The sound a plasma carries on its electrons' heat

Sound in air is a contest between inertia and pressure. Compress a region and its pressure rises and pushes the neighbouring air away; the neighbouring air has mass and overshoots; the compression moves on. The speed that results is about the square root of pressure over density, and since in a gas both come from the same molecules — their mass is the density, their thermal motion is the pressure — the speed of sound comes out close to the molecules’ own thermal speed. The note too high for thin air followed that fact to its consequence, a gas that stops carrying sound when its wavelength approaches the molecules’ free path.

A plasma has two kinds of particle and can split the job. The ions are heavy: a proton is 1,836 times an electron, an argon ion seventy-three thousand times. The electrons are light and, in most laboratory plasmas and much of space, hot — heated by an electric field that pushes on electrons and ions alike but hands energy mainly to the light ones, as the bias no battery supplies found in a radio-frequency discharge, where the electrons sit at tens of thousands of kelvin and the ions near room temperature. Compress the ions in such a plasma and the electrons, tied to them by the requirement of near-neutrality, are compressed too, and their pressure pushes back. The inertia belongs to one species and the pressure to the other. The result is a sound wave whose speed

$$c_s = \sqrt{\frac{kT_e}{m_i}}$$

has the electrons’ temperature on top and the ions’ mass underneath. It is called the ion-acoustic wave, and its speed has no counterpart in a neutral gas.

Sound with someone else’s pressure #

The wave’s mechanism is a chain of three links, and each is a plasma property described elsewhere in this collection. The ions bunch, as the molecules of air bunch in a sound wave. The electrons, being light and fast, respond to the bunching almost instantly, crowding into it to keep the plasma neutral — the screening that the long-range force that does not reach found cutting off the field of any charge beyond a Debye length. And the crowded electrons have a pressure, $n kT_e$, which they cannot exert directly on the ions; they exert it through a small electric field, which forms where the electron pressure is high and points outwards, pushing the ions back out of the bunch.

The electric field is the coupling. Its size is whatever is needed to hold the electrons in place against their own pressure gradient — the same balance the wall a plasma builds against itself found in a sheath, where the electrons’ tendency to escape sets up the field that holds them back — and the ions feel the same field with the opposite sign. So the ions are pushed by the electrons’ pressure, transmitted through a field, and they respond with their own mass. The ions’ own thermal motion adds a little pressure of its own; with cold ions it adds nothing, and the sound speed is $\sqrt{kT_e/m_i}$ exactly.

That speed can be far above the ions’ thermal speed. In an argon discharge with electrons at 3 electronvolts — thirty-five thousand kelvin — and ions near room temperature, the ion-sound speed is 2.69 kilometres a second. Sound in argon gas at the ions’ temperature travels at 323 metres a second. The plasma carries sound 8.3 times faster than the gas its ions would make on their own.

The same speed appears in the sheath at a plasma’s edge, where Bohm’s criterion requires the ions to arrive at the speed $\sqrt{kT_e/m_i}$ before the sheath can form. That is not a coincidence. The ions entering a sheath are crossing a region where the electrons’ pressure is pushing them, and the speed at which a disturbance carried by the electrons’ pressure can travel through the ions is the speed at which the ions must be moving for the edge to be stationary in their frame: the presheath accelerates them to the sound speed for the same reason a nozzle accelerates gas to the speed of sound at its throat.

Why the electrons count once and the ions three times #

The sound speed of a neutral gas carries a factor that the formula above appears to have lost: $c = \sqrt{\gamma kT/m}$, with $\gamma = 5/3$ for a monatomic gas, because a sound wave compresses the gas too quickly for heat to flow out of each compression, and an adiabatic compression stiffens the gas more than an isothermal one. Pressure is a rate of arrival built the pressure from molecules striking a wall; $\gamma$ says how much faster they strike after a quick squeeze than after a slow one. The medium decides the speed made the general point that a wave’s speed is a ratio of a stiffness to an inertia, and $\gamma$ is a statement about the stiffness.

In the plasma the two species answer the question differently, and the full expression is

$$c_s^2 = \frac{kT_e + 3kT_i}{m_i}.$$

The electrons count once — their $\gamma$ is one — because they are so fast that they cross many wavelengths of the wave in one of its periods and carry heat from compressions to rarefactions as they go, keeping every part of the wave at one electron temperature. Their compression is isothermal however quickly the wave squeezes them. The ions count three times because the opposite holds: they are slow, they cannot carry heat across a wavelength in a period, and in a collisionless plasma they cannot share the compression among their three directions of motion either, so a wave squeezing them along one direction raises their temperature along that direction alone. A one-dimensional adiabatic compression of an ideal gas has $\gamma = 3$, since only one of the three directions of motion takes up the energy. The speeds in a still room made the three directions of motion equal by collisions; here nothing equalises them on the wave’s timescale.

So the electrons give the sound its speed and are kept from giving it more by their own speed, and the ions add three times their temperature because they are too slow to do anything else. With cold ions the second term vanishes and the formula reduces to the one at the top. With equal temperatures it gives $c_s = 2\sqrt{kT/m_i}$, twice the cold-ion value — which is the 2.02 $c_s$ the kinetic calculation finds for the phase velocity at equal temperatures, and the reason the wave then sits where the ions are steepest.

The same splitting of roles runs through the waves a magnetised plasma carries. The wave that does not know what the gas is made of found that in a magnetised plasma the field supplies a stiffness of its own, and the fast magnetosonic wave combines the field’s stiffness with exactly this ion-acoustic one: along the field the plasma carries ion sound, across it the field’s pressure and the electrons’ add.

Until the electrons stop keeping up #

The neutrality on which the mechanism depends is not perfect, and the departure from it sets the wave’s dispersion. The electrons cancel the ions’ bunching only over distances longer than a Debye length. For a wave whose wavelength is long compared with $\lambda_D$, the electrons crowd into each compression almost fully, the plasma stays neutral, and the wave is sound. For a wave whose wavelength approaches the Debye length, the electrons cannot pile into each compression closely enough, and the ions’ bunching is left partly unshielded.

The arithmetic is short. Electrons in equilibrium with a potential follow Boltzmann’s law, so their density changes by $e\phi/kT_e$ of itself; Poisson’s equation ties the potential to the difference between the ion and electron densities; and solving the two together gives the electrons’ perturbation as the ions’ divided by $1 + k^2\lambda_D^2$.

At long wavelengths the electrons cancel 99 per cent of the ions’ charge and the restoring force is the electrons’ pressure; at $k\lambda_D = 3$ they cancel 10 per cent and the restoring force is the ions’ own unscreened electric field. That second regime is an oscillation, not a sound: the ions’ charge pulls them back at a frequency set by their density and mass alone, the ion plasma frequency, which is the electrons’ plasma frequency of the frequency below which nothing gets in scaled down by the square root of the mass ratio — forty-three times lower for hydrogen. The dispersion relation joins the two regimes:

$$\omega^2 = \frac{k^2c_s^2}{1 + k^2\lambda_D^2},$$

a straight line at long wavelengths that bends over towards the ion plasma frequency at short ones.

The kinetic curves are not drawn from the fluid formula. Each point is a complex root of the full dispersion relation of a plasma of Maxwellian electrons and ions — the susceptibility of each species written with the plasma dispersion function, the integral over its velocity distribution that carries every particle’s response — solved numerically for the frequency at which the plasma can sustain a disturbance unforced. The fluid formula is the limit of that relation when the ions are cold and the electrons hot. The kinetic result agrees with it at long wavelengths and large temperature ratios, rises above it where the ions are warm, and comes with something the fluid formula has no way to supply: an imaginary part.

The ions that ride the wave #

The imaginary part is the damping, and it has the same origin as the collisionless damping that the wave that dies with nothing to rub against found for the electrons’ plasma oscillation. Particles moving at nearly the wave’s speed see an almost steady field and exchange energy with it over many cycles; those moving slightly slower than the wave are pushed forward and take energy, those slightly faster are held back and give it. In a Maxwellian distribution there are more of the slower than of the faster at any speed above zero, so the net transfer is from the wave to the particles, at a rate proportional to the slope of the distribution at the wave’s speed.

For the electron plasma oscillation the particles that did this were electrons. For ion sound there are two candidates. The electrons are far faster than the wave — their thermal speed is $\sqrt{m_i/m_e}$ times the sound speed — so the wave’s speed sits near the middle of their distribution, where it is nearly flat, and they take only a little: a damping of order $\sqrt{\pi/8}\sqrt{m_e/m_i}$ per radian, which the full calculation puts at 1.3 per cent for hydrogen. The ions are the ones that matter. Their thermal speed is $\sqrt{kT_i/m_i}$, which is the sound speed multiplied by $\sqrt{T_i/T_e}$. When the two temperatures are equal, the ions’ thermal speed is the sound speed, and the wave is sitting right on the flank of the ions’ distribution.

At equal temperatures the ions that can ride the wave are numerous and their distribution is steep there, and the wave is destroyed within a cycle. Raise the electron temperature tenfold and the ions are cold beside the wave: its speed is out in the far tail of their distribution, where almost no ion is fast enough to keep pace with it, and the ions stop taking energy. What is left is the electrons’ small tax.

The fall is steep — two orders of magnitude between equal temperatures and a ratio of twenty — because the number of ions at the wave’s speed falls as a Gaussian in the ratio. Beyond about twenty the curve flattens on the electrons’ floor, which is lower for heavy ions because the electrons are then still faster relative to the sound speed and still flatter at it: argon’s floor is 0.20 per cent of the amplitude per radian, hydrogen’s 1.3. The weak-damping formula, the textbook approximation that treats the damping as a small correction to the fluid wave, follows the kinetic curves wherever the damping is small and overestimates it by up to 60 per cent at ratios between five and ten, where the damping is no longer small and the approximation is being asked for more than it can give.

Why ion sound needs a hot-electron plasma #

The consequence is that ion-acoustic waves are a property of particular plasmas, not of plasma in general. A plasma in thermal equilibrium, with electrons and ions at one temperature, cannot carry ion sound over more than a cycle; the collisionless damping takes it. A plasma in which the electrons are kept far hotter than the ions — and that is most plasmas driven by electric fields, from fluorescent tubes to the processing reactors that etch microchips — carries it easily, and in such plasmas it is the principal low-frequency wave.

The first clean measurements, by Alexander Wong, Robert Motley and Nicola D’Angelo in 1964, used a plasma of caesium or potassium made by contact ionisation on a hot plate, in which the electrons and ions came off the plate at nearly the same temperature. They launched waves from a grid and followed them down the column. The waves travelled at the speed kinetic theory gives for that temperature ratio, faster than the cold-ion sound speed by the ions’ own pressure, and died away over a few wavelengths at the rate kinetic theory gives. It was the first laboratory demonstration of Landau damping by ions, and it worked because the plasma had been made in the one condition — equal temperatures — in which the damping is large enough to measure over a short distance.

In space the same condition decides where ion sound can be seen. In the solar wind the electrons are typically a few times hotter than the protons, enough for ion-acoustic fluctuations to survive near spacecraft, and they are seen there; in regions where the protons have been heated to match, they are not. Around the Earth’s bow shock and in the turbulent regions behind it, ion-acoustic waves are thought to be one of the ways the plasma converts the energy of its bulk flow into heat in the absence of collisions.

What the figures leave out #

The figures treat a uniform, unmagnetised, collisionless plasma with Maxwellian electrons and ions, one ion species, and waves small enough to be linear. Each simplification matters somewhere. A magnetic field leaves the wave unchanged along the field but makes it a different wave across it, coupled to the ions’ gyration. Collisions add ordinary viscous damping and, at high density, change the electrons from isothermal to adiabatic, changing the sound speed by a factor close to one. A second, lighter ion species — a few per cent of helium or hydrogen in an argon plasma — can dominate the damping, because its ions are faster and lie closer to the wave’s speed. And waves of large amplitude steepen, as sound in air does, and in a plasma form ion-acoustic solitons and collisionless shocks, which no linear figure can show.

The Debye length and temperatures used are those in which the fluid and kinetic pictures can be compared directly. In each figure the wave’s frequency is low enough that the electrons follow Boltzmann’s law, which fails when the wave approaches the electrons’ own plasma frequency, a regime not drawn.

Still open: how plasmas turn flow into heat without collisions #

The linear wave is understood. What is not settled is the role ion-acoustic waves play when they are driven hard — when a current flows through a plasma fast enough that the electrons’ drift past the ions exceeds the sound speed, and the waves grow instead of damping. That ion-acoustic instability was proposed in the 1960s as a source of “anomalous resistivity”, a resistance far larger than collisions provide, in plasmas from current sheets in the solar corona to the region where magnetic field lines reconnect. Whether it is the dominant mechanism anywhere, whether its waves saturate at amplitudes large enough to matter, and how they hand their energy to the ions, are questions that spacecraft measuring the fields and particles in reconnection sites at high time resolution are now in a position to answer, and so far the answers differ from one site to the next.

The linear statement is short. In a plasma the ions carry the inertia and the electrons the pressure, so ion sound travels at $\sqrt{kT_e/m_i}$ — 2.69 km/s in an argon discharge with 3 eV electrons, eight times the gas’s own sound speed — bending towards the ion plasma frequency once its wavelength approaches the Debye length; and it survives only while $T_e \gg T_i$, losing 0.45 of its amplitude per radian when the two are equal and 0.031 when the electrons are ten times hotter, because only then are the ions too slow to ride it. A sound needs someone to push and someone to be pushed, and in a plasma the two can be different particles at different temperatures.