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An emission-limited thermionic current is set by the rate at which the heated surface supplies Electrons, with nearly complete collection. At fixed temperature, increasing the attracting electric potential difference then produces an approximate collection plateau. That plateau is the useful regime for fitting the Richardson-Dushman law, provided the electric field has not significantly lowered the surface barrier.

The emitting surface still matters in this regime. Adsorption changes the work function; different crystal faces can have different barriers. Heating may desorb contaminants or change a coating, so repeated measurements at the same temperature need not agree unless the surface preparation is controlled. For fixed $A,T$, a barrier change $\Delta W$ multiplies $J$ by $e^{-\Delta W/(k_{B}T)}$. In the chosen model at $1000\mathrm{K}$, raising the barrier by $0.10\mathrm{eV}$ reduces $J$ to about 0.313 of its previous value.

The Richardson-Dushman law models the magnitude of emitted electric current density asHere $T$ is absolute thermodynamic temperature, $W$ is the work function, $k_B$ is the Boltzmann constant, and $A$ is the Thermionic Richardson constant. The signed electric current density points opposite to the outgoing Electron motion; $J$ here is positive. For uniform emission over area $S$, the emitted electric current magnitude is $I=SJ$.

The Richardson-Dushman law assumes an emitter close to thermal equilibrium, a surface with a well-defined and approximately constant work function over the fitted temperature range, and predominantly thermal escape over the barrier. Interpreting a measured collector electric current with this law additionally requires an emission-limited thermionic current, known emitting area, and negligible return emission from the collector. Strong electric fields can lower the surface barrier or enable quantum tunnelling; then this zero-field expression is insufficient.

For a deliberately chosen teaching model, take $W=2.0\mathrm{eV}$ and $A=120\mathrm{A}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{K}}^{-2}$, expressing the barrier in electronvolts. These are illustrative parameters, not measured properties of a named material. Using $k_B=8.617333262\times 10^{-5}\mathrm{eV}{\mathrm{K}}^{-1}$, at $1000\mathrm{K}$ the Boltzmann factor is about $8.33\times 10^{-11}$, yetA 20 percent increase in temperature raises the predicted electric current density by about 69 times. For a chosen emitting area $S=0.010\mathrm{c}{\mathrm{m}}^2$, these correspond to approximately $0.100\mathrm{mA}$ and $6.89\mathrm{mA}$ if collection is complete. The small escaping fraction is replenished by the external electrical circuit.

The sensitivity is especially clear from the logarithmic derivative at fixed $A,W$:In the chosen model at $1000\mathrm{K}$, a $1\mathrm{K}$ rise changes $J$ by about 2.5 percent to first order. Accurate temperature measurement matters even when electric current measurement is easy.

Choose a fixed positive reference $A_{\mathrm{ref}}$ with the same units as $A$. Taking the natural logarithm of a dimensionless ratio givesThus the Richardson plot is a straight line when $A$ and $W$ are constant, with slope $-W/k_B$ and intercept $\ln (A/A_{\mathrm{ref}})$. The familiar notation $\ln (J/T^2)$ silently chooses units; the reference makes that choice explicit. Changing $A_{\mathrm{ref}}$ shifts the intercept and leaves the inferred work function unchanged. If the horizontal axis instead uses $1000\mathrm{K}/T$, its slope is $-W/(1000\mathrm{K}{k}_B)$.

For the chosen $2.0\mathrm{eV}$ example, choose $A_{\mathrm{ref}}=120\mathrm{A}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{K}}^{-2}$. Then $y(1000\mathrm{K})=-23.2090$ and $y(1200\mathrm{K})=-19.3409$. Their slope is approximately $-23209\mathrm{K}$, giving $W=2.0\mathrm{eV}$. Two temperatures determine a line algebraically; several temperatures allow curvature and drift to be detected. These values are calculated, rather than experimental observations.

A straight trace alone does not establish a unique surface work function. Adsorption, changing surface coverage, or a mixture of regions with different barriers can change the inferred slope. For independent emitting patches with area fractions $f_i$, constant $A_i$ and barriers $W_i$,Lower barriers contribute disproportionately. Even with fixed patches, the slope of the Richardson plot gives a temperature-dependent weighted mean of the $W_i$, rather than their simple area average. If $A$ or $W$ themselves vary with temperature, the apparent slope-derived barrier becomesCheck collection, temperature calibration, and surface stability before assigning curvature to a new emission mechanism.

Release of Electrons from a heated material; one source of Electrons in an X-ray tube.

Heating a metal gives some Electrons enough energy to escape its surface. A heated diode cathode can therefore supply an electric current across a vacuum gap to a collecting diode anode. This makes temperature a control for the supply of Electrons. Conversely, measuring that supply at several temperatures lets us estimate the surface work function: the energy barrier measured from the Fermi level to the nearby vacuum level. A Richardson plot turns the strong exponential function dependence into a slope.

Three questions must be kept separate: how many Electrons cross the surface, how many traverse the gap, and how many the collecting diode anode records. The Richardson-Dushman law answers the first under specified surface assumptions. Space charge can obstruct the second, while incomplete collection can alter the third.