Emittance Calculation Progress and Plans
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Emittance Calculation Progress and Plans
Chris RogersMICE CM
24 September 2005
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• About the Beard…• It could have been worse…
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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
PID
Vmeas
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• Assign each muon some statistical weight w
• Matching Condition is usually ( = (333 mm, 0) in the upstream tracker solenoid (MICE Stage VI)– Note =400 mm for pz = 240 MeV/c
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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
• Should be possible to do better– Some algorithms planned but not implemented
measn n2
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Momentum Amplitude Correlation
• At least three definitions of the amplitude exist – “Palmer” (FSII)
– “Balbekov” (muc258)
– “Ecalc9” (muc280) - note units are [mm]
• A prescription for generating the correlation exists– Generate transverse phase space in a gaussian as normal– Generate E or pz in a gaussian as normal
– But add a term to make E or pz like• Nothing more than a handwaving justification in the literature• A prescription we can follow I guess• But many questions remain
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mceBr
mcpA trans
B
2
222
rAP
EAEE )1( 20
))((2)(2)()(1 22222
xyyxzyxz
ypxpLypxpyxppppm
A
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More on P-A correlation
• Build grid in phase space all with pz 200 MeV/c e.g.– (x,px) = (0,5) (0,10) (0,15) … (5,5) (5,10) (5,15) …
• Fire it through MICE magnetic fields– No RF/LH2
• We introduce a momentum amplitude correlation!!!
<A2> 15 pi
<A2> 6 pi
(FS2)
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meas
• 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
– And is the matrix with elements ij
muonsj
muonsi
muonsjiij wu
wwu
wuwu
w1112
Nn m
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Nzn m
p 2
or
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true
• 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– Mis-PID also introduces an offset– If we can characterise our PID and beam well, we can in principle
correct this offset
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Measurement Error• The expression for meas in terms of the error in the
measurement of the phase space variable is given by
• 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– Error on emittance not only dependent on the resolution in a single
phase space variable– Worry about whether an error in x introduces and error in px
– Worry about whether the error in x is greater at different px
– E.g. worry that the TOF resolution at the reference plane is highly dependent on the pz resolution of the tracker
),(),(),( jtrueiij
truej
trueiij
measj
measiij uuuuuu
),(),( jiijtruejiij uuuu
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Uncorrected 4D Emittance (Ellis)
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Corrected 4D Emittance (Ellis)
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Calculating the Error• Calculate the values of R, C using G4MICE
– Verify G4MICE in stage I & II– Knowing the error is more important to the baseline analysis than the actual size
of the error
• Requires some care– Once we are beyond MICE stage I & II it will be difficult to re-verify G4MICE– If the detector errors change we will be blind
• E.g. the spectrometer field drifts, a fibre dies, etc…
• So we should understand the errors in detail– So, for example, if the B-field drifts during the experiment we can spot it
• We should be actively checking sources of error– Check spectrometer field between runs, etc…
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PID• Error introduced on covariance measurement by mis-PID is
something like
– bs is background identified as signal, sb is signal identified as background– This should be after the bunch sampling
• We should be able to estimate this offset– We can measure the distribution of incoming particles – We can calculate the probability of mis-identification (from e.g. Monte
Carlo simulation)
• In the case that Vijbs, Vij
sb ~ Vijtrue emittance is not changed
– Nmeas=Ntrue+Nbs-Nsb
• We should also worry about measurement of transmission– Perhaps this is more important for PID– We need to understand what analysis is required for scraping
• These ideas need to be verified by simulation
sbijsb
bsijbs
trueijtrue
measijmeas VNVNVNVN
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Useful 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 bucket?
• 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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Useful 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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Useful 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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Useful Quantities - SPA• SPA single particle amplitude ~ Ecalc9f amplitude above
– Calculate optical functions Oc – SPE-like quantity independent of bunch measurement– Can be calculated in G4MICE Analysis– Note Oc is not uniquely defined (depends on input beam)
• 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
nAA bunch2
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UOUA c12
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Example use - nonlinear optics
• Build grid in phase space as above• Fire it through MICE magnetic fields
– Examine change in amplitude upstream vs downstream– No RF/LH2
• Show nice features– Dynamic aperture?– Emittance growth vs (x,px,y,py)?
Initial A2
A2 /A
2
Initial x
A
2 )/A
2
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Useful Quantities - Holzer Emittance• 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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Unanswered Questions• How do we do the offline bunching?
– I have some ideas– This is the next thing to tackle
• Analysis of longitudinal dynamics– We need to understand the TOF resolution ito 6D emittance
measurement– How do we do the 6D emittance measurement?
• How good is the PID in terms of emittance?– Do we need the correction/will it really work?
• Does there exist a serious understanding of momentum-amplitude correlation?– We cannot really talk about this without understanding– We need a detailed analysis of the non-linear beam dynamics of the
cooling channel
• More detail on the scraping/transmission analysis