Two-dimensional charge densities in intense rectangular ion beams with space-charge neutralization.

Abstract

The physics of an intense (current density \ensuremath{\gtrsim}1 mA/${\mathrm{cm}}^{2}$), positively charged, high aspect ratio rectangular ion beam is explored theoretically in a beam plasma containing a large density of space-charge neutralizing electrons. Before entering the plasma, the beam traverses an electron-free drift region in which it rapidly expands due to the mutual interion Coulomb repulsion. Within the beam plasma, the divergence rate is arrested by electron screening. Residual transverse velocities are small compared to the beam velocity and the beam ion density profile at the drift-plasma interface can be taken as longitudinally constant throughout the beam plasma. It is assumed that collisions between beam ions and residual gas molecules, producing a steady generation of electrons and slow residual gas ions, is the dominant mechanism sustaining the beam plasma. The cross-sectional profile of the beam ion distribution is of known form. To obtain the electron and slow-ion densities, charge is conserved and the energy balance of the plasma examined. Electron, slow-ion, and beam ion densities are then introduced into the two-dimensional Poisson equation to produce a second order partial integrodifferential equation requiring a numerical solution. This solution is obtained by expanding the density and potential functions in a complete set of orthogonal (Chebyshev) functions and reducing the differential equation to a system of linear algebraic equations. Applying the theory to a 100 keV, 10 mA, high aspect ratio arsenic beam, the electron density profile is predicted to display a shape similar to that of the beam ions. As with previous results for cylindrical beams, both the slow-ion and electron densities and, hence, the degree of space-charge neutralization are shown to depend strongly on the residual gas density, with the beam potential and electric field decreasing with increasing residual gas density. Unique to this analysis, it is predicted that the beam potential possesses substantial cylindrical symmetry even for very high aspect ratio rectangular beams, in complete agreement with recent experimental observations.