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Chris Rogers

MICE Collaboration Meeting

11th Feb 2005

Tracking and Cooling performance of G4MICE

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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 LH2

• Need windows and RF

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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

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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

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Number Of Sheets R

Radial field Br 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 10Used a solenoid with inner radius = 200 mm, thickness = 100 mm , length = 200 mm, current density = 60 A mm-2

Br Br

r/mm Number sheets

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Number Of Sheets Z

Longitudinal field Bz 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 mmUsed the same solenoid

Bz

r/mm

Bz

Number of Sheets

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Interpolation Algorithm1. Perform spline fit along z at r1 and r2 for Bz, Br

2. Take linear interpolation across r

z1 z2

r2

r1

Use spline fit to get (Bz, Br) at (z,r1) and (z,r2)(z,r)

z1 z2

r2

r1

Use linear interpolation to get (Bz, Br) at (z,r)

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Interpolation Algorithm - qualityFractional error of interpolated field vs analytical calculation

Bz - peak error ~ 0.4% Br - peak error ~ -1% Interpolation error - default grid spacing of 10 mm - well within tolerances

- e.g. to good approximation px ~ Bz

- largest errors are far off axis - r~ 20 cm - large central error in Br is on edge of spline’s validity

z/cm

r/cm r/cm

z/cm

dBr/BrdBz/Bz

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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, px(z = 0) = 30 MeV, pz(z = 0) = 200 MeV

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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 z1

Linear interpolation between beginning of step and end of step

Error

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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

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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

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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.

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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%

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LH2 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)

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Longitudinal Effects - Energy• Energy distribution well known and both packages give

very similar distributions– Variance 1.05 (G4MICE), 1.11 (ICOOL) /MeV2

Red - ICOOLBlue - G4MICE

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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 /ns2

– Covariance(E,t) G4MICE , ICOOL -0.171 /MeV ns

Red - ICOOLBlue - G4MICE

Left G4MICE

Right ICOOL

t t

E

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Transverse Effects - x & px• Gaussian-like distributions for MSc

– Gaussian in distribution centre

– ICOOL has more events in tails• Pulls out variance

– G4MICE V(x) = 7.24 mm2; V(px) = 7.52 MeV2

– ICOOL V(x) = 9.09 mm2; V(px) = 8.56 MeV2

• Theoretical & Statistical uncertainty in this regime

Red - ICOOLBlue - G4MICE

Red - ICOOLBlue - G4MICE

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Transverse Correlation

• As expected, strong transverse correlation in MSc– ICOOL: V(x,px) = 7.51

– G4MICE: V(x,px) = 6.14

G4MICE ICOOL

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Covariance Matrices (for reference)

• ICOOL

2.21256e-06 -0.0010117 0.000284663 0.000329624 0.000184366 0.000649116 -0.0010117 1.03225 -0.0146321 -0.0260422 0.0392652 0.0088467 0.000284663 -0.0146321 9.08972 7.51744 -0.482846 -0.369532 0.000329624 -0.0260422 7.51744 8.55547 -0.388884 -0.135323 0.000184366 0.0392652 -0.482846 -0.388884 10.1381 8.57693 0.000649116 0.0088467 -0.369532 -0.135323 8.57693 9.92383

• G4MICE 1.42203e-06 -0.0010605 1.49702e-05 2.79679e-05 7.87676e-06 2.88317e-05 -0.0010605 1.05356 -0.022305 -0.0304444 -0.00996861 -0.0115705 1.49702e-05 -0.022305 7.24267 6.14538 0.093047 0.0214795 2.79679e-05 -0.0304444 6.14538 7.52094 0.0308814 -0.0602627 7.87676e-06 -0.00996861 0.093047 0.0308814 7.12071 6.05957 2.88317e-05 -0.0115705 0.0214795 -0.0602627 6.05957 7.45958

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Emittance in Absorbers

• Fire various emittance beams through absorbers– Work with matched beams in constant 4T Bz

– 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

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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 ICOOLBlue G4MICEGreen G4MICETorispherical windows

Red & Yellow ICOOL

Blue & Blue G4MICE

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Emittance Performance - magnets only (1 mm)

ICOOLG4MICE

• 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

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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

?

ICOOLG4MICE

ICOOLG4MICE

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Summary - Default values

• B-Field– Br ~ 1%; Bz ~ 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