Centroid theory of transverse electron-proton two-stream instability in a long proton bunch

This paper presents an analytical investigation of the transverse electron-proton (e-p) two-stream instability in a proton bunch propagating through a stationary electron background. The equations of motion, including damping effects, are derived for the centroids of the proton beam and the electron cloud by considering Lorentzian and Gaussian frequency spreads for the particles. For a Lorentzian frequency distribution, we derive the asymptotic solution of the coupled linear centroid equations in the time domain and study the e-p instability in proton bunches with nonuniform line densities. Examples are given for both uniform and parabolic proton line densities.