Space-charge-limited and temperature-limited electron flow in the vicinity of edges and conical points

Classes of similarity solutions (or solutions by separation of variables) are found for space-charge-limited and temperature-limited flow in the neighborhood of a cathode in the shape of a corner and in the neighborhood of a conical point, both of arbitrary acute angle. These solutions exhibit characteristic exponents for the behavior of density, flow, electrostatic potential, and current density as functions of the distance from the corner (or conical point). These solutions were used to obtain the scaling of beam emittance with surface roughness, assuming that the surface can be approximated by a series of conical points. The results show that the scaling is insensitive to the exact shape of the surface profile (even if there are points, as in the conical model). Identical scalings are obtained if the surface roughness is two dimensional, i.e. grooves