[MAP] Emittance simulation experiments

[Kirk T McDonald]

Tom,

Great job!

I put your .gif files in the web directory
http://puhep1.princeton.edu/~mcdonald/mumu/Roberts/

This confirms past efforts of Juan, Rick and Scott, MuCool note 288.

In practice, we don't use many free-space drifts, as our beams are almost 
always "confined" in a solenoid channel.

If you have the energy, a next numerical step would be to include a constant 
axial magnetic field, B, with vector potential A_phi = B r /2.
Probably, the particles should be "born" in this field.

It would be interesting to see the difference in the rms emittance with and 
without including the vector potential in the momenta.

Hopefully, rms emittance will be more constant when the vector potential is 
included, but I doubt that it will be perfectly constant.

A constant magnetic field contributes nothing to p_z (or p_t = - E_mech + q 
V), so it is of interest to report also the 4-d transverse subemittance and 
the 2-d longitudinal subemittance.

------------
After this, perhaps you could study a spatially varying magnetic field.

One possibility is to chose an analytic form for B_z along the axis, and use 
series expansions for B_z, B_r and A_phi off axis.
http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf

It would be a favor to the target simulation effort to consider a 20-T 
field, say 1 m long around the volume where the beam is "born", followed by 
a Kevin-Paul taper down to 1.5 T over the next 15 m.
See sec. 2.2 of
http://www.hep.princeton.edu/~mcdonald/mumu/target/taper.pdf
with exponent p = 1.   Set d B_z / d z = 0 at the beginning and end of the 
taper.

------------------
Eventually, it would be interesting to pass the beam through an rf cavity. 
A simple cylindrical pillbox cavity would be a good start.

Juan and I are struggling to give forms for the vector and scalar potentials 
for this.   We haven't quite converged, but if people have the energy to 
look, see
http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf

-----------------------------
-----------------------------
A clue as to the rough road ahead is the following.

Consider an ideal toroidal magnet, whose axis is the beam axis.

If the magnetic field is varying, this is a primitive induction linac.

But, just consider a DC field in the toroid.

The beam, of course, passes through the hole in the toroid, and never 
encounters the magnetic field.   The beam is unaffected by the DC toroid.

However, the vector potential is nonzero outside the toroid where the B 
field is zero.    Recall that the line integral of A_tangent around a loop = 
magnetic flux through loop.

So, strictly speaking, we should include this vector potential in the beam 
emittance calculation -- even though the beam is completely unaffected by 
the magnet!

However, I predict that the rms emittance will be different with and without 
the vector potential (even though phase volume is not changed by the 
inclusion of the vector potential).

This highlights the strategic issue: do we or don’t we include the vector 
(and scalar) potential in the emittance calculations.  For time-dependent 
fields there is the additional ambiguity as to which gauge to use for the 
potentials.

--Kirk


-----Original Message----- 
From: Tom Roberts
Sent: Wednesday, March 16, 2011 11:49 AM
To: MAP List
Subject: [MAP] Emittance simulation experiments

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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