Use is made of the macroscopic cold-fluid Poisson equations to investigate the electrostatic stability properties of nonrelativistic, non-neutral electron flow in a cylindrical diode with applied magnetic field B₀e^z. The cathode is located at r=a and the anode is located at r=b. Space-charge-limited flow with Eᵣ⁰(r=a)=0 is assumed. Detailed stability properties are investigated analytically and numerically for electrostatic flute perturbations with {∂}/{∂}z=0. Particular emphasis is placed on the influence of the neutral anode plasma on stability behavior assuming uniform cathode electron density (n^b) extending from the cathode (r=a) to r=rb, and uniform anode plasma density (n^ₑ=Zᵢn^ᵢ) extending from r=rₚ to the anode (r=b). Depending on the cathode electron density (as measured by sb={ω}^ pb²/ωce²), the anode plasma density (as measured by sₑ={ω}^ ₚₑ²/ωce²), the diode aspect ratio, etc., it is found that there can be a strong coupling of the anode plasma to the cathode electrons, and a concomitant large influence on detailed stability behavior for both the high-frequency (electron-driven) and low-frequency (ion-driven) branches. Detailed stability properties are investigated over a wide range of cathode electron density, anode plasma density, diode aspect ratio, etc.
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Davidson et al. (1985) studied this question.