Microwave signal amplification and Pierce instability on radial electron flows in cylindrical and spherical diodes

Linear space charge perturbations of focused electron beams flowing between cylindrical and spherical electrodes on convergent or divergent trajectories are studied, and the amplification of high-frequency signals when the flow is modulated at one electrode is computed. It is shown that divergent beams give the largest amplification effect. The instability of electron beams drifting through grounded grids (Pierce instability in cylindrical or spherical diodes) is also considered. The instability threshold occurs at higher critical currents when the curvature of the electrodes is large. Results for planar electrodes are recovered in the limit of zero curvature devices. Spherical configurations have better signal amplification and stability properties than similar planar or cylindrical systems