1. Effect of nonlinearity on Head-Tail instability 3/18/04

2. contents 2-particle model and head-tail instability Nonlinear force –One particle under nonlinear force –Decomposition of the results –Nonlinear resonance under sinusoidal driving –Is this a way to simplify the dynamics of 2- particle model under nonlinear force Conclusion

3. 2-particle model A bunch made of two macro-particles, each of them executes synchrotron oscillation. The tail-particle sees the wake field produced by head- particle. After each half synchrotron period, the two particles switch longitudinal positions

4. Equations of motion: Where, is the wake force, and is related to by: * is a dimensionless parameter Solutions stable condition:, when * See A. Chao “physics of collective beam instabilities in high energy accelerator”

5. Modes of the model tune

6. Sextupole and damping Can be unstable when is smaller than 2

7. Head particle motion in phase space

8. FFT of equation-1 solution

9. Tune shift vs. Xmax

10. Ratio of amplitudes vs. Xmax

11. Ratio of amplitudes vs. tune shift

12. Sinusoidal driving nonlinear oscillation

13. Fold over effect Driving frequency Xmax

14. Transition point Xmax X_initial

15. Bi-stability

16. In phase space

17. In phase space with larger driving

19. Compare the approximation with the total wake case Total wake Fundamentalcomponent only

20. 2-particle model

21. Conclusion The wake force from head particle is approximated with a simple harmonic solution. The equation of motion of tail particle becomes a sinusoidal driving nonlinear oscillation. Based on understanding of motion of the single particles, further work instability for 2-particle model under nonlinear force can be better investigated.