Transcript Emittance Calculation Progress and Plans Chris Rogers Analysis PC 18 August 2005 Overview • Talk in detail about how we can do the emittance calculation – Sample.

Emittance Calculation Progress
and Plans
Chris Rogers
Analysis PC
18 August 2005
1
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
2
Emittance Calculation Roadmap
Uimeas
PID
Understood, tools exist
Roughly understood
Not really understood
Sampled bunch
I’ll run through each box in
this talk
Vmeas
Monte Carlo
~ Calculate
Vtrue
Emittance(z)
+/- error
3
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 (b, a) = (333 mm, 0) in the
upstream tracker solenoid (MICE Stage VI)
– Need to check for different MICE stages
4
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 2 n nmeas
– With 106 particles and 10 bins/dimension we
have ~ 1 particle in each bin
– Atrocious precision
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Sampling a Bunch - clever
algorithm
1.
Bin ito single particle emittance/amplitude (I.e. 1D)
–
2.
Use the covariance matrix of the desired distribution
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 “Gaussianness”
–
There’s >~ 3 months work here!
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Clever algorithm - cartoon
0
1
2
3
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Vmeas
• We then calculate the covariance matrix using
1
1
1
Vij 
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 
V
n 
V
or
m
m
– Where the additional factor of <pz> is required if we use x’ type
variables to normalise the emittance
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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:
u
meas
i
u
– Where di is a small error.
true
i
 di
• The expression for Vtrue can then be written in terms of the
measured parameters:
meas
T
V2true

V

R

R
C
n
2n
• 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
10
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 di 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
14
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 p mm rad
– This is because the area of an ellipse in 2D phase space is
Area  p | V2 |
– But a 2n dimensional hyperellipsoid has volume given by
– With
Volume g 2n | V2n |
g 2n

n
volume
pn
(1  n)
– So strictly speaking the units p mm rad are wrong, and we should
n
instead use
g 2 n mm rad
• For simplicity I ignore this
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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
qAy 

qAx
q

U   t , x, y, E 
, px 
, py 
c
c
c 

• 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
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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
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
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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
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Cooling ito SPE
• Now add RF (electrostatic?) and LH2
– Note <SPE> decreases by ~10% … Cooling!! (~<SPE>/2n)
21
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
A
2
bunch
 A2 

2n
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Example use - nonlinear optics
DA2/A2
A2
• Build grid in phase space
• Fire it through MICE magnetic fields
– Examine change in amplitude upstream vs
downstream
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Some more DA2 plots
Final pz
Initial px
Initial x
• Show D(SPE) independent of the rest of
phase space
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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 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
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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
26