Theory of coupled-bunch longitudinal instabilities in a storage ring for arbitrary rf potentials

We present a theory of coupled-bunch longitudinal instabilities that is based on the coupled set of Vlasov equations governing the particle distribution function, and can be applied to arbitrary longitudinal potentials. We find that the coupled-bunch growth rate is given by a dispersion relation that is parametrized by the eigenvalues of the linear (harmonic) coupled-bunch matrix problem. Our theory therefore treats the wakefield-driven source of the instability and the effect of Landau damping together and on equal footing, and also indicates that the stabilizing effects of Landau damping can approximately be compared to the instability growth rates in a harmonic potential that has the same bunch length. We then apply the theory to a weakly nonlinear oscillator and to a quartic potential that is relevant for ultralow emittance storage rings that employ bunch-lengthening systems. In the latter case we find that the theory compares quite favorably with particle tracking simulations for the parameters of the planned upgrade of the Advanced Photon Source.

Figure 1

Theoretical coupled-bunch growth rate as a function of the HOM frequency difference from a revolution harmonic. (a) plots the theoretical growth rate as we increase the matrix growth rate $\lambda$ from being equal to the nonlinearity to four times that. (b) shows the effect of Landau damping by comparing two cases from panel (a) to those os a simple harmonic oscillator.

Figure 2

Comparison of theory (blue) to elegant simulations (red) of the predicted and observed multi-bunch growth rate for no synchrotron radiation damping and longitudinal potential $V_0\propto z^4$. Panel (a) illustrates the effective Landau damping by plotting the threshold matrix growth rate as a function of normalized HOM detuning; the Landau damping rate is $\sim ({\alpha }_c{\sigma }_{\delta }/{\sigma }_t)/10\sim ⟨f_s⟩$. Panel (b) plots the simulated and theoretical growth rate for the 921 MHz HOM in Table 1, and compares them to the theory of a simple harmonic oscillator (SHO) of same bunch duration ${\sigma }_t\approx 51\mathrm{ps}$.

Figure 3

Comparison of theory (blue) to elegant simulations (red) of the predicted and observed multibunch growth rate including synchrotron radiation. Panel (a) plots the results for the 921 MHz HOM using parameters from Tables 1 and 2, i.e., is identical to Fig. 2 but for the inclusion of synchrotron emission, which we model in the theory by simply subtracting off the synchrotron damping rate of 49 $1/\mathrm{s}$. The agreement is quite good. Panel (b) shows that the agreement becomes less impressive as one approaches the instability threshold. The red line shows that halving the shunt impedance (and, hence $\lambda$) agrees reasonably well; on the other hand, while subtracting the damping time from the theory predicts stability when $R_s$ is cut by one-third, the simulations show an instability near resonance.