Influence of space charge on field emission of electrons from spheric points

Poisson´s equation in spherical symmetry is solved numerically as a function of the cathode field E1 and the cathode electrostatic field E0 (i.e. the field pertaining to the absence of current flow). This solution is combined with the Fowler-Nordheim field emission equation to give the cathode current density J as a function of Eo, the cathode work function ¿¿, the cathode radius r1, and the anode radius r2. Computations are carried out over the following ranges of these parameters: 2.0 ¿¿ ¿¿ ¿¿ 6.0 eV, 10¿¿8 ¿¿ r1 ¿¿ 10¿¿5 m, 104 ¿¿ r1/r2 ¿¿ 107, 0.9 ¿¿ E1/E0 ¿¿ 1.0; which is sufficient to study space charge effects in field emission guns and, in particular, in the field emission microscope. It is shown that previous analyses, based on the application of parallel plane theory to this spherical problem, give current densities for the onset of space charge effects that are an order of magnitude too great. The influence of space charge on the measurement of the work function and on image contrast in the field emission microscope is considered in detail.