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

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