Illustrated Physics
physics, drawn
Concept

Plasma oscillation — where it appears

The collective oscillation of a plasma's electrons about its ions, at a frequency fixed by the density alone. It is the reason a plasma reflects radiation below that frequency and passes it above, and short-wavelength oscillations are damped away by resonance with the particles moving at their own phase speed.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A click that arrives sorted by pitch. Arrival time against frequency for a broadband pulse travelling 40 megametres along a magnetic field line through a plasma of 1e+8 electrons per cubic metre in a field of 300 nanotesla — the magnetosphere, roughly. The gyrofrequency is 8.4 kilohertz and the plasma frequency 90 kilohertz, so this is the regime where a right-hand circularly polarised wave travels happily below both of them. It does not travel at one speed. The group velocity is 2c√ω(ω_c − ω)^(3/2)/(ω_p ω_c), which is zero at zero frequency and zero again at the gyrofrequency and peaks in between — at 2.10 kilohertz here, which is a quarter of the gyrofrequency, found by searching the drawn function rather than quoted. Everything either side of that peak is slower, so the curve has a nose: 3.21 s at 500 Hz, 2.20 s at 2000 Hz, 3.59 s at 5000 Hz. The branch below the nose is the classic whistler — a lightning stroke in one hemisphere reaching a receiver in the other as a note gliding downward over about a second, which is what gave the phenomenon its name — and the branch above it arrives as a rising tone at the same time. Both are observed, and a recording that shows the two joined at the nose is how the gyrofrequency along the path is read off directly. Read the other way it is an instrument: the product of the delay and the square root of the frequency is 79 s·Hz^½ here, nearly constant across the low end of the band, and it measures the electron content of a path through the magnetosphere that nothing else could reach.

The whistle that arrives sorted

A lightning stroke in one hemisphere reaches a receiver in the other as a note gliding downward over about a second. Nothing dispersed the sound; there was no sound. A radio pulse travelled forty megametres along a magnetic field line through a plasma whose group velocity depends on frequency, and arrived with its frequencies separated by up to three seconds.

astrophysics · Plasma oscillation A wave that dies with nothing to rub against. The electric field of a plasma wave at kλ = 0.5, against time in plasma periods, on a logarithmic scale, obtained by integrating the collisionless kinetic equation as an initial-value problem. There are no collisions in the equation, no viscosity and no resistance; the only operator acting on the distribution is a rotation of phase whose rate depends on the particle's speed. The field nevertheless falls exponentially, at 0.1534 per plasma period, against the published root of the kinetic dispersion relation at this wavenumber, 0.1534, and Landau's asymptotic formula's 0.1514. Meanwhile the free energy of the perturbation — the weighted norm of the distribution plus the field energy, which the equation conserves exactly — moves by 2.4e-10. So nothing has been dissipated: every joule the field loses is still in the distribution, and the accounting closes to a part in ten thousand million. The energy has gone into the particles' ordered motion, and the information about the wave is wound into structure at finer and finer scales in velocity.

The wave that dies with nothing to rub against

Every damping in this collection so far removes energy from a wave and puts it somewhere warmer. This one removes it and produces no heat at all: there are no collisions in the equation, the entropy is unchanged, the whole thing runs backwards perfectly, and the wave still dies exponentially. What it dies into is structure in velocity too fine for a field to see.

astrophysics · Plasma oscillation The count that works for everything with a mass. How many beams a Stern-Gerlach analyser splits a particle into, for four spins, with the photon on the bottom row. For anything with a mass the answer is 2j+1 — go to the particle's rest frame, where its spin can point in any direction, and count the projections along whichever axis the magnet defines. A spin-one particle gives three: up, down, and a middle beam that is not deflected at all. The photon has spin one and gives two. The middle state does not exist, and it is not that it is hard to produce or weakly coupled — there is no such state of the electromagnetic field. A light wave has two polarisations and the third one, in which the field would oscillate along the direction of travel, is not a solution of Maxwell's equations at all.

Two states where the counting says three

A particle of spin one has three states, and the photon has two. The missing one is not rare or weakly coupled — there is no such state of the electromagnetic field. What removes it is that a massless particle has no rest frame, so the rotations that would turn one projection into another are not available; and the state comes back the moment the particle acquires a mass, which is what a photon does in a plasma.

quantum · Spin A relativistic mass on a spring, at four amplitudes, scaled to one size. The displacement of a mass on an ideal spring against time, over two periods, with the mass's inertia growing with its energy as relativity requires, at amplitudes of 0.2, 1, 3, 10 times c/ω₀ — the distance light travels in one radian of the slow oscillation. Each curve is divided by its own amplitude and its own period so that shape alone is compared. The small swing is the familiar sine. The large one is a triangle: the mass spends almost the whole cycle moving at nearly the speed of light, turning round abruptly at each end. Periods: 1.007, 1.175, 2.132, 6.431 times the slow period.

The spring that becomes a light-clock

A mass on an ideal spring keeps the same period however far it swings — that is what makes it a clock. Give the mass the inertia relativity demands, pull it back far enough, and the clock breaks. The speed can never pass c, so a large swing spends almost all its time moving at nearly the speed of light, turning round abruptly at each end: the sine becomes a triangle, the period grows until it is simply the time light takes to cross the swing four times, and a clock riding on the mass ages a small fraction of what one beside it does.

relativity · Relativistic dynamics Two beams that each stay straight, and wind each other into whirlpools. Phase space of a one-dimensional particle simulation: every dot is an electron, plotted at its position along a periodic box one fastest-growing wavelength long (10.26 v₀/ωₚ) and its velocity, in units of the beam speed. One beam starts at +v₀, the other at −v₀, with a 0.0001 ripple in position. At t = 18/ωₚthe beams are visibly rippled; by 26 the ripple has grown into overturning, and by 40 the two beams have wound round each other into a single vortex, filamented into strands. When growth first stops, at ωₚt = 25.2, the wave's field is 0.56 m v₀ωₚ/e; electrons trapped in it bounce at 0.59 ωₚ, faster than the 0.354 ωₚ at which it was growing, and it traps any electron within ±1.92 v₀ of the wave's speed — zero, by symmetry — so both beams are inside its separatrix; total energy was conserved to 0.31%.

The two beams that unravel each other

A cold beam of electrons, alone, is perfectly stable: disturb it and it rings at its plasma frequency and carries the ringing downstream. Send a second beam through it the other way and the pair is not stable at all. A ripple of the right length grows by a factor of e in less than half a plasma period, takes a third of the beams' drift energy into an electric field, and then winds both beams into a single whirlpool in which their drift is gone. Neither beam can do it. The growth exists only in the pair, and a beam a millionth as dense as the plasma it crosses is enough to start it.

astrophysics · Plasma oscillation

Named alongside it

The objects these essays reach for when they reach for this one.

Landau dampingPlasma frequencyTwo stream instabilityAnharmonicityCircular polarisationCollisionless plasmaCountingDebye lengthDegeneracyDispersionDispersion relationDistribution function

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