Theory on Electron Cooling

1. Theory on Electron Cooling He Zhang CASA Journal Club Talk, 12/03/2012

2. Theory on Electron Cooling This talk is based on the following references: • YA. S. Derbenev and A. N. SkrinskyThe Kinetics of Electron Cooling of Beams in Heavy Particle Storage Rings • YA. S. Derbenev and A. N. Skrinsky The Effect of an Accompanying Magnetic Field on Electron Cooling • YA. S. Derbenev and A. N. SkrinskyThe Physics of Electron Cooling • A. H. Sorensen and E. BonderupElectron Cooling • H. Poth Electron Cooling: Theory, Experiment, Application • V. V. Parkhomchuk and A. N. SkrinskyElectron Cooling: Physics and Prospective Applications • V. V. Parkhomchuk and A. N. Skrinsky Electron Cooling: 35 years of development • J. D. Jackson Classical Electrodynamics • F. Yang Atomic Physics (in Chinese) • R. O. DendyPlasma Dynamics He Zhang

3. Theory on Electron Cooling Basic Idea He Zhang

4. Theory on Electron Cooling Two models: • Binary collision model: • Collisions between ions and electrons • Statistical effect • Dielectric plasma model: • Electromagnetic wave travelling through the plasma • Response of the plasma He Zhang

5. Binary Collision Model Coulomb scattering formula Momentum lost Mean energy lost through electron gas He Zhang

6. Binary Collision Model Friction force If electrons are moving with If electrons have a velocity distribution He Zhang

7. Binary Collision Model Diffusion coefficients If electrons have a velocity distribution Relation between Friction and Diffusion coefficients He Zhang

8. Binary Collision Model An example: a spherical Maxwellian electron velocity distribution Rewrite the friction force formula He Zhang

9. Binary Collision Model Now plug in the electron velocity distribution: with Using the error function and integrate by parts He Zhang

10. Binary Collision Model The friction force: Similarly one can calculate the diffusion coefficients: He Zhang

11. Binary Collision Model He Zhang

12. Binary Collision Model Another important case: disk-like velocity distribution • Deeper potential and larger friction force in longitudinal direction • Force can be calculated using the following approximation He Zhang

13. Binary Collision Model Under a longitudinal magnetic field • Larmor resonance • Two classes of the collisions • Fast collision • Adiabatic collision He Zhang

14. Binary Collision Model No-magnetic component (same as before) Magnetic or adiabatic component • Cannot use the same formula due to the loss of transverse freedoms for the electrons • Diffusion coefficients can be calculated He Zhang

15. Binary Collision Model He Zhang

16. Binary Collision Model • When and • When and He Zhang

17. Dielectric Plasma Model • Electron beam is treated as a continuous fluid (plasma) • A moving ion inside the electron plasma will induce a field • Define the dielectric function as He Zhang

18. Dielectric Plasma Model From Poisson Equation We get For point charge He Zhang

19. Dielectric Plasma Model Electron plasma at rest with no magnetic field Because of the symmetry, is directed along Using He Zhang

20. Dielectric Plasma Model with is determined by the minimum impact parameter Comparing with the bi-collision formula Agree! He Zhang

21. Dielectric Plasma Model Electron gas at rest with finite magnetic field Dielectric function Friction force He Zhang

22. Dielectric Plasma Model He Zhang

23. Dielectric Plasma Model Thermal electron gas with finite magnetic field He Zhang

24. Cooling of Positive and Negative Ions by Magnetized electrons • When electrons will be push back by negative ions • Extra • Extra friction force He Zhang

25. Fokker-Planck equation • probability for a particle at velocity to have a change of velocity during time . • distribution function at velocity space using the Taylor expansion of , and , we get • Knowing and , can be solved. He Zhang

26. Summary • Friction and diffusion • Binary collision model • Dielectric plasma model • Solve Fokker-Plank equation to get He Zhang