# Why don't we use Fermi-Dirac statistics for deriving Richardson equation for thermionic emission?

Why does the Richardson-Dushman equation need Maxwell-Boltzmann statistics instead of Fermi-Dirac statistics to model thermionic emission, though only fermions are our only interest in those emissions?

In the original derivation published by Richardson in 1901, he assumed that only electrons in the thermal or Maxwell–Boltzmann tail of the distribution function $$f(E)$$ contribute to the current density. In other words, the electrons with higher energies $$(E \gg E_F)$$ can escape from metal.
Fermi-Dirac statistics approaches Maxwell-Boltzmann statistics for high temperature, low densities and also when $$E \gg E_f$$ often referred to as the classical limit. $$f(E) = \dfrac{1}{1+\exp\left(\dfrac{E-E_F}{kT}\right)} \approx \exp\left(-\frac{E-E_F}{kT}\right)$$
The efforts to understand statistical methods and the MB distribution enable an understanding of first formulation of thermionic emission, in which electrons were modeled as evaporating off the surface of a metal. Richardson’s original derivation applied a Carnot cycle to the electron gas in equilibrium with a metal. Dushman reconsidered Richardson’s equation and his expression for the coefficient agrees with more modern treatments apart from a factor of $$2$$ associated with electron spin (a purely quantum mechanical feature of the electron revealed by the Stern–Gerlach experiment in 1922). It is a pleasingly different method of determining current density from thermionic emission using thermodynamic arguments without invoking the quantum mechanical ideas that form the backbone of most modern treatments.