Longitudinal Motion in High Current Ion Beams - A Self-Consistent Phase Space Distribution with an Envelope Equation

Many applications of particle acceleration, such as heavy ion fusion, require longitudinal bunching of a high intensity particle beam to extremely high particle currents with correspondingly high space charge forces. This requires a precise analysis of longitudinal motion including stability analysis. Previous papers have treated the longitudinal space charge force as strictly linear, and have not been self-consistent; that is, they have not displayed a phase space distribution consistent with this linear force so that the transport of the phase space distribution could be followed, and departures from linearity could be analyzed. This is unlike the situation for transverse phase space where the Kapchinskij-Vladimirskij (K-V) distribution can be used as the basis of an analysis of transverse motion. In this paper we derive a self-consistent particle distribution in longitudinal phase space which is a solution of the Vlasov equation and derive an envelope equation for this solution. The solution is developed in Section II from a stationary solution of the Vlasov equation derived in Section I.