The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010 1 Update on...

The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

1

Update on Parametric-resonance Ionization Cooling (PIC)

V.S. MorozovOld Dominion University

V. Ivanov, R.P. Johnson, M. NeubauerMuons, Inc.

A. AfanasevHampton University and Muons, Inc.

A. S. Bogacz, Y.S. DerbenevThomas Jefferson National Accelerator Facility

K. YoneharaFermi National Accelerator Laboratory

Muons, Inc.

2

PIC Concept

• Parametric resonance induced in muon cooling channel

• Muon beam naturally focused with period of free oscillations

• Wedge-shaped absorber plates combined with energy-restoring RF cavities placed at focal points (assuming aberrations corrected)– Ionization cooling maintains constant angular spread– Parametric resonance causes strong beam size reduction– Emittance exchange at wedge absorbers produces longitudinal cooling

• Resulting equilibrium transverse emittances are an order of magnitude smaller than in conventional ionization cooling

Muons, Inc.


• Varying dispersion– small at absorbers to minimize energy straggling– non-zero at absorbers for emittance exchange– large between focal points for compensating chromatic and spherical

aberrations

• Correlated optics– one free oscillations’ period low-integer multiple of the other - / + = 1 or 2– dispersion magnitude oscillation period D factor of 2 shorter than +

• Required features can be produced by epicyclic magnetic field configuration– solenoid with two superimposed different-period transverse helical fields

▫ uniform smoothly-varying fringe-field-free configuration 3

PIC RequirementsMuons, Inc.


4

Epicyclic Channel• Two transverse helical fields with wave numbers k1 and k2

• Equation of motion

• Analytic solution under approximation kc = const (pz = const)

• Dispersion function containing two oscillating terms

• Condition for dispersion to periodically return to zero

1 2( ), /T c T c z zp ik p i b b k eB p c

1 21 2 1 2

ik z ik zb b B e B e

1 2 1 1 2 2

1 2 1 2

/ /,T

c c z c c

b b b k b kip u x iy

k k k k p k k k k

Muons, Inc.

1 22 2

1 2( ) ( )c c

b buD p

p k k k k

2

221 2

1 1

,c

c

k kBk k

B k k


5

Approach to DesigningCorrelated Optics

• Consider second helix perturbation

• Adjust desired free-oscillation period ratio - /+ = 1 or 2 in primary helix

• By choosing wave number k2 of second helix, set dispersion oscillation period D = |2/(k2-k1)| such that + /D = 2

• Adjust strength of second helix to create oscillating dispersion

• Iteratively adjust - /+ and + /D by changing helices’ parameters until correlated optics is achieved

Muons, Inc.


6

Single Helix• Equilibrium condition

• Orbit stability condition

• Betatron tunes

• For given r = Q+/Q- , one can solve for ∂b/∂a if

2, , 1

1 1cT

z

kpb qq

B q p k

221 2

2

1ˆ0 ( ) 14 1

qG q g D R

2 2 2 3/2

12 2

(1 ) (1 )ˆ ,1

q bD g g

pk a

Muons, Inc.

2 2Q R R G

42 2 2 2 2 2 2 2 2 2 2( 1) [2(1 ) ( 1) ] ( )( 1) 0

4r q r r q q r r q


• No solution for Q+ = Q-

• Two solution regions for Q+ = 2Q-

– | | << 1, -2 < q < -1, B2/B1 ~ 1

– | | >> 1Choose:

= -5.4q = -1.54Bsol = 2 Tbd = -0.154 Tbq = 0.065 T/mkc = 32.9 mk = -61.0 mQ- = 0.464, Q+ = 0.929B2/B1 ~ 0.04

7

Adjusting Betatron TunesMuons, Inc.


• Using one-period linear transformation matrix

• Track particle over many periodsand take Fourier transform of coordinate vector component

1 1

11 1

2 2

2 21

1 2

0, ,0

1 1cos(2 ) tr , cos(2 ) tr

2 2

n n

u uu u A E AM M X M X Mu u F B Bu u

Q A Q B

8

Determining Betatron TunesMuons, Inc.


9

Finding Periodic Orbit• No exact analytic solution in case of two helices

• Stable periodic orbit does not always exit

• Begin with single helix where stable periodic orbit is knows to exist

• Use one or combination of the following to find periodic orbit when second helix is present– Adiabatically increase strength of second helix while tracking orbit– Use “friction” force making particle trajectory converge to periodic orbit

– Increase second helix’s strength from zero in steps finding periodic orbit iteratively on each step

Muons, Inc.


[ ]( ) z

z

p e B Q Q ee

10

Dispersion in Epicyclic ChannelMuons, Inc.


Second helix strength

|D| = const |D| oscillates not reaching 0 |D| oscillates reaching 0

11

Periodic Orbit in Epicyclic Channel

Muons, Inc.


• Consider: no solenoid, two helices of equal strengths with equal-magnitude and opposite-sign wave numbers

• Field periodic with = 2/k, • Vertical field only at any point in horizontal plane• Periodic orbit lies in horizontal plane

12

Another Option for PIC ChannelMuons, Inc.


1 2 1 20 0, ( , ) ( , )ikz ikz

z zb b e b e b b

1 2

1 2

1m0.74 T0

d d

q q

b bb b

• More conventional orbital dynamics problem• Horizontal and vertical motion uncoupled• Magnetic structure accommodates both muon charges• Transverse motion stable in both dimensions• Dispersion has oscillatory behavior

13

Periodic Orbit and DispersionMuons, Inc.


• Eliminates scattering on pressurizing gas and cavity wall while enhancing accelerating voltage

14

RF Cavity Concept for EPIC & REMEXMuons, Inc.


Open Cell EPIC/REMEX Cell

Closed Pillbox Cell

BEAM

Beryllium grids

IrisesBe wedge Thermal stabilizer

max

2

4accel

surf

E

E

max

3

4accel

surf

E

E

max

4

4accel

surf

E

E

The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010 1 Update on...

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