Oscillation modes, Decoherence, Beam Transfer Functions, and Instabilities for a Beam Subjected to Space-Charge Forces

Based on the oscillation modes of a bunch subject to the space-charge force, we discuss the decoherence of the dipole oscillation, the beam transfer function, and the beam instability. The longitudinal phase space is discretized into $n_l\times n_J$ grids in the azimuthal and radial directions, respectively. The transverse dipole moment in the longitudinal phase space is represented by a vector of length $n_l{n}_J$, and the transformation given by the Vlasov equation is represented by a matrix transformation of size $n_l{n}_J\times n_l{n}_J$. Solving the eigenvalue problem of this matrix gives the $n_l{n}_J$ eigen-oscillation modes of the bunch. Using the resulting eigenvectors within a standard linear-algebraic treatment, we analyze these three phenomena. The analysis was performed for the Rapid Cycling Synchrotron (RCS) of the China Spallation Neutron Source (CSNS) using $n_l=41$ and $n_J=40$, and it visualizes the features related to the continuous distribution of the tune, Landau damping, and isolated modes separated from the continuous tune region.