Three-dimensional kinetic stability theorem for high-intensity charged particle beams

Global conservation constraints obtained from the nonlinear Vlasov–Maxwell equations are used to derive a three-dimensional kinetic stability theorem for an intense non-neutral ion beam (or charge bunch) propagating in the z direction with average axial velocity $v_b=\mathrm{const}$ and characteristic kinetic energy ${\mathrm{(\gamma }}_b{\text{−1)mc}}^2$ in the laboratory frame. Here, ${\gamma }_b{\text{=(1−v}}_b^2{\mathrm{/c}}^2{)}^{\text{−1/2}}$ is the relativistic mass factor, and a perfectly conducting cylindrical wall is located at radius ${\mathrm{r=r}}_w,$ where ${\mathrm{r=(x}}^2{\mathrm{+y}}^2{)}^{\text{1/2}}$ is the radial distance from the beam axis. The particle motion in the beam frame (“primed” coordinates) is assumed to be nonrelativistic, and the beam is assumed to have sufficiently high directed axial velocity that $v_b\mathrm{\gg |}{{v}}^{\prime }\mathrm{|.}$ Space-charge effects and transverse electromagnetic effects are incorporated into the analysis in a fully self-consistent manner. The nonlinear Vlasov–Maxwell equations are Lorentz-transformed to the beam frame, and the applied focusing potential is assumed to have the (time-stationary) form ${\psi }_{\mathrm{sf}}^{\prime }({{x}}^{\prime }{\mathrm{)=(\gamma }}_b{\text{m/2)[ω}}_{\mathrm{\beta \perp }}^2{\mathrm{(x}}^{\text{′2}}{\mathrm{+y}}^{\text{′2}}{\mathrm{)+\omega }}_{\mathrm{\beta z}}^2{z}^{\text{′2}}\mathrm{],}$ where ${\omega }_{\mathrm{\beta \perp }}$ and ${\omega }_{\mathrm{\beta z}}$ are constant focusing frequencies. It is shown that a sufficient condition for linear and nonlinear stability for perturbations with arbitrary polarization about a beam equilibrium distribution $f_{\mathrm{eq}}({{x}}^{\prime },{{p}}^{\prime })$ is that $f_{\mathrm{eq}}$ be a monotonically decreasing function of the single-particle energy, i.e., ${\mathrm{\partial f}}_{\mathrm{eq}}{\mathrm{(H}}^{\prime }{\mathrm{)/\partial H}}^{\prime }\text{⩽0.}$ Here, $H^{\prime }={{p}}^{\text{′2}}{\text{/2m+ψ}}_{\mathrm{sf}}^{\prime }({{x}}^{\prime }{\mathrm{)+\phi }}_{\mathrm{eq}}({{x}}^{\prime }\mathrm{),}$ where ${\phi }_{\mathrm{eq}}({{x}}^{\prime })$ is the space-charge potential.