Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S16 / 2

For weak potentials, Boltzmann electrons give $\delta n_e=n_{e}e\varphi /(k_{B}T)$ against fixed ions. Poisson's equation becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$ outside a test charge, giving a Yukawa potential with $\boxed{{\lambda }_D=\sqrt{{ϵ}_0{k}_{B}T/(n_e{e}^2)}}$. For a small electron displacement $x$ relative to fixed ions, charge separation produces restoring field $E=n_{e}ex/{ϵ}_0$, so $m\ddot{x}=-n_e{e}^{2}x/{ϵ}_0$. Hence $\boxed{{\omega }_p=\sqrt{n_e{e}^2/({ϵ}_0{m}_e)}}$. The two formulas respectively encode thermal screening and electron inertia.

Choose $N_D=(4\pi /3)n_e{\lambda }_D^3$. The stipulated estimate then givesThus density cancels and conductivity grows as $T^{3/2}$ at fixed ion charge. Physically faster electrons are less effectively deflected by Coulomb encounters. Numerical factors depend on the Debye-volume convention, and a realistic collision calculation introduces a Coulomb logarithm and evolving ionization; the cancellation is approximate rather than exact for every real plasma.

In the cold solid, phonon scattering makes $\tau \propto T^{-1}$ and therefore conductivity falls as temperature rises, at roughly fixed carrier density. After melting and ionization, the character of scattering changes; a hot Coulomb plasma eventually follows an increasing $T^{3/2}$ branch. A minimum in the crossover region is natural, but a weakly coupled plasma formula is not quantitatively justified throughout the warm dense regime.

Room temperature corresponds to about 0.026 eV, or $2.6\times 10^{-5}$ keV. Read the left-hand branch there and draw a horizontal line to the rising hot branch. The original graph gives a return temperature of order $\boxed{k_B{T}^{\prime }\sim 1\text{–}2\mathrm{keV}}$, with graph-reading uncertainty, not the temperature of minimum conductivity.

At full ionization $n_e=13\rho /(27m_u)\simeq 7.8\times 10^{29}$ m${}^{-3}$. Inserting the inferred temperature into $N_D=(4\pi /3)n_e[{ϵ}_0{k}_B{T}^{\prime }/(n_e{e}^2){]}^{3/2}$ gives a Debye-sphere occupancy of order $60$–$170$. This is much greater than one and supports a collective weak-coupling description more than the conductivity-minimum region does. The approximate collision-time formula and omitted Coulomb logarithm do not warrant forcing the entire plotted conductivity curve to have its exact prefactor.