Transverse Multibunch Modes for Non - Rigid Bunches , Including Mode Coupling ∗

A method for computing transverse multibunch growth rates and frequency shifts in rings, which has been described previously [1, 2], is applied to the PEP-II B factory. The method allows multibunch modes with different internal-bunch oscillation modes to couple to one another, similar to single-bunch mode coupling. Including coupling between the multibunch modes gives effects similar to those seen in single-bunch mode coupling. These effects occur at currents that are lower than the single-bunch mode coupling threshold. 1 Physical Motivation Instability due to transverse mode coupling cannot occur unless two requirements are met. First, there must be a mechanism for the rigid (m = 0) motion to drive the head-tail (m = 1) motion, or vice-versa (only considering coupling between these two modes). In the case of a single bunch, this driving comes about because the head of the bunch sees no wakefield, whereas the tail of the bunch sees the wakefield of the entire bunch. The second requirement is that the frequencies of the two types of motion must be similar so that one mode can resonantly drive the other. In the case of a single bunch, this comes about because the average transverse wake in the bunch usually acts as an effective defocussing force on the bunch centroid, reducing the oscillation frequency of the m = 0 mode to the point where it eventually equals the frequency of one of the m = 1 modes. Now consider multibunch modes. A transverse multibunch mode is a mode where each bunch in the train executes identical types of oscillations: for example, rigid oscillations (m = 0), or head-tail oscillations (m = 1). Calculations up to this point have typically treated these multibunch modes as uncoupled. This paper shows that important effects are missed when coupling between these modes is ignored. One expects some coupling between the multibunch modes for the reasons outlined in the first paragraph. Consider an m = 0 multibunch oscillation. Such an oscillation will induce a wakefield, which in general has a nonzero slope in most places. This nonzero slope means that each bunch sees a different wakefield at the head and the tail. Thus, an m = 0 multibunch oscillation can drive an m = 1 multibunch oscillation. If the current is high enough and/or the bunches are close enough together so that the wakefields extend from one bunch to the next, the difference in wake seen across one bunch due to previous bunches can be significant, even compared to the difference in wake seen across the bunch due to its own wakefield. This can occur even when the wavelength of the wakefield in question is much longer than the length of the bunch. B factories such as PEP-II at SLAC [3] operate at high currents with a large number of bunches, and thus one might expect this driving to be significant. A broadband impedance corresponds to a wakefield that is short range; the wakefields do not typically extend from one bunch to the next. Therefore, when only a broadband impedance exists, mode coupling is adequately described by looking at a single bunch. But for narrow-band impedances, such as cavity higher order modes, which correspond to wakefields that extend over long distances, a bunch can create wakefields that are visible to several bunches behind it. Thus, these narrow-band impedances can easily be the mechanism through which the m = 0 and m = 1 multibunch modes drive one another. The decay time for the cavity higher order modes in the PEP-II B factory is much longer than the time between bunches [3, 4], and thus this driving can be significant. ∗Work supported by Department of Energy contract DE–AC03–76SF00515. Presented at the International Workshop on Collective Effects and Impedance for B-Factories, Tsukuba, Japan, 12–17 June, 1995. Since these narrow-band impedances couple the various bunches together, they also may cause frequency shifts and growth rates in the multibunch modes that are comparable to the synchrotron frequency. Thus, there are multibunch modes whose frequencies are shifted in such a way that the corresponding m = 0 and m = 1 multibunch mode frequencies coincide at currents that are smaller than the current at which the two modes coincided if only a single bunch was considered. Multibunch mode coupling is therefore expected to give sharp increases in growth rates at currents that are lower than the corresponding current at which single-bunch mode coupling occurs.