Influence of profile shape on the extraordinary-mode stability properties of relativistic non-neutral electron flow in a planar diode with applied magnetic field

Abstract

The cold-fluid extraordinary-mode eigenvalue equation is solved numerically to determine the influence of equilibrium profile shape on the detailed stability properties of relativistic non-neutral electron flow in a planar diode with cathode located at x=0 and anode at x=d. Stability properties are investigated for the class of equilibrium energy profiles γb(x) specified by γb(x)=λ cosh α1x+(1−λ) {[1−α22(b2−x2)]1/2/[1−α22b2]1/2} over the interval 0≤x≤b. Here α1 and α2 are constants (with α22b2<1), x=b is the outer edge of the electron layer, and λ is a constant parameter in the range of 0≤λ≤1. The corresponding equilibrium profiles for Bz(x), nb(x), and Ex(x) are determined self-consistently from the steady-state (∂/∂t=0) cold-fluid-Maxwell equations. As the parameter λ is varied from unity to zero there is a large change in the equilibrium profile for nb(x)/γb(x), ranging from nb(x)/γb(x)=const over the interval 0≤x<b when λ=1, to monotonic decreasing profiles for nb(x)/γb(x) when λ<1. The numerical analysis of the extraordinary-mode eigenvalue equation shows that the detailed stability properties are very sensitive to the shape of the equilibrium profiles. As λ is reduced from unity, and the profile for nb(x)/γb(x) becomes monotonic decreasing, it is found that the instability growth rate Im ω is reduced. Moreover, the more rapid the decrease in nb(x)/γb(x) (i.e., the smaller the value of λ), the more the growth rate is reduced. Indeed, in some parameter regimes, the instability growth rate can be reduced to zero over the range of wavenumber k examined numerically.