Kinetic analysis of intense sheet beam stability properties for uniform phase-space density

This paper makes use of the Vlasov-Maxwell equations to investigate collective excitations in an intense sheet beam, infinite in the y and z directions, propagating in the z direction with directed kinetic energy (γ_b-1)m_bc². The beam is confined in the x direction by the smooth-focusing force F→_foc=-γ_bm_bω_β⊥²xe→_x, and perfectly conducting walls are located at x=±x_w. A self-consistent water bag equilibrium f_b⁰ satisfying the steady-state (∂/∂t=0) Vlasov-Maxwell equations is shown to be exactly solvable for the beam density n_b⁰(x) and electrostatic potential φ⁰(x). A closed Schrödinger-like eigenvalue equation is derived, assuming small-amplitude perturbations (δf_b,δφ) about the self-consistent water bag equilibrium, and the eigenfrequency spectrum is shown to be purely real. The WKB approximation is employed to determine the eigenfrequency spectrum as a function of the normalized beam intensity s_b=ω-^ _pb²/γ_b²ω_β⊥², where ω-^ _pb²=4πn-^ _be_b²/γ_bm_b and n-^ _b=n_b(x=0) is the on-axis number density of beam particles.