1

Status Update

Chris RogersAnalysis PC6th 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

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/ICOOL 25 MeV/RF 90o

25 MeV/RF 40o

1 MeV/RF 90o

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(pz) (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?

9

10

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

11

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)

12

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

13

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

14

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

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: 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

16

17

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?

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

19

20

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

21

22

Emittance Measurement at TOF II

23