Title: Linear Theory of Ionization Cooling in 6D
1Linear 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
2- Theory development . . . . . . . . . . . . . . .
. . . . Kwang-Je Kim - Examples and asymmetric beams . . . . . . . .
Chun-xi Wang
3Ionization 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
4Equation of Motion
5Emittance Exchange
6Hamiltonian Under Consideration
Solenoid dipole quadrupole RF absorber
Goal theoretical framework and possible solution
solenoid
dipole
quadrupole
r.f.
,
7Equations for Dispersion Functions
In Larmor frame
Dispersion function decouples the betatron motion
and dispersive effect
8Coordinate Transformation
- Rotating (Larmor) frame
- Decouple the transverse and longitudinal motion
via dispersion - x xb Dxd, Px Pxb Dx?d
- Dispersion vanishes at rf
-
9Wedge Absorbers
qw
10Natural ionization energy loss is insufficient
for longitudinal cooling
Transverse cooling
slope is too gentle for effective
longitudinal cooling
Will be neglected
11Model for Ionization Processin Larmor Frame
?
M.S.
straggling
Average loss replenished by RF
12Equation for 6-D Phase Space Variables
- x x? ? Dx, Px Px? ?
- z z? -
- Dispersion vanishes at cavities
- Drop suffix ?
13Equilibrium 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.
14Quadratic Invariants
- Three Courant-Snyder invariants
-
- (?, ?, ?), (?z, ?z , ?z) Twist
parameters for ? and - Two more invariants when ?x ?y
-
- These are complete set!
15Beam Invariants, Distribution,and Moments
- Beam invariants (emittances)
- Distribution
-
-
-
16Beam Invariants, Distribution, and Moments
(contd.)
- Non-vanishing moments
-
-
- These are the inverses of Eq. (a).
(b)
17Evolution Near Equilibrium
- ?i are slowly varying functions of s.
-
-
-
-
- Insert
- Use Eq. (b) to convert to emittances.
18Evolution Near Equilibrium (contd.)
- Diffusive part straggling ?? and multiple
scattering ?. - x(sDs) x(s)-Dx?d.
- Px(sDs) Px(s)-Dx????
- lt ?dgt lt ?gt 0
- lt ?d2gt ?dDs, lt ?2gt ? Ds, lt?d ?gt 0
19Emittance Evolution Near Equilibrium
- ??s -(?-ec-) ?sec?aes?xyb?L?s,
- ??a -(?-ec-) ?aec?s ?a,
- ??xy -(?-ec-) ?xyes?s ?xy,
- ??L -(?-ec-) ?Lb?s ?L,
- ??z -(???2ec-) ?z ?z,
-
- C cos(qD-qw), s sin(qD qw), s-? sin
(qD?-qw) - b ?xb aes- be?s-?
20The Excitations
21Remarks
- 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
- Robinsons Theorem
- Numerical examples and comparison with
simulations are in progress. -
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