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Status Update

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

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Status Update

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  1. Status Update Chris Rogers Analysis PC 6th April 06

  2. 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

  3. Emittance Growth 25 MeV/RF 40o 25 MeV/G4MICE 25 MeV/ICOOL 25 MeV/RF 90o • 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 1 MeV/RF 90o

  4. 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…

  5. Beta(pz) (Palmer) (Bob Palmer)

  6. Emittance(z) (Palmer) Bob Palmer Dashed is for all m Full is for m which make it to end

  7. 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?

  8. 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) = MU(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) = Sjmijuj(zi) • 2nd Moment Transport: • Say we have a bunch with second moment Sparticlesui (zfin)uj(zfin)/n • Then at some point zfin, moments are Sparticles(Simik ui(zin)) (Sjmjk 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

  9. 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)

  10. Application to Solenoids - leading order • Use U = (Q,t,P,Pt;z) and Q = (x/l, y/l); P = (px/p0, py/p0), t = Dt/l, Pt = Dpt/cp0 • H2(U,z) = P2/2l - B0(QxP).zu/2l + B02Q2/8l + Pt2/2(b0g0l)2 • H3 (U,z) = Pt H2/b0 • 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: = Si{[(B02qi/4 - B0(qiuxP).zu/2) / pi] - [(pi-B0Qxpiu.zu/2) / qi]}dz/l + pt/(b0g0) dz/l  / t • :f2::f2: = 0 in limit dz->0 • Remember if U is the phase space vector, Ufin=M2Uin, with uj/ ui = dij • 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? • B0Qxpiu.zu/2 term looks fishy

  11. Next to leading order • f3 is given by f3 = - H3(M2U, z)dz • H3(M2U,z)dz = Pt H2(M2 U, z)/b0 dz = Pt H2(U, z)dz in limit dz->0 • Then :f3: = :Pt H2:dz = Pt :H2:dz/b0 + H2 :Pt:dz/b0 = Pt :f2: /b0 + H2 dz /b0 d/dt • 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/b0) • In longitudinal phase space the transfer map becomes • ptfin = ptin • tfin = tin + pt/(b0g0) dz + (pt2/(b02g0) + H2/b0)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

  12. 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 = <M2uiM2uj> • Repeat but take M to 3rd order • <uiuj>fin = <M2(1 + pt/b0)uiM2(1 + pt/b0)uj> = <M2uiM2uj(1+pt/b0)2> • Assume a nearly Gaussian distribution and pt independent of Q,P • Broken assumption but I hope okay for den<<en • <M2uiM2uj(1+pt/b0)2> = <M2uiM2uj(pt/b0)2>+ <M2uiM2uj> • <M2uiM2uj(pt/b0)2>= <M2uiM2uj><(pt/b0)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

  13. 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: • ptfin = ptin • tfin = tin + ptin/(b0g0) dz + ( (ptin)2 /(b02g0) + H2in/b0)dz • Then in lim dz -> 0 (is this right? Only true if variables are independent?) • <ptpt> = const • <ptt>fin = <ptt>in + <ptpt>in/(b0g0) dz + <t(pt2/(b02g0) + H2/b0)>dz • <tt>fin = <tt>in + 2<tpt>in/(b0g0) dz + 2<t(pt2/(b02g0) + H2/b0)>dz • Longitudinal emittance (squared) is given by • efin2= <ptpt>in(<tt>in + 2<tpt>in/(b0g0) dz) - ( (<ptt>in)2 +2 <tpt>in <ptpt>in/(b0g0) dz) + 2(<ptpt>in - <ptt>in)<t(pt2/(b02g0) + H2/b0)>dz = ein2 + 2(<ptpt>in - <ptt>in)<t(pt2/(b02g0) + H2/b0)>dz • Growth term looks at least related to amplitude momentum correlation • Need to check against tracking to fix/test algebra

  14. 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?

  15. 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

  16. Scraping Analysis (1D) Aperture 1 Transport as beta function Aperture 2 Initial beam • 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 Transport of apertures Aperture 2 Aperture 1 • 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

  17. Emittance Measurement at TOF II

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