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

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Linear theory of ionization cooling in 6d

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



Ionization cooling theory in linear approximation
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
Equation of Motion . Kwang-Je Kim

  • Phase space vector


Emittance exchange
Emittance Exchange . Kwang-Je Kim


Hamiltonian under consideration

Lab . Kwang-Je Kim

frame

rotating frame with symmetric focusing

Hamiltonian Under Consideration

Solenoid + dipole + quadrupole + RF + absorber

Goal: theoretical framework and possible solution

solenoid

dipole

quadrupole

r.f.

,


Equations for dispersion functions
Equations for Dispersion Functions . Kwang-Je Kim

In Larmor frame

Dispersion function decouples the betatron motion and dispersive effect


Coordinate transformation
Coordinate Transformation . Kwang-Je Kim

  • Rotating (Larmor) frame

  • Decouple the transverse and longitudinal motion via dispersion:

    • x = xb + Dxd, Px = Pxb + Dxd

  • Dispersion vanishes at rf


  • Wedge absorbers
    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


    Model for ionization process in larmor frame

    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
    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
    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
    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 and moments
    Beam Invariants, Distribution, . Kwang-Je Kimand Moments

    • Beam invariants (emittances):

    • Distribution:


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

    • Non-vanishing moments:

      These are the inverses of Eq. (a).

    (b)


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

    • Diffusive part: straggling  and multiple scattering .

      • x(s+Ds) = x(s)-Dxd.

      • Px(s+Ds) = Px(s)-Dx+

      • < d> = < > = 0

      • < d2> = dDs, < 2> =  Ds, <d > = 0


    Emittance evolution near equilibrium
    Emittance Evolution Near Equilibrium . Kwang-Je Kim

    s = -(-ec-) s+ec+a+es+xy+bL+s,

    a = -(-ec-) a+ec+s+ a,

    xy = -(-ec-) xy+es+s+ xy,

    L = -(-ec-) L+bs+ L,

    z = -(+2ec-) z+ z,

    C± = cos(qD-qw), s± = sin(qD ±qw), s- = sin (qD-qw)

    b = xb + aes- + bes-


    The excitations
    The Excitations . Kwang-Je Kim


    Remarks
    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.


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