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

[Tom Roberts]

Along those lines, I must ask: is emittance really what we need to consider? In 
particular, as we prepare to evaluate, compare, and down-select cooling 
techniques over the next year or so.

Remember that the primary goal is to maximize the luminosity of a muon collider 
for a given number of protons on target. (There are secondary goals related to 
neutrino factories, costs, and other aspects such as real estate.)

In the cooling channel, if phase space becomes filamented, then the beams are 
not Gaussian, and emittance does not truly reflect the luminosity of the 
collider. A devil's advocate can easily construct gedankens in which the actual 
luminosity is either less than or greater than the computation using transverse 
emittance and the IP's beta*:

Imagine that the phase space filamentation and the focusing at the IP conspire 
to make each beam be a hollow tube with an empty center. Then if the radii of 
the two beam tubes are equal the actual luminosity will be larger than the 
computation (computed transverse sigmas will reflect the diameter of the tube, 
not its wall thickness); if one beam fits inside the tube of the other, the 
luminosity will be less than the computation. (I do not claim this is likely, 
but I don't think that this sort of thing is impossible, either.)

Indeed, in most accelerators the beams are not Gaussian, as the tails have been 
deliberately removed to reduce halo.

How important is this effect?

To accurately assess cooling techniques, do we need to simulate all the way to 
the IP and compute the simulated luminosity? Or can we use emittance at the end 
of the cooling channel as a proxy for luminosity?


Tom Roberts



On 3/14/11 3/14/11 - 9:46 AM, Chris Rogers wrote:
> 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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