Coherent synchrotron radiation and bunch stability in a compactstorage ring

We examine the effect of the collective force due to coherent synchrotron radiation (CSR) in an electron storage ring with small bending radius. In a computation based on time-domain integration of the nonlinear Vlasov equation, we find the threshold current for a longitudinal microwave instability induced by CSR alone. The model accounts for suppression of radiation at long wave lengths due to shielding by the vacuum chamber. In a calculation just above threshold, small ripples in the charge distribution build up over a fraction of a synchrotron period, but then die out to yield a relatively smooth but altered distribution with eventual oscillations in bunch length. The instability evolves from small noise on an initial smooth bunch of r.m.s.length much greater than the shielding cutoff. The paper includes a derivation and extensive analysis of the complete impedance function Z for synchrotron radiation with parallel plate shielding. We find corrections to the lowest approximation to the coherent force which involve ''off-diagonal'' values of Z, that is, fields with phase velocity not equal to the particle velocity