Kinetic equilibrium of a periodically twisted ellipse-shaped charged-particle beam

A Vlasov equilibrium of the Kapchinskij-Vladimirskij form is obtained for a periodically twisted ellipse-shaped charged-particle beam in a nonaxisymmetric periodic magnetic focusing field. The single-particle Hamiltonian dynamics is analyzed self-consistently. A constant of motion analogous to the Courant-Snyder invariant is found. The equilibrium distribution function is constructed. The statistical properties of the beam equilibrium are studied. In the zero-temperature limit, the generalized envelope equations derived from the kinetic equilibrium theory recover the generalized envelope equations obtained in the cold-fluid equilibrium theory. Examples of periodically twisted elliptic beam equilibria are presented, and potential applications are discussed. For ribbon-beam amplifier and ribbon-beam klystron applications, the kinetic equilibrium theory predicts that the effect of beam temperature on the beam envelopes is negligibly small.