Lawrence Berkeley National Laboratory

Because of tight requirements on beam quality longitudinal single-bunch instabilities are a serious concern for the damping rings of the next generation of linear colliders. Unlike multi-bunch instabilities they cannot be damped using feed-back systems and need to be avoided altogether. We present an analysis of these instabilities for the current Feb. ~;03 NLC main damping ring design, with attention paid to coherent synchrotron radiation and vacuum chamber effects, with the latter including the main components (RF cavities, BPM's, and resistive wall). The study is carried out by solving the Vlasov-Fokker-Planck equation for the longitudinal motion numerically. Comparison is made, whenever possible, with linear theory. We find that collective effects are dominated by coherent synchrotron radiation and estimate the instability threshold to be safely above 6 times the design current.