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

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

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

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”

Modes of the model tune

Sextupole and damping Can be unstable when is smaller than 2

Head particle motion in phase space

FFT of equation-1 solution

Tune shift vs. Xmax

Ratio of amplitudes vs. Xmax

Ratio of amplitudes vs. tune shift

Sinusoidal driving nonlinear oscillation

Fold over effect Driving frequency Xmax

Transition point Xmax X_initial

Bi-stability

In phase space

In phase space with larger driving

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

2-particle model

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.