1 Chris Rogers MICE Collaboration Meeting 11th Feb 2005 Tracking and Cooling performance of G4MICE

2 Tracking in G4MICE - Version 1 Summary of MICE note 93 to compare ICOOL and G4MICE: –G4MICE solenoidal field model –G4MICE transport model –G4MICE dE/dx and Multiple Scattering models in LH 2 Need windows and RF

3 Magnetic Field Algorithm 1.Calculate B-field from a sheet model 2.Save B-field to uniform grid 3.Interpolate from grid Alternatively, read in an external map Just need to consider interpolation

4 Sheet Model No analytical solution exists for a thick solenoid Analytical solution does exist for a sheet carrying some current density Model many sheets… in limit of infinite number of sheets carrying a small current, we have a continuous current carrying solenoid Assume that the current density is constant throughout solenoid t t/n t/2n Sheet Current J/n n = number of sheets

5 Number Of Sheets R Radial field B r of a BeamTools solenoid - as a function of r, plotting solenoids constructed from different Nos of sheets - as a function of the number of sheets at r = 180 mm - default no. sheets is 10 Used a solenoid with inner radius = 200 mm, thickness = 100 mm, length = 200 mm, current density = 60 A mm -2 BrBr BrBr r/mm Number sheets

6 Number Of Sheets Z Longitudinal field B z of a BeamTools solenoid - as a function of z, plotting solenoids constructed from different Nos of sheets - as a function of the number of sheets at z = 45 mm Used the same solenoid BzBz r/mm BzBz Number of Sheets

7 Interpolation Algorithm 1.Perform spline fit along z at r 1 and r 2 for B z, B r 2.Take linear interpolation across r z1z1 z2z2 r2r2 r1r1 Use spline fit to get (B z, B r ) at (z,r 1 ) and (z,r 2 )(z,r) z1z1 z2z2 r2r2 r1r1 Use linear interpolation to get (B z, B r ) at (z,r)

8 Interpolation Algorithm - quality Fractional error of interpolated field vs analytical calculation B z - peak error ~ 0.4%B r - peak error ~ -1% Interpolation error - default grid spacing of 10 mm - well within tolerances - e.g. to good approximation p x ~ B z - largest errors are far off axis - r~ 20 cm - large central error in B r is on edge of spline’s validity z/cm r/cm z/cm dBr/Br dBz/Bz

9 Particle Position Go on to compare the tracking with ICOOL Should remember ICOOL is not perfect I intend to run some comparisons against analytical models in the future I used ICOOL’s field algorithm rather than a common map We will go on to compare the downstream positions given certain initial conditions - below, p x (z = 0) = 30 MeV, p z (z = 0) = 200 MeV

10 A word of caution I am using G4MICE’s new virtual plane code –Very much under development –Several known bugs, very little testing –Currently uses a linear interpolation across the step –Accuracy very dependent on step size –Relatively straightforward to improve Virtual plane at z 1 Linear interpolation between beginning of step and end of step Error

11 Grid Size - x Fire particles with different px, track them through a solenoid, examine resulting x and the error on x Default grid spacing 10 mm (yellow) <~ 1 e -2 error on x

12 Grid Size - px Repeat the exercise, this time examine px downstream of the solenoid Default grid spacing 10 mm (yellow) <~ 1 e -4 error on px

13 Step Size - x Repeat the exercise, but now change step size. This time track the particles through the entire MICE lattice. Again, yellow is default step size (40 mm) Maximum error ~ 2% Both simulations use same grid spacing.

14 Step Size - px Repeat the exercise, but now change step size. This time track the particles through the entire MICE lattice. Again, yellow is default step size (40 mm) Maximum error on px ~ 1%

15 LH 2 Absorbers Use Cylindrical absorber –Thickness 350 mm –No windows Start with a 10,000 event sample –Pz 200 MeV, Px = Py = 0 –No B-Field Use Restricted Bethe-Bloch with Density effect & Vavilov distribution (i.e. best simulation)

16 Longitudinal Effects - Energy Energy distribution well known and both packages give very similar distributions –Variance 1.05 (G4MICE), 1.11 (ICOOL) /MeV 2 Red - ICOOL Blue - G4MICE

17 Longitudinal Effects - Time Time distribution is less well known, and the packages give quite different results –Variance(t) G4MICE 1.42e-6, ICOOL 0.52e-6 /ns 2 –Covariance(E,t) G4MICE, ICOOL /MeV ns Red - ICOOL Blue - G4MICE Left G4MICE Right ICOOL tt E

18 Transverse Effects - x & p x Gaussian-like distributions for MSc –Gaussian in distribution centre –ICOOL has more events in tails Pulls out variance –G4MICE V(x) = 7.24 mm 2 ; V(p x ) = 7.52 MeV 2 –ICOOL V(x) = 9.09 mm 2 ; V(p x ) = 8.56 MeV 2 Theoretical & Statistical uncertainty in this regime Red - ICOOL Blue - G4MICE Red - ICOOL Blue - G4MICE

19 Transverse Correlation As expected, strong transverse correlation in MSc –ICOOL: V(x,p x ) = 7.51 –G4MICE: V(x,p x ) = 6.14 G4MICEICOOL

20 Covariance Matrices (for reference) ICOOL e G4MICE e e e e e e e e e

21 Emittance in Absorbers Fire various emittance beams through absorbers –Work with matched beams in constant 4T B z –Examine change in emittance, change in beta function, as a function of distance through absorber –Examine longitudinal emittance and transverse emittance separately assume no longitudinal-transverse coupling

22 Emittance through the absorber D(Emittance) vs Emittance Longitudinal See broadly similar behaviour between the two simulations  (ICOOL) -  (G4MICE) < 0.5% transverse  (ICOOL) -  (G4MICE) < 0.3% longitudinal Significant discrepancy in equilibrium emittance (beta = 320 mm) Transverse Vary  (t) Vary  (E) Red ICOOL Blue G4MICE Green G4MICE Torispherical windows Red & Yellow ICOOL Blue & Blue G4MICE

23 Emittance Performance - magnets only (1 mm) ICOOL G4MICE Systematic difference ~ 0.2 % Some idea of reasons 1 mm step size, ~ 1000 events These plots need work - to come later with full cooling analysis

24 Cooling Performance - absorbers and electrostatic fields Hopefully will examine dE/dx model and multiple scattering in the future For now, only show cooling plots - at 2.5  and 5.5  emittance Still need to understand these ICOOL has no windows, G4MICE has windows Emittance growth in rf/equilibrium emittance beam? 1 mm step size, ~ 800 events ? ICOOL G4MICE ICOOL G4MICE

25 Summary - Default values B-Field –B r ~ 1%; B z ~ 0.4% Grid Spacing –X ~ 1%; Px ~ 0.01% Step Size –X ~ 2%; Px ~ 1% Single Particle through absorber –  (E) ~ 1%,  (t) ~ 25%;  (px) ~ 10% –(But time spread is negligible factor anyway) Bunch through absorber –  ~ 0.5% transverse, 0.3% longitudinal