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Linear Theory of Ionization Cooling in 6D

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KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source ... Solenoidal focusing and angular momentum. Emittance exchange ... – PowerPoint PPT presentation

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Title: Linear Theory of Ionization Cooling in 6D


1
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

2
  • Theory development . . . . . . . . . . . . . . .
    . . . . Kwang-Je Kim
  • Examples and asymmetric beams . . . . . . . .
    Chun-xi Wang

3
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

4
Equation of Motion
  • Phase space vector

5
Emittance Exchange
6
Hamiltonian Under Consideration
Solenoid dipole quadrupole RF absorber
Goal theoretical framework and possible solution
solenoid
dipole
quadrupole
r.f.
,
7
Equations for Dispersion Functions
In Larmor frame
Dispersion function decouples the betatron motion
and dispersive effect
8
Coordinate Transformation
  • Rotating (Larmor) frame
  • Decouple the transverse and longitudinal motion
    via dispersion
  • x xb Dxd, Px Pxb Dx?d
  • Dispersion vanishes at rf

9
Wedge Absorbers
qw
10
Natural ionization energy loss is insufficient
for longitudinal cooling
Transverse cooling
slope is too gentle for effective
longitudinal cooling
Will be neglected
11
Model for Ionization Processin Larmor Frame
  • Transverse
  • Longitudinal

?
M.S.
straggling
Average loss replenished by RF
12
Equation for 6-D Phase Space Variables
  • x x? ? Dx, Px Px? ?
  • z z? -
  • Dispersion vanishes at cavities
  • Drop suffix ?

13
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.

14
Quadratic Invariants
  • Three Courant-Snyder invariants
  • (?, ?, ?), (?z, ?z , ?z) Twist
    parameters for ? and
  • Two more invariants when ?x ?y
  • These are complete set!

15
Beam Invariants, Distribution,and Moments
  • Beam invariants (emittances)
  • Distribution

16
Beam Invariants, Distribution, and Moments
(contd.)
  • Non-vanishing moments
  • These are the inverses of Eq. (a).

(b)
17
Evolution Near Equilibrium
  • ?i are slowly varying functions of s.
  • Insert
  • Use Eq. (b) to convert to emittances.

18
Evolution 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

19
Emittance 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-?

20
The Excitations

21
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
  • Robinsons Theorem
  • Numerical examples and comparison with
    simulations are in progress.
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