Chromaticity effects on head-tail instabilities for broadband impedance using two particle model, Vlasov analysis, and simulations

Effects of the chromaticity on head-tail instabilities for broadband impedances are comprehensively studied, using the two particle model, the Vlasov analysis and computer simulations. We show both in the two particle model and the Vlasov analysis with the trapezoidal (semiconstant) wake model that we can derive universal contour plots for the growth factor as a function of the two dimensionless parameters: the wakefield strength, ϒ , and the difference of the betatron phase advances between the head and the tail, χ . They reveal how the chromaticity affects strong head-tail instabilities and excites head-tail instabilities. We also apply the LEP (Large Electron-Positron Collider) broadband resonator model to the Vlasov approach and find that the results are in very good agreement with those of the trapezoidal wake model. The theoretical findings are also reinforced by the simulation results. The trapezoidal wake model turns out to be a very useful tool since it significantly simplifies the time domain analysis and provides well-behaved impedance at the same time.