Beam Divergence From Sharp Emitters In A General Longitudinal Magnetic Field

We calculate the trajectories of electrons emitted from a sharp protrusion with corners, where the electric field is intense and the radius of curvature tends to zero. The calculations include the effects of a longitudinal magnetic field but ignore the space charge effects. We find that the arbitrarily large electric field at a mathematically sharp corner does not necessarily impart to the electrons an excessive amount of transverse momentum, whether or not the external magnetic field is present. Scaling laws are derived for the beam's transverse displacement in terms of macroscopic quantities, such as anode-cathode voltage drop, gap separation, magnetic field, and the protrusion dimensions. The implication of these findings on the electron sources generated from micro-tips is addressed.