= Richardson plot
{c}
{title2=$\ln[J/(A_{\rm ref}T^2)]$ versus $1/T$}
Choose a fixed positive reference $A_{\rm ref}$ with the same units as $A$. Taking the <natural logarithm> of a dimensionless ratio gives
$$
x=\frac1T,\qquad
y=\ln\!\left(\frac{J}{A_{\rm ref}T^2}\right)
=\ln\!\left(\frac{A}{A_{\rm ref}}\right)-\frac{W}{k_B}x.
$$
Thus the <Richardson plot> is a <straight line> when $A$ and $W$ are constant, with slope $-W/k_B$ and intercept $\ln(A/A_{\rm ref})$. The familiar notation $\ln(J/T^2)$ silently chooses units; the reference makes that choice explicit. Changing $A_{\rm ref}$ 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 $A_{\rm ref}=120\,\mathrm{A\,cm^{-2}\,K^{-2}}$. 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.
\Image[/thermionic-emission-temperature-richardson-plot.png]
{title=One chosen emission model as a temperature curve and a dimensionless Richardson plot}
{description=The marked values are calculations with constant $W=2.0\,\mathrm{eV}$ and $A=A_{\rm ref}=120\,\mathrm{A\,cm^{-2}\,K^{-2}}$. They illustrate the <Richardson-Dushman law>, rather than a particular material or a historical measurement.}
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 $W_i$,
$$
J=T^2\sum_i f_i A_i e^{-W_i/(k_BT)},\qquad \sum_i f_i=1.
$$
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 becomes
$$
W_{\rm app}=-k_B\frac{dy}{d(1/T)}
=W-T\frac{dW}{dT}+k_BT^2\frac{d\ln A}{dT}.
$$
Check collection, <temperature> calibration, and surface stability before assigning curvature to a new emission mechanism.
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