1. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 1 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 1 st 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

2. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 2 -25 -20 -15 -10 -5 0 5 10 15 20 25 0123456789101112131415 Half width x / cm Half width 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%

3. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 3 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

4. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 4 Beam line parameters table from Kevin Kevin’s dataTrace space transfer matrix approximation ElementPosition Effective Length Field Strength s k = (e/p)*dB/dx [p=(250–11–3)~235MeV] Omega (phase advance) = s * Sqrt Mag k mmT/mmm -2 TOF0 centre20.8116 Drift 24.9637 – 20.8116 – 0.33 = 3.8221 Drift Space20.8624 CKOV121.0624 Drift Space21.5674 Q35 Qd - Q724.9637 0.66 0.88758 QD0.661.1330.748 Drift Space25.6237 Drift26.1237 – 24.9637 – 0.66 = 0.5 Q35 Qd - Q826.1237 0.66 -1.34275 QF0.66-1.7141.131 Drift Space26.7837 Drift27.2837 – 26.1237 – 0.66 = 0.5 Q35 Qd - Q927.2837 0.66 1.14749 QD0.661.4640.966 Drift Space.27.9437 Drift 28.8437 – 27.2837 – 0.33 = 1.23 TOF1 centre 28.8437 Q35 dimensions: Pole tip radius (the radial distance between the central axis of the quadrupole and its pole tip) 17.82 cm Vertical ½ aperture 23.6 cm, Horizontal ½ aperture 23.6 cm

5. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 5 Mathematica output: trace space transfer matrices

6. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 6 Gaussian beam 7.5mm MC truth G4MICE detector simulation of TOF hits with x’ reconstructed using MC transfer matrix TOF 0TOF 1 x / m x’ x / m x’ x / m x’ x / m x’

7. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 7 Extra slides

8. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 8 TOF 0  TOF 1 PDG calculations

9. Mark Rayner 14/8/08Analysis Meeting: Emittance measurement using the TOFs 9 Transverse covariance matrix before TOF 0