Landau damping of the weak head-tail instability for beams with quadratic amplitude-dependent betatron tunes and binomial amplitude distributions

The simple theory of the weak head-tail instability does not exhibit a threshold current. Experimentally, the instability does not always appear, due to the readily available stabilizing mechanism known as Landau damping. Here we extend the theory to include Landau damping that arises from a quadratic dependence of betatron tune on amplitude, as occurs when octupoles are present in the synchrotron lattice. The binomial density distribution, which is appropriate for proton beams, is chosen as the steady-state for both the transverse and longitudinal phase-space distributions. Using Sacherer´s formalism we look for solutions by expanding the perturbation in a set of basis functions. The method involves evaluating the dispersion integral, and finding the eigenvalues of the interaction matrix. The results can be represented in a “stability diagram” with the standard interpretation but with the axes representing the real and imaginary parts of the eigenvalues. As a numerical example, we determine the octupole strength needed to damp the reported high-order modes observed in the CERN PS for bunches with “LHC characteristics”