status update
DESCRIPTION
Status Update. Chris Rogers Analysis PC 6th April 06. Threads. I have many different threads on the go at the moment Emittance growth & non-linear beam optics Momentum acceptance & resonance structure Work continues, nothing firm yet - PowerPoint PPT PresentationTRANSCRIPT
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Status Update
Chris RogersAnalysis PC6th April 06
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Threads I have many different threads on the go at the
moment Emittance growth & non-linear beam optics Momentum acceptance & resonance structure
Work continues, nothing firm yet Noticed a mistake in the results shown at the collaboration
meeting Scraping analysis (related to tracker window and diffuser
position, also shielding) A few comments
TOF II justification(TOF digitisation, TOF reconstruction) See note
Beam reweighting algorithms Investigating general method for associating phase space
volume with each particle using so-called “Voronoi diagrams”
Need this by EPAC
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Emittance Growth
The bottom line in the plot above It would be difficult but not impossible to make the same
mistake in both ICOOL and G4MICE Ecalc9 reproduces the same emittance calculation
Try to understand why I see this emittance growth
1 MeV/G4MICE
25 MeV/G4MICE
25 MeV/ICOOL 25 MeV/RF 90o
25 MeV/RF 40o
1 MeV/RF 90o
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Comments from Bob Palmer
Paraphrased but I hope accurate Bob uses a beam with several beta functions He selects the beta function at each momentum so that it is
periodic over a MICE lattice “…if there are particles at other momenta in the sample, then those at
other momenta will experience different betas and different beta beats…
“…The momentum dependence of the matching was designed to match the beam from one lattice to the next for all momenta (with their different initial and final betas) at the same time…
“…In practice one can do that only for 2 or 3 momenta, but that is far better than doing a match just at the central momentum…
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Beta(pz) (Palmer)
(Bob Palmer)
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Emittance(z) (Palmer)
Bob Palmer
Dashed is for all Full is for which make it to end
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Comment (Me)
Bob sees periodic emittance growth if he uses multiple beta functions
Bob sees less emittance growth even if he doesn’t use multiple beta functions
But he didn’t include the tracker/matching section In principle it should be possible to choose a beam such that the
beta function is periodic over the full MICE lattice Then the emittance change should also be periodic to first order
But what about resonances? Next steps:
(I) Can I reproduce Palmer’s results in previous slides? (II) Can I reproduce these results or similar in full MICE
Because MICE is not symmetric about the centre of a half cell the resonant structure may be different
Need to verify I would like to understand emittance growth in terms of
generalised non-linear beam optics (we need this to show cooling)
Beam reweighting?
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Beam Optics & Emittance Definition of linear beam optics:
Say we transport a beam from zin to zfin
Define an operator M s.t. U(zfin) = M U(zin) and M is called a transfer map In the linear approximation the elements of U(zfin) are a linear combination of
the elements of U(zin) e.g. x(zf) = m00 x(zin) + m01 y(zin) + m02 px(zin) + m03 py(zin) where mij are
constants Then M can be written as a matrix with elements mij such that ui(zf) =
jmijuj(zi)
2nd Moment Transport: Say we have a bunch with second moment particlesui (zfin) uj(zfin)/n Then at some point zfin, moments are particles(imik ui(zin)) (jmjk uj(zin))/n But this is just a linear combination of input 2nd moments
Emittance conservation: It can be shown that, in the linear approximation, so long as M is symplectic,
emittance is conserved (Dragt, Neri, Rangarajan; PRA, Vol. 45, 2572, 1992) Symplectic means “Obeys Hamilton’s equations of motion” Sufficient condition for phase space volume conservation
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Non-linear beam optics Expand Hamiltonian as a polynomial series
H=H2+H3+H4+… where Hn is a sum of nth order polynomials in phase space coordinates ui
Then the transfer map is given by a Lie algebra M = … exp(:f4:) exp(:f3:) exp(:f2:)
Here :f:g = [f,g] = ( ( f/ qi)( / pi) - ( f/ pi)( / qi) ) g exp(:f:) = 1 + :f:/1! + :f::f:/2! + :f::f::f:/3! + …
And fi are functions of (Hi, Hi-1 … H2)
fi are derived in e.g. Dragt, Forest, J. Math. Phys. Vol 24, 2734, 1983 in terms of the Hamiltonian terms Hi for “non-resonant H”
For a solenoid the Hi are given in e.g. Parsa, PAC 1993, “Effects of the Third Order Transfer Maps and Solenoid on a High Brightness Beam” as a function of B0
Or try Dragt, Numerical third-order transfer map for solenoid, NIM A Vol298, 441-459 1990 but none explicitly calculate f3, etc
“Second order effects are purely chromatic aberrations” Alternative Taylor expansion treatment exists
E.g. NIM A 2004, Vol 519, 162–174, Makino, Berz, Johnstone, Errede (uses COSY Infinity)
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Application to Solenoids - leading order
Use U = (Q,,P,Pt;z) and Q = (x/l, y/l); P = (px/p0, py/p0), = t/l, P = pt/cp0
H2(U,z) = P2/2l - B0(QxP).zu/2l + B02Q2/8l + Pt
2/2(l)2
H3 (U,z) = Pt H2/
H4 (U,z) = … B0 = eBz/p0
zu is the unit vector in the z direction H2 gives a matrix transfer map, M2
Use f2 = -H2 dz M2 = exp(:f2:) = 1+:f2:+:f2::f2:/2+…
:f2: = i{[(B02qi/4 - B0(qi
uxP).zu/2) / pi] - [(pi-B0Qxpiu.zu/2) / qi]}dz/l
+ pt/(00) dz/l / :f2::f2: = 0 in limit dz->0 Remember if U is the phase space vector, Ufin=M2 Uin, with uj/ ui = ij
Ignoring the cross terms, this reduces to the usual transfer matrix for a thin lens with focusing strength (eBz/2p0)2
Cross terms give the solenoidal angular momentum? B0Qxpi
u.zu/2 term looks fishy
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Next to leading order
f3 is given by f3 = - H3(M2 U, z)dz H3(M2U,z)dz = Pt H2(M2 U, z)/0 dz = Pt H2(U, z)dz in limit dz->0 Then :f3: = :Pt H2:dz = Pt :H2:dz/0 + H2 :Pt:dz/0 = Pt :f2: /0 + H2 dz /0
d/d Again :f3:n = 0 in limit dz -> 0
The transfer map to 3rd order is M3= exp(:f3:) exp(:f2:)=(1+:f3:+…)(1+:f2:+…) =1+:f2:+:f3: in limit dz->0
In transverse phase space the transfer map becomes M3 = M2(1 + pt/0)
In longitudinal phase space the transfer map becomes pt
fin = ptin
fin = in + pt/(00) dz + (pt2/(0
2 0) + H2/0)dz
Longitudinal and transverse phase space are now coupled It may be necessary to go to 4th/5th order to get good agreement
with tracking
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2nd Moment Transport As before, 2nd moments are transported via <uiuj>fin = <MuiMuj>in
Formally for some pdf h(U) it can be shown that (Janaki & Rangarajan, Phys Rev E, Vol 59, 4577, 1999)
<uiuj>fin = int(hfin(U) ui’ uj’ ) d2nU = int( hin(U) (Mui’) (Muj’) ) Take M to 2nd order; only consider transverse moments i.e. Q and P
<uiuj>fin = <M2ui M2uj> Repeat but take M to 3rd order
<uiuj>fin = <M2(1 + pt/0)ui M2(1 + pt/0)uj>
= <M2uiM2uj(1+pt/0)2> Assume a nearly Gaussian distribution and pt independent of Q,P
Broken assumption but I hope okay for n<<n
<M2uiM2uj(1+pt/0)2> = <M2uiM2uj(pt/0)2>+ <M2uiM2uj> <M2uiM2uj(pt/0)2>= <M2uiM2uj><(pt/0)2>
Need to test prediction now with simulation Expect to find the (probably many) flaws in my algebra
M4 terms should prove interesting also Spherical aberrations independent of energy spread
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Longitudinal Emittance Growth
This was all triggered by a desire to see emittance growth from energy straggling so need to understand longitudinal emittance growth
Use: pt
fin = ptin
fin = in + ptin/(00) dz + ( (pt
in)2 /(02
0) + H2in/0)dz
Then in lim dz -> 0 (is this right? Only true if variables are independent?) <ptpt> = const <pt>fin = <pt>in + <ptpt>in/(00) dz + < (pt
2/(02
0) + H2/0)>dz <>fin = <>in + 2<pt>in/(00) dz + 2< (pt
2/(02
0) + H2/0)>dz Longitudinal emittance (squared) is given by
fin2= <ptpt>in (<>in + 2<pt>in/(00) dz) -
( (<pt>in)2 +2 <pt>in <ptpt>in/(00) dz ) +
2(<ptpt>in - <pt>in)< (pt2/(0
2 0) + H2/0)>dz
= in2 + 2(<ptpt>in - <pt>in)< (pt
2/(02
0) + H2/0)>dz Growth term looks at least related to amplitude momentum correlation Need to check against tracking to fix/test algebra
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Resonant Structure
Pass 1M muons from -2750 to +2750 Look at change in emittance for bins with different
central momenta Try using different bin sizes (not sure this worked)
Should I bin in p, E, pz?
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Resonant structure II
I’m also working on integrating Fourier transform with MICE optics
I’m also working on delta calculation in MICE optics
Transfer matrix
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Scraping Analysis (1D)
Initial beam
Aperture 1
Transport as beta function
Aperture 2
It is necessary to transport the aperture through MICE in 2D phase space to get the true beam width that is seen downstream
The analysis which uses the beta function for transport is analogous to transporting the yellow blob only and ignores the blue particles
Aperture 1
Transport of apertures
Aperture 2
This 1D analysis using the beta function will always underestimate the amount of beam that is transferred through MICE and hence underestimate the apertures required in the tracker
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Emittance Measurement at TOF II
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