FIGURE OF MERIT FOR MUON IONIZATION COOLING

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FIGURE OF MERIT FOR MUON IONIZATION COOLING

• Ulisse Bravar
• University of Oxford
• 28 July 2004

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100 m cooling channel

• Channel structure from Study II
• Cooling
• de / dx -e/l e,equil./l
• Goal 4-D cooling. Reduce transverse emittance
  from initial value e to e,equil.
• Accurate definition and precise measurement of
  emittance not that important

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MICE

• Goal measure small effect with high precision,
  i.e. De/e 10 to 10-3
• Full MICE (LH RF)
• Empty MICE (no LH, RF)
• Software ecalc9f
• De/e does not stay constant in empty channel

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The MICE experiment
The MICE experiment

• Measure a change in e4 with an accuracy of 10-3.
  
• Measurement must be precise !!!

Coupling Coils 12
Spectrometer solenoid 1
Matching coils 12
Spectrometer solenoid 2
Matching coils 12
Focus coils 1
Focus coils 2
Focus coils 3
m
Beam PID TOF 0 Cherenkov TOF 1
RF cavities 1
RF cavities 2
Downstream particle ID TOF 2 Cherenkov Calorimet
er
Diffusers 12
Liquid Hydrogen absorbers 1,2,3
Incoming muon beam
Trackers 1 2 measurement of emittance in and
out
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Quantities to be measured in MICE
cooling effect at nominal input emittance 10
Acceptance beam of 5 cm and 120 mrad
rms
equilibrium emittance 2.5 p mm rad
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Emittance measurement
Each spectrometer measures 6 parameters per
particle x y t x dx/dz Px/Pz y
dy/dz Py/Pz t dt/dz E/Pz Determines,
for an ensemble (sample) of N particles, the
moments Averages ltxgt ltygt etc Second moments
variance(x) sx2 lt x2 - ltxgt2 gt etc
covariance(x) sxy lt x.y - ltxgtltygt gt
Covariance matrix
M
Getting to e.g. sxt is essentially
impossible with multiparticle bunch
measurements
Compare ein with eout
Evaluate emittance with
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Emittance in MICE (1)

• Trace space emittance
• etr sqrt (ltx2gt ltx2gt)
• (actually, etr comes from the determinant of the
  4x4 covariance matrix)
• Cooling in RF
• Heating in LH
• Not good !!!

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Emittance in MICE (2)

• Normalised emittance
• (the quantity from ecalc9f)
• e sqrt (ltx2gt ltpx2gt)
• (again, from the determinant of the 4x4
  covariance matrix)
• Normalised trace space emittance
• etr,norm (ltpzgt/mmc) sqrt (ltx2gt ltx2gt)
• The two definitions are equivalent only when spz
  0 (Gruber 2003) !!!
• Expect large spread in pz in cooling channel

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Muon counting in MICE

• Alternative technique to measure cooling
• fix 4-D phase space volume
• count number of muons inside that volume
• Solid lines
• number of muons in x-px space increases in MICE
• Dashed lines
• number of muons in x-x space decreases
• Use x-px space !!!

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Emittance in drift (1)

• Problem Normalised emittance increases in drift
• (e.g. Gallardo 2004)
• Trace space emittance stays constant in drift
• (Floettmann 2003)

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Emittance in drift (2)

• x-px correlation builds up initial final

Emittance increase can be contained by
introducing appropriate x-px correlation in
initial beam
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Emittance in drift (3)

• Normalised emittance in drift stays constant if
  we measure e at fixed time, not fixed z
• For constant e, we need linear eqn. of motion
• a) normalised emittance
• x2 x1 t dx/dt x1 t px/m
• b) trace space emittance
• x2 x1 z dx/dz x1 z x
• Fixed t not very useful or practical !!!

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Solenoidal field

• Quasi-solenoidal magnetic field
• Bz 4 T within 1
• Initial b within 1 of nominal value
• b fluctuates by less than 1

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Emittance in a solenoid (1)

• Normalised 4x4 emittance ecalc9f
• Normalised 2x2 emittance
• Normalised 4x4 trace space emittance
• Normalised 2x2 emittance with canonical angular
  momentum

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Muon counting in a solenoid

• In a solenoid, things stay more or less constant
• This is 100 true in 4-D x-px phase space
• solid lines
• Approximately true in 4-D x-x trace space
• dashed lines

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Emittance in a solenoid (2)

• Use of canonical angular momentum
• px px eAx/c, Ax vector
  potential
• to calculate e
• Advantages
• a) Correlation sx,y s1,4 ltlt 1
• b) 2-D emittance exx constant
• Note Numerically, this is the same as
  subtracting the canonical angular momentum L
  introduced by the solenoidal fringe field
• Usually sx,y s1,4 in 4x4 covariance matrix
  takes care of this 2nd order correlation
• We may want to study 2-D ex and ey separately
  see next page !!!

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MICE beam from ISIS

• Beam in upstream spectrometer
• Beam after Pb scatterer

y
y
x
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How to measure e (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 !!!

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How to measure e (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 !!!

e increase in drift
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Quick fix x px correlation
Close spectrometers Far spectrometers
Far spectrometers with x-px correlation
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Gaussian beam profiles

• Real beams are non-gaussian
• Gaussian beams may become non-gaussian along the
  cooling channel
• When calculating e from 4x4 covariance matrix,
  non-gaussian beams result in e 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

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Conclusions

• Use normalised emittance x-px as figure of merit
• Accept increase in e 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