[MAP] Is rms emittance gauge invariant?

alex dragt dragtnb at comcast.net
Mon Mar 14 23:45:05 EDT 2011


Dear Kirk,

Gauge transformations are symplectic maps.  See Exercises 6.2.8 and  
6.5.3 in the book "Lie Methods ...", which can be downloaded from the  
Web site

http://www.physics.umd.edu/dsat/

By construction, eigen emittances are invariant under linear  
symplectic transformations, and therefore invariant under gauge  
transformations in the linear approximation.  Note the words "eigen  
emittances"!  Eigen emittances are not the same as rms emittances.   
See the papers

A. Dragt, F. Neri, et al., "LIE ALGEBRAIC TREATMENT OF LINEAR AND  
NONLINEAR BEAM DYNAMICS",
Annual Review of Nuclear and Particle Science 38, p. 455 (1988).

A. Dragt, R. Gluckstern, et al., "THEORY OF EMITTANCE INVARIANTS",   
Lecture Notes in
Physics 343: Proceedings of the Joint US-CERN Capri School on  
Accelerator Physics,
Springer Verlag (1989).

A. Dragt, F. Neri, et al., "GENERAL MOMENT INVARIANTS FOR LINEAR  
HAMILTONIAN SYSTEMS",
Physical Review A, 45, p. 2572 (1992).


Rather, eigen emittances tell you what rms emittances can be achieved  
by proper beam manipulation (using symplectic elements in the linear  
approximation).  See Chapter 26 of Lie Methods ....

Every symplectic map can be factorized into linear and nonlinear  
parts, all of which are symplectic.  The linear part of a symplectic  
map is described by a symplectic matrix.  See Sections 7.6 through 7.8  
of Lie Methods ....  [Every map (in an even number of variables) can  
be factorized into symplectic and nonsymplectic parts.  See Chapter 22  
of Lie methods.]  As related in Section 1.1.2 of Lie Methods...,  
accelerators were originally designed using only linear linear  
symplectic maps (symplectic matrices).  In fact, it was often not  
recognized that even the so called linear beam-line elements such as  
drifts, quads, and bends also have nonlinear parts.  Karl Brown was  
the first to include quadratic effects, which made it possible to  
treat sextupoles in the lowest nonlinear approximation.  With the  
advent of Lie methods and Truncated Power Series Algebra (TPSA) it is  
now possible to work to quite high order, at least for idealized  
elements excluding s dependences and fringe fields.  We are currently  
working (see below) to extend these methods to realistic beam-line  
elements including fringe fields and high-order multipole effects.

Although an approximation, it is  always good to begin with a linear  
(symplectic matrix) design.  But then one must recognize that  
nonlinear corrections can be important.  For example, for many years,  
SLAC failed to included the third-order nonlinear terms in drifts when  
trying to simulate the B-factory dynamic aperture.  To their  
amazement, they found that the agreement between simulation and  
experiment improved when they finally did so.  In the cases of  
solenoids and quadrupoles it is important to recognize that third- 
order fringe-field effects can be important.  Indeed, in the case of  
electron microscopes, solenoid third-order fringe-field effects are  
the chief source of aberrations.  And, in the case of microprobes and  
beam telescopes, quadrupole third-order fringe-field effects are the  
chief source of aberrations.  Due to their complicated nature, not  
much is known about the effects of fringe fields for realistic  
dipoles.  This will change with the fruition of the work on curved  
elements, also sketched below.

With regard to the effect of vector potential terms, it is always  
possible to select a gauge such that the vector potential vanishes in  
the field-free regions.  Indeed, when B=0, we must have curl A=0, and  
therefore A is a gradient of a scalar field, and this scalar field can  
be used to gauge transform A to zero, etc.  We have developed surface  
methods which take E and B field data as input and produce as output a  
vector potential.  This vector potential has the property that it  
decays to zero outside the element.  See the paper

Phys. Rev. ST Accel. Beams 13, 064001 (2010) [17 pages]
Accurate transfer maps for realistic beam-line elements: Straight  
elements

which deals with straight magnetic elements.  This paper also cites  
the work of Dan Abell, which does analogous things for realistic RF  
cavities.  For the beginning of work on curved elements, see the  
attachment





I hope these comments answer some of your questions.

Best regards,

Alex Dragt

***************************************************

On Mar 14, 2011, at 12:33 PM, Kirk T McDonald wrote:

> Alex,
>
> Phase volume is independent of gauge, because a gauge transformation  
> is a canonical transformation.
>
> This does NOT immediately imply that rms emittance is gauge  
> invariant --  since rms emittance is (unfortunately) not an exact  
> measure of phase volume.
>
> Did Alex Dragt really claim that rms emittance is gauge invariant?
>
> --------------
> A gauge transformation
> A -> A + grad f
> V -> V + d f / d t
> leaves the fields
> E = - grad V - d A / dt
> B = curl A
> unchanged.
>
> But, the terms grad V, d A / d t and curl A do not appear in an rms  
> emittance calculation, which involves A and V (in case we use  
> coordinates
> x
> y
> t
> p_x = p_mech_x + q A_x
> p_y = p_mech_y + q A_y
> p_t = E_mech + q V
>
> So, it appears to me that the differences in 2nd moments of these  
> quantities, which form the rms emittance, do not result in the kind  
> of cancellation associated with gauge invariance.
>
> If so, it becomes rather questionable what is the physical  
> significance of rms emittance when electromagnetic fields are  
> present (as in any particle accelerator).
>
> --Kirk
>
>
>
>
> -----Original Message----- From: Alexey Burov
> Sent: Monday, March 14, 2011 8:00 AM
> To: map-l at lists.bnl.gov
> Subject: Re: [MAP] Liouville's theorem, kinematic invariants,and  
> dynamic invariants
>
> It was already mentioned by Alex Dragt that emiitances are independent
> on the gauge transformations, since they are canonical.
>
> On 3/13/11 10:57 PM, Valeri Lebedev wrote:
>> Dear All,
>> I was impressed with intensity of the discussion and a large number  
>> of e-mails and would like to add a few more words.
>> 1. First, there is no uncertainty with choice of the vector  
>> potential in the real applications. One has to keep in mind that  
>> the reason we would like to know the emittances is that we want to  
>> use this beam in a collider and we need to know the emittances and  
>> Twiss parameters of the beam out-coming the cooling section. That  
>> means that the computed emittances have to coincide with usual  
>> emittances in the regions where magnetic field is zero. For obvious  
>> reason the vector potential has to be equal to zero in these  
>> regions and uncertainty disappears.
>> 2. For some reason a necessity to know the Twiss parameters of out- 
>> coming beam was not discussed, but, I would like to note, that the  
>> knowledge of Twiss parameters is the same important as knowledge of  
>> emittances if one wants to prevent the emittance growth in the  
>> course of beam transfer to the collider and to minimize required  
>> apertures and, consequently, non-linearities in the course of beam  
>> transport and acceleration from cooling section to the collider.
>>
>> These problems are addressed in my and Alex Bogacz paper and I  
>> cannot agree that it is too complicated to be understood by a  
>> general folks. As far as I understand all problems are addressed  
>> there. Otherwise we do not have a correct language to discuss  
>> cooling.
>>
>> Valeri
>>
>>
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