A high-order moment simulation model

We consider a high-current beam transport system in which the particle motion in the two transverse directions (x and y) are coupled. Chernin¹ has studied the beam envelope, with linear transverse spatial dependence of magnetic field, for a monoenergetic particle beam. We are interested in a beam which is distributed in energy, and a transport system with nonlinear transverse dependence of the magnetic field. For this type of system the spatial coordinate (z) along the beam motion can no longer be equivalenced to the time variable; faster particles will overtake the slower particles as time evolves. We present relativistically covariant moment equations for modelling beam transport, based on the work of Newcomb² and Amendt and Weitzner³. The beam is described by a set of partial differential moment equations, instead of a set of ordinary differential envelope equations. (This aspect of our approach distinguishes it from the work of Chernin¹ and of Channell and coworkers^{4, 5}.) The code formalism is based on transverse averages of the moment equations obtained from the relativistic Vlasov equation. The spatial coordinate along the beam motion is discretized. (The theory by Channel et al. uses moment equations to model a three-dimensional beam bunch. For long bunches, however, it is impractical to carry high enough longitudinal moments to model the oscillations within the bunch.) The moment equations are closed by setting higher order correlation functions to zero. A non-linear space charge model, similar to the one used in the BEDLAM code⁴, has been developed. The formalism for the code as well as results of the model will be presented. Comparisions will be made between simulations keeping up to second, third and fourth order correlations.