How does the Richardson-Dushman equation describe thermionic emission?

The Richardson-Dushman equation describes the rate of thermionic emission by relating the emission current density ($J$) to the absolute temperature ($T$) of the emitter material.12

The equation demonstrates that the rate of electron emission increases rapidly with temperature and decreases exponentially with the electronic work function ($W$ or $\varphi$), which is the energy required for an electron to escape the surface of the material. Mathematically, the current density is expressed as:13

$$J=A{T}^2{e}^{-W/kT}$$

In this formula:
* $J$: The emission current density in amperes per square metre.34
* $T$: The absolute temperature of the emitter in kelvins.13
* $W$: The work function of the material in electron volts.34
* $k$: Boltzmann’s constant.13
* $A$: A constant characteristic of the emitter material and its surface finish.34

Because emission is highly sensitive to temperature and the work function, practical thermionic power converters are designed using materials with low work functions that can operate reliably at high temperatures.13