1. FIGURE OF MERIT FOR MUON IONIZATION COOLING Ulisse Bravar University of Oxford 28 July 2004

2. 100 m cooling channel Channel structure from Study II Cooling: d   / dx =    +   equil. / Goal: 4-D cooling. Reduce transverse emittance from initial value   to   equil. Accurate definition and precise measurement of emittance not that important

3. MICE Goal: measure small effect with high precision, i.e.     ~ 10% to 10 -3 Full MICE (LH + RF) Empty MICE (no LH, RF) Software: ecalc9f     does not stay constant in empty channel

4. The MICE experiment Measure a change in e4 with an accuracy of 10-3. Measurement must be precise !!!  Incoming muon beam Diffusers 1&2 Beam PID TOF 0 Cherenkov TOF 1 Trackers 1 & 2 measurement of emittance in and out Liquid Hydrogen absorbers 1,2,3 Downstream particle ID: TOF 2 Cherenkov Calorimeter RF cavities 1RF cavities 2 Spectrometer solenoid 1 Matching coils 1&2 Focus coils 1 Spectrometer solenoid 2 Coupling Coils 1&2 Focus coils 2Focus coils 3 Matching coils 1&2 The MICE experiment

5. Quantities to be measured in MICE equilibrium emittance = 2.5  mm rad cooling effect at nominal input emittance ~10% Acceptance: beam of 5 cm and 120 mrad rms

6. Emittance measurement Each spectrometer measures 6 parameters per particle x y t x’ = dx/dz = P x /P z y’ = dy/dz = P y /P z t’ = dt/dz =E/P z Determines, for an ensemble (sample) of N particles, the moments: Averages etc… Second moments: variance(x)  x 2 = 2 > etc… covariance(x)  xy = > Covariance matrix M = M = Evaluate emittance with: Compare  in with  out Getting to e.g.  x’t’ is essentially impossible is essentially impossible with multiparticle bunch with multiparticle bunch measurements measurements

7. Emittance in MICE (1) Trace space emittance:  tr ~ sqrt ( ) (actually,  tr comes from the determinant of the 4x4 covariance matrix) Cooling in RF Heating in LH Not good !!!

8. Emittance in MICE (2) Normalised emittance (the quantity from ecalc9f):   ~ sqrt ( ) (again, from the determinant of the 4x4 covariance matrix) Normalised trace space emittance  tr,norm ~ ( /m  c) sqrt ( ) The two definitions are equivalent only when  pz = 0 (Gruber 2003) !!! Expect large spread in p z in cooling channel

9. Muon counting in MICE Alternative technique to measure cooling: a)fix 4-D phase space volume b)count number of muons inside that volume Solid lines number of muons in x-p x space increases in MICE Dashed lines number of muons in x-x’ space decreases Use x-p x space !!!

10. Emittance in drift (1) Problem: Normalised emittance increases in drift (e.g. Gallardo 2004) Trace space emittance stays constant in drift (Floettmann 2003)

11. Emittance in drift (2) x-p x correlation builds up: initial final Emittance increase can be contained by introducing appropriate x-p x correlation in initial beam

12. Emittance in drift (3) Normalised emittance in drift stays constant if we measure   at fixed time, not fixed z For constant , we need linear eqn. of motion: a)normalised emittance: x 2 = x 1 + t dx/dt = x 1 + t p x /m b)trace space emittance: x 2 = x 1 + z dx/dz = x 1 + z x’ Fixed t not very useful or practical !!!

13. Solenoidal field Quasi-solenoidal magnetic field: B z = 4 T within 1% Initial   within 1% of nominal value   fluctuates by less than 1 %

14. Emittance in a solenoid (1) Normalised 4x4 emittance – ecalc9f Normalised 2x2 emittance Normalised 4x4 trace space emittance Normalised 2x2 emittance with canonical angular momentum

15. Muon counting in a solenoid In a solenoid, things stay more or less constant This is 100% true in 4-D x-p x phase space solid lines Approximately true in 4-D x-x’ trace space dashed lines

16. Emittance in a solenoid (2) Use of canonical angular momentum: p x p x + eA x /c, A x = vector potential to calculate   Advantages: a)Correlation  x,y’ =  1,4 << 1 b)2-D emittance  xx’ ~ constant Note: Numerically, this is the same as subtracting the canonical angular momentum L introduced by the solenoidal fringe field Usually  x,y’ =  1,4 in 4x4 covariance matrix takes care of this 2 nd order correlation We may want to study 2-D  x and  y separately… see next page !!!

17. MICE beam from ISIS Beam in upstream spectrometer Beam after Pb scatterer x yy

18. How to measure   (1) Standard MICE MICE with LH but no RF Mismatch in downstream spectrometer We are measuring something different from the beam that we are cooling !!!

19. How to measure   (2) Spectrometers close to MICE cooling channel Spectrometers far from MICE cooling channel with pseudo-drift space in between If spectrometers are too far apart, we are again measuring something different from the beam that we are cooling !!!   increase in “drift”

20. Quick fix: x – p x correlation Close spectrometers Far spectrometers with x-p x correlation

21. Gaussian beam profiles Real beams are non-gaussian Gaussian beams may become non-gaussian along the cooling channel When calculating   from 4x4 covariance matrix, non-gaussian beams result in   increase Can improve emittance measurement by determining the 4-D phase space volume In the case of MICE, may not be possible to achieve 10 -3 Cooling that results in twisted phase space distributions is not very useful

22. Conclusions Use normalised emittance x-p x as figure of merit Accept increase in   in drift space Consider using 2-D emittance with canonical angular momentum Make sure that the measured beam and the cooled beam are the same thing Do measure 4-D phase space volume of beam, but do not use as figure of merit