Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Emittance measurement using the TOFs• The question: can we use position measurements from two TOFs to measure

transverse emittance?– Question: x-emittance, y-emittance, transverse emittance, and the trace space

(geometric) or phase space (normalized) definition?• Supplementary question: If muons come from pion decays in the PSI solenoid

where there is a strong magnetic vector potential, will the muon beam have angular momentum? If so, then 1-d emittance is not conserved. Ignore today.

– Alpha, beta, gamma also interesting. Anything else?• Proposed method

– Step 1: Do a simulation to determine the transfer matrix between the TOFs using MC truth

• Caveat: Equivalent to relying on a 1st order Taylor expansion. Cannot express the increase in emittance and decrease in energy expected in the Cherenkov

– Step 2: Measure the positions in both TOFs• Error = slab width / root 12 ~ 2 cm

– Step 3: Deduce the conjugate momentum in both planes using the simulated transfer matrix and the measured positions

– Step 4: Get whichever optical beam properties you like in the plane of either TOF from the new phase space distributions

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ha

lf w

idth

x /

cm

Ha

lf w

idth

y /

cm

TOF0 Diffuser

norm. em. 7.1 mm after the diffuser

z / m

Clues on the probable beam just before TOF 0• Kevin’s assumptions at the target

– Pions have mean Pz 444 MeV/c– Each variable is assume to have a

top hat distribution due to scraping

• x 5.1 mm, x’ 0.033• y 2.0 mm, y’ 0.014• Pz 2.5%

– Could use G4MICE to figure out the muon optical functions

– Haven’t done this yet• Average muon momentum / MeV?

– Tune dipoles for 208.58 after diffuser– 222.87 before diffuser– 250 before TOF0

• -11 in each TOF• -3 in the Cherenkov• -2 in the 8 m air

• CM15 Transport half width plot

– Cov x’x’ = cov xx * (beta/Pz)2

– Marco: beta before diffuser 83 cm• (Half width)2 / beta is constant• Beta x TOF0 190 cm• Beta y TOF0 332 cm

– Gradients ~ 0 so alphas ~ 0• Kevin’s muon beam assumption

– dp/p ~ 10%

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Use G4MICE to simulate the entire process• Which beam?

– Gaussian/Top-hat (?) muons starting just before TOF0 in the Stage 6 setup• Switch to real setup when choice/position of two detectors decided

– Initial mean phase space vector: Pz 250 MeV/c, otherwise zero– Initial moments matrix: L = 0, alphas = 0, beta x = 1.9 m, beta y = 3.3 m, RMS Pz = 10% [Kevin]

• Sigma({x, y}) = {28, 37} (mm)½ * Sqrt( normalized emittance )– Will fill the TOFs [Sigma(y) = 200 mm] at 30 mm normalized emittance– Will only fill a single slab width [Sigma(y) = 20 mm] at 0.3 mm normalized emittance

– Vary initial normalized emittance: 0.5mm, 1mm, 2.5mm, 5mm, 7.5mm, 10mm, 15mm, 20mm, 30mm, 40mm

• Trial analysis– Step 1: Use MC truth to get the transfer matrix

• Problem: G4MICE step length only approximates path through quadrupoles leading to a spread in the G4MICE calculated transfer matrix elements even in a linear channel. Use mean of these and compare to pen and paper prediction.

– Step 2: Use G4MICE digitization of the simulated detector response for the positions– Step 3: Solve for conjugate momenta as before

• Caveat: Using the same simulation for Step 1 and Step 2 is cheating! Errors in e.g. G4MICE quadrupole model won’t show up

– Step 4: Get the optical beam parameters by fitting bivariate Gaussians and also using standard statistical formulae

• Compare with same analysis of truth data

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Beam line parameters table from Kevin

Kevin’s data Trace space transfer matrix approximation

Element PositionEffective Length

Field Strength

sk = (e/p)*dB/dx

[p=(250–11–3)~235MeV] Omega (phase advance)

= s * Sqrt Mag k

m m T/m m m-2

TOF0 centre 20.8116

Drift24.9637 – 20.8116 – 0.33

= 3.8221

Drift Space 20.8624

CKOV1 21.0624

Drift Space 21.5674

Q35 Qd - Q7 24.9637 0.66 0.88758 QD 0.66 1.133 0.748

Drift Space 25.6237 Drift 26.1237 – 24.9637 – 0.66 = 0.5

Q35 Qd - Q8 26.1237 0.66 -1.34275 QF 0.66 -1.714 1.131

Drift Space 26.7837 Drift 27.2837 – 26.1237 – 0.66 = 0.5

Q35 Qd - Q9 27.2837 0.66 1.14749 QD 0.66 1.464 0.966

Drift Space. 27.9437Drift

28.8437 – 27.2837 – 0.33

= 1.23TOF1 centre 28.8437

Q35 dimensions: Pole tip radius (the radial distance between the central axis of the quadrupole and its pole tip) 17.82 cmVertical ½ aperture 23.6 cm, Horizontal ½ aperture 23.6 cm

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Mathematica output: trace space transfer matrices

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Gaussian beam 7.5mmMC truth G4MICE detector simulation of TOF hits with x’ reconstructed using MC transfer matrix

TOF 0 TOF 1x / m

x’

x / m

x’

x / m

x’

x / m

x’

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Extra slides

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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TOF 0 TOF 1 PDG calculations

Energy and momentumdetails thickness density dE/dx (min I) dE mass E before p before E after p after dp

cm g cm-3 MeV g-1 cm2 MeV MeV c-2 MeV MeV MeV MeV MeVTOF0 scintillator polyvinyltoluene 5 1.03 1.97 10.12 105.66 271 249.5535 260.88 238.5253 11.02819Ckov aerogel silica aerogel 8 0.2 1.74 2.78 105.66 260.88 238.5253 258.1 235.4816 3.043762Air air dry, 1 atm 730 1.20E-03 1.82 1.6 105.66 258.1 235.4816 256.5 233.7268 1.754785TOF1 scintillator polyvinyltoluene 5 1.03 1.97 10.12 105.66 256.5 233.7268 246.38 222.5737 11.15306

Lorentz and timeE average p average beta gamma dtMeV MeV microseconds

TOF0 265.94 244.0394361 0.917648477 6.3322692 1.816236508Ckov 259.49 237.0034609 0.913343331 6.031219 2.91967607Air 257.3 234.6041871 0.91179241 5.9299811 266.8736115TOF1 251.44 228.1502661 0.907374587 5.660227 1.83680113

ScatteringX0 X0 RMS theta RMS thetag cm-2 cm mrad degrees

TOF0 43.9 42.62135922 27.02095823 1.5481882Ckov 27.25 136.25 19.21043782 1.1006779Air 36.62 30516.66667 11.9337395 0.6837535TOF1 43.9 42.62135922 29.23004715 1.6747598

Mark Rayner 14/8/08 Analysis Meeting: Emittance measurement using the TOFs

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Transverse covariance matrix before TOF 0

mass Pz beta x beta yMeV MeV mm mm

105.66 250 1900 3300

Norm em Cov xx Cov PxPx Cov yy Cov PyPymm (mm)2 (MeV)2 (mm)2 (MeV)2

0.5 401.508 6.951316 697.356 4.0022731 803.016 13.90263 1394.712 8.0045452 1606.032 27.80526 2789.424 16.009095 4015.08 69.51316 6973.56 40.02273

10 8030.16 139.0263 13947.12 80.0454520 16060.32 278.0526 27894.24 160.090930 24090.48 417.0789 41841.36 240.136440 32120.64 556.1053 55788.48 320.1818