Linear Theory of Ionization Cooling in 6D
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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