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 PIDTOF 0

CherenkovTOF 1

Trackers 1 & 2 measurement of emittance in and out

Liquid Hydrogen absorbers 1,2,3

Downstreamparticle ID:

TOF 2 Cherenkov

Calorimeter

RF cavities 1 RF cavities 2

Spectrometer solenoid 1

Matching coils 1&2

Focus coils 1 Spectrometer solenoid 2

Coupling Coils 1&2

Focus coils 2 Focus coils 3Matching coils 1&2

The MICE experiment

Quantities to be measured in MICE

equilibrium emittance = 2.5 mm rad

cooling effect at nominal inputemittance ~10%

Acceptance: beam of 5 cm and 120 mrad rms

Emittance measurementEach 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 <x> <y> etc… Second moments: variance(x) x

2 = < x2 - <x>2 > etc… covariance(x) xy = < x.y - <x><y> >

Covariance matrix

M =M =

2't

't'y2

'y

't'x2

'x

'tt2t

'yt2y

'xt'xy'xxxtxy2x

...............

............

............

............

............

2'y'xyx

D4

't'y'xytxD6

)Mdet(

)Mdet(

Evaluate emittance with: CompareCompare in in withwith outout

Getting to e.g.Getting to e.g. x’t’x’t’ is essentially impossibleis essentially impossible with multiparticle bunch with multiparticle bunch measurementsmeasurements

Emittance in MICE (1)

• Trace space emittance:

tr ~ sqrt (<x2> <x’2>)

(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 (<x2> <px2>)

(again, from the determinant of the 4x4 covariance matrix)

• Normalised trace space emittance

tr,norm ~ (<pz>/mc) sqrt (<x2> <x’2>)

• The two definitions are equivalent only when pz = 0 (Gruber 2003) !!!

• Expect large spread in pz in cooling channel

Muon counting in MICE• Alternative technique to

measure cooling: a) fix 4-D phase space volumeb) 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 !!!

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-px correlation builds up: initial final

Emittance increase can be contained by introducing appropriate x-px 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:

x2 = x1 + t dx/dt = x1 + t px/mb) trace space emittance:

x2 = x1 + z dx/dz = x1 + z x’

• Fixed t not very useful or practical !!!

Solenoidal field

• Quasi-solenoidal magnetic field:

Bz = 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-px 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:

px px + eAx/c, Ax = vector potential

to calculate

• Advantages:a) Correlation x,y’ = 1,4 << 1b) 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 2nd 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

y y

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 – px correlation

Close spectrometersFar spectrometers

Far spectrometerswith

x-px 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-px 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