Microinstability threshold of an axially focused neutralized ion beam

Abstract

The response to electrostatic perturbations, φ=φ̂(r,z)ei(lθ−ωt), is investigated for the Migma Exyder, a proposed disk-shaped device where ions are trapped in the periphery and focused towards the axis [Nucl. Instrum. Methods A 271, 214 (1988)]. The ions are modeled as a relativistic monoenergetic distribution with a small spread in the impact parameter (the distance of closest approach to the axis) and with the axial extent ΔZ much shorter than the radial extent R0. It is found that odd l modes do not give rise to a low-frequency response. For l=0, an energy principle for low frequency ion–ion interactions is formulated, and it is shown that the extremizing perturbation is located in a narrow region surrounding the axis. The threshold is proportional to the total number of stored particles, N=4ε0(π−2)γ(γ2−1) ×mic2R0/q2i, where γ is the usual relativistic factor, and this stability condition therefore allows for a high density on axis in highly focused systems. Compared to a cylindrical shape, a disk-shaped plasma gives an improvement in the density by the factor R0/ΔZ. The threshold in particle number for the relativistic ion–ion two-stream instability scales as γ3 and thus with relativistic effects the instability is further suppressed. Inclusion of the electron response increases the threshold even further. Preliminary investigations indicate that finite frequency l=0 modes can give a lower threshold than the zero frequency case, at least for nonrelativistic beam energies. These finite frequency instabilities can arise from coupling between positive and negative energy modes. Finite l modes arise if electrons are magnetized at the axis of the device. A balance between the electron field energy and the cross field electron convection establishes discrete modes. When these modes resonate with the ion bounce motion or azimuthal drift motion, bands of instability arise with relatively low density thresholds compared to the l=0 case.