Emittance definition and MICE staging

U. Bravar Univ. of Oxford

1 Apr. 2004

• Topics: a) Figure of merit for MICEb) Performance of MICE stages

Figure of merit for MICE• MICE is designed to measure something to 10-3 • We have yet to decide what…

• Current quantity: normalised transverse emittance

as calculated by ecalc9f.for (G. Penn, 2001)from the 4x4 covariance matrix

• 4 definitions of rms emittance (K. Floettmann, 2003):i) normalised emittance ii) trace space emittance iii) geometric emittanceiv) normalised trace space emittance

• Past wisdom (prior to Abingdon meeting): normalised emittance = sqrt (<x2><px

2>) is Liouville compliant

Rms emittance and cooling

• Rms emittance, four definitions (K. Floettmann, 2003):

a) normalised emittance n calculated from <x2>, <(px)2>

b) beam emittance beam = n / <pz>

c) trace space emittance tr calculated from <x2>, <(x’)2>

d) normalised trace space emittance n,tr = <pz> tr

• MICE cooling measured by decrease in 4-D transv. n: in–outin

where is obtained from the 4X4 covariance matrix.

Emittance in MICE

• from ecalc9f.for

• Full MICE (LH + RF)• Empty MICE (no LH, RF)

• Emittance is not constant in empty channel

• Does Liouville’s theorem still apply?

• Implications for MICE?

Emittance conservation

• J. Gallardo (2004) showed that 6-D normalised emittance calculated by ecalc9f.for is not constant in drift!

• K. Floettmann’s paper (2003): 4-D trace space emittance tr = sqrt {<x2><(x’)2>} stays constant in drift!

• What does this mean?

Solution• Liouville’s theorem: df/dt = 0 • Where:

f(qj,pj;t) = state function; pj = canonical conjugate momenta;

t = independent variable.

• Calculate emittance at fixed t, not z! • Use correct conjugate momenta:

in drift: x, px, not x, x’ in B-field x, px+eAx/c

• Calculate 6-D emittance! • and L are constant only if transverse and longitudinal motions are completely decoupled. • This is not the case in MICE!

• ecalc9f.for calculates , L, 6 at fixed z;variables are x, y, t, px, py, E.

• These emittances are not constant in drift or in an empty channel!

Simple example

• Drift space. • Non-relativistic beam • Initial spread: p / p = 10%. • Top: 6-D n at fixed t. • Bottom: 6-D n at fixed z. • This true in general;

theoretical proof available; simulations in progress.

• Questions:

a) can we find n at fixed t? b) is it useful for MICE?

Emittance measurement• We now have a quantity that is Liouville-compliant. • Can we measure it to 10-3? • Beam rms: x = 3.3 cm, px = 20 MeV/c• Tracker resolution: x << x • Relative error on x 0.5 (x / x )2

• Relative error on 6-D emittance 6 1.2 (x / x )2 = 10-3

• Therefore x 0.03 x

• In other words, we need x 0.01 cm, px 0.5 MeV/c• Current resolutions from SciFi: x 0.05 cm, px 2.5 MeV/c. Way too big!!!• Plus, statistical error on x = 1/sqrt (#) = 10-3, so # = 106

• Things get better if we want to measure 4-D transverse emittance• Worse if we want to measure cooling 10% • Question: is this the right figure of merit for MICE? • We may calculate rms emittance… but we can’t deal with more advanced stuff.

• THESE FIGURES ARE PRELIMINARY. WORK IN PROGRESS!!!

Alternative: event counting• Procedure:

a) count number of muons in 2-D, 4-D or 6-D phase-space ellipsoids; b) show that #(+) increases from the upstream tracker to the downstream tracker.

• Problems: a) need to determine n first;b) particle ID prior to upstream spectrometer.

• Advantages: a) straightforward;b) this is the quantity that matters in a real cooling channel.

Alternative: muon counting • i.e. measurement of the increase in

phase space density

• MICE channel, ecalc9f.for, 4-D ellipsoid

• Initial = 6000 mm mrad10,000 events.

• Full MICE channel (LH & RF). • Empty channel (no LH, no RF) &

same beam.• Empty channel (no LH, no RF) &

different beam.

• Still, need to compute 6 prior to counting.

• Same old problem: quantity does not stay constant in empty channel!!!

Gaussian beam profiles• Real beams are non-gaussian• Gaussian input beams may become non-gaussian along the MICE

channel (see e.g. study on magnet alignment tolerances)

• When calculating e from 4x4 matrix with 2nd order moments, non-gaussian beams result in e increase.

• Can improve emittance computations and measurement of phase space volume. May not be possible to achieve 10-3.

• However, cooling that results in twisted phase space volume is not very useful.

• May need new figure of merit, in addition to , something to measure Gaussian shape of the beam.

• e.g. use 3rd order moments to measure skewness of beam

MICE stages

Questions

• Beam optics: can we use the same cooling channel solutions in all six stages?

• Can we do all the physics of MICE with stages IV or V?

Beam optics

• Software provided by Bob Palmer.

• Based on ICOOL. • Tuning of coil currents

assumes: a) ideal beam;b) long channel, 100 m;c) empty channel;d) constant momentum pz = 200 MeV/c.

Note: potential problem with stay-clear area in match coil.

Actual MICE channel• When running MICE stages IV,

V and VI, things are different.

• Stage VI (full MICE)• Stage V (2 LH + 1 RF)• Stage IV (1 LH + 0 RF)

• Two problems: a) is not minimum in the centre of LH absorbers; b) is not flat in downstream spectrometer.

• Fine tuning of coil currents necessary! Work in progress.

= 42 cm in the centre of LH

• We get maximum cooling when this is true!

• Cooling formula: equilibrium emittance n,equilibrium =

• May do some tuning of focus coil currents.

• Unlikely solutions: a) move LH; b) additional match coils.

Performance of stages IV, V and VI

• Stage VI (full MICE)• Stage V (2 LH + 1 RF)• Stage IV (1 LH + 0 RF)

• Input emittances: in 3,000; 6,000; 9,000 & 12,000 mm mrad.

• Start with 10,000 muons. Count number of muons that are left as a function of z along the MICE channel.

• Note: z = 0 in the middle of the upstream spectrometer. ICOOL runs all the way to the middle of the down-stream spectrometer, to z = 4.10 m (Stage IV), 6.85 m (Stage V) & 9.60 m (Stage VI).

Cooling

• Stage VI (full MICE)• Stage V (2 LH + 1 RF)• Stage IV (1 LH + 0 RF)

• Input emittances: in 3,000; 6,000; 9,000 & 12,000 mm mrad.

+ beam, <pz> 200 MeV/c.

• Note: fluctuations due to tracker resolution are not included.

in = 6,000 mm mrad

• Stage VI (full MICE)• Stage V (2 LH + 1 RF)• Stage IV (1 LH + 0 RF)

• Questions: why is in >> in ?

• Shouldn’t in be 6,000 mm mrad in all cases?

Summary of results

MICE Stage IV MICE Stage V MICE Stage VI

Input emittance Transmission Transmission Transmission

,in = 3,000 mm mrad 1.7% 9,999/10,000 3.9% 9,990/10,000 2.8% 9,977/10,000

,in = 6,000 mm mrad 4.6% 9,955/10,000 10.9% 9,899/10,000 12.5% 9,865/10,000

,in = 9,000 mm mrad 4.9% 9,665/10,000 12.9% 9,430/10,000 16.6% 9,412/10,000

,in = 12,000 mm mrad 5.4% 9,107/10,000 13.7% 8,694/10,000 18.1% 8,609/10,000

1) At all z locations, only muon tracks that make it all the way to the downstream spectrometer are used to calculate .

2) Transmission = number of muons that reach the middle of the downstream spectrometer, out of 10,000 initial. Muon decay is disabled.

3) / = (upstream – downstream) / upstream = cooling; is measured in the centre of the upstream and downstream spectrometers.

Statistical errors• Question: what are the errors in the

table on the previous page?

• Answer: the figure shows (y-axis) vs. transmission (x-axis) for Stage VI.

i) Each point is a different ICOOL run with a different random beam.

ii) Total of 20 input beams, each beam contains 10,000 events.

iii) All beams are Gaussian, in 6000 mm mrad. x-axis, actually shows #(+) lost in the MICE channel, i.e. Transmission = 10,000 – #(+).

• Standard full MICE.• MICE channel with LH only, no Al and

Be windows.

• In short: 10% • Errors 0.5% statistic + 0.5%

systematic.

MICE stage VI• MICE stage IV can prove ionisation cooling, but cannot prove the feasibility

of a long cooling channel, since it has no RF.

• MICE stage V can demonstrate the feasibility of a muon ionisation cooling channel.

• MICE stage VI: a) central absorber: much more representative of real channel:

i) beam optics same as in long channel; ii) e.g. determine cooling at = 42 cm.

b) in addition to flip and no-flip modes, can run in semi-flip;c) represents one full flip element.d) …..

• Does all this justify stage VI?

Beta functions – flip mode

= 42 cm

= 25 cm

= 17 cm

= 7 cm

Conclusions• For the time being, use ecalc9f.for and as the figure of merit for

MICE.• Measure out, not , to 10-3 absolute!

• Need to reach consensus on appropriate figure of merit. URGENT!!!

• Fine tuning of MICE channel optics is necessary. • Need solutions for all stages of MICE, including stage I. • Study additional channel configurations, no-flip & semi-flip,

additional absorbers.

• Investigate all advantages of having stage VI.