[MAP] Emittance simulation experiments

[Tom Roberts]

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
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