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Linear Theory of Ionization Cooling in 6DPowerPoint Presentation

Linear Theory of Ionization Cooling in 6D

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### 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

- Theory development . . . . . . . . . . . . . . . . . . .Kwang-Je Kim
- Examples and asymmetric beams . . . . . . . . Chun-xi Wang

Ionization Cooling Theory in Linear Approximation .Kwang-Je Kim

- 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

Equation of Motion .Kwang-Je Kim

- Phase space vector

Emittance Exchange .Kwang-Je Kim

Lab .Kwang-Je Kim

frame

rotating frame with symmetric focusing

Hamiltonian Under ConsiderationSolenoid + dipole + quadrupole + RF + absorber

Goal: theoretical framework and possible solution

solenoid

dipole

quadrupole

r.f.

,

Equations for Dispersion Functions .Kwang-Je Kim

In Larmor frame

Dispersion function decouples the betatron motion and dispersive effect

Coordinate Transformation .Kwang-Je Kim Dispersion vanishes at rf

- Rotating (Larmor) frame
- Decouple the transverse and longitudinal motion via dispersion:
- x = xb + Dxd, Px = Pxb + Dxd

Wedge Absorbers .Kwang-Je Kim

qw

momentum gain .Kwang-Je Kim

net loss

momentum loss

Natural ionization energy loss is insufficient for longitudinal cooling

Transverse cooling

slope is too gentle for

effective longitudinal

cooling

Will be neglected

wedge .Kwang-Je Kim

Model for Ionization Processin Larmor Frame

Transverse:

Longitudinal:

M.S.

straggling

: Average loss replenished by RF

Equation for 6-D Phase Space Variables .Kwang-Je Kim

- x = x + Dx, Px = Px +
- z = z -
- Dispersion vanishes at cavities
- Drop suffix

Equilibrium Distribution .Kwang-Je Kim

- Linear stochastic equation Gaussian distribution
- For weak dissipation, the equilibrium distribution evolves approximately as Hamiltonian system.
I is a quadratic invariant with periodic coefficients.

Quadratic Invariants .Kwang-Je Kim

- Three Courant-Snyder invariants:
(, , ), (z, z , z); Twist parameters for and ||

- Two more invariants when x = y:
These are complete set!

Beam Invariants, Distribution, .Kwang-Je Kimand Moments

- Beam invariants (emittances):
- Distribution:

Beam Invariants, Distribution, and Moments (contd.) .Kwang-Je Kim

- Non-vanishing moments:
These are the inverses of Eq. (a).

(b)

Evolution Near Equilibrium .Kwang-Je Kim

- i are slowly varying functions of s.
- Insert
- Use Eq. (b) to convert to emittances.

Evolution Near Equilibrium (contd.) .Kwang-Je Kim

- Diffusive part: straggling and multiple scattering .
- x(s+Ds) = x(s)-Dxd.
- Px(s+Ds) = Px(s)-Dx+
- < d> = < > = 0
- < d2> = dDs, < 2> = Ds, <d > = 0

Emittance Evolution Near Equilibrium .Kwang-Je Kim

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(qD-qw), s± = sin(qD ±qw), s- = sin (qD-qw)

b = xb + aes- + bes-

The Excitations .Kwang-Je Kim

Remarks .Kwang-Je Kim

- 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.