Chris Rogers, Analysis Parallel, MICE CM17 Progress in Cooling Channel Simulation

Overview Aim is to simulate and understand errors on tracking in G4MICE to level of much less than detector resolution Solenoid modelling accuracy RF field maps RF field map modelling accuracy Absorber window GEANT4 parameters? Physics material model (GEANT4.8.2) Then examine G4MICE 6D cooling performance (still in progress) Match beam using full 6D Linear Beam optics package for RF, Quads, Solenoids Longitudinal matching & optics (still no amplitude momentum correlation) Mostly updated material, some new stuff

Solenoid Tracking Accuracy Reminder: Solenoid modelled using infinitely thin concentric current carrying sheets Analytical solution written to a grid During simulation, field at some point is taken by interpolation from the grid Dominant error is grid spacing Plot grid spacing vs tracking error through MICE Assume high precision for a very fine grid z spacing r spacing z spacing r spacing  (r)  (pt)

RF Modelling Implemented RF field maps Read in a file generated by SuperFish Thanks to Rick Fernow for providing SuperFish files

RF Tracking Accuracy Look at grid size vs tracking accuracy Dominant error is in  (r) (!) Even a very fine map introduces noticeable errors (compare with detector resolution ~ 0.3 mm)

Multiple Scattering G4MICE MSc model (GEANT 4.8.2) Points are ICOOL Fano model Shown to agree well with ELMS/MuScat Curves are Moliere model Histo is G4MICE LH 2 Al Be

dEdX Bethe Bloch curve has random looking fluctuations (dashed line is calculated Bethe Bloch) Each point represents 1e4 muons so not statistical Also note the energy straggling curve (200 MeV/c muon) Points ICOOL Curve is Landau fit - Vavilov is a better model Histo is G4MICE

Window Model Implemented arbitrary shaped windows (polycones) Plots are of actual track hits in the simulation Left is hits in absorber window Right is thickness of absorber window vs r G4 implements z vs r_inner, r_outer so where dz/dr is small, interpolation will be less accurate

GEANT4 Parameters Started studying G4 Parameters Delta intersection gives error on finding volume boundaries Delta one step gives stepping error Max step size gives step size Some strange behaviour (no plots) Delta one step seems to control the chance that a track is stepped “accurately” Max step size seems to control the actual stepping accuracy More work needed here

Beam Heating from Window Polycone (realistic) window Cylindrical window Points show beam heating (change in SPE) vs radius for two models - cylindrical window and “realistic” window Windows have similar heating in centre (a little thinner) But realistic window shows slightly more heating on the edge Note the different scales Surprising how modest the emittance growth is (5x thickness) Histogram shows number of muons vs radius (6 pi beam)

Optics Updates Updated optics package to do full 6D integration of first order transfer map Several applications for this in mind Automatic beam matching Fast estimation of input beam quality / cooling performance Calculation of particle amplitudes w/o requiring a beam (Also interesting to understand physics of the beam optics) Started working on second order map Still WIP

Physics Principle Take hamiltonian H Expand as a Taylor series in u, H=H 0 +H 1 + H 2 +H 3 +H 4 +… Use equations of motion du i /dz = [H,u i ] where u i is an element of the phase space vector u and [f,g] is the poisson bracket q i are position variables, p i are momentum variables Then use equation of motion dM 2 /dz = [H 2,M 2 (z)] to integrate M 2 Equivalent to dM 2 /dz = M M 2 where M is M // = M perp = K =K c +B 2 /4p RF focussing

Transfer Map Determinant A convenient test for the correct integration is to check that the transfer map has determinant 1 This means emittance is conserved Required for all first order transfer maps Converges on 1 as numerical integration precision increases

Transverse beta Beta function with magnets only Beta function with full EM & materials

Longitudinal beta Linear calculation (full line) Longitudinal beta with small transverse emittance RF field map (dashed) vs pill box (dotted) Longitudinal beta with 6 mm rad transverse emittance Need to look at amplitude momentum correlation here What do we do at 90 o where non-linearities are much worse?  perp 6 mm rad

Resonances - Tr(M) Trace of transfer map When trace > 4, resonance Full line is 2.75 m lattice; dashed line is transfer map from to m; dotted line is transfer map from m to m What does “resonance” mean for a non-periodic beta function?

Resonances - Tracking Repeating 10.4 m latticeRepeating 2.75 m lattice Resonance structure vs momentum Good agreement with Tr(M) from previous slide Much reduced transmission on the resonances

Conclusions Next step is to look at cooling performance in the knowledge that the simulation is accurate to a high precision Paves the way for the data challenge Really starting to have a tool worthy of the experiment in G4MICE Understanding the cooling channel to some precision Need to worry about amplitude momentum correlation and longitudinal emittance growth