Linear theory of ionization cooling in 6d
This presentation is the property of its rightful owner.
Sponsored Links
1 / 21

Linear Theory of Ionization Cooling in 6D PowerPoint PPT Presentation


  • 152 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Linear Theory of Ionization Cooling in 6D

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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


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

  • Phase space vector


Emittance Exchange


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

In Larmor frame

Dispersion function decouples the betatron motion and dispersive effect


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

    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

    Model for Ionization Processin Larmor Frame

    Transverse:

    Longitudinal:

    M.S.

    straggling

    : Average loss replenished by RF


    Equation for 6-D Phase Space Variables

    • x = x +  Dx, Px = Px + 

    • z = z -

    • Dispersion vanishes at cavities

    • Drop suffix 


    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

    • 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 (emittances):

    • Distribution:


    Beam Invariants, Distribution, and Moments (contd.)

    • Non-vanishing moments:

      These are the inverses of Eq. (a).

    (b)


    Evolution Near Equilibrium

    • i are slowly varying functions of s.

    • Insert

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


    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

    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


    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.


  • Login