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In his Nobel lecture, delivered on 12 December 1929 for the 1928 prize, Owen Willans Richardson recalled heating wires for weeks while hand-pumping away gas released by the apparatus. With platinum, he found a finite emission per unit area that rose rapidly with temperature. His initial 1901 formula used $T^{1/2}$; he preferred the $T^2$ form in 1911.^[ref]

The early apparatus surrounded a heated platinum wire with a metal cylinder connected to earth through a galvanometer. Wire electrical resistance supplied the temperature readout. After checking collection saturation, Richardson measured at a wire potential of $-120\mathrm{V}$, over roughly $1300$–$1600\mathrm{K}$; reversing polarity suppressed the negative discharge.^[ref]

The inference rests on several controls. A temperature sweep probes thermal escape; reversing the collecting electric field tests the sign of the mobile electric charge; varying collection voltage tests whether transport is limiting. Prolonged pumping and heating test whether a transient gas release is responsible. These controls address different alternatives, and none by itself establishes a clean, uniform surface.

For a modern repetition, stabilize each temperature, measure the collector electric current against collecting electric potential difference, and record the pressure and surface preparation. Repeat selected temperatures after cooling to reveal drift. Use the collection plateau and the actual emitting area to construct a Richardson plot. The resulting slope estimates the work function only within the stated surface and transport assumptions; it does not turn an uncontrolled wire into a reference material.

Consider a planar surface with outward normal along $z$. For an auxiliary equilibrium electron gas just outside it, let $n_v$ be the total number density, counting both signs of $v_z$. Its Maxwell-Boltzmann velocity distribution has normal-component densityAn Electron with $v_z>0$ crosses more area per unit time when it moves faster. The outgoing number flux therefore uses $v_z{f}_z$, rather than just the probability of $v_z>0$:This integral contributes only $T^{1/2}$. Holding a classical bulk number density fixed and imposing a normal kinetic energy threshold $W$ would instead give $\Gamma =n\sqrt{k_{B}T/(2\pi m_e)}{e}^{-W/(k_{B}T)}$; that argument alone does not produce $T^2$.

For the ideal surface model, measure external Electron energies from the metal's chemical potential $\mu$. A freely propagating external state has $E-\mu =W+p^2/(2m_e)$. When $W\gg k_{B}T$, its Fermi-Dirac distribution reduces to the Maxwell–Boltzmann statistics tail. Counting two spin states per phase space cell of volume $h^3$ givesEach of the three Gaussian integrals supplies $T^{1/2}$. Combining this $T^{3/2}$ state count with the $T^{1/2}$ crossing factor givesThe auxiliary atmosphere supplies an equilibrium state-counting argument; an operating collector removes the outgoing Electrons. It need not maintain a real equilibrium gas throughout the gap. Unit transmission and the ideal free Electron state count are substantive assumptions, so this calculation does not determine an arbitrary real surface's measured $A$.

In particular, Electrons inside a metal generally form a degenerate Fermi gas. A classical distribution for the escaping tail does not imply a classical distribution for the whole interior. Richardson himself emphasized this distinction in his lecture.^[ref]