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 ${\ensuremath{\gamma}}_{b}{m}_{b}{\ensuremath{\beta}}_{b}c$ 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 ${\mathbf{F}}_{\mathrm{foc}}^{b}=\ensuremath{-}{\ensuremath{\gamma}}_{b}\phantom{\rule{0ex}{0ex}}{m}_{b}{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{{0}^{2}}\phantom{\rule{0ex}{0ex}}{\mathbf{x}}_{\ensuremath{\perp}}$ in the smooth-focusing approximation. Here, ${\ensuremath{\omega}}_{\ensuremath{\beta}b}^{0}=\mathrm{const}$ is the effective betatron frequency associated with the applied focusing field, ${\mathbf{x}}_{\ensuremath{\perp}}$ is the transverse displacement from the beam axis, $({\ensuremath{\gamma}}_{b}\ensuremath{-}1){m}_{b}{c}^{2}$ is the ion kinetic energy, and ${V}_{b}={\ensuremath{\beta}}_{b}c$ is the average axial velocity, where ${\ensuremath{\gamma}}_{b}=(1\ensuremath{-}{\ensuremath{\beta}}_{b}^{2}\phantom{\rule{0ex}{0ex}}{)}^{\ensuremath{-}1/2}$. 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 $\ensuremath{-}e$ and ${n}_{e}$. For ${Z}_{b}{n}_{b}g{n}_{e}$, the electrons are electrostatically confined in the transverse direction by the space-charge potential $\ensuremath{\varphi}$ 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, ${k}_{z}^{2}{r}_{b}^{2}\ensuremath{\ll}1$, and sufficiently high frequency that $|\ensuremath{\omega}/{k}_{z}|\ensuremath{\gg}{v}_{\mathrm{Tez}}$ and $|\ensuremath{\omega}/{k}_{z}{\ensuremath{-}V}_{b}|\ensuremath{\gg}{v}_{\mathrm{Tbz}}$, where ${v}_{\mathrm{Tez}}$ and ${v}_{\mathrm{Tbz}}$ are the characteristic axial thermal speeds of the background electrons and beam ions. In this regime, Landau damping (in axial velocity space ${v}_{z}$) by resonant ions and electrons is negligibly small. We introduce the ion plasma frequency squared defined by ${\stackrel{^}{\ensuremath{\omega}}}_{\mathrm{pb}}^{2}\phantom{\rule{0ex}{0ex}}=4\ensuremath{\pi}{\stackrel{^}{n}}_{b}{Z}_{b}^{2}{e}^{2}/{\ensuremath{\gamma}}_{b}{m}_{b}$, and the fractional charge neutralization defined by $f={\stackrel{^}{n}}_{e}/{Z}_{b}\phantom{\rule{0ex}{0ex}}{\stackrel{^}{n}}_{b}$, where ${\stackrel{^}{n}}_{b}$ and ${\stackrel{^}{n}}_{e}$ are the characteristic ion and electron densities. The equilibrium and stability analysis is carried out for arbitrary normalized beam intensity ${\stackrel{^}{\ensuremath{\omega}}}_{\mathrm{pb}}^{2}\phantom{\rule{0ex}{0ex}}/{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{{0}^{2}}$, and arbitrary fractional charge neutralization $f$, consistent with radial confinement of the beam particles. For the moderately high beam intensities envisioned in the proton linacs and storage rings for the Accelerator for Production of Tritium and the Spallation Neutron Source, the normalized beam intensity is typically ${\stackrel{^}{\ensuremath{\omega}}}_{\mathrm{pb}}^{2}\phantom{\rule{0ex}{0ex}}/{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{{0}^{2}}\ensuremath{\lesssim}0.1$. For heavy ion fusion applications, however, the transverse beam emittance is very small, and the space-charge-dominated beam intensity is much larger, with ${\stackrel{^}{\ensuremath{\omega}}}_{\mathrm{pb}}^{2}\phantom{\rule{0ex}{0ex}}/{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{{0}^{2}}\ensuremath{\lesssim}2{\ensuremath{\gamma}}_{b}^{2}$. The stability analysis shows that the instability growth rate $\mathrm{Im}\ensuremath{\omega}$ increases with increasing normalized beam intensity ${\stackrel{^}{\ensuremath{\omega}}}_{\mathrm{pb}}^{2}\phantom{\rule{0ex}{0ex}}/{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{{0}^{2}}$ and increasing fractional charge neutralization $f$. In addition, the instability is strongest (largest growth rate) for perturbations with azimuthal mode number $\ensuremath{\ell}=1$, corresponding to a simple (dipole) transverse displacement of the beam ions and the background electrons. For the case of overlapping step-function density profiles for the beam ions and background electrons, corresponding to monoenergetic ions and electrons, a key result is that there is no threshold in beam intensity ${\stackrel{^}{\ensuremath{\omega}}}_{\mathrm{pb}}^{2}\phantom{\rule{0ex}{0ex}}/{\ensuremath{\omega}}_{\ensuremath{\beta}b}^{{0}^{2}}$ or fractional charge neutralization $f$ for the onset of instability. Finally, for the case of continuously varying density profiles with parabolic profile shape, a semiquantitative estimate is made of the effects of the corresponding spread in (depressed) betatron frequency on stability behavior, including an estimate of the instability threshold for the case of weak density nonuniformity.