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. 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. ,

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. 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

11. wedge 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(s+Ds) = x(s)-Dxd. • Px(s+Ds) = Px(s)-Dx+ • < d> = < > = 0 • < d2> = dDs, < 2> =  Ds, <d > = 0

19. 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-

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 “Robinson’s” Theorem • Numerical examples and comparison with simulations are in progress.