Illustrated Physics
physics, drawn
Concept

Phonon — where it appears

A quantised sound wave, the elementary excitation of a vibrating lattice or liquid. Its energy density goes as the fourth power of temperature, which gives the low-temperature heat capacity its cubic law and makes a phonon gas behave in many respects like a gas of photons.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

The average position of something that is only shaking. The mean displacement of an oscillator against temperature, taken as a Boltzmann average over each well rather than from any expansion of it, with the temperature measured against each well's own depth so that unlike bonds can share an axis. A symmetric well gives exactly zero at every temperature — heating a harmonic solid makes it vibrate harder and does not make it larger. The others drift outward, because the outward side is the shallower one, and the measured slopes are a pendulum 0.000, a chemical bond 0.766, a pair of atoms 0.150. Thermal expansion is not a property a spring has; it is one a spring lacks.

Why heating a perfect spring changes nothing

A harmonic solid vibrates harder when heated and does not get any longer. Thermal expansion lives entirely in the term that the harmonic approximation throws away — and so does the fact that a solid conducts heat at a finite rate, which is the same discarded term doing a second job nobody would have connected to the first.

thermodynamics · Harmonic approximation The speed of a wave that carries no pressure. Second-sound speed against temperature, computed from the two-fluid equations with the normal component treated as a phonon gas — which it is below about six-tenths of a kelvin. The upper line is ordinary sound at 238 m/s, which moves the two components together. The lower curve is the other mode, and at low temperature it sits at 137.4 m/s, which is 238/√3 to three figures: a result with no adjustable constant in it, and the reason to believe the two-fluid model rather than merely to use it. What oscillates in this wave is not the density — the two components move in opposite directions and their sum stays put — but the fraction that is normal, which is a temperature. So a temperature disturbance in helium II propagates, with a speed, a reflection and a resonance, where in every ordinary liquid it diffuses and has none of those. Above a kelvin the rotons take over from the phonons and the measured curve falls to about 20 m/s; the model here is the low-temperature one and it is drawn only where it holds.

The heat that arrives as a wave

Two fluids with two velocities give two wave equations, not one. In the first the components move together and the density oscillates, which is ordinary sound. In the second they move oppositely, the density stays put, and what oscillates is the temperature — so a heat pulse in liquid helium has a speed, a front and a reflection.

fluids · Superfluidity The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all.

The frequency a lattice cannot carry

A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

waves · Periodic media The edge that is a straight line, not a step. The absorption of gallium arsenide near its own band gap of 1.424 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 5.6 millielectronvolts at 10 kelvin to 7.5 at 300, a factor of 1.33. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's.

Below the gap, where there is nothing to absorb

A semiconductor is supposed to be transparent below its band gap, and it is not. The absorption falls exponentially instead, over seven decades, along a straight line whose slope is an energy of a few millielectronvolts — and the description that produces that line has no states in the gap at all. What the slope measures is how much the gap itself is moving about.

waves · Attenuation One dimensionless group between an engine and Carnot. The efficiency of a thermoelectric couple against the temperature of its hot side, with the cold side at 300 kelvin, for 4 values of the figure of merit, and the Carnot ceiling drawn above them. The expression has exactly one material quantity in it — the dimensionless group formed from the Seebeck coefficient squared, the electrical conductivity, the temperature and the thermal conductivity — and everything else is the two temperatures. At 600 kelvin, a figure of merit of one gives 10.8 per cent against a Carnot ceiling of 50.0, and a figure of merit of four gives 22.6. The approach to the ceiling is slow: every doubling of the group buys less than the last, so the difference between a good material and a perfect one is smaller than the difference between a poor material and a good one.

An engine with one number in it

A thermoelectric couple has no moving part and no working fluid, and its efficiency is the Carnot value multiplied by a factor containing exactly one dimensionless group of material properties. Sixty years of effort have moved that group from about one to about two, and the reason it is hard is that its three ingredients are not independent: raising the conductivity ruins the coefficient it is squared against, and the only lever that is really free is the heat the lattice carries.

thermodynamics · Heat engines Helium's excitations, and the line that decides how fast it can flow. The energy of an elementary excitation in superfluid helium-4 against its momentum, as a smooth fit through published neutron-scattering values: a phonon branch rising at the speed of sound, 238 m/s, a maximum near 13.9 K, and the roton minimum of 8.62 K at 1.92 Å⁻¹. A straight line from the origin has a slope that is a velocity. The steepest line that just touches the spectrum is Landau's critical velocity, 57 m/s, and it touches at 1.98 Å⁻¹, just past the roton minimum. A body moving slower than that cannot make any excitation at all and so cannot lose energy to the liquid; the dashed line is the speed of sound, far above it.

The speed below which nothing can be made

An object moving through an ordinary liquid always feels drag, because it can always pass some of its energy to the liquid. In a superfluid there may be nothing it can pass it to. Landau's argument turns the question into geometry: an excitation of momentum p and energy ε can be made only by a body moving faster than ε/p, so a liquid flows without friction below the smallest value of that ratio. For helium the smallest value is not the speed of sound but 57 m/s, set by the dip in its spectrum called the roton — and an ideal gas, with no dip and no sound, has no such speed at all.

quantum · Superfluidity How well a diamond-like crystal conducts heat, from 1 K to 1000 K. The thermal conductivity of a model crystal with diamond's sound speed and Debye temperature, from kinetic theory applied to its phonons, against temperature on logarithmic axes: natural carbon in a sample 1 mm across, isotopically enriched carbon, and natural carbon in a sample 0.1 mm across; copper's room-temperature 400 W/m K is marked for comparison. The model is calibrated to natural diamond's 2,200 W/m K at room temperature. At low temperature the conductivity rises as the cube of the temperature, limited by the sample's walls; at high temperature it falls as umklapp collisions shorten the phonons' paths. Between, it peaks: 3.7·10⁴ W/m K at 100 K for the natural crystal; 7.8·10⁴ W/m K at 94 K for the enriched crystal; 1.2·10⁴ W/m K at 126 K for the thin natural crystal. The model gets the shape — the rise as the cube of the temperature, a peak near 100 K, the fall — but overestimates the heights, because it leaves out the phonon collisions that conserve momentum and every defect but isotopes; measured enriched diamond peaks near 4 × 10⁴ W/m K. An insulator with no free electrons still conducts heat five times better than copper at room temperature.

The gas of sound that carries a diamond's heat

Diamond has no free electrons and conducts heat five times better than copper. What carries the heat is a gas — not of molecules but of quantised sound waves, phonons — and the kinetic theory that gives the conductivity of air gives the conductivity of a crystal with the same formula: a third of heat capacity times speed times mean free path. The gas is strange in two ways. At low temperature its particles cross the whole crystal without meeting anything, so the conductivity depends on how big the sample is. And at room temperature one atom in a hundred being a neutron heavier is enough to scatter its fastest-oscillating members.

thermodynamics · Kinetic theory The two lines a vibration adds to scattered light. The light scattered by silicon's optical vibration at 520 cm⁻¹, against its shift from the 532 nm laser line in cm⁻¹ (negative, lost to the crystal; positive, gained from it), on a logarithmic intensity axis, at 100 K, 300 K and 600 K; the strong unshifted line and its wings are cut out at the centre. The Stokes line, where the light gives up a quantum of vibration, barely changes: 1.00, 1.09 and 1.40 in relative strength. The anti-Stokes line, where it takes one, needs a vibration already excited and grows from 7·10⁻⁴ of the Stokes line at 100 K to 0.103 at 300 K and 0.36 at 600 K. The light comes back bluer only as often as the crystal is warm enough to pay for it.

The light that comes back bluer when it is warm

Shine a green laser on a crystal and a tiny fraction of the light comes back with a different colour: a little redder, having given a quantum of energy to one of the crystal's vibrations, or a little bluer, having taken one. Losing energy is always possible. Gaining it needs a vibration already excited, so the bluer light is fainter, and fainter still the colder the crystal. The ratio of the two is a Boltzmann factor, fixed by quantum statistics and nothing else. It is a thermometer that needs no contact and no calibration — and in a long glass fibre, it is thousands of thermometers at once.

optics · Scattering

Named alongside it

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

Thermal conductivityApproximationBand gapThe Boltzmann factorDispersion relationSemiconductorAbsorptionAnharmonicityAttenuationBloch waveBose einstein distributionCarnot efficiency

All concepts