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

Assume dilute, nondegenerate ideal electrons, protons and neutral atoms in thermal and chemical equilibrium, with charge neutrality $n_p=n_e$. Set ${\mu }_H={\mu }_p+{\mu }_e$, using each ideal species' translational factor $n{\lambda }_{\mathrm{th}}^3/g$ and hydrogen ground-state binding energy $-R$. Neglect excited states and take the heavy proton/atom thermal-wavelength ratio as one. With electron spin and neutral-atom degeneracies treated in the convention of the source, the remaining factor isDifferent partition-degeneracy conventions must be propagated consistently. Screening, dense-plasma level dissolution and departures from local thermodynamic equilibrium lie outside this derivation.

For total nuclear density $n$, $n_e=n\zeta$ and $n_H=n(1-\zeta )$. Saha's equation becomes $n{\zeta }^2/(1-\zeta )=C{T}^{3/2}{e}^{-R/(k_{B}T)}$. At $\zeta \ll 1$,The factor two in the exponent follows from taking a square root; using the full ionization temperature would double the predicted logarithmic slope.

Write $u=1/T$. Then $\log \zeta =\mathrm{constant}-(3/4)\log u-T_{H}u$, giving local slope $d{\log }_{10}\zeta /du=-(T_H+3T/4)/\log 10$. At 5000–6000 K its magnitude is about $3.6\times 10^4$ K. The low-ionization tail in the original graph drops roughly one decade per $2.8\times 10^{-5}$ K${}^{-1}$, consistent with this value.

At $u=2.0\times 10^{-4}$ K${}^{-1}$, the graph gives $T\simeq 5000$ K and $\zeta \simeq 0.004$. Invert the full equation to get $n=C{T}^{3/2}{e}^{-R/(k_{B}T)}(1-\zeta )/{\zeta }^2$, yielding $\boxed{n\sim 10^{18}{\mathrm{m}}^{-3}}$. The neighboring tail markers give the same order; plotting precision and the squared dependence on the read ionization fraction preclude claiming many significant digits.

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.

Take electrostatic perturbations proportional to $e^{i(kz-\omega t)}$, with ions immobile on the fast electron time scale. Linearized Vlasov gives $f_1=-i(q{E}_1/m)(\partial f_0/\partial v_z)/(\omega -k{v}_z)$. Integrating over transverse velocities gives $F_0(v_z)=n_0\sqrt{m/(2\pi k_{B}T)}{e}^{-m{v}_z^2/(2k_{B}T)}$. Combine the perturbed charge density with Poisson's equation:Since $F_0^{\prime }=-m{v}_z{F}_0/(k_{B}T)$, this is preciselyThe pole selects particles moving at phase speed. A causal initial-value prescription specifies how the contour passes it and produces resonant energy exchange, including Landau damping for a decreasing Maxwellian. A bare real improper integral without that prescription is incomplete. The unclosed brace in the printed Maxwellian exponent is typographical; the normalized Gaussian above fixes its meaning.

When $|k{v}_z/\omega |\ll 1$ over the populated velocities, expand the denominator in powers of $k{v}_z/\omega$. Odd Gaussian moments vanish. The first two nonzero terms use $⟨v_z^2⟩=v_{th}^2$ and $⟨v_z^4⟩=3v_{th}^4$, yielding $1-{\omega }_p^2/{\omega }^2-3{\omega }_p^2{k}^2{v}_{th}^2/{\omega }^4+\cdots =0$. Solving consistently to order $k^2$ givesThis is a long-wavelength/high-phase-speed approximation, not an exact Maxwellian dispersion relation at arbitrary $k$.

Differentiate the approximate dispersion: $v_g=3k{v}_{th}^2/\omega$, while $v_{\varphi }=\omega /k$. Therefore $\boxed{v_g{v}_{\varphi }=3v_{th}^2}$. In the high-phase-speed regime that justified the approximation, the group speed is correspondingly small. Trying to make $v_g$ much larger than $v_{th}$ would put the phase speed into the thermally populated distribution, where resonant Landau damping is strong and the undamped-fluid estimate becomes unreliable. This explains the stated tendency for these electron electrostatic waves, not a universal prohibition on fast electromagnetic waves in plasma.