[MAP] Liouville's theorem and electromagnetic fields

[Kirk T McDonald]

Alexander,

Thanks for your comments on these intriguing issues.

Your are another whom I don't know, and whom apparently some of my MAP 
colleagues feel should be shielded from comments by me, and vice versa.

Your Fermilab TM with Sergei Naigetsev is very nice.

I see that you quote ref. 4 = book of Reiser as noting that phase volume is 
the same whether or not a vector potential is included in the calculation.

If I follow your paper, you use this result to justify ignoring the vector 
potential in your calculations.

It would appear that you are at odds with Alexey Burov who seems to insist 
that you must use a vector potential.

--Kirk

-----Original Message----- 
From: Alexander Shemyakin
Sent: Friday, March 11, 2011 12:25 PM
To: map-l at lists.bnl.gov
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields

Kirk,

just a couple of comments about the cooler.
The cathode is in a longitudinal magnetic field, and it is a DC gun.

To analyze non-linear contribution in the most sensitive, low-energy
portion of the electrostatic accelerator, we used the trick based on
axial symmetry of the system, calculating 4D emittance according to
http://lss.fnal.gov/archive/test-tm/2000/fermilab-tm-2107.pdf
In this case variation of the longitudinal magnetic field did not change
the value in a linear approximation.

It does not address the fundamentals you are discussing, but is simple
to implement and was helpful after inclusion into SAM code that we used
for the simulation.
Sasha


On 3/11/2011 10:52 AM, Kirk T McDonald wrote:
> Alexey,
>
> I have no "favorite" scheme for calculation of "emittances".
>
> Rather, I am dismayed by persistent, apparent numerical instabilities in 
> the
> schemes used by the MAP collaboration.
>
> A bit of history: In the precusor to the MAP project in the 1980s, I was 
> the
> first to analyze the emittance of an "rf gun", which is, I believe, what 
> you
> refer to as the source of the e-beam in your e-cooler.
>
> Before me, people calculated "emittances" in accelerating structures only
> using mechanical momenta.  I suggested that it would be better to 
> calculate
> them using canonical momenta, based on a particular vector potential for 
> the
> rf gun (which had no DC solenoid around it in the early days).
>
> This recipe definitely improved the numerical stability of the emittance
> calculations, and people have been using it ever since.
>
> However, it is clear that at some basic level this recipe should not be
> needed.  Phase volume (and subvolumes thereof) are the same whether or not
> the vector potential is included in the calculation.
>
> So, the better empirical results obtained by including the vector 
> potential
> (in some gauge) are just a sort of "numerical accident" that we don't
> understand.
>
> Of course, we all want to have the benefit of such "accidents", but if we
> don't understand how this benefit arises, we may find ourselves in
> situations where the "accidents" make things worse rather than better.
>
> Indeed, this is mostly what happens in "accidents".
>
> --Kirk
>
> --------------------------------------------------
> From: "Alexey Burov"<burov at fnal.gov>
> Sent: Friday, March 11, 2011 9:18 AM
> To: "Kirk T McDonald"<kirkmcd at Princeton.EDU>
> Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
>
>> 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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