Numerical simulation of intense-beam experiments at LLNL and LBNL

Analytical theory and nonlinearδfperturbative simulations of temperature anisotropy instability in intense charged particle beams

“…Therefore, for r w =r b * 3, we can approximate fD n;n 0 !g by a tridiagonal matrix. In this case, for the lowest-order radial modes (n 1 and n 2), the matrix dispersion relation (20) can be approximated by…”

“…(9) and (12)] calculated numerically using the equilibrium space-charge potential obtained from Poisson's equation (2). The dispersion matrix obtained in this way can be used to determine the unstable mode frequencies and growth rates by numerically searching for the solutions to the dispersion equation (20) for the complex frequency ! Re!…”

“…Of particular importance at the high beam currents and charge densities of practical interest are the effects of the intense self-fields produced by the beam space charge and current on determining the detailed equilibrium, stability, and transport properties. While considerable progress has been made in understanding the self-consistent evolution of the beam distribution function, f b x; p; t, and self-generated electric and magnetic fields, E s x; t and B s x; t, in kinetic analyses based on the nonlinear Vlasov-Maxwell equations [1,6 -10], in numerical simulation studies of intense beam propagation [11][12][13][14][15][16][17][18][19][20][21], and in macroscopic warm-fluid models [22 -25], the effects of finite geometry and space-charge effects often make predictions of detailed stability behavior difficult. It is therefore important to develop an improved understanding of fundamental collective stability properties, including the case where a large temperature anisotropy T ?b T kb can drive a Harris-like instability [26,27], familiar in the study of electrically neutral plasmas.…”

“…At the same time, the transverse temperature may increase due to nonlinearities in the applied and selffield forces, nonstationary beam profiles, and beam mismatch. These processes provide the free energy to drive collective instabilities and may lead to a deterioration of beam quality [20,28,29]. The instability may also lead to an increase of longitudinal velocity spread, which will make the focusing of the beam difficult and may impose a limit on the minimum spot size achievable in heavy ion fusion experiments.…”

Overview of the WARP code and studies of transverse resonance effects

“…The code was developed for Heavy-Ion beamdriven inertial confinement Fusion (HIP) accelerator studies (3)(4)(5)(6)(7)(8). In this application, and particularly in the US approach based on induction technology, space charge forces dominate over the beam thermal pressure WARP (9)(10)(11)(12)(13)(14) incorporates detailed descriptions of various accelerator and beamline elements, and was designed to follow beams efficiently over long path lengths A hierarchy of models affords increasing levels of detail in the description of the accelerator lattice While some of these models are based on an expansion in powers of the off-axis separation, the code can use models which are good to "all orders" (such as field specification on a full 3-D grid) when sufficient knowledge of the field is available Several Poisson equation solvers are available, optionally incorporating internal conducting elements from "first principles" with subgrid-scale resolution of the conductor boundary locations WARP uses time (rather than path length) as the independent coordinate for particle motion, this facilitates the treatment of longitudinally-extended beams The code gets its name from its use of "warped" coordinates, that is, a sequence of Cartesian grid sections aligned with the local beamline (so that each lattice element is described in its own natural coordinate system) linked by sections of "polar coordinate" grid. Coordinate transformations account for the bends while preserving the symplectic nature of the underlying advance The fields from neighboring elements can overlap, even when P r those elements are separated by a bend and so are described with reference to different frames The user specifies a coordinate system which describes the beamline as it is laid out "in the laboratory," and need not specify a reference orbit In the 3-D and r,z models, no paraxial approximation is invoked.…”