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

[Kirk T McDonald]

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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