Physics · Electricity and magnetism · Grade 10

Plasma — the fourth state

How does the gas in a lightning channel or a neon lamp turn into conducting plasma?

The situation

In a summer thunderstorm the lightning channel heats up in an instant to about 30 000 K. In a neon sign tube on an evening street, electrons ionise a thin gas and turn it into a plasma that gives off light. The Sun, the stars, interstellar gas and the Earth's ionosphere are also plasma — most of the visible matter in the universe is in this state. In tokamaks, people try to hold plasma at a hundred million degrees with a magnetic field in order to obtain thermonuclear energy. Why does an ionised gas have properties completely different from an ordinary gas, and how can it be held in a “magnetic cage”?

Topic

Plasma — the fourth state of matter. Neon lamp, lightning channel and plasma oscillations

Key ideas

Debye screening
Plasma frequency
Larmor gyration and drift

Formulas

\frac{n_{i} n_{e}}{n_{0}} = \frac{2 g_{i}}{g_{0}} (\frac{2 π m_{e} k T}{h^{2}})^{3 / 2} e^{- E_{i} / k T}

Saha equation: the degree of ionisation of a gas in thermal equilibrium.

λ_{D} = \sqrt{\frac{ε_{0} k T_{e}}{n_{e} e^{2}}}, N_{D} = \frac{4 π}{3} n_{e} λ_{D}^{3} ≫ 1

Debye radius and Debye number: if there are many particles in the sphere, the plasma is ideal.

f_{p} = \frac{1}{2 π} \sqrt{\frac{n_{e} e^{2}}{ε_{0} m_{e}}} \approx 8.98 \sqrt{n_{e}} Hz

Plasma (Langmuir) frequency; $n_{e}$ is in m⁻³.

r_{L} = \frac{m v_{⟂}}{| q | B}, {\vec{v}}_{E} = \frac{\vec{E} \times \vec{B}}{B^{2}}, \sin^{2} θ_{l c} = \frac{1}{R_{m}}

Larmor radius, E×B drift (independent of charge) and the loss cone of a magnetic trap.

β = \frac{p}{B^{2} / 2 μ_{0}}, v_{A} = \frac{B}{\sqrt{μ_{0} ρ}}, n T τ_{E} \geq 3 \cdot 10^{21} {keV·s/m}^{3}

Ratio of plasma pressure to magnetic pressure, Alfvén speed, and the Lawson criterion for D–T fusion.

Theory

What plasma is: ionisation and quasineutrality

Plasma is a gas made of free electrons and ions that is nevertheless neutral overall (quasineutral, $n_{e} \approx n_{i}$ ). If a gas is heated, or struck by electrons accelerated in an electric field, its atoms are ionised; an electron and an ion can meet and recombine — recombination. In thermal equilibrium their ratio is given by the Saha equation: the degree of ionisation depends exponentially on temperature and increases as the density falls. In the laboratory plasma is made in gas discharges: non-self-sustained discharges, in which current flows only while an external ionising source works, and self-sustained discharges, which ionise the gas themselves (glow, arc, spark, corona); also by powerful lasers, microwave and induction heating. Thermionic emission from a hot cathode makes a discharge easier (in detail — in the “Electric current in vacuum and gases” experiment). In nature: lightning, the aurora, the ionosphere, the solar wind, the stars and the interstellar medium.

Debye screening and collisions

A “cloud” of opposite charge gathers around a charge placed in a plasma and cancels its field beyond a distance $λ_{D}$ : $φ = \frac{q}{4 π ε_{0} r} e^{- r / λ_{D}}$ — Debye screening. If the gas is much larger than $λ_{D}$ and there are many particles in the Debye sphere ( $N_{D} ≫ 1$ ), it is an ideal plasma: collective effects dominate, and the plasma parameter $g = 1 / (n λ_{D}^{3}) ≪ 1$ . In thermodynamics this gives the Debye–Hückel correction: the pressure is lower than the ideal value by $Δ p / p = - κ^{3} / (24 π n)$ (equation of state). Coulomb collisions between charges are mostly small-angle: their sum is accounted for by the Coulomb logarithm $\ln Λ \approx \ln (12 π n λ_{D}^{3})$ ( $\sim 10$ – $20$ ); the collision frequency is $ν_{e i} \propto n \ln Λ / T^{3 / 2}$ . So a hot plasma is an excellent conductor: the Spitzer conductivity $σ \propto T^{3 / 2} / \ln Λ$ is about 20 times that of copper at 10 keV. Electrons diffuse faster than ions, but the electric field that results holds them back — ambipolar diffusion; heat conduction too is mainly due to the electrons.

Plasma oscillations and waves

If a layer of electrons is displaced, the field between it and the ions pulls it back: the electrons oscillate at the plasma frequency $f_{p} \approx 8.98 \sqrt{n_{e}}$ (Langmuir oscillations) — measure the period in the cold plasma of “Waves” mode. Thermal motion turns this oscillation into a propagating Langmuir wave: $ω^{2} = ω_{p}^{2} + 3 k^{2} v_{T}^{2}$ (Bohm–Gross). Heavy ions give ion-acoustic waves: $ω \approx k c_{s}$ , $c_{s} = \sqrt{k T_{e} / m_{i}}$ . An electromagnetic wave propagates in a plasma only if $ω > ω_{p}$ : $ω^{2} = ω_{p}^{2} + c^{2} k^{2}$ , and $f_{p}$ is the cut-off frequency. In the ionosphere $f_{p}$ ~ 10 MHz: short waves below it are reflected and the radio signal goes round the Earth, while higher ones (FM, TV, GPS) pass through the ionosphere. In a magnetised plasma there are more kinds of wave: Alfvén waves along the field ( $v_{A} = B / \sqrt{μ_{0} ρ}$ , the field lines vibrate like a “string”) and magnetosonic waves ( $\sqrt{v_{A}^{2} + c_{s}^{2}}$ ).

Kinetic description, Landau damping, instabilities

An exact description needs the distribution function $f (\vec{r}, \vec{v}, t)$ . The chain of linked many-particle distributions — the Bogoliubov (BBGKY) hierarchy — breaks off at a small plasma parameter: the Vlasov equation in a self-consistent field without collisions, $\partial_{t} f + \vec{v} \cdot \nabla f + \frac{q}{m} (\vec{E} + \vec{v} \times \vec{B}) \cdot \partial_{\vec{v}} f = 0$ , with collisions added through the Landau collision integral. “Waves” mode solves the Vlasov equation by the “particle” method (PIC): the upper panel is the phase space $(x, v)$ , that is, $f$ itself. Landau damping: particles with speeds close to the wave's speed exchange energy with it; in a Maxwell distribution there are more slower particles — so the wave is damped without collisions. The opposite case — two counter-streaming beams — gives the two-stream instability: the field grows exponentially and “vortices” form in phase space (a kinetic instability).

Particles in fields: drifts, traps, the magnetosphere

In a uniform $B$ a charge moves along a Larmor circle: an electron and an ion in opposite directions, with the ion's radius $\sqrt{m_{i} / m_{e}}$ times larger (at equal energy). If a transverse $E$ is added, the centre of the circle moves with speed ${\vec{v}}_{E} = \vec{E} \times \vec{B} / B^{2}$ — a drift, independent of charge and mass. If $B$ is non-uniform there is a gradient drift $v_{\nabla B} = \frac{v_{⟂}^{2}}{2 Ω} \frac{\nabla B}{B}$ , and if the field lines are curved, a curvature drift; their direction depends on the sign of the charge, so they produce a current. Where the field grows stronger, a particle is reflected from a magnetic mirror: if $\sin^{2} θ > 1 / R_{m}$ it stays trapped, while particles with a small pitch angle escape through the loss cone. The Earth's magnetosphere is such a trap: particles of the radiation belts “bounce” between the poles, the plasma of the solar wind compresses the magnetosphere, and the aurora appears when particles in the loss cone fall into the atmosphere (in detail — in the “Aurora and the Lorentz force” experiment). Of the terrestrial planets only the Earth has a strong global field; Mercury's is weak, and Venus and Mars have no global field — the solar wind interacts directly with their ionospheres.

MHD, pinch and thermonuclear fusion

A plasma can be described as two fluids (electrons and ions) or as a single conducting fluid — magnetohydrodynamics (MHD). In it the field provides a magnetic pressure $B^{2} / 2 μ_{0}$ , and $β = p / (B^{2} / 2 μ_{0})$ . In a well-conducting plasma the field lines are “frozen into” the matter — they move with it (the Lundquist number is huge). A current compresses the plasma with its own magnetic field — the pinch effect; equilibrium follows the Bennett relation $I^{2} \propto N k T$ . But a pinch is unstable: where it narrows the field is stronger — the sausage instability ( $m = 0$ ), and where it bends the field is stronger on the inner side — the kink ( $m = 1$ ); when a heavy fluid “presses” on a light one — the Rayleigh–Taylor instability. A longitudinal field suppresses them. In a tokamak a strong toroidal field and the plasma current together create helical field lines; the safety factor must be $q ≳ 2$ – $3$ . In a stellarator the helical field is produced only by complex coils, while in inertial fusion a fuel capsule is compressed by lasers. The condition for getting energy is the Lawson criterion. In an MHD generator a hot conducting gas flows through a magnetic field and separates ions and electrons onto opposite electrodes: $U = u B d$ — electricity without a turbine. Plasma technologies: welding and cutting, “etching” of microcircuits, lamps and surface treatment.

Did you know?

An ionised gas was first called “plasma” in 1928 by the American chemist and physicist Irving Langmuir.
The word “tokamak” is an abbreviation of the Russian for “toroidal chamber with magnetic coils”; the world's first tokamak, T-1, was started in 1958 in Moscow, at what is now the Kurchatov Institute.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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