Intense sheet beam stability properties for uniform phase-space density

A self-consistent one-dimensional waterbag equilibrium f_b⁰(x,p_x) for a sheet beam propagating through a smooth focusing field is shown to be exactly solvable for the beam density n_b⁰(x) and space-charge potential φ⁰(x). A closed Schrodinger-like eigenvalue equation is derived for small-amplitude perturbations, and the WKB approximation is employed to determine the eigenfrequency spectrum as a function of the normalized beam intensity s_b = ω&capped;_pb²/γ²_bω²_β⊥, where ω&capped;_pb² = 4πn&capped;_be_b²/γ_bm_b is the relativistic plasma frequency-squared and n&capped;_b = n_b(x = 0) is the on-axis number density of beam particles.