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

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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


Linear theory of ionization cooling in 6d

  • Theory development . . . . . . . . . . . . . . . . . . .Kwang-Je Kim

  • Examples and asymmetric beams . . . . . . . . Chun-xi Wang


Ionization cooling theory in linear approximation

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


Equation of motion

Equation of Motion

  • Phase space vector


Emittance exchange

Emittance Exchange


Hamiltonian under consideration

Lab

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

In Larmor frame

Dispersion function decouples the betatron motion and dispersive effect


Coordinate transformation

Coordinate Transformation

  • 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

    qw


    Linear theory of ionization cooling in 6d

    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


    Model for ionization process in larmor frame

    wedge

    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

    • x = x +  Dx, Px = Px + 

    • z = z -

    • Dispersion vanishes at cavities

    • Drop suffix 


    Equilibrium distribution

    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.


    Quadratic invariants

    Quadratic Invariants

    • 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,and Moments

    • Beam invariants (emittances):

    • Distribution:


    Beam invariants distribution and moments contd

    Beam Invariants, Distribution, and Moments (contd.)

    • Non-vanishing moments:

      These are the inverses of Eq. (a).

    (b)


    Evolution near equilibrium

    Evolution Near Equilibrium

    • i are slowly varying functions of s.

    • Insert

    • Use Eq. (b) to convert to emittances.


    Evolution near equilibrium contd

    Evolution Near Equilibrium (contd.)

    • 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

    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


    Remarks

    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

      “Robinson’s” Theorem

    • Numerical examples and comparison with simulations are in progress.


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