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Emittance definition and MICE staging

Emittance definition and MICE staging. U. Bravar Univ. of Oxford 1 Apr. 2004 Topics: a) Figure of merit for MICE b) Performance of MICE stages. Figure of merit for MICE. MICE is designed to measure something to 10 -3 We have yet to decide what…

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Emittance definition and MICE staging

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  1. Emittance definition and MICE staging U. Bravar Univ. of Oxford 1 Apr. 2004 • Topics: a) Figure of merit for MICE b) Performance of MICE stages

  2. Figure of merit for MICE • MICE is designed to measure something to 10-3 • We have yet to decide what… • Current quantity: normalised transverse emittance e^ as calculated by ecalc9f.for (G. Penn, 2001) from the 4x4 covariance matrix • 4 definitions of rms emittance (K. Floettmann, 2003): i) normalised emittance ii) trace space emittance iii) geometric emittance iv) normalised trace space emittance • Past wisdom (prior to Abingdon meeting): normalised emittance e^ = sqrt (<x2><px2>) is Liouville compliant

  3. Rms emittance and cooling • Rms emittance, four definitions (K. Floettmann, 2003): a) normalised emittance en calculated from <x2>, <(px)2> b) beam emittance ebeam = en / <pz> c) trace space emittance etr calculated from <x2>, <(x’)2> d) normalised trace space emittance en,tr = <pz> etr • MICE cooling measured by decrease in 4-D transv. en: De^ / e^ = (e^,in–- e^,out) / e^,in where e^ is obtained from the 4X4 covariance matrix.

  4. Emittance in MICE • e^ from ecalc9f.for • Full MICE (LH + RF) • Empty MICE (no LH, RF) • Emittance is not constant in empty channel • Does Liouville’s theorem still apply? • Implications for MICE?

  5. Emittance conservation • J. Gallardo (2004) showed that 6-D normalised emittance calculated by ecalc9f.for is not constant in drift! • K. Floettmann’s paper (2003): 4-D trace space emittance e^,tr = sqrt {<x2><(x’)2>} stays constant in drift! • What does this mean?

  6. Solution • Liouville’s theorem: df/dt = 0 • Where: f(qj,pj;t) = state function; pj = canonical conjugate momenta; t = independent variable. • Calculate emittance at fixed t, not z! • Use correct conjugate momenta: in drift: x, px, not x, x’ in B-field x, px+eAx/c • Calculate 6-D emittance! • e^ and eL are constant only if transverse and longitudinal motions are completely decoupled. • This is not the case in MICE! • ecalc9f.for calculates e^, eL, e6 at fixed z; variables are x, y, t, px, py, E. • These emittances are not constant in drift or in an empty channel!

  7. Simple example • Drift space. • Non-relativistic beam • Initial spread: sp / p = 10%. • Top: 6-D en at fixed t. • Bottom: 6-D en at fixed z. • This true in general; theoretical proof available; simulations in progress. • Questions: a) can we find en at fixed t? b) is it useful for MICE?

  8. Emittance measurement • We now have a quantity that is Liouville-compliant. • Can we measure it to 10-3? • Beam rms: sx = 3.3 cm, spx = 20 MeV/c • Tracker resolution: dx << sx • Relative error on sx@ 0.5 (dx / sx )2 • Relative error on 6-D emittance e6@ 1.2 (dx / sx )2 = 10-3 • Therefore dx@ 0.03 sx • In other words, we need dx@ 0.01 cm, dpx@ 0.5 MeV/c • Current resolutions from SciFi: dx@ 0.05 cm, dpx@ 2.5 MeV/c. Way too big!!! • Plus, statistical error on sx = 1/sqrt (#m) = 10-3, so #m = 106 • Things get better if we want to measure 4-D transverse emittance • Worse if we want to measure cooling De/e @ 10% • Question: is this the right figure of merit for MICE? • We may calculate rms emittance… but we can’t deal with more advanced stuff. • THESE FIGURES ARE PRELIMINARY. WORK IN PROGRESS!!!

  9. Alternative: event counting • Procedure: a) count number of muons in 2-D, 4-D or 6-D phase-space ellipsoids; b) show that #(m+) increases from the upstream tracker to the downstream tracker. • Problems: a) need to determine en first; b) particle ID prior to upstream spectrometer. • Advantages: a) straightforward; b) this is the quantity that matters in a real cooling channel.

  10. Alternative: muon counting • i.e. measurement of the increase in phase space density • MICE channel, ecalc9f.for, 4-D ellipsoid • Initial e^ = 6000 mm mrad 10,000 events. • Full MICE channel (LH & RF). • Empty channel (no LH, no RF) & same beam. • Empty channel (no LH, no RF) & different beam. • Still, need to compute e6 prior to counting. • Same old problem: quantity does not stay constant in empty channel!!!

  11. Gaussian beam profiles • Real beams are non-gaussian • Gaussian input beams may become non-gaussian along the MICE channel (see e.g. study on magnet alignment tolerances) • When calculating e from 4x4 matrix with 2nd order moments, non-gaussian beams result in e increase. • Can improve emittance computations and measurement of phase space volume. May not be possible to achieve 10-3. • However, cooling that results in twisted phase space volume is not very useful. • May need new figure of merit, in addition to e, something to measure Gaussian shape of the beam. • e.g. use 3rd order moments to measure skewness of beam

  12. MICE stages

  13. Questions • Beam optics: can we use the same cooling channel solutions in all six stages? • Can we do all the physics of MICE with stages IV or V?

  14. Beam optics • Software provided by Bob Palmer. • Based on ICOOL. • Tuning of coil currents assumes: a) ideal beam; b) long channel, 100 m; c) empty channel; d) constant momentum pz = 200 MeV/c. Note: potential problem with stay-clear area in match coil.

  15. Actual MICE channel • When running MICE stages IV, V and VI, things are different. • Stage VI (full MICE) • Stage V (2 LH + 1 RF) • Stage IV (1 LH + 0 RF) • Two problems: a) b^ is not minimum in the centre of LH absorbers; b) b^ is not flat in downstream spectrometer. • Fine tuning of coil currents necessary! Work in progress.

  16. b^ = 42 cm in the centre of LH • We get maximum cooling when this is true! • Cooling formula: equilibrium emittance en,equilibrium = kb^ • May do some tuning of focus coil currents. • Unlikely solutions: a) move LH; b) additional match coils.

  17. Performance of stages IV, V and VI • Stage VI (full MICE) • Stage V (2 LH + 1 RF) • Stage IV (1 LH + 0 RF) • Input emittances: e^in@ 3,000; 6,000; 9,000 & 12,000 mm mrad. • Start with 10,000 muons. Count number of muons that are left as a function of z along the MICE channel. • Note: z = 0 in the middle of the upstream spectrometer. ICOOL runs all the way to the middle of the down-stream spectrometer, to z = 4.10 m (Stage IV), 6.85 m (Stage V) & 9.60 m (Stage VI).

  18. Cooling • Stage VI (full MICE) • Stage V (2 LH + 1 RF) • Stage IV (1 LH + 0 RF) • Input emittances: e^in@ 3,000; 6,000; 9,000 & 12,000 mm mrad. m+ beam, <pz> @ 200 MeV/c. • Note: fluctuations due to tracker resolution are not included.

  19. e^in = 6,000 mm mrad • Stage VI (full MICE) • Stage V (2 LH + 1 RF) • Stage IV (1 LH + 0 RF) • Questions: why is • e^in >> e^in ? • Shouldn’t e^in be @ 6,000 mm mrad in all cases?

  20. Summary of results 1) At all z locations, only muon tracks that make it all the way to the downstream spectrometer are used to calculate e^. 2) Transmission = number of muons that reach the middle of the downstream spectrometer, out of 10,000 initial. Muon decay is disabled. 3) De^ / e^ = (e^upstream – e^downstream) / e^upstream = cooling; e^ is measured in the centre of the upstream and downstream spectrometers.

  21. Statistical errors • Question: what are the errors in the table on the previous page? • Answer: the figure shows De^/e^ (y-axis) vs. transmission (x-axis) for Stage VI. • Each point is a different ICOOL run with a different random beam. • Total of 20 input beams, each beam contains 10,000 events. • All beams are Gaussian, e^in@ 6000 mm mrad. x-axis, actually shows #(m+) lost in the MICE channel, i.e. Transmission = 10,000 – #(m+). • Standard full MICE. • MICE channel with LH only, no Al and Be windows. • In short: De/e @ 10% • Errors @ 0.5% statistic + 0.5% systematic.

  22. MICE stage VI • MICE stage IV can prove ionisation cooling, but cannot prove the feasibility of a long cooling channel, since it has no RF. • MICE stage V can demonstrate the feasibility of a muon ionisation cooling channel. • MICE stage VI: a) central absorber: much more representative of real channel: i) beam optics same as in long channel; ii) e.g. determine cooling at b^ = 42 cm. b) in addition to flip and no-flip modes, can run in semi-flip; c) represents one full flip element. d) ….. • Does all this justify stage VI?

  23. Beta functions – flip mode b^ = 42 cm b^ = 25 cm b^ = 17 cm b^ = 7 cm

  24. Conclusions • For the time being, use ecalc9f.for and e^ as the figure of merit for MICE. • Measure eout, notDe/e, to 10-3 absolute! • Need to reach consensus on appropriate figure of merit. URGENT!!! • Fine tuning of MICE channel optics is necessary. • Need solutions for all stages of MICE, including stage I. • Study additional channel configurations, no-flip & semi-flip, additional absorbers. • Investigate all advantages of having stage VI.

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