Every minimum is a parabola

The list of things that oscillate sinusoidally is absurdly long and has no obvious common member. A mass on a spring. A pendulum. A ship rolling. A molecule vibrating. An atom in a crystal. A charge sloshing in a circuit. A star pulsating. A wave on a string, at every point along it. There is no shared mechanism there, and the usual explanation — that each has a restoring force proportional to displacement — is true and explains nothing, because it is exactly the coincidence that wants explaining.

The answer is that there is no coincidence to explain. Take any system with a stable equilibrium, which means any system whose potential energy has a minimum, and expand that energy about the minimum:

$$V(x) = V(x_0) + V'(x_0)\,(x-x_0) + \tfrac{1}{2}V''(x_0)\,(x-x_0)^2 + \dots$$

The constant can be thrown away because energy has no zero. The linear term is zero — that is what minimum means, and it is the whole trick. So the first term that survives is quadratic, the force is its derivative and therefore linear, and the motion is sinusoidal with

$$\omega = \sqrt{\frac{V''(x_0)}{m}}.$$

One number determines everything: the curvature at the bottom of the well. Not the depth, not the shape further out, not what the system is made of.

What survives the expansion, and what does not #

The strength of the argument is what it does not require. It does not need the potential to be symmetric, or smooth beyond a second derivative, or bounded, or to have any particular functional form. Two of the three wells in the figure above are wildly asymmetric — a bond has a hard wall on one side and dissociates on the other — and their behaviour near the bottom is nonetheless identical to a spring’s.

What the expansion does require is that the second derivative is not zero. A potential like $x^4$, which has a minimum with $V''=0$, is not harmonic at any amplitude however small, and its period falls as the amplitude grows rather than staying constant. Those are rare and they matter: a critical point in a phase transition is precisely a place where the quadratic term vanishes, and the behaviour there is different in kind rather than in degree.

The claim that could fail #

The approximation makes a testable promise: that the period of a small oscillation is $2\pi\sqrt{m/V''}$ and does not depend on how large the oscillation is. That is a strong claim and it is worth measuring against the exact answer rather than assuming it.

So the promise is kept, in the limit, and broken everywhere else in a way that can be computed. What is worth noticing in that figure is which curve departs fastest. The Morse and Lennard-Jones wells are asymmetric, which means their expansions have a cubic term; the pendulum’s is symmetric and does not. A cubic term costs a first-order error in the period and a symmetric well only a second-order one, which is the reason a pendulum is a usable clock and a diatomic molecule is not even approximately one.

The pendulum’s own correction curve is the cleanest place to watch the claim fail. The leading amplitude correction to the period is $\theta_0^2/16$: a quarter of a per cent at 10°, four per cent at 45°, and rising without bound as the swing approaches the vertical. A pendulum clock is therefore not a device that is independent of amplitude — it is a device for holding the amplitude constant, which is what an escapement is for, and the distinction is the difference between a claim about a system and a claim about how it is driven.

The approximation underneath the approximation is the replacement of $\sin\theta$ by $\theta$, made exactly once in the harmonic treatment of a pendulum, and responsible for the whole of the difference between the true period and the nominal one. The error is third order in the angle rather than second, which is not luck: it is the symmetry of the well appearing in another disguise, since a symmetric restoring force has no even terms in it to be wrong by.

The number that spans fifteen decades #

If the argument is right then $\omega = \sqrt{V''/m}$ should hold for systems with nothing else in common, and the range over which it does is the best evidence that the derivation is doing real work.

The copper entry is the one worth staring at. An atom in a metal sits in a well whose stiffness is a couple of hundred newtons per metre — the same as a stiff laboratory spring, a number that could be measured with a ruler and a weight if the atom were large enough to hang things from. What makes its frequency $10^{13}$ rather than a few per second is not the stiffness at all, which is ordinary; it is the mass, which is $10^{-25}$ kg. The square root turns twenty-five orders of magnitude of mass into twelve and a half of frequency, and that is the whole of the difference between a laboratory and a lattice.

What it costs #

The harmonic approximation is so useful that it is easy to forget it throws away every phenomenon that depends on the terms it discards. Three of them are not small.

A harmonic solid cannot expand when heated. The well is symmetric, so the average position at any energy is the bottom of the well, whatever the energy. Thermal expansion is entirely due to the cubic term — the well being softer on the outward side — and a material’s expansion coefficient is a direct measurement of the asymmetry of the interatomic potential — the same asymmetry that decides how much heat a solid takes to warm once the count of modes is settled. Perfectly harmonic matter would keep its dimensions from absolute zero to infinity.

A harmonic solid has infinite thermal conductivity. Independent normal modes do not exchange energy, so a phonon launched at one end arrives at the other undisturbed and heat travels ballistically. The finite conductivity of a real insulator comes from phonons scattering off one another, and that scattering is an anharmonic term.

A harmonic bond cannot break. A parabola goes up for ever. Dissociation energy, melting, evaporation and every chemical reaction are properties of the part of the potential where the approximation has failed, which is exactly why the Morse potential exists: it is the cheapest function that is harmonic at the bottom and dissociates at the top.

So the approximation is excellent for the dynamics of small displacements and useless for the thermodynamics of them. That is an unusual division and it catches people out, because the two are usually taught in the same fortnight.

Where it becomes something else #

Two extensions turn the same expansion into subjects of their own, and both are worth naming because the transition is invisible if the expansion is treated as a trick for pendulums.

The first is many dimensions. A system with $N$ coordinates has a matrix of second derivatives rather than a single number, and diagonalising it gives $N$ independent harmonic oscillators — the normal modes. Every vibrational spectrum, every crystal’s phonon branches, every building’s response to an earthquake and every molecule’s infrared absorption is that diagonalisation. It is also where the modes of a string come from, in the limit where the number of coordinates becomes a continuum. The physics is entirely contained in the sentence above; what is added is linear algebra.

The extension that matters most is what a harmonic well does when the thing sitting in it is small enough to be a wave. The allowed energies of a bound state come out equally spaced for a parabola — $E_n = (n + \tfrac12)\hbar\omega$ — and that even spacing is the same fact as the amplitude-independent period, restated in a language where amplitude has become quantum number. A well that is not parabolic has levels which crowd together as they rise, and that is how a molecular spectrum announces its own anharmonicity: the first overtone of a real bond sits not at twice the fundamental but a per cent or two below it.

The quantum version keeps the classical result’s most peculiar feature. Because the level spacing does not depend on the level, a harmonic oscillator absorbs and emits at one frequency however excited it is, which is why the quantum and classical descriptions of a vibrating molecule agree about the frequency while disagreeing about everything else. It also has a zero-point energy of $\tfrac12\hbar\omega$, which the classical treatment cannot produce and which is what the uncertainty relation costs for confining anything to a well at all.

The mode that goes soft #

The caution that a well with vanishing curvature is different in kind is not a hypothetical, and the case where it happens is one of the more elegant measurements in solid-state physics.

Some crystals change structure as they are cooled — the atoms shift slightly from one arrangement to another, without melting or reorganising. Approaching such a transition, one particular vibrational mode becomes easier and easier to excite: the restoring force for the specific pattern of displacement that carries the crystal from the old structure to the new one weakens, and its frequency falls.

At the transition itself the curvature for that pattern reaches zero. There is no restoring force at all along that direction in the space of displacements, the frequency vanishes, and the crystal simply moves along it into the new arrangement. Below the transition the curvature is positive again about the new minimum, and the frequency rises from zero.

That is a soft mode, and the prediction is quantitative: since the curvature is going linearly through zero, the frequency squared should be proportional to the distance from the transition temperature. Measuring the frequency means measuring how much energy a neutron loses when it scatters off the crystal and excites that particular vibration, which is what an inelastic neutron scattering instrument does.

The prediction holds, and the check is the confirmation that the structural change is a curvature going through zero rather than something more complicated. It also gives the transition a mechanism a static picture cannot: the crystal is not choosing between two structures but riding down a direction that has stopped resisting.

The same language transfers. Whenever a system’s stability is lost, some second derivative has gone through zero, and the vibration associated with it slows down first — a beam approaching its buckling load rings lower and lower, a chemical bond about to break has a stretching frequency that falls, and a mechanical structure’s lowest natural frequency is the standard warning that it is approaching an instability. Listening for a frequency going soft is a way of watching a curvature disappear.

Two halves of a kT #

There is one more thing the quadratic well delivers, and it is what makes the approximation the workhorse of thermal physics as well as of dynamics.

A harmonic oscillator’s energy is a sum of two quadratic terms — one in the position and one in the momentum. Equipartition gives each of them half a $k_BT$ on average, so an oscillator in thermal equilibrium carries a whole $k_BT$: half kinetic, half potential, exactly equal on average, at every temperature.

The equality of the two halves is worth noticing on its own. It is not obvious that the energy of a thermally agitated spring should be shared evenly between its motion and its stretch, and it is a consequence of both entering the energy as squares with no other structure.

Applied to a solid it gives the oldest quantitative result in the subject. A crystal of $N$ atoms has $3N$ vibrational modes, each a harmonic oscillator; each holds $k_BT$; the total energy is $3Nk_BT$; and the heat capacity is $3Nk_B$, which per mole is about 25 joules per kelvin regardless of what the solid is made of. That is the Dulong–Petit value, measured in 1819 and unexplained for a century, and it is this essay’s expansion plus a count of coordinates.

Its failure is as informative as its success. Measured heat capacities fall below it at low temperature and go to zero — which equipartition cannot produce, since the argument has no temperature in it. What is missing is that the modes are quantised, so a mode whose quantum exceeds $k_BT$ cannot be excited and contributes nothing. The classical result is therefore the high-temperature limit of a quantum one, and the temperature at which it breaks down is a measurement of the stiffness — which is the curvature again, read out of a calorimeter instead of a spectrometer.

Run backwards, it is an instrument #

The expansion is usually run forwards: a potential is known, its curvature is taken, a frequency comes out. Run backwards it is one of the most productive measuring techniques in physics, because a frequency is the easiest quantity in the world to measure accurately and a curvature is one of the hardest.

A vibrational frequency measured to four figures gives $V''$ to four figures, and $V''$ at the bottom of a bond is the closest thing chemistry has to a direct reading of bond strength.

Carbon monoxide absorbs at 2,143 reciprocal centimetres; that number, a reduced mass and a square root give 1,902 N/m, and the shift of that absorption when the molecule binds to a metal atom is how the strength of the new bond is read off. The same argument at a much larger scale gives a ship’s metacentric height from its roll period, a bridge’s stiffness from the frequency it rings at after a lorry crosses, the sharpness of a resonance from how long the ringing lasts, and a star’s internal structure from the periods of its pulsations.

The technique has a limitation that follows directly from the derivation, and it is the reason the reading is trusted so far and no further. What is measured is the curvature at the bottom, and nothing else about the well. Two potentials with the same curvature and quite different depths give identical frequencies, so a vibrational spectrum cannot on its own say how strong a bond is in the sense of how much energy it takes to break. That has to come from the anharmonicity — from how the overtones deviate from even spacing — which is exactly the information the harmonic approximation throws away.

Hooke’s law, which is not a law #

The history is short and slightly embarrassing for the usual telling. Hooke published his spring result in 1678, having first deposited it as an anagram — ceiiinosssttuv, for ut tensio sic vis, “as the extension, so the force” — in the manner of the time, to establish priority while he worked out what to do with it. It is presented ever since as an empirical law about springs.

It is not really a law and it is not really about springs. Nothing in nature is exactly linear in the extension, and the linearity that Hooke found in metal wire is the same first term of the same expansion, with $V''$ being a property of the interatomic potential integrated over a cross-section. The reason a steel wire obeys it so beautifully — to a part in ten thousand over its whole elastic range — is not that steel is special but that the atomic displacements involved are a fraction of a per cent of a lattice spacing, so the expansion is being asked for very little indeed. Stretch the same wire until the displacements are one per cent and it stops obeying Hooke and starts obeying metallurgy.

The system Hooke’s contemporaries actually had was not a spring but a pendulum, and resolving its restoring force shows why the law looked like a law. At 12° the component along the arc is $mg\sin\theta$, and replacing it with $mg\theta$ is an error of two parts in a thousand — invisible to any seventeenth-century measurement. The effective spring constant is $mg/L$, a curvature divided by nothing more exotic than a length, and the fact that it contains $g$ is exactly why a pendulum measures gravity and a spring does not.

That reading also disposes of the question of why the constant is called a spring constant when the same equation describes a molecule. It is the curvature of a potential in both cases; the word “spring” is an accident of which system was measured first.

What the picture cannot show #

The comparison at the top of this page divides each well by its own curvature, which is what makes them lie on top of one another — and it hides the thing that matters most in practice, which is that the curvatures differ by a factor of seventy-two between the first and last. Rescaled plots are always making an argument about shape and never about size, and the size is what decides whether an oscillation at a given energy is small.

The figures also draw one-dimensional wells. A real minimum in more than one dimension has directions along which it is stiff and directions along which it is soft, and the softest direction is almost always the one that matters — a molecule bends far more easily than it stretches, a beam buckles rather than crushes, a protein’s motion is dominated by a handful of low-frequency modes out of thousands. Drawing the well as a curve rather than as a surface makes that invisible.

And nothing here says how large an amplitude is allowed. That question has no general answer: it depends on the ratio of the cubic term to the quadratic one, which is a property of the particular potential and has to be worked out each time. The honest form of the approximation always carries the next term along with it, if only to be discarded knowingly.

The ladder from here #

Later rungs on this anchor: the anharmonic corrections computed properly, by perturbation theory, giving the amplitude dependence of the period as a series; normal modes in many dimensions and the diagonalisation that produces them; the damped and driven cases, where the same quadratic well is met with dissipation and a forcing term; the quantum harmonic oscillator built from ladder operators, where the algebra replaces the differential equation entirely; and the places the expansion is not allowed — critical points, where $V''$ vanishes, and inverted potentials, where it is negative and the exponential answer replaces the sinusoidal one.

The neighbouring ladders are the pendulum, which is the worked example of the correction, energy landscapes, which is where the potential curve comes from, and the particle in a box, whose level spacing is the same statement about a different well.