Richardson's Law

$$J=\frac{4\pi me}{h^3}(k_{B}T{)}^2{e}^{-\frac{\varphi }{k_{B}T}}$$

This is the Richardson-Dushman equation for thermionic emission — the current density of electrons emitted from a heated surface.

• $J$ is the emission current density. $[J]=A/m^2$
• $m$ is the electron mass. $m=9.109\times 10^{-31}$ kg
• $e$ is the elementary charge. $e=1.602\times 10^{-19}$ C
• $h$ is Planck’s constant. $h=6.626\times 10^{-34}$ J·s
• $k_B$ is the Boltzmann constant. $k_B=1.381\times 10^{-23}$ J/K
• $T$ is the absolute temperature. $[T]=K$
• $\varphi$ is the work function of the surface — the minimum energy an electron needs to escape the material. $[\varphi ]=eV$ or $J$

The group $\frac{4\pi me}{h^3}{k}_B^2$ is often lumped together as the Richardson constant $A^{\ast }$, simplifying to $J=A^{\ast }{T}^2{e}^{-\varphi /k_{B}T}$.

The exponential term is the Boltzmann factor — fraction of electrons with enough thermal energy to overcome the work function barrier. The $T^2$ prefactor comes from integrating the Fermi-Dirac distribution over all electrons with sufficient perpendicular momentum to escape.

Richardson’s Law Example Question