[MAP] Emittance simulation experiments

Pavel Snopok snopok at gmail.com
Wed Mar 16 19:16:49 EDT 2011


Hi Tom and Kirk,

I did a similar thing to what Tom has just shown: I took the beam we 
usually use for MICE Step IV wedge simulations (p_ref=200 MeV/c, norm. 
longitudinal emittance 90 mm, norm. transverse emittance 6 mm, 
sigma_x=sigma_y=37 mm, sigma_px=sigma_py=17 MeV/c, sigma_pz=29 MeV/c, 
sigma_T=1.25 ns, no dispersion).

I used {x,px,y,py,t,E} to calculate emittances.

I ran the beam through 3.3 meters of drift in g4beamline and compared 
the result with COSY simulations where the transfer maps are calculated 
to first order (linear transfer map, red line) as well as orders 5 
(green line), 9 (blue line) and 13 (magenta line). The results for 
orders 9 and 13 are very close to the g4beamline results as more 
high-order nonlinearities are taken into account.

What I observe is that all three emittances are constant as long as the 
linear approximation is used. With any nonlinearity added, the 
approximation based on the second moment matrix shows significant growth 
of all three emittances.

At the same time, if I check the determinant of the Jacobian of the 
nonlinear transfer map, it stays constant and equal to 1 anywhere in the 
area of interest, which means that the 6d emittance does not change.

So, the conclusion I can draw is that the 6d phase space volume stays 
constant in the drift, but the approximation based on the second moment 
matrix is not valid if nonlinearities are taken into account.

Pavel

On 3/16/2011 08:49, Tom Roberts wrote:
> Stimulated by the recent discussion, plus knowledge that ecalc9
> emittances are not conserved, I performed some simple experiments
> simulating emittance. I had not remembered how huge an effect the
> non-conservation of ecalc9 emittance is....
>
> BEAM SIMULATION
> ---------------
> G4beamline simulated a beam of 10,000 mu+, with decays disabled,
> propagating for 10 meters in vacuum with no fields. This is just about
> as "clean" a sample as can be imagined, in a regime similar to our
> cooling channels (moderately relativistic, only moderately paraxial):
> sigmaX = sigmaY = 10 mm (~ size of initial beam)
> sigmaXp = sigmaYp = 0.020 (dx/dz, dy/dz; angular spread of initial beam)
> meanP = 200 MeV/c (total 3-momentum)
> sigmaP = 10 MeV/c (3-momentum spread of initial beam)
> sigmaT = 2 ns (time spread of initial beam)
> beamZ = -0.1 mm (z position of initial beam)
> These variables have UNCORRELATED Gaussian distributions, and all except
> P have zero mean. The reference track is in the +z direction,
> x=y=x'=y'=t=0; all tracks are generated at z=-0.1 mm, and all have
> Weight=1. Three different samples were made at approximately 1-meter
> intervals in the 10-meter drift:
> A) virtualdetector format=FOR009.DAT, for output to ecalc9
> B) virtualdetector format=ascii, for output to EmitA MODE_Z
> C) timentuple format=ascii, for output to EmitA MODE_T
> All three samples were taken in a single simulation run, and all
> emittance calculations below use the same set of 10,000 events.
>
> UNITS: G4beamline/Geant4 units are used throughout: mm, ns, MeV; ecalc9
> values are converted to these units.
>
>
> PLOT 1 - ECALC9
> ---------------
> The first plot is ecalc9 6-d emittance. No sigma-cut was applied, nor
> were any other cuts or corrections used. As you can see, emit6D DOUBLES
> over these 10 meters of drift. From Rick's MuCool note 280, ecalc9 uses
> the following variables and formula:
> x,y,t,Px,Py,Etot
> emit6D = c/Mass/Mass/Mass*sqrt(determinant(covar6D))
> (covar6D is the 6-D covariance matrix for the 6 track variables)
> (c = 299792458 m/s)
> The ecalc9 value of emit6D is multiplied by 1E9 (meters^3 -> mm^3).
> Applying a sigma-cut of 5 changed the values by less than 1%.
>
>
> INTRODUCING EmitA
> -----------------
> I then built an entirely new program, EmitA, intended to serve as a
> platform for testing various definitions of emittance. It is clean and
> well-commented C++ code using the Gnu Scientific Library (GSL) matrix
> routines; it can use either t or z as the independent variable, and will
> be able to handle EM fields and compute eigen-emittances in the near
> future. This code is completely independent from ecalc9; for the above
> beam it outputs exactly the same values as ecalc9, giving confidence
> that both programs compute what they claim to compute.
>
>
> PLOT 2 - EmitA
> --------------
> The second plot shows the EmitA 6-d emittance. In MODE_Z, z is the
> independent variable and the following track variables are used:
> x,Px,y,Py,t,Etot (reference value subtracted from each)
> emit6D = c_light/Mass/Mass/Mass*sqrt(determinant(covar6D))
> (c_light = 299.792458 mm/ns)
> These are the same as ecalc9 except for order (which does not affect the
> computation) -- indeed the computed values are exactly the same as
> ecalc9, to the 4 significant digits printed. Changing the 6th variable
> to -Etot does not affect the values.
>
> In MODE_T, t is the independent variable, and the following track
> variables are used:
> x,Px,y,Py,z,Pz (reference value subtracted from each)
> emit6D = 1.0/Mass/Mass/Mass*sqrt(determinant(covar6D))
> Note that z<2000 cannot be plotted, as some events are generated with t
> greater than the corresponding value (i.e. at times when the reference
> track has z<2000, some events would be at z<-0.1 and thus are not yet in
> the simulated world).
>
>
> OPENING DISCUSSION
> ------------------
> For this physical situation with no fields, the mechanical momenta used
> in both programs are the same as the canonical momenta.
>
> The fact that ecalc9 emittances are not at all conserved in a simple
> drift is worrysome. It's not clear how useful this is for evaluating
> cooling channels, or indeed for accelerator physics in general.
>
> The fact that using t and z as independent variable gives emittances
> differing by a factor of ~2 is also worrysome.
>
> I welcome comments and discussion on these points, and any others that
> come up.
>
>
> Tom Roberts
>
>
>
> _______________________________________________
> MAP-l mailing list
> MAP-l at lists.bnl.gov
> https://lists.bnl.gov/mailman/listinfo/map-l
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