Macroscopic extraordinary-mode stability properties of relativistic non-neutral electron flow in cylindrical geometry

Abstract

Use is made of a macroscopic model based on the cold-fluid-Maxwell equations to investigate the extraordinary-mode stability properties of relativistic non-neutral electron flow in a cylindrical diode with the cathode located at r=a and the anode located at r=b. The equilibrium radial electric field E0r (r)êr and axial magnetic field B0z (r)êz induce an azimuthal flow of electrons with velocity V0θb (r)=ωb(r)r. Assuming flute perturbations (∂/∂z=0) with extraordinary-mode polarization, the eigenvalue equation is derived for the effective potential Φ(r)=(ir/l)δ ̂Eθ(r). Here, l is the azimuthal mode number, and perturbations are about the general class of equilibrium profiles E0r (r), B0z (r), ωb(r), and n0b (r), consistent with the steady-state cold-fluid-Maxwell equations. For prescribed equilibrium profiles, the extraordinary-mode eigenvalue equation is solved numerically for the eigenfunction Φ(r) and complex eigenfrequency ω. As a general remark, the numerical results show that detailed stability properties exhibit a sensitive dependence on cylindrical effects. For example, at low values of the mode number l, the properties of the eigenfunction Φ(r) are qualitatively different from the planar case, and from the cylindrical case for large l values. Furthermore, it is found that the instability growth rate Im ω exhibits a sensitive dependence on the layer aspect ratio A=a/(rb−a), particularly when the electron flow is relativistic and centrifugal effects play an important role in modifying the equilibrium profiles. (Here, r=rb denotes the outer edge of the electron layer.)