Transcript Emittance Calculation Progress and Plans

Emittance Calculation Progress
and Plans
Chris Rogers
MICE CM
24 September 2005
1
• About the Beard…
• It could have been worse…
2
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
3
Emittance Calculation Roadmap
Uimeas
PID
Understood, tools exist
Roughly understood
Not really understood
Sampled bunch
Vmeas
I’ll run through each box in
this talk
Vtrue
Emittance(z)
+/- error
4
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 (b, a) = (333 mm, 0) in the
upstream tracker solenoid (MICE Stage VI)
– Note b=400 mm for pz = 240 MeV/c
5
Sampling a bunch - stupid
algorithm
• Stupid algorithm already exists but fails
– Bin particles
– Density, rbin = nbin/(bin area)
– Apply statistical weight to all particles in bin
• Wbin= rrequired/rbin
• Fails because number events in each bin goes as
2n
nmeas
– 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
6
Momentum Amplitude
Correlation
• At least three definitions of the2 amplitude exist
– “Palmer” (FSII)
AP   2 
2
r
b2
2
2
p
eBr




2
– “Balbekov” (muc258) AB   trans   
2 
 mc   2mc 
– “Ecalc9” (muc280) - note units are [mm]
A 
2
1
m
b

2
2
2
2
(
p

p
)


p
(
x

y
)

2
a
(
xp

yp
)

2
(
b


L
)(
xp

yp
)
x
y
z
x
y
y
x 

p
 z

• A prescription for generating the correlation exists
– Generate transverse phase space in a gaussian as normal
– Generate dE or dpz in a gaussian as normal
2
– But add a term to make E or pz like E  E0 ( 1  A )  dE
• Nothing more than a handwaving justification in the literature
• A prescription we can follow I guess
• But many questions remain
7
More on P-A correlation
(FS2)
<A2> 15 pi
<A2> 6 pi
• 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!!!
8
Smeas
• We then calculate the covariance matrix using
1
1
1
2
s ij 
wui u j 
wu i
wu j



 w muons
 w muons  w muons
– 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
1 2N
 pz  2 N
n 
S
n 
S
or
m
m
– Where the additional factor of <pz> is required if we use x’ type
variables to normalise the emittance
– And S is the matrix with elements sij
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Strue
• 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
10
Measurement Error
• The expression for Smeas in terms of the error in the
measurement of the phase space variable is given by
s ij (ui meas , u j meas )  s ij (ui true, u j true )  s ij (ui true, du j )
 s ij (dui , u j
true
)  s ij (dui , du j )
• 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
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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 reverify 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…
14
PID
• Error introduced on covariance measurement by mis-PID
is something like
NmeasVijmeas  NtrueVijtrue  NbsVijbs  NsbVijsb
– 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, Vijsb ~ 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
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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?
16
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
1
1
 i   nUV U  U O U
– V is the matrix of covariances
– U is particle position
– O is the matrix of measured optical functions a, b, etc
• V=nO
– Can be calculated in G4MICE Analysis SPE is area of this
ellipse
Position of particle
RMS contour of
bunch
18
Constancy of SPE
• Fire a 5 p beam through MICE stage VI with only
magnetic fields
– No RF/liquid Hydrogen
– Individual SPE’s change by ~10 %, <SPE> ~ constant
19
Cooling ito SPE
• Now add RF (electrostatic?) and LH2
– Note <SPE> decreases by ~10% … Cooling!! (~<SPE>/2n)
20
Useful Quantities - SPA
• SPA single particle amplitude ~ Ecalc9f amplitude above
–
–
–
–
1
A2  U Oc U
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
A2 bunch
 A2 

2n
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DA2/A2
D(A2)/A2
Example use - nonlinear optics
Initial A2
• Build grid in phase space as above
• Fire it through MICE magnetic fields
Initial x
– Examine change in amplitude upstream vs downstream
– No RF/LH2
• Show nice features
– Dynamic aperture?
– Emittance growth vs (x,px,y,py)?
22
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 hyperellipsoid 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
23
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 momentumamplitude 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
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