[MAP] Liouville's theorem, kinematic invariants, and dynamic invariants

[Chris Rogers]

In MICE the principal measurements are even made in a constant 4 T field
so one can calculate the vector potential pretty easily (or just use
kinematic variables).

I don't think the vector potential is non-zero and changing outside the
cavities because there would have to be a corresponding field for that.
Don't forget MICE cavities have a surface current on the Beryllium
windows so the usual rules for Maxwell in vacuum don't apply.

--

RF cavities are different and I don't think well treated in the standard
 texts. They have time and z-dependent fields which are difficult to
work with because you can't transform symplectically to reference
particle coordinates - so you can't make a power law expansion and
accelerator physics breaks.

The only way around it that I have come across is to use proper time as
the independent variable and take both z and t as dynamic variables.
Maybe it's possible to work in lab frame... but that's not what I've
seen done most of the time.

Chris

Kirk T McDonald wrote:
> Chris,
> 
> Interesting!
> 
> I am now convinced that if one is able to deduce actual phase volume,
> rather than rms emittance approximations to this, then one is free to
> leave out the vector and scalar potentials.
> 
> I also worry that the computation of the phase volume will be tricky,
> and that it may be more practical if one does include the scalar and
> vector potentials -- in some favored gauge.
> 
> You might think that in MICE things will be simpler since measurements
> are made only outside the rf cavities (although inside DC magnetic fields).
> 
> Most people would agree that for DC fields the "favored" vector
> potential is that of the Coulomb gauge.
> 
> However, it appears to me that A and V of rf cavities are nonzero
> OUTSIDE those cavities, and fall off only as 1/r from the cavity.
> So, if measurements are made fairly close to rf cavities, and one
> includes A and V in the computation of phase volume, it may be necessary
> to include the A and V of the rf cavities!
> 
> --Kirk
> 
> 
> -----Original Message----- From: Chris Rogers
> Sent: Monday, March 14, 2011 10:46 AM
> Cc: map-l at lists.bnl.gov
> Subject: Re: [MAP] Liouville's theorem, kinematic invariants, and
> dynamic invariants
> 
> A completely different approach that we are looking at in MICE is to
> calculate the true 6d phase space volume of the beam. I think when you
> start worrying about non-linear effects the power law expansions rather
> quickly stop working anyway, so this is an interesting alternative.
> 
> We have shown that MICE especially needs to worry about details of the
> emittance calculation because the optical emittance growth is not always
> small compared to the scattering.
> 
> Presumably for Kirk he is worried about targetry stuff. A funny thing
> about this is that he has particle production on the target and particle
> decay (pions->muons), both of which are not emittance preserving
> processes. Also canonical momentum may not be conserved in this instance
> which can introduce mismatch.
> 
> Chris
> 
> Alexey Burov wrote:
>> 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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>>> MAP-l at lists.bnl.gov
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> 
> 


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