Warm-fluid stability properties of intense non-neutral charged particle beams with pressure anisotropy

The macroscopic warm-fluid model developed by Lund and Davidson [Phys. Plasmas 5, 3028 (1998)] is used in the smooth-focusing approximation to investigate detailed electrostatic stability properties of an intense charged particle beam with pressure anisotropy. The macroscopic fluid-Maxwell equations are linearized for small-amplitude perturbations, and an eigenvalue equation is derived for the perturbed electrostatic potential $\mathrm{\delta \phi (}{x}\mathrm{,t),}$ allowing for arbitrary anisotropy in the perpendicular and parallel pressures, $P_{\perp }^0\mathrm{(r)}$ and $P_{\Vert }^0\mathrm{(r).}$ Detailed stability properties are calculated numerically for the case of extreme anisotropy with $P_{\Vert }^0\text{(r)=0}$ and $P_{\perp }^0\text{(r)≠0,}$ assuming axisymmetric wave perturbations $\text{(∂/∂θ=0)}$ of the form $\mathrm{\delta \phi (}{x}\mathrm{,t)=\delta \phi ̂(r)}\exp {\mathrm{(ik}}_z\mathrm{z-i\omega t),}$ where $k_z$ is the axial wavenumber, and $\text{Imω>0}$ corresponds to instability (temporal growth). For $k_z\text{=0,}$ the analysis of the eigenvalue equation leads to a discrete spectrum ${\mathrm{\{\omega }}_n\}$ of stable oscillations with ${\mathrm{Im\omega }}_n\text{=0,}$ where n is the radial mode number. On the other hand, for sufficiently large values of $k_z{r}_b,$ where $r_b$ is the beam radius, the analysis leads to an anisotropy-driven instability $\text{(Imω>0)}$ provided the normalized Debye length ${\mathrm{(\Gamma }}_D{\mathrm{=\lambda }}_{\mathrm{D\perp }}{\mathrm{/r}}_b)$ is sufficiently large and the normalized beam intensity ${\mathrm{(s}}_b{\mathrm{=\omega ̂}}_{\mathrm{pb}}^2{\text{/2γ}}_b^2{\omega }_{\mathrm{\beta \perp }}^2)$ is sufficiently below the space-charge limit. Depending on system parameters, the growth rate can be a substantial fraction of the focusing frequency ${\omega }_{\mathrm{\beta \perp }}$ of the applied field.