We have described how to construct the beam transfer function and corresponding stability diagrams for transverse oscillations under certain approximations. Other authors (for example, [4, 5]) have addressed this same issue; the significant difference in our approach is considering tune spread in two planes.
The beam transfer function gives the actual response of the beam to an oscillating driving force; it ignores effects of the impedance, or equivalently, it assumes that the beam contains an arbitrarily small amount of current. This can in principle be measured in a real machine by applying an oscillating driving force and measuring the dipole moment of the beam. The amplitude and phase relationship of the response with respect to the driving force will give the full complex beam transfer function.
A stability diagram, essentially the inverse of the beam transfer function, allows one to predict whether a given effective impedance will result in instability or will be ``Landau damped.'' Essentially, if an impedance is not strong enough, an excitation of the beam will filament, causing the dipole moment to be reduced. The rate at which this filamentation occurs depends on the range of tunes in the beam. It also depends on what direction the impedance attempts to shift the tune. If the impedance attempts to shift the tune in a direction in which a significant fraction of the beam can absorb the excitation energy through filamentation, then one will have strong Landau damping. If the attempted shift is in a direction in which the beam has little tune spread, the Landau damping will be less effective.
Here we have analyzed the case where the beam has tune spread in both of the transverse planes. The tune spreads will not act symmetrically, since the symmetry is broken by the fact that the oscillation under consideration is in only one of the planes. We have demonstrated that, as one would expect, a tune spread in both positive and negative directions will allow a wide range of real impedance-induced tune shifts in both directions, whereas if the tune spread is in only one direction, then there is only a limited impedance-induced tune shift allowed in one of the directions.
The practical meaning of this is complicated. First of all, one requires a sufficient stability domain along the real axis to encompass the real coherent tune shifts. Secondly, multibunch instabilities will lead to effective impedances with corresponding non-zero growth rates. Thus, not only does one need tune spread sufficient to deal with the real tune shifts, but also such that the stability diagrams allow the imaginary parts of the multibunch modes to fall within the diagram. Finally, note that there are minor gains to be obtained in the allowable imaginary part for a given tune spread if the tune spreads are chosen to be in the same direction or the opposite direction; however, whether it is more beneficial to have the tune spreads in the same or opposite direction seems to depend on the distribution being considered. Whether one needs tune spreads in both directions is determined by whether all the modes are shifted in the same direction or if they are shifted in opposite directions.
Certain types of distributions can lead to stability diagrams which do not seem to make sense, as demonstrated by our example of a truncated Gaussian. These are related to discontinuities in the derivative of the distributions.
Note from the stability diagrams for the quasi-parabolic distribution that when compared to a Gaussian distribution, the quasi-parabolic distribution is significantly less effective in Landau damping. The reason is that the Gaussian distribution has a much larger range of tune spread; thus, an excitation will decohere more quickly due to the larger range of frequencies involved. Therefore, Landau damping will be more effective for the Gaussian distribution than for the quasi-parabolic distribution.
One also needs to consider that the tune shift with amplitude that one attempts to induce with nonlinear magnets such as octupoles may be partially compensated by already existing nonlinear errors in the machine.
Finally, we mention that there is further work being done on this problem to address various aspects which are not covered in this analysis. In particular, we are addressing the issue of higher-order head-tail modes, as well as adding tune spread in the longitudinal direction. Also, we are considering how to make sense of the stability diagrams such as the ones we generated for the truncated Gaussian distribution.