Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S16 / 3

Linearize the electron Boltzmann distribution, $n_e(\varphi )\simeq n_{e0}[1+e\varphi /(k_{B}T)]$, about a uniform fixed-ion background. Poisson's equation gives ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$, where the electron Debye length isDefine the plasma parameter as the number of electrons in a Debye sphere:Some authors omit $4\pi /3$ in this definition. Mobile ions alter the total screening length by contributing their own susceptibility; the electron-only convention is used consistently here.

Use quasineutrality, $n_e=Z{n}_i$. Multiplying the supplied collision time by the electron plasma frequency and using the preceding definition givesThus ${\tau }_{ei}\gg 2\pi /{\omega }_p$ requires $N_D\gg (2\pi /9.6)Z\ln \Lambda$; for ordinary $Z$ and $\ln \Lambda \sim 1$–$30$, this is a large plasma parameter. Many particles then collectively screen a charge, fluctuations in a Debye sphere are relatively small, and mean-field electrostatics descriptions become meaningful. A large plasma parameter is also associated with weak Coulomb coupling, well-separated screening and close-collision scales, and collective plasma oscillations surviving many oscillations before collisionally relaxing. Roughly, relative counting fluctuations scale as $N_D^{-1/2}$ and the ratio of interaction to thermal energy decreases as $N_D$ increases. Small $N_D$ signals that strong correlations can invalidate the simple weakly coupled plasma description.

Introduce the electron momentum-relaxation rate $\nu$ through $m_e(\dot{\mathbf{v}}+\nu \mathbf{v})=-e\mathbf{E}$. For a harmonic field $\mathbf{E}=\mathrm{Re}({\mathbf{E}}_0{e}^{-i\omega t})$, the real part of the electrical conductivity is $\mathrm{Re}\sigma =n_e{e}^2\nu /[m_e({\omega }^2+{\nu }^2)]$. At weak damping, the mean heating is $Q=\frac{1}{2}\mathrm{Re}\sigma |E_0{|}^2$ and the beam intensity is $I=\frac{1}{2}{ϵ}_{0}cn|E_0{|}^2$, where $n=\sqrt{1-{\omega }_p^2/{\omega }^2}$. ThereforeWith the relaxation convention $\nu =1/(2{\tau }_{ei})$, this becomes the printed resultConvention qualification: a $90^{\circ }$ deflection time does not by itself specify a momentum-relaxation rate without a collision model. If instead ${\tau }_{ei}$ is defined by $\nu =1/{\tau }_{ei}$, the intensity absorption coefficient is twice the printed one. Amplitude and intensity damping also differ by two. The result above states the extra convention needed rather than silently identifying these quantities. It assumes $\omega >{\omega }_p$, $\nu \ll \omega$ and sufficiently weak damping away from the cutoff; the apparent divergence near cutoff is outside that approximation.

Quasineutrality gives $n_e=4\times 10^{26}{\mathrm{m}}^{-3}$. Using the supplied $90^{\circ }$ collision-time expression and $\ln \Lambda =10$ givesThe printed absorption coefficient then yields $\kappa =1.62\times 10^2{\mathrm{m}}^{-1}$. Since the unabsorbed fraction is $e^{-\kappa L}$,This uses exactly the coefficient specified by the paper and neglects reflection and changing plasma conditions. With $\nu =1/{\tau }_{ei}$ instead, the corresponding thickness is $2.14\mathrm{mm}$, illustrating the convention dependence discussed in part (c).