[MAP] Liouville's theorem and electromagnetic fields
Kirk T McDonald
kirkmcd at Princeton.EDU
Fri Mar 11 17:38:25 EST 2011
Alexander,
I may be misreading your paper.
Through sec. 4 it seems to me that you calculate "emittance" ignoring the
magnetic field (and its vector potential in the calculation).
If you had included the vector potential, the effect would have been to
replace variable theta' by
theta' + q B_0 / 2 p c
for particles of charge q. If the particles are electrons with charge q = -
e, this would be written as
theta' -e B_0 / 2 p c,
which form does appear in your eq. (11) [although you do not seem to
associate the possible merit of this expression with either the vector
potential or with canonical momentum].
Maybe your sec. 5 is meant to tell us that you actually did make this
change, but I am not sure of this.
You do seem to imply that making this change would leave the "value of the
matrix determinant" unchanged. I think by this you mean the determinant of
the matrix Sigma_r of eq. (9), which you identify with the emittance that
you call epsilon_2 in eq. (10).
So it seems to me that you are agreeing with your wording that
"It also can be demonstrated that the determinant of a 4 matrix, with the
matrix elements containing canonical momenta instead of x , is identical to
Eq. (9), at least in the case of a uniform magnetic field."
It might then seem that whether or not one includes the vector potential in
the calculation is only a matter of taste.
It may be the you agree with Alexey that it's in good taste to include the
vector potential in emittance calculations.
However, I am worried less about taste than about NUMERICAL accuracy of the
emittance calculation, where there can be considerable freedom in the choice
of gauge for the vector potential, as well as the choice to ignore it
altogether.
This issue will require further study.
--Kirk
-----Original Message-----
From: Alexander Shemyakin
Sent: Friday, March 11, 2011 2:34 PM
To: Kirk T McDonald
Cc: map-l at lists.bnl.gov
Subject: Re: [MAP] Liouville's theorem and electromagnetic fields
Kirk,
actually, I agree with what Alexey says.
My understanding is that EM fields rotate the plane in 6D space where
the particles move, but leave the total 6D volume constant. The
coupled-motion formalism finds the new position of this plane,
indicating a mode that can move through the structure unchanged, while
motion in X and Y sill looks complicated. This method includes EM fields
in a an indirect way, through Courant-Snyder parameters.
We did not find a simple way to implement this formalism into the code
simulating motion in continuous fields, but we had a much simpler
question that you seems to have: What is the effect of a specific
configuration on transverse emittance in an axially symmetrical case?
Because of the symmetry and neglect of longitudinal - transverse
coupling, we essentially calculated 4D emittance, which stays constant
and can be easily calculated in a program.
For you, it would answer only about changes of the effective 6D
emittance (in the sense that Sergey mentioned).
Sasha
On 3/11/2011 1:04 PM, Kirk T McDonald wrote:
> 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?
>>>>>>>>> _______________________________________________
>>>>>>>>> MAP-l mailing list
>>>>>>>>> MAP-l at lists.bnl.gov
>>>>>>>>> https://lists.bnl.gov/mailman/listinfo/map-l
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