When one only considers a single bunch, one finds that the growth rates sharply increase at the current where the modes frequencies coincide. If the growth rates for multibunch modes are plotted versus current, they display similar behavior. In some cases, however, the current where they sharply increase is lower than the current calculated for a single bunch (multiplied by the number of bunches).
Figures 8 and 9 show the growth rates of the m=1 modes plotted versus current for two different subsets of the multibunch modes.
Figure 8: Growth rates of m=1 multibunch modes at injection
plotted versus total beam current. This is a subset of all the
m=1 multibunch modes. The worst mode in this set is driven by
the resistive wall impedance.
Figure 9: Growth rates of m=1 multibunch modes at injection
plotted versus total beam current. This is a subset of all the
m=1 multibunch modes. The worst mode in this set is driven by
one of the cavity higher order modes.
The resistive wall impedance decreases what one might call the mode coupling threshold by a significant amount, as can be seen from Fig. 8. The threshold now seems to be around 1.5 A or so, which would correspond to 1.2 A for the actual filling.
If feedback is included, these results get somewhat worse. I will use the 1460 kHz half-bandwidth feedback described above. Figures 10 and 11 show the results.
Figure 10: Growth rates of m=1 multibunch modes at injection
with feedback plotted versus total beam current. These are the
same subset of modes as were shown in
Fig. 8.
Figure 11: Growth rates of m=1 multibunch modes at injection
with feedback plotted versus total beam current. These are the
same subset of modes as were shown in
Fig. 9.
The point where the growth rates start to increase in Fig. 10 is about the same as it was without feedback as shown in Fig. 8. However, in Fig. 11, notice that the sharp increase occurs at a significantly lower current than it did without feedback (compare Fig. 9). The reason for this is that the transfer function for the feedback described in Eq. (9) has a non-zero imaginary part. This imaginary part causes a shift in the frequencies of the multibunch modes, causing the modes to intersect at currents that are lower than usual. A different set of modes gives worst behavior in the presence of feedback; they are shown in Fig. 12.
Figure 12: Growth rates of m=1 multibunch modes at injection
with feedback plotted versus total beam current.
This is the
worst group of modes in the presence of feedback. Note that the
vertical scale is a factor of 10 smaller than in previous
figures, so as to show more detail of the region where the
growth rates are beginning to increase.