Parametric Resonance Ionization Cooling of Muons

Operated by the Southeastern Universities Research Association for the U.S. Depart. Of Energy

Thomas Jefferson National Accelerator Facility

Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

Parametric Resonance Ionization Cooling of Muons Alex Bogacz*

in collaboration withKevin Beard*, Slava Derbenev* and Rol Johnson

*Jefferson Laboratory Muons Inc.

7-th International Workshop on Neutrino Factories and Superbeams, LNF Frascati, June 21, 2005




Overview

Final transverse ionization cooling - Parametric resonance enhancement

Resonant transport channels - lattice prototypesquadrupole based

solenoid based

Transverse beam dynamics in the cooling channel – tracking studies‘soft-edge’ solenoid

linear transfer matrix

nonlinear corrections (in tracking)

thin ‘ideal’ absorber model

Chromatic aberrations - compensation of the detuning effects with RF tracking studies




Transverse parametric resonance cooling

Transport channel (between consecutive absorbers) designed to replenish large angular component, x’, sector of the phase-space, ‘mined’ by ionization cooling process.

Parametric resonance in an oscillating system - perturbing frequency is equal to the harmonic of the characteristic (resonant) frequency of the system, e.g half-integer resonance

Normal elliptical motion of a particle’s transverse coordinate in phase space becomes hyperbolic – resulting beam emittance has a wide spread in x’ and narrow spread in x – sector of the phase-space where ionization cooling is most effective




Transfer matrix of a periodic resonant lattice

Symplectic transfer matrix, M(s), for a beamline (in x or y)

-λs-λs

λs λs

e cosψ g sinψe 0

M(s) = 1- sinψ e cosψ 0 eg

Lattice period can be designed in such a way that sin = 0 n , n = 1, 2.

-λs

λs

e cosψ g sinψM(s) = 1- sinψ e cosψ

g

Coordinate and angle are uncoupled - resulting beam emittance has a wide spread in x’ and narrow spread in x.

0 0

0 0' '

ss s s

ss s s

x e x

x e xx x’ = const




Symmetrized double cell (x = 3 = y)

14.820

Mon Oct 04 00:33:59 2004 OptiM - MAIN: - D:\6Dcooling\PIC\trip_sing_trip_cell.opt

600

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]

BETA_X BETA_Y DISP_X DISP_Y

14.820

Mon Oct 04 00:35:21 2004 OptiM - MAIN: - D:\6Dcooling\PIC\trip_sing_trip_cell.opt

0.5

0PHASE_X

&Y

Q_X Q_Y

14.820

Mon Oct 04 11:31:19 2004 OptiM - MAIN: - D:\6Dcooling\PIC\inv_trip_sing_trip_cell.opt

600

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]


14.820

Mon Oct 04 11:32:12 2004 OptiM - MAIN: - D:\6Dcooling\PIC\inv_trip_sing_trip_cell.opt

0.5

0P

HA

SE

_X&

Y

Q_X Q_Y




Angular ‘shearing’ of the transverse phase-space

0.3-0.3 X [cm] View at the lattice beginning

3-3

X`[m

rad]

14.820

Mon Oct 04 00:33:59 2004 Op tiM - MA IN: - D: \6Dcoo ling\PIC\trip _sin g_trip_cel l.o pt

600

50

BETA

_X&

Y[m

]

DISP

_X&Y

[m]


0.3-0.3 X [cm] View at the lattice end3

-3X

`[mra

d]

14.820

Mon Oct 04 11:31:19 2004 O ptiM - MAIN: - D:\6Dcooling\PIC\inv_trip _sing_trip_ce ll.opt

600

50

BE

TA

_X&

Y[m

]

DIS

P_X

&Y

[m]





4-cell resonant channel

59.280

Mon Oct 04 00:55:41 2004 OptiM - MAIN: - D:\6Dcooling\PIC\trip_sing_trip_4cell.opt

600

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]


Uniform triplet lattice resonantly perturbed by a singlet

Absorber placed half way between triplets

absorber absorber absorberabsorber absorber




Solenoid cell (x = = y)

c1 L[cm]=130 B[kG]=32.4 Aperture[cm]=10 c2 L[cm]=80 B[kG]=-34.1 Aperture[cm]=10

7.20

Fri Apr 08 10:24:49 2005 OptiM - MAIN: - D:\6Dcooling\Sol chann - summ\sol_no_cav_cell.opt

200

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]


7.20


0.5

0P

HA

SE

_X&

Y

Q_X Q_Y




sol

1 +cos(kL) sin(kL) sin(kL) 1 - cos(kL)2 k 2 k

k sin(kL) 1 +cos(kL) 1 - cos(kL) sin(kL)k4 2 4 2=

sin(kL) 1 - cos(kL) 1 +cos(kL) sin(kL)2 k 2 k

1 -cos(kL) sin(kL) k sin(kL) 1 +cos(kL)k4 2 4 2

M

‘soft-edge’ solenoid model

Zero aperture solenoid - ideal linear solenoid transfer matrix:

0k = eB /pc




‘soft-edge’ solenoid – edge effect

Non-zero aperture - correction due to the finite length of the edge :

It decreases the solenoid total focusing – via the effective length of:

It introduces axially symmetric edge focusing at each solenoid end:

axially symmetric quadrupole

22 2

edge z 0-

1 k a= B (s) ds B L2 8 0k = eB /pc

z

0 -

1L = B (s) dsB

0 00

M edgeedge

edge

1 0 0 01 0 0

=1 0

0 1

M M M Msoft sol edge sol edge=




‘soft-edge’ solenoid – nonlinear effects

Nonlinear focusing term F ~ O(r2) follows from the scalar potential:

Scalar potential in a solenoid

Solenoid B-fields

(r,z) = 0 Bz zz

Bzd

d

2 z2 r24

2zBz

d

d

2

z 2 z2 3 r2 12

3zBz

d

d

3

8 z4 24 z2 r2 3 r4192

Bz(r,z) = Bz 2zBz

d

d

2

r2

4

Br(r,z) = r2

z

Bzd

d

r3

16 3zBz

d

d

3




‘soft-edge’ solenoid – nonlinear effects

In tracking simulations the first nonlinear focusing term, F ~ O(r2) is also included:

Nonliner focusing at r = 20 cm for 1 m long solenoid with 25 cm aperture radius

aL

rpceB

LdsBrdsBpce

F 31

2221 22

022

22

22 2/ 4, 0.8

20.07

2 3L r r a

B dsF r rF aLB ds




Thin absorber with re-acceleration

Ionization cooling due to energy loss (-p) in a thin absorber followed by immediate re-acceleration (p) can be described as:

The corresponding canonical transfer matrix can be written as

pp

x x

1 0 0 00 1 2 0 ˆ0 0 1 02 0 0 1

x x

y y

x xpk

Ky ypk

/zk eB pc

1

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

abs

pp

M K K

pp x xˆ K




Final cooling – initial beam parameters

after helical cooling channel rms

normalized emittance: x/y mmmrad 30

longitudinal emittance: l lp z/mc) momentum spread: p/p bunch length: z

mm

mm

0.8

0.0130

p = 287 MeV/c




Solenoid cell, with absorber – 6D tracking

7.20


200

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]


20-20 S [cm] View at the lattice beginning

150

-150

dP/P

* 1

000,

1-1 X [cm] View at the lattice beginning

75-7

5X

`[mra

d]

1-1 X [cm] View at the element 14

75-7

5X

`[mra

d]

20-20 S [cm] View at the element 14

150

-150

dP/P

* 1

000,

each absorber: Dp/p = 0.054 cm Be p = 14 MeV/c

detuning effect - the momentum-dependent betatron frequency causes off- momentum particles to be out of resonance with the focusing lattice




Solenoid cell, with absorber and RF– 6D tracking

7.20

Fri Apr 08 12:45:48 2005 OptiM - MAIN: - D:\6Dcooling\Sol chann - summ\sol_cav_cell.opt

200

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]



150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,

synchrotron phase advance of 2/8 per cell – two RF cavities at zero-crossing (cavity gradient: 17.3 MeV/m at 400 MHz)

each absorber: Dp/p = 0.054 cm Be p = 14 MeV/c

by choosing suitable synchrotron motion parameters, the resonance condition can be maintained




Chromatic aberration compensation with RF– cells 1-4

top plots: NO RF, bottom plots: with the RF


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


-75

X`[m

rad]


150

-150

dP/P

* 1

000,


75-7

5X

[̀mra

d]


150

-15

0dP

/P *

100

0,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

[̀mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000

,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,




Chromatic aberration compensation with RF– cells 5-8

top plots: NO RF, bottom plots: with the RF


75-7

5X

[̀mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-15

0dP

/P *

100

0,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

5X

`[mra

d]


150

-150

dP/P

* 1

000,


75-7

2X

`[mra

d]


150

-150

dP/P

* 1

000,




Solenoid cell – G4BL view




Summary

Present status…Prototype PIC lattices - quadrupole and solenoid channels

Lattices with absorbers studied via transfer matrix code

Beam dynamics studies vie multi-particle tracking

Building and testing G4BL tools

Chromatic aberration compensation with synchrotron motionProof-of-principle transport code tracking (solenoid triplet channel)

Future work…G4beamline simulation of a qudrupole/solenoid channel with absorbers

G4beamline simulation with absorbers followed by RF cavities - include multiple scattering and energy straggling effects

Emittance calculation - implementation of ecalc9

Parametric Resonance Ionization Cooling of Muons

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