KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 1 Linear Theory of Ionization Cooling in 6D Kwang-Je Kim & Chun-xi Wang University.

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KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source

Linear Theory of Ionization Cooling in 6D

Kwang-Je Kim & Chun-xi Wang

University of Chicago and Argonne National Laboratory

Cooling Theory/Simulation Day

Illinois Institute of Technology

February 5, 2002

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• Theory development . . . . . . . . . . . . . . . . . . .

Kwang-Je Kim

• Examples and asymmetric beams . . . . . . . .

Chun-xi Wang

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Ionization Cooling Theory in Linear Approximation

• Similar in principle to radiation damping in electron storage rings, but needs to take into account:- Solenoidal focusing and angular momentum

- Emittance exchange

• Slow evolution near equilibrium can be described by five Hamiltonian invariants

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Equation of Motion• Phase space vector

•

•

0

0Tyx p

pp;),z,p,y,p,x(

X

motionnHamiltonia;JH,XdsdX

H H

HXX21 TH

MH dsdX

dsdX

dsdX

0100

1000

0001

0010

J

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Emittance Exchange

Dipole (bend)

+p

0

-p

x xop/pDipole introduces dispersion

Wedge Absorber reduces energy spread

beam

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Hamiltonian Under ConsiderationSolenoid + dipole + quadrupole + RF + absorber

Goal: theoretical framework and possible solution

Lab frame

dipole quadrupole r.f.

rotating frame with symmetric focusing

,

solenoid

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Equations for Dispersion Functions

Dispersion function decouples the betatron motion and dispersive effect

In Larmor frame

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• Rotating (Larmor) frame

• Decouple the transverse and longitudinal motion via dispersion:

x = x + Dx, Px = Px + Dx

• Dispersion vanishes at rf

Coordinate Transformation

ˆ,PDyDPDxDzz yyyxxx

22222y

2x zVI

21

yx2K

PP21 H

sinDcosD

1)s(I yx

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Wedge Absorbers

w

y

yx

xss,y,x

wW sin,cosy

,x

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Natural ionization energy loss is insufficient for longitudinal cooling

slope is too gentle for effective longitudinalcooling

momentum gain

momentum loss net loss

Transverse cooling

Will be neglected

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Model for Ionization Processin Larmor Frame

Transverse:

Longitudinal:

xePP zds

d

M.S.

yy

xxds

d

wedge

straggling : Average loss replenished by RF

y

,x

v,u

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Equation for 6-D Phase Space Variables• x = x + Dx, Px = Px +

• z = z -

• Dispersion vanishes at cavities

• Drop suffix

y)(xD

yyyxxx PDyDPDxD 0VDVD xx

δDPx xx xxx

2x ξPηδDxκP

δDPy yy

yyyy2

y ξPηδDPκP

yyxx PDPDηδI(s)z

δyx ξδ)DvD(uvy)(uxV(s)zδ

xyxxx ξDκDδκyPP

yxxyy ξPχDδκxPP

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Equilibrium Distribution• Linear stochastic equation Gaussian distribution

• For weak dissipation, the equilibrium distribution evolves approximately as Hamiltonian system.

I is a quadratic invariant with periodic coefficients.

s;xINes,xf periodic:sI,xxsIs;xI ijjiij

0f,sf

H

0dsdI

I,sI

H

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Quadratic Invariants• Three Courant-Snyder invariants:

(, , ), (z, z , z); Twist parameters for and ||

• Two more invariants when x = y:

These are complete set!

yx,PxP2xI 2xx

2x

2zz

2zz z2zI

xyL yPxPI

yxxyxy PP)yPxP(xyI

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Beam Invariants, Distribution,and Moments

• Beam invariants (emittances):

• Distribution:

)a(L,xy,z,y,xi,I21

ii

2L

2xyyxD4zD4D6 ,

z2zI

D42zLL2xyIxy2yIxxIy

exp2

1s,xf

D63

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Beam Invariants, Distribution, and Moments (contd.)

• Non-vanishing moments:

These are the inverses of Eq. (a).

zyx,γα,β,εP,Px,x x2

xx

2

γα,β,PP,2

PyPx,xy xyyx

xy

Lxy ε2PyPx

(b)

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Evolution Near Equilibrium i are slowly varying functions of s.

•

•

• Insert

• Use Eq. (b) to convert to emittances.

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Mx

Mx xxx2x

dsd

21

dsd

dsd

xdsd

x2xdsd

.Diff

2

.Diss

.Dissdsd

.Diff.DissM dsd

dsd

dsd

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Evolution Near Equilibrium (contd.)• Diffusive part: straggling and multiple scattering .

x(s+s) = x(s)-Dx.

Px(s+s) = Px(s)-Dx+

< > = < > = 0

< > = s, < > = s, <> = 0

2

x2 Dx

s

xxx DDxPs

2

x2

x Dps

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Emittance Evolution Near Equilibrium

s = -(-ec-) s+ec+a+es+xy+bL+s,

a = -(-ec-) a+ec+s+ a,

xy = -(-ec-) xy+es+s+ xy,

L = -(-ec-) L+bs+ L,

z = -(+2ec-) z+ z,

C± = cos(D-w), s± = sin(D ± w), s- = sin (D-w)

b = x + es- + es-

D21

e,21

e,21

yxa,s D

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The Excitations

221

,22

1yyxx HH

z2

y2

xzz DD21

xyxy H

xyyxL DDDD

yx,DDD2D 2xxx

2xx H

yxxyyxyxxy DDDDDDDD H

yxa,s

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Remarks• Reproduces the straight channel results for D = 0.• Damping of the longitudinal emittance at the expense of the transverse damping.• 6-D phase spare area

“Robinson’s” Theorem

• Numerical examples and comparison with simulations are in progress.

2L

2xyyxzD6

D6D6D6 22dsd

KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 1 Linear Theory of Ionization Cooling in 6D Kwang-Je Kim & Chun-xi Wang University.

Documents

ionization cooling theory

rf slide

longitudinal cooling

solenoid slide

wang slide

suffix slide

p x d x dispersion

larmor frame slide