Macroscopic cold-fluid equilibrium properties of relativistic non-neutral electron flow in a cylindrical diode

The cold-fluid Maxwell equations are used to investigate detailed equilibrium properties ({∂}/{∂}t=0) of relativistic non-neutral electron flow in cylindrical diode geometry with applied magnetic field. There is considerable latitude in specifying the equilibrium profiles. In particular, the profiles for any one of the density nb⁰(r), axial magnetic field Bz⁰(r), radial electric field Eᵣ⁰(r), or azimuthal flow velocity Vthetab⁰(r)=βb(r)c can be specified, and the remaining three profiles calculated self-consistently from Poisson's equation, the {∇}×{}B⁰ Maxwell equation, and the radial force balance equation. As a first example, assuming that the functional form of βb(r) is specified, formal expressions are derived for Bz⁰(r), nb⁰(r), and Eᵣ⁰(r). As a second example, it is assumed that the radial electric field is prescribed by Eᵣ⁰(r)=const×{}sinh[{κ}(r-a)] over the radial extent of the electron layer extending from r=a (the cathode) to r=rb (the outer edge of the electron layer). In this case, a closed-form expression for the density profile nb⁰(r) is obtained, but the profiles for the magnetic field Bz⁰(r) and flow velocity βb(r)c must generally be determined numerically when the layer aspect ratio A=a/(rb-a) is finite. The influence of cylindrical effects (e.g., centrifugal effects) and finite aspect ratio A on detailed equilibrium properties is calculated, treating the planar diode limit (A={∞}) as a reference case where the equilibrium profiles are known in closed analytic form. It is found that cylindrical effects and finite aspect ratio can have a large influence on profile shape, particularly when the electron flow is relativistic.