Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S16 / 1

A weak positive potential perturbs Boltzmann electrons as $n_e\simeq n_0[1+e\varphi /(k_{B}T)]$ against nearly fixed ions. Poisson's equation outside the test charge becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$, withThe resulting potential is proportional to $e^{-r/{\lambda }_D}/r$. The Debye length is the screening distance over which charge imbalance is suppressed, within this classical weak-perturbation model.

Define the plasma parameter as the number in a Debye sphere, $N_D=(4\pi /3)n_e{\lambda }_D^3$; some conventions omit $4\pi /3$. A good classical plasma has many particles in a screening volume, $N_D\gg 1$, and a system size much larger than ${\lambda }_D$, so collective fields dominate individual encounters.

A collision here means cumulative Coulomb deflection that significantly changes a particle's velocity, not necessarily a literal hard-sphere impact. A collisionless description requires the phenomenon's timescale to be short compared with the appropriate collision time, and its length scale short compared with the mean free path. Parametrically $\boxed{{\omega }_p{t}_c\sim N_D/\log \Lambda }$ up to convention-dependent constants, with Coulomb logarithm $\log \Lambda$ representing the range of impact parameters. Many oscillations can then occur before collisional relaxation. The formula itself presupposes the weak-coupling regime.

Rapid longitudinal oscillations in a weakly collisional plasma compress the velocity distribution primarily along one direction. There is insufficient time for collisions to redistribute that energy among three directions; the effective one-dimensional adiabatic exponent is $1+2/1=3$. Ordinary monatomic-gas sound with local collisional equilibrium has three translational degrees of freedom, giving $1+2/3=5/3$. Thus $\Gamma =3$ belongs to the stated kinetic/time-scale limit, not to every thermodynamic process involving ionized gas.

The ion density is $\rho /(9m_u)$ and double ionization gives $n_e\simeq 2.48\times 10^{29}$ m${}^{-3}$. Therefore $\hslash {\omega }_p\simeq 18.5$ eV. The photon energy loss gives $\hslash \omega =30$ eV, while momentum conservation givesInverting the stipulated dispersion law givesAn angular frequency expressed “in eV” means $\hslash \omega$, not the frequency itself. The temperature is a formal inference conditional on the dispersion model, whose validity is tested next.

The inferred parameters give ${\lambda }_D\simeq 6.46\times 10^{-11}$ m and $\boxed{N_D\simeq 0.28}$, or $n_e{\lambda }_D^3\simeq 0.067$ without the spherical factor. Neither is much larger than one. Thus a good classical weakly coupled plasma approximation is not justified. Correlations, collisions and potentially degeneracy matter at this density; treating the preceding 18.7 eV as a reliable temperature measurement would overstate what this simple model establishes.