[MAP] Emittance simulation experiments

[Tom Roberts]

On 3/16/11 3/16/11 - 3:11 PM, Kirk T McDonald wrote:
> You latest computation shows that the "nonlinearity" associated with
> transport in zero field is related to nonzero momentum/energy spread.

Yes.

> In MICE, where the energy spread will be large, this effect will be large,
> and care will be needed to see "cooling" as a small correction to the growth
> of RMS emittance in the absence of cooling.
>
> The SciFi trackers of MICE are inside solenoids, which hopefully suppresses
> the apparent emittance growth, particularly when the vector potential is
> included in the calculation.   Good field maps will be needed to do a good
> job on this.

Yes to all that. Not only for MICE, but for any cooling channel.

The reason solenoids reduce this effect is that in this momentum range, 
higher-speed particles have longer path lengths (on average); I think that a 
quadrupole channel (without skew quads) would make it worse (higher-speed 
particles have shorter path lengths on average). Indeed, one can design a 
helical-solenoid channel to be isochronous over a reasonably large momentum 
spread; eventually I'll try one of them, too.


> --------------------
> The (E,t) emittance has different units than the (p_z,z) emittance.

Not im EcalcA - refer to the equations in my initial email (quoted far below). 
When using {x,Px,y,Py,t,E} there is a factor of c_light in front, which is 
replaced by 1.0 for the variables {x,Px,y,Py,z,Pz}. The factor 1/Mass^3 cancels 
the (MeV/c)^3 from the Px*Py*Pz in the sqrt(determinant), resulting in units 
(mm^3 rad^3), the same for both variable sets.


> In particular, 1 ns ~ 30 cm, so even if you shifted a decimal point, there
> is likely a factor of 3 lurking.

In one case I saw a factor of 2, in another, a factor of 2.6. I did not "shift 
any decimal points", I computed with the equations given below.


> A smaller effect is that the particles are not fully relativistic, so
> with p_z ~ 200 MeV/c, you have E ~ 225 MeV.

Yes. I will run again with 100 GeV muons, so momentum variance in the beam will 
not significantly vary velocity.


BTW I'm not too concerned with details of the vector potential inside RF 
cavities, as I don't think we need to compute emittance there. But we definitely 
do need to understand the effects of solenoid magnetic fields, and probably 
helical dipoles/quadrupoles as well (which complicate things considerably).


Tom Roberts

>
> --Kirk
>
> ---Original Message-----
> From: Tom Roberts
> Sent: Wednesday, March 16, 2011 2:47 PM
> To: Kirk T McDonald
> Cc: MAP List ; Alex Dragt
> Subject: Re: [MAP] Emittance simulation experiments
>
> 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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