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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

- 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

- Phase space vector

Lab

frame

rotating frame with symmetric focusing

Solenoid + dipole + quadrupole + RF + absorber

Goal: theoretical framework and possible solution

solenoid

dipole

quadrupole

r.f.

,

In Larmor frame

Dispersion function decouples the betatron motion and dispersive effect

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

qw

momentum gain

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

Transverse:

Longitudinal:

M.S.

straggling

: Average loss replenished by RF

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

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

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

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

- Beam invariants (emittances):
- Distribution:

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

(b)

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

- 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

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-

- 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.