Two-stream sausage and hollowing instabilities in high-intensity particle beams

Axisymmetric two-stream instabilities in high-intensity particle beams are investigated analytically by making use of the Vlasov-Maxwell equations in the smooth-focusing approximation. The eigenfunctions for the axisymmetric radial modes are calculated self-consistently in order to determine the dispersion relation describing collective stability properties. Stability properties for the 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 ω in terms of the axial wavenumber k and other system parameters. The eigenfunctions obtained self-consistently for the sausage and hollowing modes indicate that the perturbations exist only inside the beam. Therefore, the location of the conducting wall does not have an effect on stability behavior. The growth rates of the sausage and hollowing modes are of the same order of magnitude as that of the hose (dipole-mode) 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 presented.