Filamentation instability of a relativistic hollow electron beam

The filamentation instability properties of a relativistic hollow electron beam confined in axial flow by a uniform magnetic field, in the absence of background plasma, in a pipe are investigated via the Vlasov–Maxwell equations. The instability is found to be have two sidebands, one with a spectrum of positive wavenumbers k and the other with a spectrum of negative wavenumbers. The spectral point k = 0, associated with the diokotron instability, is excluded from the two unstable sidebands of the filamentation instability. Only in the limit of zero axial beam flow (γ→1), does the diokotron instability become asymptotically part of the filamentation instability spectrum. In this limit, the two sidebands of the filamentation instability merge asymptotically and symmetrically toward the diokotron instability spectral point, k = 0, in agreement with the basic driving physical mechanisms and geometry configurations for these two distinct and different instabilities.