Multibunch Instability Investigations for a Tau-Charm Factory

In the design of high-luminosity colliders for high-energy physics, it has become clear that multibunch instabilities will be one of the primary effects that limit beam intensity, and hence luminosity. This paper reports on a series of calculations of multibunch growth rates, using the LBL accelerator physics code ZAP, that illustrate the seriousness of the effect for typical design parameters of a Tau-Charm Factory. A common feature of high-luminosity machines is the requirement of a small beta function at the interaction point. To maintain the advantages of a low beta function, however, requires that the rms bunch length, {sigma}{sub {ell}}, be smaller than {beta}*. This leads, in general, to several inconvenient aspects: (1) The requirement for short bunches leads to the need for a substantial amount of RF hardware-introducing just the narrow-band (high-Q) impedance that generates multibunch instabilities in the first place. (2) The need for short bunches means that bunch lengthening from the longitudinal microwave instability must be avoided. Since the longitudinal impedance Z{sub {parallel}}/n cannot be reduced indefinitely, there is a clear benefit to using many bunches, with lower current per bunch. (3) The short bunches have a Fourier spectrum extending up to very high frequencies, thus effectively sampling impedances in this regime that would be essentially invisible to longer bunches. This aspect can be seen in the exponential cutoff factor, proportional to ({sigma}{sub {ell}/R}){sup 2}, in the expressions for the effective impedance given. In practice, it is difficult to achieve a high luminosity without having a high average beam current in the rings. Because the multibunch growth rates scale linearly with average current, the resulting-rates tend to be very high. It might be imagined that, for sufficient bunch separation and low enough Q values for the higher-order cavity modes, the wake fields have time to die away between successive bunches, thus reducing the bunch-to-bunch coupling. For most cases of interest, however, it is hard to reduce the Q values sufficiently to achieve this condition. Because the details of higher-order modes of the RF cavities are only a guess at present, the results contained herein should not be interpreted quantitatively. However, experience has shown that the magnitudes of multibunch growth rates estimated as is done here are in reasonable agreement with observed growth rates under comparable conditions. Thus, although the particular modes that grow will depend on details of the impedance that are not well known at this time, the predicted growth rates are expected to reflect the requirements of a feedback system with good accuracy