Illustrated Physics
physics, drawn
Thermodynamics

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.
14 min read 5 figures The same equation againWhere the model stops

Assumes: The viscosity that does not care how much gas there is · How far a molecule gets

The speeds in a still room found the molecules of air moving at hundreds of metres a second, and pressure is a rate of arrival turned their collisions with a wall into the gas law. How far a molecule gets found the mean free path, sixty-eight nanometres in air, and the viscosity that does not care how much gas there is used it to compute how a gas carries momentum and heat across a gap: the thermal conductivity is a third of the heat capacity per volume, times the molecules’ speed, times their mean free path. Later essays carried the same counting to neutrinos, to light diffusing out of the Sun and to the first correction to the gas law.

This essay applies the formula to something with no molecules at all. A crystal of diamond is a rigid lattice of carbon atoms that do not travel anywhere, and it has no free electrons. Yet it carries heat better than any metal at room temperature. The carriers are the vibrations of the lattice themselves, quantised into particles — phonons — that move at the speed of sound, collide, scatter and diffuse, and form a gas inside the solid. The kinetic theory of gases describes them with the same formula, and the ways the phonon gas differs from air are what make crystals conduct as they do.

Sound in packets

A crystal’s atoms are held in place by springs, the bonds between them, and every minimum is a parabola found that near equilibrium the springs are harmonic. A harmonic lattice has normal modes — waves of displacement running through it at the speed of sound, with wavelengths from the size of the crystal down to twice the spacing of the atoms — and quantum mechanics makes the energy of each mode come in whole quanta. A quantum of lattice vibration is a phonon. A warm crystal is full of them, distributed over the modes as photons are distributed over the modes of a cavity, and a temperature gradient makes more of them travel from hot to cold than from cold to hot. That net flow is the heat current.

So the phonons are a gas, and kinetic theory applies to them: the conductivity is κ=13Cvℓ\kappa = \frac{1}{3}C v \ell, with CC the heat capacity of the phonon gas per unit volume, vv the speed of sound, and ℓ\ell the distance a phonon travels between being scattered. The model in this essay takes that formula seriously, frequency by frequency, for a crystal with diamond’s sound speed, thirteen kilometres a second, and its Debye temperature, 2,230 kelvin, and with three ways of scattering a phonon: hitting the crystal’s walls, hitting an atom of the wrong mass, and colliding with other phonons. One constant, the strength of the phonon collisions, is fixed so that the natural crystal conducts 2,200 watts per metre per kelvin at room temperature, the measured value for diamond. The rest follows.

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.
Fig. 1 The conductivity of a diamond-like crystal from kinetic theory of its phonons, against temperature, for natural carbon in a 1 mm sample, enriched carbon, and natural carbon in a 0.1 mm sample, with copper’s 400 W/m K marked. It rises as T3T^3, peaks near 100 K — at 3.7×1043.7 \times 10^4 W/m K for the natural crystal — and falls at high temperature.

The curve has three parts, and each is a different factor in 13Cvℓ\frac{1}{3}Cv\ell taking control. At low temperature the conductivity rises steeply, as the cube of the temperature. Near a hundred kelvin it peaks, tens of thousands of watts per metre per kelvin, a hundred times copper’s room-temperature value. At high temperature it falls, passing diamond’s 2,200 at room temperature and continuing down. The model reproduces the shape that measurements on diamond show. It overestimates the height of the peak — measured enriched diamond peaks near forty thousand, natural lower — because it leaves out some of the physics; the shape is the part to trust.

How many phonons there are

The heat a phonon gas can hold. The heat capacity per volume of the model crystal, as a fraction of the Dulong–Petit value 3nk, against temperature on logarithmic axes, from Debye's theory with Θ = 2230 K. Far below Θ only the longest-wavelength phonons are excited and the capacity rises as T³ (dashed): 7·10⁻⁶ of the classical value at 10 K, 0.0070 at 100 K. At 300 K diamond is still well below its plateau, at 0.166, because its Debye temperature is so high. The heat capacity is the C in κ = Cvℓ/3: it is why the conductivity rises steeply at low temperature, where every doubling of the temperature puts eight times as many phonons to work carrying heat, while their path stays fixed by the walls.
Fig. 2 The Debye heat capacity of the model crystal as a fraction of the classical 3nk, against temperature. Far below the Debye temperature, 2,230 K, it rises as T3T^3: 7×10−67 \times 10^{-6} of the classical value at 10 K, 0.007 at 100 K, and still only 0.166 at 300 K.

The heat capacity of the phonon gas is the first factor. At high temperature every mode holds its classical share of energy, half a kT for every way of moving, and the crystal reaches the Dulong–Petit value, three times Boltzmann’s constant per atom. At low temperature only the modes with quanta smaller than kTkT are excited, and those are the long-wavelength ones, whose number grows as the cube of the temperature, so the heat capacity falls as T3T^3, Debye’s law. Diamond’s bonds are so stiff that its Debye temperature is 2,230 kelvin, and at room temperature it is still far below its classical heat capacity, at a sixth of it: most of its modes are frozen, and the heat is carried by the long-wavelength minority.

That is the rising part of the conductivity curve. At low temperature the number of phonons grows as T3T^3, their speed is fixed, and, as the next section shows, their mean free path is fixed too; so the conductivity grows as T3T^3, doubling the temperature putting eight times as many carriers to work.

A gas that does not meet itself

How far a phonon gets, and what stops it. The mean free path of a typical thermal phonon — one at the energy that carries most of the heat, about 3.8 kT — in the natural crystal 1 mm across, against temperature on logarithmic axes, for each way of being scattered and combined. The walls (dashed) stop every phonon within a millimetre; isotopes (dotted) scatter high-frequency phonons as ω⁴, and so matter more as the crystal warms; umklapp collisions between phonons (thin) switch on exponentially as e^(−Θ/3T). At 10 K the combined path is 999.3 μm: the walls decide. At 100 K, 102.0 μm; at 300 K, 178 nm, set by umklapp. A gas's mean free path is set by its molecules hitting each other; a phonon gas's, by the crystal's edges, its imperfections and the phonons' own collisions, each winning in its own range of temperature.
Fig. 3 The mean free path of a typical heat-carrying phonon in the natural 1 mm crystal against temperature: set by the walls below about 50 K (0.999 mm at 10 K), by isotopes and collisions near the peak (0.10 mm at 100 K), and by umklapp collisions at room temperature (178 nm at 300 K).

The mean free path is where the phonon gas departs most from air. A molecule of air travels sixty-eight nanometres before hitting another. A phonon in a cold, perfect crystal hits nothing: there are few other phonons to collide with, the crystal is perfectly regular, and the phonon travels until it reaches the crystal’s surface. Below about fifty kelvin, in a crystal a millimetre across, the mean free path is the size of the crystal. As the temperature rises, two other processes take over. Atoms of the wrong mass — carbon-13 among the carbon-12, about one atom in ninety in natural carbon — scatter phonons like specks of dust scattering light, as the fourth power of frequency, the same Rayleigh law that makes the sky blue. And phonons begin to collide with one another.

Not every collision between phonons resists heat flow. Two phonons that merge into one carry their total momentum into it, and a gas whose collisions all conserve momentum has no reason to stop flowing: those “normal” collisions only reshuffle energy among the phonons. The collisions that create thermal resistance are the ones Peierls identified in 1929, in which the merged phonon’s wavevector would land outside the range a lattice can support and is folded back by a vector of the crystal’s reciprocal lattice, reversing the momentum. These umklapp collisions need phonons with large wavevectors, which are scarce until the temperature is a sizeable fraction of the Debye temperature, so they switch on exponentially, as e−Θ/3Te^{-\Theta/3T}. At room temperature in diamond they limit a typical phonon to about two hundred nanometres, and they are why the conductivity falls at high temperature.

Conductivity that depends on the sample

The wall-limited regime has a consequence no gas of molecules at ordinary pressure shows.

A conductivity set by the size of the sample. The thermal conductivity of the natural model crystal at low temperature, for samples 0.1, 1 and 10 mm across, on logarithmic axes. Below the peak the phonons cross the sample without colliding, and the walls set their mean free path: the conductivity is the heat capacity times the speed times the sample's width, rising as T³ and proportional to the size. At 5 K the three samples conduct 3.02, 30.15, 300.56 W/m K — each ten times the last — and their peaks lie at 126, 100, 79 K. Casimir pointed this out in 1938: a crystal's conductivity at low temperature is not a property of the material but of the specimen, and de Haas and Biermasz had already measured it rising with the crystal's thickness. It is the phonon gas's version of a gas so thin that its molecules cross the container without meeting.
Fig. 4 The low-temperature conductivity of the natural crystal for samples 0.1, 1 and 10 mm across: at 5 K, 3.0, 30 and 301 W/m K, each ten times the last, rising as T3T^3; the peaks lie at 126, 100 and 79 K.

If the phonons cross the crystal without colliding, their mean free path is the crystal’s width, and the conductivity is proportional to it. A crystal ten times thicker conducts ten times better at five kelvin: three, thirty and three hundred watts per metre per kelvin for samples a tenth of a millimetre, a millimetre and a centimetre across. The thermal conductivity has stopped being a property of the material and become a property of the specimen. Casimir explained it in 1938, after de Haas and Biermasz had measured crystals of potassium chloride and found their low-temperature conductivity rising with their thickness. The phonon gas in that regime is like a molecular gas so rarefied that its molecules cross the container without meeting each other, the regime the viscosity that does not care how much gas there is found where Maxwell’s pressure-independence breaks down.

The surface also matters. A phonon that hits a perfectly smooth surface reflects specularly and keeps going, and in a smooth enough crystal the mean free path can exceed its width; a rough surface scatters phonons diffusely and holds the path to the width. Polishing a crystal raises its low-temperature conductivity, a fact that would be absurd for a metal and is routine for an insulator. The size effect also reaches room temperature when the sample is small enough. A silicon wire fifty nanometres thick, narrower than the paths of many of the phonons that carry heat in bulk silicon, conducts several times less than bulk silicon, and wires with rough surfaces less still; the effect is used deliberately in thermoelectric devices, which need materials that conduct electricity well and heat badly, and is a nuisance in the transistors of a microchip, whose nanometre-scale channels shed their heat less readily than the bulk value promises.

One atom in ninety

Which phonons carry a diamond's heat at room temperature. The contribution of phonons of each energy to the thermal conductivity of the model crystal at 300 K, against the phonon energy in units of kT, for natural carbon and for carbon enriched to 99.9 per cent carbon-12, scaled to the larger peak. Isotope scattering falls hardest on the high-frequency phonons, as ω⁴ — the Rayleigh law, for sound off a mass defect instead of light off a molecule — and removing most of the carbon-13 lets more of them carry heat; but at room temperature umklapp collisions already limit them, and in this model the enriched crystal conducts only 2306 W/m K against 2200, 5 per cent more. Measured, enriched diamond conducts about half as much again as natural at room temperature, 3,300 against 2,200 W/m K. The difference is the part of the physics the model leaves out: normal collisions between phonons, which conserve momentum and so do not resist heat flow themselves, keep handing energy to the high-frequency phonons that isotopes scatter, so removing isotopes helps far more than the simple sum of rates allows.
Fig. 5 Which phonon energies carry the heat at 300 K in the model crystal, for natural and for enriched carbon. The model gives the enriched crystal 2,306 W/m K against 2,200; measured, enriched diamond reaches about 3,300.

The isotope effect is the most surprising part, and the model’s failure to capture its size is instructive. Carbon-13 is chemically identical to carbon-12; its bonds are the same strength. It differs only in mass, by one part in twelve, and a heavier atom in a lattice of lighter ones vibrates differently from its neighbours and scatters the waves that pass through it. Short wavelengths, high frequencies, feel it most. Diamond grown from carbon enriched to 99.9 per cent carbon-12 was first made in 1990, and at room temperature it conducts about half as much again as natural diamond: about 3,300 watts per metre per kelvin, the highest of any known material at that temperature. Near its peak the enriched crystal is several times better.

The simple model, which adds the isotope scattering rate to the umklapp rate for each phonon separately, gets only five per cent. What it misses is the role of the normal collisions, which conserve momentum and so resist nothing themselves, but which constantly hand energy from low-frequency phonons to high-frequency ones and back. In natural diamond those high-frequency phonons are scattered by isotopes, and the normal collisions keep refilling them, so the isotopes drain heat from the whole phonon gas through them; remove the isotopes and the drain closes. Callaway’s 1959 model of phonon conductivity, which includes normal collisions as a separate term, captures this, and is what is used to fit real measurements. The difference between the two models is a reminder that in a gas whose collisions sometimes conserve momentum, adding scattering rates is not enough.

Two gases in a metal, and none to speak of in a glass

A metal carries heat with two gases at once: its phonons and its free electrons. In copper the electrons win by a wide margin, because they are fast — the electrons that carry heat sit at the top of a Fermi sea and move at a thousand kilometres a second, a hundred times the speed of sound — and because, as the collisions the exclusion principle forbids found, they rarely collide with one another. The electrons carry charge as well as heat, and the ratio of a metal’s thermal to its electrical conductivity is the same for every good metal at a given temperature, the Wiedemann–Franz law, because both are carried by the same particles with the same mean free path. Diamond has no free electrons at all and beats copper anyway, with its phonons alone.

A glass is the opposite extreme. Its atoms are bonded as stiffly as a crystal’s, but they are not arranged in a lattice, so a vibration cannot travel far as a well-defined wave before the disorder scrambles it: the mean free path of most of its phonons is only a few atomic spacings. Window glass conducts about one watt per metre per kelvin, two thousand times less than diamond, and the difference is almost entirely in ℓ\ell. Glasses also behave strangely when cold. Below about a kelvin their conductivity rises as T2T^2 rather than T3T^3, and their heat capacity is larger than Debye’s law allows, because of tunnelling defects — atoms or groups of atoms with two nearly equal positions — that absorb phonons and store heat; their existence was inferred from exactly these thermal measurements in 1971 and is still not understood in microscopic detail.

A ring that tells diamonds from glass

The conductivity is used in jewellery shops. A diamond tester is a heated metal tip pressed against a stone, and it measures how fast the stone draws heat away from the tip. Glass and the cubic zirconia used to imitate diamond conduct about one watt per metre per kelvin; diamond conducts two thousand. The difference is so large that a cheap thermal probe tells them apart at once, which a refractive-index measurement on a cut stone does far less reliably. The one common imitation the thermal tester fails on is moissanite, silicon carbide, itself a stiff covalent crystal with a high phonon conductivity, and testers that also measure electrical conductivity were introduced to catch it.

Diamond’s conductivity is also put to work. Slabs of synthetic diamond are used as heat spreaders under high-power laser diodes and radio-frequency transistors, where a few square millimetres of chip dissipate watts, and the diamond carries the heat sideways faster than any metal could before it is passed to a copper block. The material is chosen because, of all solids at room temperature, it combines the stiffest bonds — hence the fastest sound and the highest Debye temperature — with light atoms and an almost perfect lattice, every factor in 13Cvℓ\frac{1}{3}Cv\ell pushed to its extreme.

When the phonon gas flows like a fluid

If normal collisions dominate and umklapp and boundary scattering are rare, the phonon gas does something a gas of molecules does when its collisions conserve momentum: it flows hydrodynamically, as a fluid, and a pulse of heat can travel through it as a wave rather than diffusing. That wave, second sound, is the phenomenon the heat that arrives as a wave found in superfluid helium. In crystals it requires a narrow window of temperature and very pure material, and it was seen in sodium fluoride and solid helium in the 1960s and 1970s at a few kelvin. In 2019 it was seen in graphite above a hundred kelvin, where its strong in-plane bonds and weak interlayer coupling make normal collisions unusually dominant, and graphitic materials are now studied for heat flow that is neither diffusive nor ballistic.

What the pictures cannot show

The model treats the crystal as isotropic with a single sound speed and a Debye spectrum cut off at 2,230 kelvin, whereas a real crystal has three acoustic branches with different speeds and a spectrum that departs from Debye’s at short wavelengths; it represents the umklapp rate by a common empirical form with one fitted constant; it ignores normal collisions as a separate process, dislocations, impurities other than isotopes, and the specular reflection of phonons at smooth walls. It is calibrated to diamond’s room-temperature conductivity and reproduces the shape of the measured curve, not its heights: it overestimates the peaks and underestimates the isotope effect at room temperature for reasons the essay describes. The mean free paths are for a single representative phonon energy; the real distribution of paths is broad.

Still open: the smallest conductivity a crystal can have

Diamond is the top of the scale. At the bottom are materials engineered to conduct heat as badly as possible — for thermoelectric generators, which need a temperature difference across them to survive, and for thermal barrier coatings on turbine blades. Their designers shorten the phonon mean free path with heavy atoms, rattling guest atoms in cage-like crystals, nanostructured grain boundaries and deliberate disorder, aiming at the floor where the mean free path is as short as the spacing of the atoms and a crystal conducts no better than a glass. Whether that floor can be broken — whether a crystal can conduct worse than its own amorphous form, by phonons that interfere or localise rather than merely scatter — is being argued, with candidate materials reported and their measurements disputed.

The habit worth carrying away is to ask what the carriers are before asking how well something conducts. In an insulator heat is carried by a gas of phonons, and kinetic theory’s κ=Cvℓ/3\kappa = Cv\ell/3 applies to it: the heat capacity rises as T3T^3, the mean free path is the sample’s width when cold and a few hundred nanometres when umklapp collisions switch on, and the conductivity rises, peaks and falls — 2,200 W/m K in diamond at room temperature, five times copper’s. At low temperature the answer depends on the size of the crystal, and at room temperature on one atom in ninety being a neutron heavier.

Part 10 of 10

This essay is one argument about Kinetic theory. The others:

The pair potential, and the two things it does to a gas. The Lennard-Jones potential between two molecules, in units of its own depth and range, with the Mayer function it produces at 1, 3, 8 times the well depth in temperature. The virial coefficient is minus the integral of that function over volume, so the two parts of the potential contribute with opposite signs: the steep repulsive core makes the function minus one there, giving a positive contribution — molecules take up room — and the attractive well makes it positive, giving a negative one. At low temperature the attraction dominates and a gas is easier to compress than an ideal one; at high temperature the core dominates and it is harder. Between them is one temperature at which they cancel. The first correction to the gas law Part 9 — An ideal gas has no forces between its molecules. The first correction to what it does is computable from those forces alone — one integral over the pair potential — and its sign flips at a temperature where a real gas obeys the ideal law without being ideal at all.

Where this goes next

The viscosity of a gas, over 8 decades of pressure. The viscosity of 3 gases at 300 K against pressure, on a logarithmic pressure axis and a linear viscosity one. The lines are flat, and that is the whole figure. Viscosity is the rate at which momentum is carried across a shear, which is the density of carriers times the distance each one carries it: ⅓ρv̄λ. Doubling the pressure doubles the density and halves the free path, and the two cancel exactly, so the same gas at a hundredth of an atmosphere is exactly as viscous as at one — which is not what anybody expects of a thinner gas and is what is measured. nitrogen comes out at 17.9 μPa·s against a measured 17.9, helium comes out at 19.3 μPa·s against a measured 19.9, argon comes out at 21.7 μPa·s against a measured 22.7. The flatness ends when the free path reaches the apparatus rather than the next layer of gas: at a vessel 10 mm across that is around 0.68 Pa for nitrogen, 1.94 Pa for helium, 0.69 Pa for argon, below which there is no gas-to-gas hand-off left to make. The viscosity that does not care how much gas there is Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been. A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell. How far a molecule gets A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot. 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.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Debye modelHeat capacityIsotope effectKinetic theoryMean free pathPhononThermal conductivityUmklapp scattering