Vlasov-Maxwell description of axisymmetric two-stream instabilities in high-intensity particle beams

Two-stream instabilities in high-intensity particle beams are investigated by making use of the Vlasov-Maxwell equations in the smooth-focusing approximation. Stability properties for the axisymmetric sausage and hollowing modes, characterized by radial mode numbers n = 1 and n = 2, respectively, are investigated, and the dispersion relations are obtained for the complex eigenfrequency in terms of the axial wavenumber k and other system parameters. Detailed stability properties are calculated over a wide range of normalized beam intensity S_b and fractional charge neutralization f. The growth rates of the sausage and hollowing modes are found to be of the same order of magnitude as that of the dipole-mode two-stream instability. Therefore, it is concluded that the axisymmetric sausage and hollowing instabilities may also be deleterious to intense ion beam propagation when a background component of electrons is present