= Thermionic tail-flux derivation
{title2=$T^{3/2}\times T^{1/2}=T^2$}
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 density
$$
f_z(v_z)=\sqrt{\frac{m_e}{2\pi k_BT}}
e^{-m_ev_z^2/(2k_BT)}.
$$
An <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 $v_z>0$:
$$
\Gamma=n_v\int_0^\infty v_zf_z(v_z)\,dv_z
=n_v\sqrt{\frac{k_BT}{2\pi m_e}}.
$$
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 $\Gamma=n\sqrt{k_BT/(2\pi m_e)}e^{-W/(k_BT)}$; 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 $W\gg k_BT$, its <Fermi-Dirac distribution> reduces to the <Maxwell–Boltzmann statistics> tail. Counting two <spin> states per <phase space> cell of volume $h^3$ gives
$$
n_v=\frac{2}{h^3}\int_{\mathbb R^3}
e^{-[W+p^2/(2m_e)]/(k_BT)}\,d^3p
=2\left(\frac{2\pi m_e k_BT}{h^2}\right)^{3/2}
e^{-W/(k_BT)}.
$$
Each of the three <Gaussian integrals> supplies $T^{1/2}$. Combining this $T^{3/2}$ state count with the $T^{1/2}$ crossing factor gives
$$
J=e\Gamma
=\frac{4\pi m_e e k_B^2}{h^3}\,T^2e^{-W/(k_BT)}.
$$
The 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.<Nobel Lectures, Physics 1922–1941/Thermionic Phenomena and the Laws which Govern Them by Owen Willans Richardson>[Nobel Lectures, Physics 1922–1941; pp. 230–231, internal versus external electron statistics]{ref}
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