A statistical mechanics derivation of the empirical asymptotic thermionic emission equation

This paper will obtain a physical derivation for a very useful empirical emission formula. Thermionic emission has long been understood, by the supply side limited regime, described by Richardson-Dushman's equation and the space charge regime by Child-Langmuir's equation. These equations adequately describe thermionic emission and provide a measure of internal parameters, such as work function, over a limited range and in special designed devices. They fail in the intermediate region between the two laws where most measurements are required, as well as in practical devices. This theory places the well known empirical emission formula on a solid physical foundation.