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 γbmbβbc through a stationary population of background electrons. The ion beam has characteristic radius rb and is treated as continuous in the z direction, and the applied transverse focusing force on the beam ions is modeled by Ffocb=−γbmbωβb02x⊥ in the smooth-focusing approximation. Here, ωβb0=const is the effective betatron frequency associated with the applied focusing field, x⊥ is the transverse displacement from the beam axis, (γb−1)mbc2 is the ion kinetic energy, and Vb=βbc is the average axial velocity, where γb=(1−βb2)−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 +Zbe and nb , and the electron charge and number density by −e and ne . For Zbnb>ne , the electrons are electrostatically confined in the transverse direction by the space-charge potential φ 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=rw . In addition, the analysis assumes perturbations with long axial wavelength, kz2rb2≪1 , and sufficiently high frequency that |ω/kz|≫vTez and |ω/kz−Vb|≫vTbz , where vTez and vTbz are the characteristic axial thermal speeds of the background electrons and beam ions. In this regime, Landau damping (in axial velocity space vz ) by resonant ions and electrons is negligibly small. We introduce the ion plasma frequency squared defined by ω^pb2=4πn^bZb2e2/γbmb , and the fractional charge neutralization defined by f=n^e/Zbn^b , where n^b and n^e are the characteristic ion and electron densities. The equilibrium and stability analysis is carried out for arbitrary normalized beam intensity ω^pb2/ωβb02 , 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 ty