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

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

Kwang-Je Kim

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

Chun-xi Wang

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

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

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

Emittance Exchange

Dipole (bend)

+p

0

-p

x xop/pDipole introduces dispersion

Wedge Absorber reduces energy spread

beam

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

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

Equations for Dispersion Functions

Dispersion function decouples the betatron motion and dispersive effect

In Larmor frame

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

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

Wedge Absorbers

w

y

yx

xss,y,x

wW sin,cosy

,x

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

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

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

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

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

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

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

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

Evolution Near Equilibrium i are slowly varying functions of s.

•

•

• Insert

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

22

Mx

Mx xxx2x

dsd

21

dsd

dsd

xdsd

x2xdsd

.Diff

2

.Diss

.Dissdsd

.Diff.DissM dsd

dsd

dsd

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

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

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

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

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