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

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4 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/ICOOL25 MeV/RF 90 o 25 MeV/RF 40 o 1 MeV/RF 90 o

5 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…

6 Beta(p z ) (Palmer) (Bob Palmer)

7 Emittance(z) (Palmer) Bob Palmer Dashed is for all  Full is for  which make it to end

8 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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10 Beam Optics & Emittance Definition of linear beam optics: Say we transport a beam from z in to z fin Define an operator M s.t. U(z fin ) = M U(z in ) and M is called a transfer map In the linear approximation the elements of U(z fin ) are a linear combination of the elements of U(z in ) e.g. x(z f ) = m 00 x(z in ) + m 01 y(z in ) + m 02 p x (z in ) + m 03 p y (z in ) where m ij are constants Then M can be written as a matrix with elements m ij such that u i (z f ) =  j m ij u j (z i ) 2nd Moment Transport: Say we have a bunch with second moment  particles u i (z fin ) u j (z fin )/n Then at some point z fin, moments are  particles (  i m ik u i (z in )) (  j m jk u j (z in ))/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

11 Non-linear beam optics Expand Hamiltonian as a polynomial series H=H 2 +H 3 +H 4 +… where H n is a sum of n th order polynomials in phase space coordinates u i Then the transfer map is given by a Lie algebra M = … exp(:f 4 :) exp(:f 3 :) exp(:f 2 :) Here :f:g = [f,g] = ( (  f/  q i )(  /  p i ) - (  f/  p i )(  /  q i ) ) g exp(:f:) = 1 + :f:/1! + :f::f:/2! + :f::f::f:/3! + … And f i are functions of (H i, H i-1 … H 2 ) f i are derived in e.g. Dragt, Forest, J. Math. Phys. Vol 24, 2734, 1983 in terms of the Hamiltonian terms H i for “non-resonant H” For a solenoid the H i 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 B 0 Or try Dragt, Numerical third-order transfer map for solenoid, NIM A Vol298, but none explicitly calculate f 3, 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)

12 Application to Solenoids - leading order Use U = (Q, ,P,P t ;z) and Q = (x/l, y/l); P = (p x /p 0, p y /p 0 ),  =  t/l, P  =  p t /cp 0 H 2 (U,z) = P 2 /2l - B 0 (QxP).z u /2l + B 0 2 Q 2 /8l + P t 2 /2(     l) 2 H 3 (U,z) = P t H 2 /   H 4 (U,z) = … B 0 = eB z /p 0 z u is the unit vector in the z direction H 2 gives a matrix transfer map, M 2 Use f 2 = -H 2 dz M 2 = exp(:f 2 :) = 1+:f 2 :+:f 2 ::f 2 :/2+… :f 2 : =  i {[ (B 0 2 q i /4 - B 0 (q i u xP).z u /2)  /  p i ] - [(p i -B 0 Qxp i u.z u /2)  /  q i ]}dz/l + p t /(  0  0 ) dz/l  /   :f 2 ::f 2 : = 0 in limit dz->0 Remember if U is the phase space vector, U fin =M 2 U in, with  u j /  u i =  ij Ignoring the cross terms, this reduces to the usual transfer matrix for a thin lens with focusing strength (eB z /2p 0 ) 2 Cross terms give the solenoidal angular momentum? B 0 Qxp i u.z u /2 term looks fishy

13 Next to leading order f 3 is given by f 3 = - H 3 (M 2 U, z)dz H 3 (M 2 U,z)dz = P t H 2 (M 2 U, z)/  0 dz = P t H 2 (U, z)dz in limit dz->0 Then :f 3 : = :P t H 2 :dz = P t :H 2 :dz/  0 + H 2 :P t :dz/  0 = P t :f 2 : /  0 + H 2 dz /  0 d/d  Again :f 3 : n = 0 in limit dz -> 0 The transfer map to 3rd order is M 3 = exp(:f 3 :) exp(:f 2 :)=(1+:f 3 :+…)(1+:f 2 :+…) =1+:f 2 :+:f 3 : in limit dz->0 In transverse phase space the transfer map becomes M 3 = M 2 (1 + p t /  0 ) In longitudinal phase space the transfer map becomes p t fin = p t in  fin =  in + p t /(  0  0 ) dz + (p t 2 /(  0 2  0 ) + H 2 /  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

14 2nd Moment Transport As before, 2nd moments are transported via fin = in Formally for some pdf h(U) it can be shown that (Janaki & Rangarajan, Phys Rev E, Vol 59, 4577, 1999) fin = int(h fin (U) u i ’ u j ’ ) d 2n U = int( h in (U) (Mu i ’) (Mu j ’) ) Take M to 2nd order; only consider transverse moments i.e. Q and P fin = Repeat but take M to 3rd order fin = = Assume a nearly Gaussian distribution and p t independent of Q,P Broken assumption but I hope okay for  n <<  n = + = Need to test prediction now with simulation Expect to find the (probably many) flaws in my algebra M 4 terms should prove interesting also Spherical aberrations independent of energy spread

15 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: p t fin = p t in  fin =  in + p t in /(  0  0 ) dz + ( (p t in ) 2 /(  0 2  0 ) + H 2 in /  0 )dz Then in lim dz -> 0 (is this right? Only true if variables are independent?) = const fin = in + in /(  0  0 ) dz + dz fin = in + 2 in /(  0  0 ) dz + 2 dz Longitudinal emittance (squared) is given by  fin 2  = in ( in + 2 in /(  0  0 ) dz) - ( ( in ) 2 +2 in in /(  0  0 ) dz ) + 2( in - in ) dz =  in 2 + 2( in - in ) dz Growth term looks at least related to amplitude momentum correlation Need to check against tracking to fix/test algebra

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17 Resonant Structure Pass 1M muons from to 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, p z ?

18 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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20 Scraping Analysis (1D) Initial beam Aperture 1Transport 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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22 Emittance Measurement at TOF II

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