
>> gauge transformations are symplectic maps, and hence do not  affect  the
>> eigen emitances.  See the book Lie Methods ... available  at the Web site
>>
>> http://www.physics.umd.edu/dsat/
>>
>> Best,
>>
>> Alex
>>
>>
>> On Mar 10, 2011, at 3:55 PM, Don Summers wrote:
>>
>>> The exact reference for Swann's paper is
>>>
>>> W. F. G. Swann,  Application of Liouville's Theorem to Electron  Orbits
>>> in the Earth's Magnetic Field,
>>> Phys. Rev. 44, 224–227 (1933)
>>> http://prola.aps.org/abstract/PR/v44/i3/p224_1
>>>
>>> Best,
>>> Don
>>>
>>>
>>> On Thu, 10 Mar 2011 16:33:10 -0800, Robert D Ryne wrote
>>>> I have not yet read the papers mentioned. But here are some brief
>>>> comments. Alex Dragt and I (cc to Alex) have been thinking about
>>>> this  a lot in the past months.
>>>>
>>>> The natural quantities to be computed are called "eigen-emittances."
>>>> To compute them properly they need to be derived from a beam 2nd
>>>> moment matrix, Sigma, formed using canonical variables.
>>>> The eigen-emittances are invariant under linear symplectic
>>>> transformations.
>>>>
>>>> The eigen-emittances can be computed in various ways, but the
>>>> simplest  is to compute the eigen-values of J Sigma, where J is the
>>>> fundamental  symplectic 2-form; the eigen-emittances are the modulii
>>>> of the eigen- values of J Sigma (which are pure imaginary and in +/-
>>>> pairs). If one  is interested in calculating the symplectic matrix
>>>> that transforms  Sigma to Williamson normal form, Alex Dragt has an
>>>> algorithm to do  this and has implemented it in the MaryLie code.
>>>>
>>>> Though the entries of Sigma will depend on the choice of gauge, the
>>>> eigen-emittances themselves are gauge invariant. We can't just set
>>>> the  vector potential to zero inside elements where it is nonzero,
>>>> and  expect to calculate the correct eigen-emittances (as was
>>>> suggested  below).
>>>>
>>>>>>>> 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 [WINDOWS-1252?]“longitudinal” coordinates (E,t)  rather
>>>>>>>> than (P_z,z).
>>>>>>>>
>>>>>>>> Is there any written reference that explains this  [WINDOWS-1252?]“well
>>>>>>>> [WINDOWS-
>>> 1252?]known”
>>>>>>>> fact?
>>>>>>>>
>>>> The above follows directly from whether we use the time t as the
>>>> independent variable or the Cartesian coordinate z as the
>>>> independent  variable. When using the time, the longitudinal
>>>> variables are
>>>> (z,p_{z,canonical}). When using z, the longitudinal variables are (t,
>>>> - E) where t is arrival time at location z, and where E is the
>>>> total  energy of a particle when it reaches location z, i.e.
>>>> E=\gamma m c^2 +  q \Phi.
>>>>
>>>> Rob
>>>>
>>>> On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:
>>>>
>>>>> Hi Kirk,
>>>>>
>>>>> Thank you for digging out these interesting papers.
>>>>> Of course the Poincare invariants remain the same no matter what
>>>>> momenta are used.
>>>>> But this is not what we calculate from tracking or measurement data
>>>>> using standard definition.
>>>>> So a clarification is still needed of what and how we should
>>>>> calculate.
>>>>>
>>>>> Yuri
>>>>>
>>>>> ----- Original Message -----
>>>>> From: Kirk T McDonald<kirkmcd at Princeton.EDU>
>>>>> Date: Thursday, March 10, 2011 4:09 pm
>>>>> Subject: [MAP] Liouville's theorem and electromagnetic fields
>>>>> To: MAP List<map-l at lists.bnl.gov>
>>>>> Cc: Kirk McDonald<kirkmcd at Princeton.EDU>
>>>>>
>>>>>
>>>>>> Folks,
>>>>>>
>>>>>> There is a technical question as to how we should be calculating
>>>>>> emittance for beams in electromagnetic fields.
>>>>>>
>>>>>> The formal theory of [WINDOWS-1252?]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 [WINDOWS-1252?]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 [WINDOWS-1252?]“momentum”.
>>>>>>
>>>>>> This result has the consequence that phase-space volume (and
>>>>>> emittance) is actually gauge invariant [WINDOWS-1252?]– 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 [WINDOWS-1252?]“ideal” emittance if  the
>>>>>> phase-space
>>>>>> volume is more [WINDOWS-1252?]“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 [WINDOWS-1252?]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  [WINDOWS-1252?]Liouville’s
>>>>>> theorem for
>>>>>> cosmic rays in the [WINDOWS-1252?]Earth’s atmosphere (using
>>>>>> mechanical momentum).
>>>>>> This may well be the earliest paper about particle beams and
>>>>>> [WINDOWS-1252?]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
>>>>>> [WINDOWS-1252?]“longitudinal” coordinates (E,t) rather than  (P_z,z).
>>>>>>
>>>>>> Is there any written reference that explains this  [WINDOWS-1252?]“well
>>>>>> [WINDOWS-
>>> 1252?]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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