Kinetic description of electron-proton instability in high-intensity proton linacs and storage rings based on the Vlasov-Maxwell equations

The present analysis makes use of the Vlasov-Maxwell equations to develop a fully kinetic description of the electrostatic, electron-ion two-stream instability driven by the directed axial motion of a high-intensity ion beam propagating in the $z$ direction with average axial momentum ${\gamma }_b{m}_b{\beta }_{b}c$ through a stationary population of background electrons. The ion beam has characteristic radius $r_b$ and is treated as continuous in the $z$ direction, and the applied transverse focusing force on the beam ions is modeled by ${\mathbf{F}}_{\mathrm{foc}}^b=-{\gamma }_b{m}_b{\omega }_{\beta b}^{0^2}{\mathbf{x}}_{\perp }$ in the smooth-focusing approximation. Here, ${\omega }_{\beta b}^0=\mathrm{const}$ is the effective betatron frequency associated with the applied focusing field, ${\mathbf{x}}_{\perp }$ is the transverse displacement from the beam axis, $({\gamma }_b-1\mathrm{})m_b{c}^2$ is the ion kinetic energy, and $V_b={\beta }_{b}c$ is the average axial velocity, where ${\gamma }_b=(1-{\beta }_b^2{)}^{-1\mathrm{}/2}$. Furthermore, the ion motion in the beam frame is assumed to be nonrelativistic, and the electron motion in the laboratory frame is assumed to be nonrelativistic. The ion charge and number density are denoted by ${+Z}_{b}e$ and $n_b$, and the electron charge and number density by $-e$ and $n_e$. For $Z_b{n}_b>n_e$, the electrons are electrostatically confined in the transverse direction by the space-charge potential $\phi$ produced by the excess ion charge. The equilibrium and stability analysis retains the effects of finite radial geometry transverse to the beam propagation direction, including the presence of a perfectly conducting cylindrical wall located at radius ${r=r}_w$. In addition, the analysis assumes perturbations with long axial wavelength, $k_z^2{r}_b^2\ll 1$, and sufficiently high frequency that $|\omega /k_z|\gg v_{\mathrm{Tez}}$ and