1. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 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. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 2 Theory development...................Kwang-Je Kim Examples and asymmetric beams........ Chun-xi Wang

3. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 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. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 4 Equation of Motion Phase space vector

5. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 5 Emittance Exchange

6. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 6 Hamiltonian Under Consideration Solenoid + dipole + quadrupole + RF + absorber Goal: theoretical framework and possible solution Lab frame dipolequadrupoler.f. rotating frame with symmetric focusing, solenoid

7. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 7 Equations for Dispersion Functions Dispersion function decouples the betatron motion and dispersive effect In Larmor frame

8. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 8 Rotating (Larmor) frame Decouple the transverse and longitudinal motion via dispersion: x = x  + D x , P x = P x  + D x  Dispersion vanishes at rf Coordinate Transformation

9. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 9 Wedge Absorbers ww

10. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 10 Natural ionization energy loss is insufficient for longitudinal cooling slope is too gentle for effective longitudinal cooling momentum gain momentum loss net loss Transverse cooling Will be neglected

11. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 11 Model for Ionization Process in Larmor Frame Transverse: Longitudinal: M.S. wedge straggling : Average loss replenished by RF 

12. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 12 Equation for 6-D Phase Space Variables x = x  +  D x, P x = P x  +  z = z  - Dispersion vanishes at cavities Drop suffix 

13. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 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. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 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. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 15 Beam Invariants, Distribution, and Moments Beam invariants (emittances): Distribution:

16. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 16 Beam Invariants, Distribution, and Moments (contd.) Non-vanishing moments: These are the inverses of Eq. (a). (b)

17. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 17 Evolution Near Equilibrium  i are slowly varying functions of s. Insert Use Eq. (b) to convert to emittances.

18. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 18 Evolution Near Equilibrium (contd.) Diffusive part: straggling   and multiple scattering . x(s+  s) = x(s)-D x  . P x (s+  s) = P x (s)-D x   +  = = 0 =    s, =   s, = 0

19. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 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(  D -  w ), s ± = sin(  D ±  w ), s - = sin (  D -  w ) b =  x  +  es - +  es -

20. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 20 The Excitations

21. KJK 2/5/02 IIT Cooling Theory/Simulation Day Advanced Photon Source 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.