Threshold studies of the microwave instability in electron storage rings

We use a Vlasov-Fokker-Planck program and a linearized Vlasov solver to study the microwave instability threshold of impedance models: (1) a $Q=1$ resonator and (2) shielded coherent synchrotron radiation (CSR), and find the results of the two programs agree well. For shielded CSR we show that only two dimensionless parameters, the shielding parameter $\ensuremath{\Pi}$ and the strength parameter ${S}_{\mathrm{csr}}$, are needed to describe the system. We further show that there is a strong instability associated with CSR, and that the threshold, to good approximation, is given by $({S}_{\mathrm{csr}}{)}_{\mathrm{th}}=0.5+0.12\ensuremath{\Pi}$. In particular, this means that shielding has little effect in stabilizing the beam for $\ensuremath{\Pi}\ensuremath{\lesssim}2$; for larger $\ensuremath{\Pi}$ it is effective, with threshold current depending on shielding aperture as ${h}^{\ensuremath{-}3/2}$. We, in addition, find another instability in the vicinity of $\ensuremath{\Pi}=0.7$ with a lower threshold, $({S}_{\mathrm{csr}}{)}_{\mathrm{th}}\ensuremath{\approx}0.2$. We find that the threshold to this instability depends strongly on damping time, $({S}_{\mathrm{csr}}{)}_{\mathrm{th}}\ensuremath{\sim}{\ensuremath{\tau}}_{p}^{\ensuremath{-}1/2}$, and that the tune spread at threshold is small---both hallmarks of a weak instability.