Why heating a perfect spring changes nothing

Heat a steel rod and it gets longer, by about twelve parts in a million per degree. Every account of why says the same thing: the atoms vibrate more vigorously, so they need more room. That sentence is a description of what the rod does and it is not a mechanism, and the difference matters, because the mechanism it suggests is wrong in a way that can be checked in a line of arithmetic.

A symmetric well does not expand. However hot it is made, the average position stays at the bottom, because for every excursion to the right there is an equally probable and equally large one to the left. Vigour is not displacement.

The average that has to be taken #

The quantity that decides the length of a solid is not the amplitude of the vibration. It is the mean position, and in classical statistical mechanics the mean position of a coordinate in a potential $V(x)$ at temperature $T$ is

$$\langle x \rangle = \frac{\int x\,e^{-V(x)/kT}\,dx}{\int e^{-V(x)/kT}\,dx}.$$

Put a parabola into that expression. The exponential of a parabola is a Gaussian, a Gaussian is symmetric about its centre, and the mean of a symmetric distribution is its centre. The answer is the minimum of the well, at every temperature, exactly.

So the entire phenomenon of thermal expansion is contained in the departure from the parabola. Write the potential as

$$V(x) = \tfrac12 k x^2 - g x^3 + \dots$$

and carry the cubic term through the average to first order. The result is

$$\langle x \rangle = \frac{3 g\, kT}{k^2},$$

linear in temperature, proportional to the cubic coefficient, and zero when that coefficient is zero. The figures measure exactly that slope on two real potentials and recover it to a few per cent — the residue being the quartic term, which is the next thing the expansion drops.

Draw a harmonic well with an energy line across it and the failure is visible in one picture. The line is symmetric about the minimum, so the two turning points are equidistant from it and the midpoint of the excursion is the minimum — at every energy without exception. Raise the line and the excursion grows while its midpoint stays exactly where it was. That is thermal expansion failing to happen, and it fails for a reason about symmetry rather than about size.

What replaces that symmetry in a real solid is a competition between two exponentials. At the same displacement the outward excursion is cheaper than the inward one, because the well is shallower that way, so the Boltzmann factor weights it more heavily — and the whole of thermal expansion is the difference between two exponentials whose arguments differ in the third power of the displacement. A cubic term in the potential becomes a linear term in the mean position, which is why the expansion coefficient is roughly constant over a wide range and not constant at all near zero.

The number, for a real bond #

A Morse potential is the standard one-line model of a chemical bond: $V(r) = D\left(1 - e^{-a(r-r_0)}\right)^2$, with $D$ the dissociation energy and $1/a$ the range over which it acts. Expanding it gives $k = 2Da^2$ and $g = Da^3$, so

$$\alpha = \frac{1}{r_0}\frac{d\langle x\rangle}{dT} = \frac{3k_B}{4 D a r_0}.$$

Every quantity on the right is known independently for a real solid. For a metallic bond with a dissociation energy near an electron volt, a range of about 0.07 nm and a spacing of 0.25 nm, that gives $1.7\times10^{-5}$ per kelvin. Copper’s measured value is $1.65\times10^{-5}$.

The formula says more than the number. It says the expansion coefficient is inversely proportional to the bond energy, so tightly bound solids expand least — diamond at $10^{-6}$, tungsten at $4.5\times10^{-6}$, aluminium at $2.3\times10^{-5}$, and soft polymers at $10^{-4}$. That ordering is the same as the ordering of melting points, and for the same reason: both are ratios of a thermal energy to a bond energy. It is the same comparison that decides how much of a solid’s heat capacity is available at a given temperature, and the recurrence of one ratio across unrelated properties is the usual sign that a single energy scale is doing all the work.

The same term, doing a second job #

Here is the connection that makes the subject worth an essay rather than a footnote. In a perfectly harmonic crystal the normal modes are exactly independent. A mode excited stays excited, for ever; it cannot exchange energy with any other mode, because the equations of motion are linear and superposition is exact.

A heat current in a solid is carried by those modes. If they never scatter off one another, a phonon launched at one end arrives at the other having lost nothing, and the thermal conductivity of the crystal is not large — it is infinite, limited only by the boundaries.

The harmonic answer for a wave on a string — speed from tension and density — is an excellent one, and its excellence is the trouble. A linear medium transmits every wave at that speed independently of every other, which is what makes superposition work and what makes a musical instrument possible. It is also what would make a solid unable to reach thermal equilibrium with itself: independent modes cannot share, so a perfectly harmonic crystal heated at one end would never warm at the other.

So a real solid’s finite thermal conductivity is evidence of anharmonicity, and the same cubic term is responsible: it is the lowest-order coupling that lets one phonon decay into two, or two combine into one. That coupling is what allows a solid to come to equilibrium with itself at all, and a system that cannot redistribute energy among its degrees of freedom has no temperature to speak of — the arrow that a diffusion equation carries needs a mechanism, and in a crystal this is it. The rate of those processes rises with temperature — there are more phonons to scatter from — which is why the thermal conductivity of an insulating crystal falls as $1/T$ over a wide range above its Debye temperature.

Two facts that no intuition connects therefore have one cause. A solid expands, and a solid conducts heat at a finite rate, because its interatomic potential is not a parabola. Remove the cubic term and both consequences vanish together: an infinitely conducting crystal of fixed length.

The relationship is quantitative as well as qualitative. The Grüneisen parameter — a dimensionless number of order two for most solids — measures how much a mode’s frequency changes when the crystal is compressed, which is anharmonicity by another name, and it appears in the expansion coefficient and in the phonon scattering rate alike. A material with an unusually small Grüneisen parameter has both an unusually small expansion and an unusually high lattice conductivity, and diamond is the standard example of both at once.

The relation that ties the two together #

The Grüneisen parameter was named above and deserves the relation it sits in, because that relation is what turns “expansion and conduction have the same cause” into something a measurement can check.

Write $\gamma$ for how much a vibrational frequency falls when the crystal is expanded — strictly, minus the logarithmic derivative of a mode frequency with respect to volume. In a strictly harmonic solid the frequencies are set by the curvature of the potential and squeezing the lattice does not change them, so $\gamma$ is zero. Any non-zero value is anharmonicity, measured.

The volume expansion coefficient then comes out as

$$\alpha_V = \frac{\gamma\,C_V\,\kappa_T}{V},$$

with $C_V$ the heat capacity and $\kappa_T$ the compressibility. Everything on the right except $\gamma$ is measurable by routine means, so the relation is a way of extracting $\gamma$ — and what comes out, for most solids, is a number between one and three, with very little variation across metals, ionic crystals and semiconductors alike.

Two things make that useful. It is nearly temperature-independent, so the expansion coefficient and the heat capacity track each other over the whole range where both are measurable — including down through the freeze-out, where both fall as $T^3$ together. That parallel tracking of two apparently unrelated quantities over three decades of temperature is the strongest evidence the mechanism has.

And it puts a number on the connection to conductivity. A large $\gamma$ means strongly volume-dependent frequencies, which means strongly coupled modes, which means phonons that scatter readily and a low lattice conductivity. Diamond has $\gamma$ near one and both an exceptionally small expansion and the highest lattice conductivity of any solid; a soft, loosely bound crystal has a large $\gamma$ and the reverse of both. One dimensionless number, two properties, and the sign of the correlation fixed by the argument rather than fitted.

Where the classical calculation runs out #

The average above is classical: the Boltzmann factor treats the coordinate as continuous and the energy as freely available. That fails at low temperature for exactly the reason it fails for heat capacity.

Below the Debye temperature the classical formula overestimates the expansion, and at very low temperatures the coefficient falls as $T^3$ alongside the heat capacity, because both are proportional to the number of thermally excited modes. The proportionality between the two — expansion and heat capacity, tracking each other over decades of temperature — is one of the strongest pieces of evidence that the mechanism is the one described here, and it is called the Grüneisen relation.

Counting is what the other half of the same potential gives. Equipartition counts modes and does not care whether they are harmonic; the expansion counts asymmetry and does not care how many modes there are. Two properties of one solid, read off two different features of one curve — and the reason a material can have a large heat capacity and a small expansion coefficient, or the reverse, with no contradiction between them.

The materials that go the other way #

If expansion comes from the shallower outward side of the well, a material with the opposite behaviour needs a different mechanism, and several exist.

Water below 4 °C contracts on heating, and the cause is not a bond at all: it is that the open hydrogen-bonded network of ice-like clusters is less dense than the disordered liquid, so breaking clusters up as the temperature rises makes the liquid denser. The competition between that and ordinary expansion crosses at 4 °C, which is why lakes freeze from the top.

Water goes the other way, and it goes wrong for a reason worth naming. Its anomalies — the density maximum at 4 °C, the negative slope of its melting curve, the unusually high boiling point for its molecular mass — all come from directional hydrogen bonding, and none of them is captured by a spherically symmetric pair potential. A model built on a single bond and a single coordinate is going to be wrong about water, and knowing where it will be wrong is most of what the model is worth.

Zirconium tungstate contracts over eight hundred degrees, and the mechanism is geometric: the structure is a network of rigid polyhedra joined at corners, and the transverse vibration of a joining oxygen atom pulls the two polyhedra together even though every individual bond is lengthening. That is not anharmonicity of a bond; it is a linkage. The distinction matters because the two respond differently to pressure, and because the second can be engineered while the first cannot.

And rubber contracts on heating under load — its restoring force is a counting of configurations rather than a distortion of a bond — for a reason that is not about potentials at all: its elasticity is entropic, the retracting force comes from the number of configurations a coiled chain has, and heating increases that force. A stretched rubber band lifting a weight gets shorter when warmed, which is the cleanest demonstration in a kitchen that not every elasticity is a spring.

Why twelve parts per million is a large number #

An expansion coefficient looks negligible and is not, and the reason is that a solid restrained from expanding does not simply fail to expand: it develops a stress.

Heat a bar that is free at both ends by $\Delta T$ and it lengthens by $\alpha\Delta T$. Clamp both ends and it lengthens by nothing, which means it has been compressed by that same strain relative to where it wanted to be, and the stress that goes with a strain is the modulus times it:

$$\sigma = E\,\alpha\,\Delta T.$$

For steel, with a modulus of 200 gigapascals and a coefficient of twelve parts per million, fifty degrees of restrained heating is 120 megapascals — a substantial fraction of the yield stress of ordinary structural steel, from a temperature change a summer afternoon supplies. The strain is minute and the stress is not, because the modulus is enormous.

That single line is behind a great deal of engineering that otherwise looks like superstition. Rails are laid with gaps, or else stressed deliberately when installed so that they are in tension at the coldest expected temperature and merely unstressed at the hottest. Bridges sit on rollers. Pipework carrying hot fluid is routed with deliberate bends that have no function but to be flexible. A glass dish cracks when moved from an oven to cold water because the outside contracts against an inside that has not yet cooled, and the borosilicate version survives it by having a third of the expansion coefficient rather than by being stronger.

And it explains why matching coefficients is so often the binding constraint in choosing materials. A seal between a metal and a glass, a coating on a turbine blade, a chip bonded to a substrate: all of them are two materials held together across a temperature range, and the stress at the join is the difference of two coefficients times the modulus times the range. The materials cannot be chosen for their function alone.

The alloy that cancels its own expansion #

Since $\alpha$ is set by a bond’s asymmetry, an unusually small value ought to require an unusually symmetric bond — and the material with the smallest coefficient of any common alloy does not have one. It cheats, by adding a second mechanism of the opposite sign.

An iron–nickel alloy at about 36 per cent nickel expands about a tenth as much as either of its constituents, over a wide range around room temperature. What is happening is that the alloy is ferromagnetic, and a ferromagnet’s volume depends on its magnetisation — the ordered state occupies slightly more room than the disordered one. Heating reduces the magnetisation, which contracts the lattice, and over a range of composition that contraction very nearly cancels the ordinary anharmonic expansion.

Two consequences follow that identify the mechanism unambiguously. The cancellation only works below the Curie temperature, since above it there is no magnetisation left to lose — and indeed the alloy’s expansion coefficient jumps to an ordinary value there. And it is sensitive to composition in a way no anharmonic effect would be, because a percent or two of nickel moves the magnetic contribution appreciably while barely touching the bond.

The alloy was found in 1896 and its inventor was given a Nobel prize for it, which is a startling award for a metallurgical accident until the applications are counted: pendulum clocks that keep time through a temperature change, surveying tapes whose length is a length, and the standard metre bars of the period. A property that is normally a nuisance was removed by cancelling it against a completely unrelated one, and this essay’s argument explains why that was the only available route — the asymmetry of a metallic bond is not something a metallurgist can turn down.

Where the model stops #

The classical average is not the quantum one, and the difference is not only at low temperature. A quantum oscillator in an anharmonic well has a zero-point excursion, and because the well is asymmetric that excursion has a non-zero mean: a crystal at absolute zero is already slightly expanded relative to the minimum of its own potential, by an amount that has nothing to do with temperature. The effect is measurable, and the cleanest evidence for it is that different isotopes of the same element have different lattice constants — solid neon-20 and neon-22 differ in spacing by about a part in a thousand, entirely because the heavier one has a smaller zero-point amplitude. Nothing classical can produce a dependence of a lattice spacing on a mass, since the mass cancels out of every equilibrium condition.

One coordinate is not a solid. Everything above treats an atom in a one-dimensional well, and a real atom sits in a three-dimensional potential made by all its neighbours. In a cubic crystal the argument survives essentially intact; in a layered or chain-like one it does not, and expansion coefficients along different axes can differ by a factor of ten or have opposite signs.

The average is taken at fixed temperature and not at fixed pressure or volume, and the distinction is the difference between the coefficient a table quotes and the one the calculation gives. Correcting between them requires the bulk modulus, which is another property of the same potential.

And the cubic term is not always the leading one. In a well that is very nearly symmetric the quartic term can dominate, and a quartic term produces no expansion at first order at all — it stiffens the well and changes the frequency instead. Which term wins is a fact about a particular material and cannot be settled by the general argument.

What the pictures cannot show #

The hero figure draws a mean position against temperature and cannot show what is being averaged. What a solid at 300 K is doing is a superposition of 10²³ modes, each with a random phase, and the atom’s position is a sum of contributions from all of them; the picture of a single atom rocking in a single well is a device for getting the asymmetry right, not a description of the motion.

Nor can any of these figures show the thing this essay claims is the same term twice. Expansion is drawn as a drift and conductivity as a rate, and no drawing puts them in the same frame — the connection is algebraic, through a third derivative that appears in both calculations, and its visibility is a matter of doing the two calculations and recognising the same symbol.

Where this ladder goes next #

The ladder began with the observation that every minimum is a parabola — a statement about why so many unlike systems obey one equation. Its second rung took two such systems and coupled them, which produced normal modes and the exchange between two pendulums that swap. This third rung is the first that requires the approximation to fail: not as an error to be estimated, but as the sole source of an entire class of phenomena.

The habit worth carrying away is a way of interrogating any approximation. Ask which properties are exactly zero in it. A quantity that a model returns as identically nought is not being approximated; it is being excluded, and if it is measurably non-zero then the model is not slightly wrong about it but silent. The harmonic approximation returns zero for the expansion, zero for the thermal resistance and zero for the coupling between modes — three enormous omissions hiding inside an approximation that is otherwise accurate to a few per cent, and all three are the same omission.

What is left on this ladder is the case where the anharmonic term is not small: a well so asymmetric that no expansion in powers works, where the motion has to be solved rather than corrected. The pendulum’s exact period is the elementary version of that, and it belongs to a different anchor.