Fokker-Planck analysis of transverse collective instabilities in electron storage rings

We analyze single bunch transverse instabilities due to wakefields using a Fokker-Planck model. We first expand on the work of T. Suzuki, Part. Accel. 12, 237 (1982) to derive the theoretical model including chromaticity, both dipolar and quadrupolar transverse wakefields, and the effects of damping and diffusion due to the synchrotron radiation. We reduce the problem to a linear matrix equation, whose eigenvalues and eigenvectors determine the collective stability of the beam. We then show that various predictions of the theory agree quite well with results from particle tracking simulations, including the threshold current for transverse instability and the profile of the unstable mode. In particular, we find that predicting collective stability for high energy electron beams at moderate to large values of chromaticity requires the full Fokker-Planck analysis to properly account for the effects of damping and diffusion due to synchrotron radiation.

Figure 1

Lattice Twiss functions and parameters used in the tracking simulations.

Figure 2

Real and imaginary parts of $\mathrm{\Omega }$ for the two-mode theory with ${\xi }_x=0$ (a) and ${\xi }_x=0.75$ (b). Panel (c) compares the instability threshold predicted by theory and simulation for two different rf voltages. The two mode theory follows the theory reasonably well provided $k_{\xi }{\sigma }_z=2\pi {\xi }_x{\sigma }_z/{\alpha }_c{\mathcal{C}}_R\lesssim 0.7$.

Figure 3

(a) Maximum stable current $I_{\text{thresh}}$ as a function of chromaticity for two different rf voltages. As the chromatic head-tail shift becomes larger than unity the current threshold is higher for the lower rf voltage that has a longer bunch length. (b) shows that the $I_{\text{thresh}}$ is approximately a function of $k_{\xi }{\sigma }_z$ for both the resistive wall impedance $Z_{\mathrm{RW}}$ and the constant wake $W_{\text{const}}$

Figure 4

Comparison of the real and imaginary parts of the longitudinal distribution function ${\tilde{g}}_1$ obtained theoretically [(a) and (b)] and via simulation [(c) and (d)] at a chromaticity of 5 units. As indicated by the four pairs of lobes, the growing mode is dominated by the azimuthal mode number $m=-4$; theory indicates that $\sim 70\%$ of the mode content has $m=-4$, while $\sim 20\%$ has $m=-2$, $m=-3$, or $m=-4$. These plots map the variation in the longitudinal plane of the unstable betatron oscillation; at this turn in the simulation the betatron amplitude is about twice the rms beam size at equilibrium, while the maximum longitudinal variation is a little less than ${\sigma }_x/2$.

Figure 5

(a) Change in the resistive wall instability threshold current at ${\xi }_x=0$ as a function of the quadrupolar to dipolar impedance ratio $r_Q$. (b) Comparison of the two-mode theory and simulation for the purely dipolar impedance $Z_D$ and the sum $Z_D+Z_Q$.

Figure 6

Comparison of theory and simulation including the quadrupolar impedance. (a) plots the current limit with and without the quadrupolar impedance, which for the resistive wall elements of Table 1 has $Z_Q^{\beta }(k)\approx -Z_D^{\beta }(k)/2$. Including $Z_Q$ increases $I_{\text{thresh}}$ by 10–40% at large chromaticity. (b) plots the real part of the unstable growing mode obtained from the simulation with ${\xi }_x=5$, which should be compared to the theory in panel (c). The unstable ${\tilde{g}}_1$ has $\sim 35\%$ of the basis functions with $m=-4$ and $\sim 30\%$ with $m=-3$, and a broader range of azimuthal variation then the purely dipolar impedance example of Fig. 4.

Figure 7

Effect of adding a longitudinal impedance $Z_{\parallel }$. Panel (a) shows that both theory and simulation predict a reduction in $I_{\text{thresh}}$ at low ${\xi }_x$, which can be attributed to the decrease in the effective synchrotron frequency. At higher ${\xi }_x$ and $I$, (b) shows that the bunch lengthening due to $Z_{\parallel }$ increases the maximum stable current, and the theory agrees well with the simulation for ${\xi }_x\lesssim 4$ ($I\lesssim 3\mathrm{mA}$); for larger currents the theory overestimates $I_{\text{thresh}}$. Panel (c) shows the difference in the longitudinal potential $V_z$ from that of the assumed simple harmonic oscillator at the simulation-predicted $I_{\text{thresh}}$ for ${\xi }_x=0$, 2, 3, 4, and 5.

Figure 8

Evolution of the dipole moment $⟨x⟩$ for three different chromaticities when one includes Fokker-Planck damping and diffusion. The colored lines indicate simulation results using the parameters from Fig. 1, which has ${\omega }_s{\tau }_x\approx 53.1$, ${\omega }_s{\tau }_z\approx 61.9$, and $k_{\xi }{\sigma }_z/{\xi }_x\approx 0.37$. For this lattice, the nonlinear tune shift with amplitude is negligible at these small oscillation amplitudes.