Presentation on theme: "FIGURE OF MERIT FOR MUON IONIZATION COOLING Ulisse Bravar University of Oxford 28 July 2004."— Presentation transcript:
FIGURE OF MERIT FOR MUON IONIZATION COOLING Ulisse Bravar University of Oxford 28 July 2004
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
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
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
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
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
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 !!!
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
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 !!!
Emittance in drift (1) Problem: Normalised emittance increases in drift (e.g. Gallardo 2004) Trace space emittance stays constant in drift (Floettmann 2003)
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
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 !!!
Solenoidal field Quasi-solenoidal magnetic field: B z = 4 T within 1% Initial within 1% of nominal value fluctuates by less than 1 %
Emittance in a solenoid (1) Normalised 4x4 emittance – ecalc9f Normalised 2x2 emittance Normalised 4x4 trace space emittance Normalised 2x2 emittance with canonical angular momentum
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
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 !!!
MICE beam from ISIS Beam in upstream spectrometer Beam after Pb scatterer x yy
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 !!!
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”
Quick fix: x – p x correlation Close spectrometers Far spectrometers with x-p x correlation
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
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