  The characteristics of field electron and ion emission change when the space
charge formed by the emitted charge is sufficient to suppress the extracting
electric field. This phenomenon is well described for planar emitting diodes by
the one dimensional (1D) theory. Here we generalize for any 3D geometry by
deriving the scaling laws describing the field suppression in the weak space
charge regime. We propose a novel corrected equivalent planar diode model,
which describes the space charge effects for any geometry in terms of the 1D
theory, utilizing a correction factor that adjusts the diode's scaling
characteristics. We then develop a computational method, based on the
Particle-In-Cell technique, which solves numerically the space charge problem.
We validate our theory by comparing it to both our numerical calculations and
existing experimental data, either of which can be used to obtain the
geometrical correction factor of the corrected equivalent planar diode model.
