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Emittance Calculation Progress and Plans

Chris Rogers

Analysis PC

18 August 2005

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Overview• Talk in detail about how we can do the emittance

calculation– Sample bunch

– Remove experimental error (PID & tracking)

– Calculate Emittance

• Talk about other useful quantities– Scraping/Aperture

– Decay Losses

– Single Particle Emittance

– Single Particle Amplitude

– Holzer Particle Number

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Emittance(z)+/- error

Uimeas

Sampled bunch

Monte Carlo

PID

Vmeas

~ Calculate

Vtrue

Emittance Calculation RoadmapUnderstood, tools exist

Roughly understood

Not really understood

I’ll run through each box in this talk

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Beam Matching• The cooling channel is designed to accept a certain

distribution of particles– The beta function should be periodic over a cell of the magnetic

field

– The beta function should be a minimum in the liquid Hydrogen for optimal cooling

– The longitudinal distribution should be realistic for the appropriate phase rotation system

• My standard approach is to – Do a reasonable job with the beamline (for good efficiency)

– Then sample a Gaussian distribution from the available events for the final analysis (Algorithm not implemented in G4MICE)

• Assign each muon some statistical weight w

• Matching Condition is usually ( = (333 mm, 0) in the upstream tracker solenoid (MICE Stage VI)– Need to check for different MICE stages

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Sampling a bunch - stupid algorithm

• Stupid algorithm already exists but fails– Bin particles

– Density, bin = nbin/(bin area)

– Apply statistical weight to all particles in bin• Wbin= requiredbin

• Fails because number events in each bin goes as– With 106 particles and 10 bins/dimension we have ~ 1

particle in each bin– Atrocious precision

measn n2

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Sampling a Bunch - clever algorithm

1. Bin ito single particle emittance/amplitude (I.e. 1D) – Use the covariance matrix of the desired distribution

2. Then reweight so that the distributions inside each SPE bin are flat in phase space

• Not coded yet• Stage 2 should be easy but I haven’t quite worked out the

details yet– Rotate the bunch and take successive 1D histograms to avoid the

binning problem in the stupid algorithm

– E.g. for a 2D phase space, if the distribution is flat in x and flat in px

and flat in (x+px) then it will be flat in 2D phase space

• May be productive to develop a measure of “Gaussian-ness”– There’s >~ 3 months work here!

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Clever algorithm - cartoon0

1

2 3

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Vmeas

• We then calculate the covariance matrix using

– Where ui are the measured phase space coords, and w is the statistical weight

– I haven’t specified whether we use px or x’=px/pz type variables

• Recall emittance is related to the determinant of the 2N dimensional covariance matrix according to

– Where the additional factor of <pz> is required if we use x’ type variables to normalise the emittance

muonsj

muonsi

muonsjiij wu

wwu

wuwu

wV

111

Nn m

21V

Nzn m

p 2 V

or

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Vtrue

• This gives us the measured covariance matrix– Includes errors due to mis-PID– Includes errors due to detector resolutions

• Correct for detector resolutions– Detector resolution introduces an offset in emittance– If we can characterise our detector resolutions well, we can understand and

correct the offset

• Correct for mis-PID (new!)– Mis-PID also introduces an offset– If we can characterise our PID and beam well, we can also correct this offset– This is a new idea sparked by discussion with Rikard at the last CM - disclaimer

is I only looked at it on Tuesday after getting back from my holidays … still very immature ideas

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Measurement Error

• The measured value uimeas of a true value uitrue has the relation:

– Where i is a small error.

• The expression for Vtrue can then be written in terms of the measured parameters:

• For a bit more detail see MICE Note 90 (tracker note)

• This is similar to addition in quadrature, except that the error is not independent of the phase space coordinates

itruei

measi uu

CRRVV Tnn meas2

true2

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Uncorrected 4D Emittance (Ellis)

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Corrected 4D Emittance (Ellis)

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Calculating the Error• We can get the offset from G4MICE, but IMHO this is not enough

– Monte Carlo can be wrong

– We should be prepared to expect the unexpected

• I would be happier if we could calculate/explain the pdf of the errors i depending on the source of the error– Dead fibres

– Misalignment

– Multiple scattering

– Other stuff

• So, for example, if one of the fibres dies during the experiment we can spot it– Or whatever

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PID

• I included a brief note on how PID effects our experimental measurement– Some mathematical detail that may make this more suitable as a

document rather than slides

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Emittance Calculation

• Finally calculate the emittance according to the formula given in slide 7

• Technical note about units– Everyone quotes emittance as being something like 6 mm rad

– This is because the area of an ellipse in 2D phase space is

– But a 2n dimensional hyperellipsoid has volume given by

– With

– So strictly speaking the units mm rad are wrong, and we should instead use mm rad

• For simplicity I ignore this

|| 2VArea

|| 22 nngVolume V

)1(2 ng

n

n

volumen

nn g 2

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More on Units• Everyone says that emittance conservation is a consequence of Liouville• But Liouville’s Theorem refers to canonical phase space coordinates

such as

• I have yet to see this relationship done “properly” in the literature• Two coordinate sets seem to be in use in accelerator physics

– Trace space (t,x,y,dt/dz,dx/dz,dy/dz)• More common

– Kinetic phase space (t,x,y,E,px,py)

• Less common, Ecalc9 uses it

• I’ve been using it as the default for compatibility with ICOOL

– I can use either in G4MICE Analysis

c

qAp

c

qAp

c

qEyxt y

yx

x ,,,,,

U

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Related Quantities - Scraping• There exists a closed surface in phase space beyond which particles

strike the walls– Surface in 6D phase space!

• We should be able to measure this surface– Transmission, radiation damage, ?dynamic aperture?, ?rf aperture?

• We should be able to measure the effects on the muon of striking the walls– Are all particles lost?

• This means that we must have sufficient acceptance in the detectors etc that the entire scraped surface makes it to the first absorber

• It would also be useful to distinguish between scraping losses and decay losses– Is this possible

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Related Quantities - Decay Losses• We may also want to get at decay losses

• Expect ~ 20% or more loss in a FS2 style neutrino factory cooling channel due to decays

• But should be easily calculable

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Related Quantities - SPE• Single Particle Emittance i

– V is the matrix of covariances

– U is particle position

– O is the matrix of measured optical functions , , etc• V=nO

– Can be calculated in G4MICE Analysis

UOUUVUni11

SPE is area of this ellipse

Position of particle

RMS contour of bunch

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Constancy of SPE• Fire a 5 beam through MICE stage VI with only

magnetic fields– No RF/liquid Hydrogen

– Individual SPE’s change by ~10 %, <SPE> ~ constant

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Cooling ito SPE

• Now add RF (electrostatic?) and LH2

– Note <SPE> decreases by ~10% … Cooling!! (~<SPE>/2n)

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Related Quantities - SPA• SPA single particle amplitude

– Use calculated optical functions– SPE-like quantity independent of bunch measurement– Can be calculated in G4MICE Analysis

• One powerful use of this method is to look at phase space without requiring any bunch– Good for simulation– Possibly use as an experimental technique?– Get much higher statistics in particular regions of phase space

• Get back to “bunch amplitude” ~ bunch emittance– Use

n

AA bunch

2

22

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Example use - nonlinear optics

• Build grid in phase space

• Fire it through MICE magnetic fields– Examine change in amplitude upstream vs

downstream

A2

A2/A2

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Some more A2 plots

• Show (SPE) independent of the rest of phase space

Initial x

Final pzInitial px

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Related Quantities - Holzer

• Calculate the maximum number of particles sitting in an arbritrary hyper-ellipsoid of a given volume– Holzer suggests using a minimising algorithm to find the hyper-

ellipsoid of a particular volume that has the most particles in it

– To first approximation, this will be similar to the hyper-ellipsoid given by UTV-1U

– Then this becomes the number of particles with SPE lower than some value ~ the volume of the hyper-ellipsoid

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Overview• This is all I know about Analysis!

• Detailed outline of how to do the emittance calculation– Sample bunch

– Remove experimental error (PID & tracking)

– Calculate Emittance

• Talked about other useful quantities– Scraping/Aperture

– Decay Losses

– Single Particle Emittance

– Single Particle Amplitude

– Holzer Particle Number