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   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   = sqrt ( ) is Liouville compliant

3. Rms emittance and cooling Rms emittance, four definitions (K. Floettmann, 2003): a) normalised emittance  n calculated from, b) beam emittance  beam =  n / c) trace space emittance  tr calculated from, d) normalised trace space emittance  n,tr =  tr MICE cooling measured by decrease in 4-D transv.  n :       in –   out   in where   is obtained from the 4 X 4 covariance matrix.

4. Emittance in MICE   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   tr = sqrt { } stays constant in drift! What does this mean?

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

7. Simple example Drift space. Non-relativistic beam Initial spread:  p / p = 10%. Top: 6-D  n at fixed t. Bottom: 6-D  n at fixed z. This true in general; theoretical proof available; simulations in progress. Questions: a) can we find  n 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:  x = 3.3 cm,  px = 20 MeV/c Tracker resolution:  x <<  x Relative error on  x  0.5 (  x /  x ) 2 Relative error on 6-D emittance  6  1.2 (  x /  x ) 2 = 10 -3 Therefore  x  0.03  x In other words, we need  x  0.01 cm,  px  0.5 MeV/c Current resolutions from SciFi:  x  0.05 cm,  px  2.5 MeV/c. Way too big!!! Plus, statistical error on  x = 1/sqrt (#  ) = 10 -3, so #  = 10 6 Things get better if we want to measure 4-D transverse emittance Worse if we want to measure cooling  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 #(  + ) increases from the upstream tracker to the downstream tracker. Problems: a) need to determine  n 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   = 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  6 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 2 nd 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 , something to measure Gaussian shape of the beam. e.g. use 3 rd 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 p z = 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)   is not minimum in the centre of LH absorbers; b)   is not flat in downstream spectrometer. Fine tuning of coil currents necessary! Work in progress.

16.   = 42 cm in the centre of LH We get maximum cooling when this is true! Cooling formula: equilibrium emittance  n,equilibrium =   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:   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:   in  3,000; 6,000; 9,000 & 12,000 mm mrad.  + beam,  200 MeV/c. Note: fluctuations due to tracker resolution are not included.

19.   in = 6,000 mm mrad Stage VI (full MICE) Stage V (2 LH + 1 RF) Stage IV (1 LH + 0 RF) Questions: why is   in >>   in ? Shouldn’t   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  . 2) Transmission = number of muons that reach the middle of the downstream spectrometer, out of 10,000 initial. Muon decay is disabled. 3)   /   = (   upstream –   downstream ) /   upstream = cooling;   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     (y-axis) vs. transmission (x-axis) for Stage VI. i)Each point is a different ICOOL run with a different random beam. ii)Total of 20 input beams, each beam contains 10,000 events. iii)All beams are Gaussian,   in  6000 mm mrad. x-axis, actually shows #(  + ) lost in the MICE channel, i.e. Transmission = 10,000 – #(  + ). Standard full MICE. MICE channel with LH only, no Al and Be windows. In short:  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   = 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   = 42 cm   = 25 cm   = 17 cm   = 7 cm

24. Conclusions For the time being, use ecalc9f.for and   as the figure of merit for MICE. Measure  out, not , 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.