The usual expression for the longitudinal wake field in terms of the impedance is exact only for the model in which the source of the field is a rigid bunch. To account for a deforming bunch one has to invoke the \it complete impedance, a function of both wave number and frequency. A computation of the corresponding wake field would be expensive, since it would involve integrals over frequency and time in addition to the usual sum over wave number. We treat the problem of approximating this field in an example of current interest, the case of coherent synchrotron radiation (CSR) in the presence of shielding by the vacuum chamber.

Direct numerical methods for solving the Vlasov equation offer some advantages over macroparticle simulations, as they do not suffer from the consequences of the statistical fluctuations inherent in using a number of macroparticles smaller than thebunch population. Unfortunately these methods are more time-consuming and generally considered impractical in a full 6D phase space. However, in a lower-dimension phase space they may become attractive if the beam dynamics is sensitive to the presence of small charge-density fluctuations and a high resolution is needed. In this paper we present a 2D Vlasov solverfor studying the longitudinal beam dynamics in single-pass systems of interest for X-Ray FELs, where characterization of the microbunching instability stemming from self-field amplified noise is of particular relevance.

Direct numerical methods for solving the Vlasov equation offer some advantages over macroparticle simulations, as they do not suffer from the numerical noise inherent in using a number of macroparticles smaller than the bunch population. Unfortunately these methods are more time-consuming and generally considered impractical in a full 6D phase space. However, in a lower-dimension phase space they may become attractive if the beam dynamics is sensitive to the presence of small charge-density fluctuations and a high resolution is needed. In this paper we present a 2D Vlasov solver for studying the longitudinal beam dynamics in single-pass systems of interest for X-FEL's, where characterization of the microbunching instability is of particular relevance. The solver includes a model to account for the smearing effect of a finite horizontal emittance on microbuncing. We explore the effect of space charge and coherent synchrotron radiation (CSR). The numerical solutions are compared with results from linear theory and good agreement is found in the regime where linear theory applies.

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. Themodel 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 yielda 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 greaterthan 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.