1. Accelerator Physics Statistical and Collective Effects II
A. S. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage
Jefferson Lab
Old Dominion University
Lecture 17
G. A. Krafft
Jefferson Lab

2. Beam Temperature K-V has single value for the transverse Hamiltonian
No Temperature!
Temperature in beam introduces thermal spreads
Transverse Temperature
Debye length
Longitudinal Temperature
Landau Damping

3. Waterbag Distribution
Lemons and Thode were first to point out SC field is solved as Bessel Functions for a certain equation of state. Later, others, including my advisor and I showed the equation of state was exact for the waterbag distribution.

5. Equation for Beam Radius

6. Debye Length Picture*
*Davidson and Qin

7. Collisionless (Landau) Damping
Other important effect of thermal spreads in accelerator physics
Longitudinal Plasma Oscillations (1 D)

8. Linearized
In fluid limit plasma oscillations are undamped

9. Vlasov Analysis of Problem

10. Initial Value Problem
Laplace in t and Fourier in z

11. Dielectric function Landau (self-consistent) dielectric function
Solution for normal modes are

12. Collisionless Damping
For Lorentzian distribution
Landau damping rate

13. Negative Mass Instability
Simplified argument: assume longitudinal clump on otherwise uniform beam
Particles pushed away from clump centroid
If above transition, come back LATER if ahead of clump center and EARLIER if behind it
The clump is therefore enhanced!
INSTABILITY; particles act as if they have negative mass (they accelerate backward compared to force!) θ

14. Longitudinal Impedance

15. Frequency Domain

16. NMI Simple Analysis

17. Linearized Continuity Equation

18. Oscillation Frequency

19. NMI Growth time rc rb Δz

20. Stabilization by Beam Temperature?

21. Dispersion Relation

22. Landau Damping

24. LD from another view

25. Resonance Effect

26. Inhomogeneous Solution

27. Oscillators Similtaneously Excited

28. Oscillation Frequency

29. NMI Growth time rc rb Δz

30. Multipass BBU Instability

31. BBU Theory

33. Perturbation Theory Works

34. CEBAF (Design) Simulations
Stable
Bunch Number

35. Close to threshold
Bunch Number

36. Unstable
Bunch Number

37. Chromatic (Landau) Damping