[MAP] Liouville's theorem and electromagnetic fields

Alexander Shemyakin shemyakin at fnal.gov
Fri Mar 11 14:34:44 EST 2011


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
>>>>>>>> _______________________________________________
>>>>>>>> MAP-l mailing list
>>>>>>>> MAP-l at lists.bnl.gov
>>>>>>>> https://lists.bnl.gov/mailman/listinfo/map-l
>>>>>> -- 
>>>>>> Open WebMail Project (http://openwebmail.org)
>>>>>> Debian Project (http://www.debian.org)
>>>>>>
>>>> _______________________________________________
>>>> MAP-l mailing list
>>>> MAP-l at lists.bnl.gov
>>>> https://lists.bnl.gov/mailman/listinfo/map-l
>>>
>> _______________________________________________
>> MAP-l mailing list
>> MAP-l at lists.bnl.gov
>> https://lists.bnl.gov/mailman/listinfo/map-l
>
>
> _______________________________________________
> MAP-l mailing list
> MAP-l at lists.bnl.gov
> https://lists.bnl.gov/mailman/listinfo/map-l
>




More information about the MAP-l mailing list