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_{b}c^{2}. The beam is confined in the x direction by the smooth-focusing force F[over →]_{foc}=-γ_{b}m_{b}ω_{β⊥}^{2}xe[over →]_{x}, and perfectly conducting walls are located at x=±x_{w}. A self-consistent water bag equilibrium f_{b}^{0} satisfying the steady-state (∂/∂t=0) Vlasov-Maxwell equations is shown to be exactly solvable for the beam density n_{b}^{0}(x) and electrostatic potential φ^{0}(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[over ^]_{pb}^{2}/γ_{b}^{2}ω_{β⊥}^{2}, where ω[over ^]_{pb}^{2}=4πn[over ^]_{b}e_{b}^{2}/γ_{b}m_{b} and n[over ^]_{b}=n_{b}(x=0) is the on-axis number density of beam particles