[MAP] Emittance simulation experiments

Tom Roberts tjrob at fnal.gov
Wed Mar 16 14:47:50 EDT 2011


Kirk:

Inspired by your remarks, I re-ran the simulation, with the only change being 
sigmaP=0.1 (MeV/c). So this beam has dp/p=0.05% RMS (the original beam had 
dp/p=5% RMS). The attached plot shows that emit6D is now constant (better than 
0.1%), but there's still the question why using z and t as independent variables 
give such different answers.

All of the things you mention are on my list of things I intend to investigate. 
I specifically designed EmitA to easily accommodate sub-emittances, vector 
potential, and eigen-emittances (and any other variations).

Unfortunately, PAC11 and the SBIR proposal deadline the following week will 
greatly limit my ability to work on this for the next 3 weeks.

Tom Roberts


On 3/16/11 3/16/11 - 12:47 PM, Kirk T McDonald wrote:
> 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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