[MAP] Use of x,y,t as Hamiltonian coordinates.

Wed Mar 16 19:38:10 EDT 2011

Dan, Kirk, MAP,

Here's what I did:

1) I took the MICE Step IV magnet configuration;
2) put it in COSY to get a high-order transfer map,
3) checked that the magnetic field in COSY is consistent with g4beamline 
(see field_comparison.png attached),
4) calculated a ninth order transverse map from the symmetry point (z=0) 
to some point outside the last magnet (z=3.3 m),
5) based on that map, calculated the nonlinear determinant of the 
Jacobian (the determinant that takes into account all the nonlinearities 
of the underlying transfer map, essentially a high-order polynomial in 
x,px,y,py,t,E that allows to calculate the determinant at any point in 
phase space).

I found that the determinant is equal to 1 everywhere in the area far 
greater than the beam size under consideration. Attached determinant.png 
shows the deviation of the determinant from 1 as a function of x and y. 
You can see that the deviation is of the order of 10^(-11), and the fact 
that there is a deviation at all results from the fact that the ninth 
order approximation is not sufficient for larger amplitudes.

The fact that the determinant is 1 entails that the 6d emittance is 
constant. However, if one looks at the 6d emittance approximation 
produced by using the second moment matrix approximation, it is clearly 
not constant.

P.S. Apologies for generating traffic by attaching figures, I try to 
keep file sizes as small as possible. Since there is no end in sight to 
this discussion, should we rather establish some file repository and 
send around links rather than files?

Pavel

On 3/16/2011 14:18, Daniel Kaplan wrote:
> Kirk et al.,
>
> 	As I recall, this discussion was stimulated by Pavel showing that 6D and transverse (and, to a lesser extent, longitudinal) emittances as computed by ecalc9 are not constants of the motion in an absorber-free version of MICE Step IV. In his talk at the JLab meeting he showed a COSY calculation to the effect that the volume of the beam in phase space _is_ a constant of the motion in that configuration.
>
> 	So the issue is not so much whether emittance is conserved, but whether we know how to compute emittance.
>
> 	The thing computed by ecalc9 is not a constant of the motion, but some other "emittance" is.
>
> 	(Sorry to be "spamming" those who would rather not see this, but nobody has come forward with a more restricted email list.)
>
> 						Dan
>
> On Mar 16, 2011, at 3:53 PM, Kirk T McDonald wrote:
>
>> Greg,
>>
>> Good to hear from you!
>>
>> Your comments suggest that I may be too naïve on (at least) one thing.
>>
>> Emittance growth appears in simulations of a drift space for a beam with an
>> energy spread.
>>
>> Your .pdf file notes that there is quadratic emittance growth with time/z in
>> a drift space.   Solenoids should leave  (E,t) and (p_z,z) unchanged
>> compared to the zero field case, so the issue of growth of longitudinal
>> emittance growth should remain.
>>
>> Other folks in MAP have done lots of simulations of beams with energy
>> spreads moving down long solenoid channels.  In general, I have the
>> impression that we do not see much emittance growth in these simulations.
>>
>> I may have the wrong impression here, which has led to my naive assumption
>> that the helical particle trajectories result in a more stable RMS emittance
>> with z position.
>>
>> Comments?
>>
>> --Kirk
>>
>> -----Original Message-----
>> From: Gregory Penn
>> Sent: Wednesday, March 16, 2011 2:35 PM
>> To: MAP-l at lists.bnl.gov
>> Subject: Re: [MAP] Use of x,y,t as Hamiltonian coordinates.
>>
>> This has been a very interesting discussion.
>>
>> It's easier for me to talk about the calculations in "emitcalc.for"
>> because that was written more recently.  I think the 4D and 6D
>> calculations are very similar to those in ECALC9.
>>
>> Because all linear correlations are automatically removed in the 6D
>> calculation, and all linear transverse correlations are removed in the
>> 4D calculation, any errors arise from nonlinear corrections to the
>> scalar potential (assumed to be zero) and vector potential (assumed to
>> be static and usually from a solenoid field, but can include high
>> order nonlinear terms).  There are certainly errors calculated within
>> an RF cavity because of the dynamic fields, and the nonlinearity will
>> be strong because typical beams have a time spread comparable to the
>> RF period.  However, what's important are the nonlinear terms in the
>> transverse vector potential, and I think those terms will usually lead
>> to weak corrections, with terms like A_r~r sin(omega t) and r^3
>> sin(omega t).  It would certainly be better to correct the transverse
>> momentum if the vector potential as a function of time is known, and
>> that could be added to the code.
>>
>> Courant's invariant emittances (what Rob calls eigen emittances) are
>> calculated similarly and should have the same accuracy.  Removing the
>> nonlinear correlations of energy and transverse amplitude changes the
>> emittances, of course, but that feature can be turned off.
>>
>> The emittance growth in a drift is more fundamental, there is always
>> going to be a nonlinear time-energy (and z-pz) correlation in a drift
>> when the gamma factor is only moderately bigger than 1.  See attached
>> PDF for a calculation.  For a long enough drift, the longitudinal
>> phase space will be boomerang-shaped rather than an ellipse.  One
>> could correct the energy as a function of time to remove this
>> correlation, but it does have real consequences especially for
>> transport using RF.  To do this automatically would mean also
>> removing, for example, the quadratic droop in energy resulting from
>> accelerating a beam on crest.
>>
>> -Gregg
>>
>> On Mon, Mar 14, 2011 at 12:55 PM, Kirk T McDonald<kirkmcd at princeton.edu>
>> wrote:
>>> Rick,
>>>
>>> Thanks for this.
>>>
>>> It sounds to me like ECALC9 only includes a vector potential for DC
>>> magnets.
>>>
>>> For rf cavities, it sounds like neither the vector nor scalar potential
>>> are
>>> considered.
>>>
>>> This is inconsistent with the nominal rules of Hamiltonian dynamics.
>>>
>>> I am still trying to clarify whether we can get away with such apparent
>>> inconsistencies.
>>>
>>> --Kirk
>>>
>>> From: Fernow, Richard C
>>> Sent: Monday, March 14, 2011 1:51 PM
>>> To: MAP-l at lists.bnl.gov
>>> Subject: Re: [MAP] Use of x,y,t as Hamiltonian coordinates.
>>>
>>>
>>> Kirk,
>>>
>>>
>>>
>>> First let me say that the physics content of ECALC9 is entirely the work
>>> of
>>> Gregg Penn. I made a modified version ECALC9F that took its input from an
>>> input file instead of the keyboard and have tried to maintain it as
>>> problems
>>> arose over the years. I also wrote the Mucool note 280 to help me
>>> understand
>>> what Gregg was doing. Sometimes this isn’t very clear in the code. I don’t
>>> think FORTRAN was Gregg’s native language. Anyway looking over the code
>>> again this morning
>>>
>>>
>>>
>>> 1.      Greg calculates em6, emT, and emL using three different matrices.
>>>
>>> 2.      The variables for em6 are {x, y, t, px, py, E}. The momenta appear
>>> to be mechanical, i.e. the correction for Bz is not applied here.
>>>
>>> 3.      The variables for emT are {x, y, px, py}. The momenta also appear
>>> to
>>> be mechanical in this matrix. The statement I made about this at JLAB was
>>> apparently wrong.
>>>
>>> 4.      The variables for emL are {t, E, AT2}. The third variable is the
>>> square of the transverse amplitude.
>>>
>>> 5.      The only place where the on-axis field appears to be used is in
>>> calculating the canonical angular momentum and the transverse amplitude.
>>>
>>>
>>>
>>> Rick
>>>
>>>
>>>
>>>
>>>
>>> It is my understanding the ICOOL/ECALC presently uses the first of these
>>> transformations, I.e., it uses coordinates
>>>
>>> (x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, E_mech)
>>>
>>>
>>>
>>> [Rick Fernow: Can you comment on this?]
>>>
>>>
>>>
>>> It now seems to me that this is actually OK in principle (although perhaps
>>> a
>>> somewhat "ugly" transformation)
>>>
>>>
>>>
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>>>
>>
>>
>>
>>
>>
>>
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