= Thermionic Richardson constant
{c}
{title2=$A$}
The coefficient $A$ has dimensions of <electric current density> divided by squared <temperature>. For ideal <nonrelativistic particles> with the free <electron mass>, two <spin> states, a planar uniform surface, and unit escape transmission, the <thermionic tail-flux derivation> gives
$$
A_0=\frac{4\pi m_e e k_B^2}{h^3}
\simeq1.202\times10^6\,\mathrm{A\,m^{-2}\,K^{-2}}
=120.2\,\mathrm{A\,cm^{-2}\,K^{-2}},
$$
where $e$ is the positive <elementary charge> and $h$ is the <Planck constant>. The two area units differ by a factor of $10^4$. A fitted $A$ need not equal $A_0$: reflection at the surface, the <electronic band structure>, and uncertain emitting area affect the inferred coefficient. A constant escape transmission $\tau$ would give $A=\tau A_0$ in this simple model; an <energy>-dependent transmission generally requires an integral instead.
The name joins <Owen Willans Richardson> with <Saul Dushman>. Dushman's 1923 paper develops a thermodynamic treatment of <Electron> emission analogous to <evaporation> and evaluates the prefactor using quantum ideas.<Electron Emission from Metals as a Function of Temperature by Saul Dushman>[Abstract, general equation and theoretical evaluation of A]{ref}
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