Negative energy oscillations and instability of intense beams

The energy of transverse space-charge oscillations of a focused beam is derived as kinetic plus electric field energy of a quasi-monochromatic spectrum of oscillations. It is shown that the existence of modes of negative energy in certain beam models with non-monotonic energy distribution, like the delta-function Kapchinstij-Vladimirsky (K-V) distribution, can give rise to instability if coupling occurs between a negative and a positive energy mode. This mechanism explains the instability that is obtained from analyzing the dispersion relation of a continuously focused intense beam with a K-V distribution of particles. (author)