Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S16 / 2

The electron Debye length is the scale over which a weak electrostatic disturbance is screened. Take the ions as an immobile neutralizing background and the electrons as an isothermal Boltzmann distribution. For a small potential $\varphi$,Poisson's equation then becomes ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$ away from the perturbing source, withA test charge has a screened potential proportional to $e^{-r/{\lambda }_D}/r$. For fully static screening with mobile ions, inverse squared Debye lengths from all species add; the displayed expression is the electron scale used below.

The electron plasma frequency is the natural collective oscillation frequency when electrons are displaced from a nearly fixed ion background. Displace a uniform electron fluid by $\xi$ along a slab normal. The charge separation produces the restoring electric force $-n_e{e}^2\xi /{ϵ}_0$ on an electron. Hence $m_e\ddot{\xi }=-n_e{e}^2\xi /{ϵ}_0$ andIt sets a response time $1/{\omega }_p$ and the cold unmagnetized electromagnetic cutoff frequency. These collective effects arise from long-range electric fields, even when individual binary collisions are infrequent.

Define the plasma parameter here as the number of electrons in an electron Debye sphere,Some texts instead use $n_e{\lambda }_D^3$, or its inverse as a coupling parameter. A weakly coupled collective plasma has $N_D\gg 1$; fluctuations from individual nearby particles are small compared with the collective description.

A Coulomb collision means the cumulative deflection or relaxation produced by many small-angle encounters, not necessarily a hard contact between two particles. The Coulomb logarithm $\ln {\Lambda }_C\simeq \ln (b_{\max }/b_{\min })$ summarizes the impact-parameter range, with $b_{\max }$ of order ${\lambda }_D$. A process is collisionless when its timescale is much shorter than the appropriate collisional relaxation time, or its length scale is much smaller than a corresponding mean free path. It still contains collective interactions.

For fully ionized ions of charge $Ze$, a thermal electron–ion momentum-relaxation rate has the scalingup to numerical factors depending on the Debye-sphere and collision-time conventions. More explicitly, the Maxwellian rate used below gives ${\nu }_{ei}/{\omega }_p\simeq 0.089Z\ln {\Lambda }_C/N_D$, or $t_{\mathrm{col}}{\omega }_p\simeq 11.3N_D/(Z\ln {\Lambda }_C)$. Specifying this factor prevents mixing a small coupling parameter with a large Debye-sphere population.^[ref]

Use collisional inverse bremsstrahlung in a weakly coupled, unmagnetized plasma. The specified density is $n_e=10^{26}{\mathrm{m}}^{-3}$, and $k_{B}T=1000\mathrm{eV}$. The electron Debye length is $2.35\times 10^{-8}\mathrm{m}$, the plasma frequency is $5.64\times 10^{14}{\mathrm{s}}^{-1}$, and the laser angular frequency is $\omega =2\pi c/\lambda =1.79\times 10^{15}{\mathrm{s}}^{-1}$. Thus $X={\omega }_p^2/{\omega }^2\simeq 0.099<1$: the plasma is underdense and a propagating wave exists.

Estimate $b_{\min }$ as the larger of the classical Coulomb-deflection length $Z{e}^2/(4\pi {ϵ}_0{k}_{B}T)$ and the electron quantum length $\hslash /\sqrt{m_e{k}_{B}T}$. Both are about $9\times 10^{-12}\mathrm{m}$ here, giving $\ln {\Lambda }_C\simeq 8$. With the usual thermal collision convention,The $Z$ factor follows from $n_i{Z}^2=n_{e}Z$.^[ref]

The weakly collisional Drude model gives ${ϵ}_r=1-{\omega }_p^2/[\omega (\omega +i{\nu }_{ei})]$ for fields proportional to $e^{-i\omega t}$. Expanding at ${\nu }_{ei}\ll \omega$ yields a real refractive index $n_r=\sqrt{1-X}\simeq 0.949$ and intensity attenuation coefficientConsequently the intensity e-folding absorption length isi.e. of order a centimetre. Taking a Coulomb logarithm between about 5 and 10 gives roughly $5$–$10\mathrm{mm}$. The electric-field amplitude decay length is twice this intensity length. This estimate assumes a uniform slab thick enough to define an absorption length and neglects gradients, nonlinear quiver heating and other absorption mechanisms.

For a sinusoidal laser field of peak amplitude $E_0$, an electron's peak nonrelativistic quiver speed is $v_q=e{E}_0/(m_e\omega )$. Choose the one-component electron thermal speed $v_T=\sqrt{k_{B}T/m_e}=1.33\times 10^7\mathrm{m}{\mathrm{s}}^{-1}$. The vacuum-equivalent laser intensity is $I={ϵ}_{0}c{E}_0^2/2$, so the onset $v_q\sim v_T$ occurs atUsing $\sqrt{2k_{B}T/m_e}$ as the thermal-speed convention doubles this estimate; its order of magnitude is the physical content. Once the quiver is comparable to the thermal speed, a collision rate computed for an undriven thermal distribution is no longer adequate.

Relativistic motion begins when the normalized field $a_0=e{E}_0/(m_{e}c\omega )$ is of order one. ThusThe ratio $I_{\mathrm{rel}}/I_{\mathrm{th}}=m_e{c}^2/(k_{B}T)\simeq 511$ checks the separation of the two regimes. These estimates use peak field and cycle-averaged vacuum intensity. The small refractive-index and interface-transmission corrections in this underdense plasma change only the few-percent prefactor, below the accuracy intended here.