1. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 1 Lecture 15  Radiation Damping Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Alex Bogacz, Geoff Krafft and Timofey Zolkin

2. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 2 Lecture 15  Radiation Damping Outline Beam dynamics with synchrotron radiation: Define action-angle variables for describing symplectic motion of a particle along a beam line. Discuss the effect of synchrotron radiation on the (linear) motion of particles in storage rings. Derive expressions for the damping times of the vertical, horizontal and longitudinal emittances. Introduce the synchrotron radiation integrals (Sand’s Integrals). Discuss the effects of quantum excitation, and derive expressions for the equilibrium horizontal and longitudinal beam emittances in an electron storage ring. M. Sands, “The physics of electron storage rings, an introduction” SLAC-121. 1970 A. Wolski, University of Liverpool and the Cockcroft Institute, CAS 2009, http://cas.web.cern.ch/cas/Germany2009/Lectures/PDF-Web/Wolski-1.pdf USPAS, Fort Collins, CO, June 10-21, 2013

3. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 3 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Coordinate system

4. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 4 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Longitudinal coordinate

5. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 5 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Energy deviation

6. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 6 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Canonical variables

7. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 7 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Symplectic matrices U U U U

8. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 8 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Twiss parameters and the particle action

9. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 9 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Cartesian variables and action-angle variables

10. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 10 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Action and Emittance

11. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 11 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Action and Radiation

12. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 12 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Radiation damping of vertical emittance

13. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 13 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Radiation damping of vertical emittance

14. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 14 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Radiation damping of vertical emittance

15. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 15 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Radiation damping of vertical emittance

16. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 16 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Radiation damping of vertical emittance

17. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 17 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Synchrotron radiation energy loss U 0

18. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 18 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Synchrotron radiation energy loss

19. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 19 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 The first synchrotron radiation integral

20. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 20 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of horizontal emittance

21. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 21 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Horizontal-longitudinal coupling

22. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 22 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of horizontal emittance

23. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 23 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Radiation damping of horizontal emittance

24. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 24 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A: Damping of horizontal emittance

25. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 25 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A: Damping of horizontal emittance

26. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 26 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A: Damping of horizontal emittance

27. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 27 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A: Damping of horizontal emittance

28. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 28 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A: Damping of horizontal emittance

29. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 29 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix A: Damping of horizontal emittance

30. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 30 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

31. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 31 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

32. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 32 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

33. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 33 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

34. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 34 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

35. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 35 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

36. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 36 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

37. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 37 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

38. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 38 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Damping of synchrotron oscillations

39. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 39 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Summary: Synchrotron radiation damping

40. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 40 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Quantum excitation

41. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 41 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Quantum excitation of horizontal emittance

42. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 42 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Quantum excitation of horizontal emittance

43. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 43 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B: Quantum excitation of horizontal emittance

44. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 44 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B: Quantum excitation of horizontal emittance

45. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 45 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B: Quantum excitation of horizontal emittance

46. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 46 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Appendix B: Quantum excitation of horizontal emittance

47. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 47 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Equilibrium horizontal emittance

48. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 48 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Quantum excitation of vertical emittance

49. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 49 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Quantum excitation of synchrotron oscillations

50. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 50 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Quantum excitation of synchrotron oscillations

51. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 51 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Natural energy spread

52. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 52 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Summary: Radiation Damping

53. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 53 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Summary: Synchrotron radiation integrals

54. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 54 Lecture 15  Radiation Damping USPAS, Fort Collins, CO, June 10-21, 2013 Summary: Sand’s Integrals

55. Operated by JSA for the U.S. Department of Energy Thomas Jefferson National Accelerator Facility 55 Lecture 15  Radiation Damping Summary So far we have: discussed the effect of synchrotron radiation on the (linear) motion of particles in storage rings; derived expressions for the damping times of the vertical, horizontal and longitudinal emittances; discussed the effects of quantum excitation, and derive expressions for the equilibrium horizontal and longitudinal beam emittances in an electron storage ring. USPAS, Fort Collins, CO, June 10-21, 2013