Anharmonicity, Thermal Expansion, and Heat Conduction

The harmonic crystal is an idealization with two unphysical consequences: it does not expand when heated, and it conducts heat without resistance. Both failures trace to the same omission. Truncating the interatomic potential at second order makes the normal modes exact, independent, and eternal — phonons never scatter off one another, and the mean atomic position never shifts. Restoring the cubic and quartic terms couples the modes and displaces the potential minimum. This lesson shows how those anharmonic terms produce thermal expansion, phonon–phonon scattering, and a finite, temperature-dependent thermal conductivity.

Why the harmonic crystal fails

Two statements about a purely harmonic solid must be established before repairing them.

No thermal expansion. In a harmonic potential $U = c x^{2}$ the average displacement $⟨ x ⟩$ is zero at every temperature, because the parabola is symmetric about its minimum: positive and negative excursions are equally likely and equally weighted. Heating raises the amplitude but not the mean position, so the crystal does not expand.
Infinite thermal conductivity. The harmonic normal modes are exact eigenstates. A phonon wavepacket, once launched, propagates forever at its group velocity without decaying into other modes. A temperature gradient would drive an unimpeded phonon flux, giving infinite conductivity. Any real resistance requires a mechanism that scatters phonons, and the harmonic Hamiltonian provides none.

The cure for both is the same: keep the next terms in the expansion of $U$ .

The anharmonic potential and thermal expansion

Expand the interaction of a pair of atoms about the equilibrium separation, writing $x$ for the displacement from it:

U (x) = c x^{2} - g x^{3} - f x^{4} + \dots

The cubic term $- g x^{3}$ (with $g > 0$ ) makes the well asymmetric: it is softer on the side of increasing separation (large positive $x$ ) and stiffer on the side of compression. An atom oscillating in this well spends more time at larger separations, so its mean position drifts outward as the amplitude grows. That drift is thermal expansion.

An asymmetric interatomic potential. The harmonic parabola (light, symmetric) is compared with the true well (accent), softer on the large-separation side. As temperature rises the thermal average of the position shifts outward along the dashed midline, producing expansion.

The mean displacement follows from a classical Boltzmann average. For small anharmonicity the cubic and quartic terms are corrections, and expanding the Boltzmann factor to first order in $g$ and $f$ gives

⟨ x ⟩ = \frac{\int x e ^{- U (x) / k_{B} T} d x}{\int e ^{- U (x) / k_{B} T} d x} = \frac{3 g}{4 c ^{2}} k_{B} T .

The mean position grows linearly with temperature, so the linear thermal- expansion coefficient $α = (1/ x_{0}) d ⟨ x ⟩ / d T = 3 g k_{B} / (4 c^{2} x_{0})$ is a constant at high temperature, set entirely by the cubic coefficient $g$ . A symmetric potential ( $g = 0$ ) gives $⟨ x ⟩ = 0$ and no expansion, confirming that thermal expansion is an anharmonic effect. The full quantum treatment replaces $k_{B} T$ by the mode energy and reproduces the same link to the heat capacity through the Grüneisen parameter.

Phonon–phonon scattering

The cubic term does more than shift the minimum: it couples the normal modes. A term proportional to $x^{3}$ , written in normal coordinates, is a product of three mode amplitudes, so it connects three phonons. To lowest order it allows a phonon to decay into two, or two phonons to merge into one — a three-phonon process. These are the collisions that give phonons a finite lifetime and a mean free path.

Every three-phonon process conserves energy and crystal momentum:

ω_{1} = ω_{2} + ω_{3}, k_{1} = k_{2} + k_{3} + G,

for the decay of phonon $1$ into $2$ and $3$ (merging is the reverse). The reciprocal-lattice vector $G$ distinguishes the two kinds of collision, and the distinction controls whether the process resists heat flow at all.

Two three-phonon collisions. In the normal process the outgoing wavevectors sum, within the zone (dashed), to the incoming one, so the momentum flow is unchanged. In the Umklapp process the sum exceeds the zone and a reciprocal-lattice vector G returns it, flipping the direction of the heat- carrying momentum.

An Umklapp process requires phonons with wavevectors large enough that their sum reaches the zone boundary — of order $k_{D} /2$ . Exciting such phonons costs an energy of order $k_{B} Θ_{D} /2$ , so the number of Umklapp-active phonons carries a Boltzmann factor $e^{- Θ_{D} / b T}$ (with $b$ of order $2$ ) that becomes tiny at low temperature. This freeze-out of Umklapp scattering is the key to the temperature dependence of the conductivity.

Thermal conductivity

Treat the phonons as a gas carrying heat down a temperature gradient. Kinetic theory gives the lattice thermal conductivity

κ = \frac{1}{3} C v_{s} ℓ,

with $C$ the phonon heat capacity per unit volume, $v_{s}$ the mean phonon speed, and $ℓ = v_{s} τ$ the phonon mean free path set by the scattering time $τ$ . The temperature dependence of $κ$ is the competition between $C (T)$ and $ℓ (T)$ , and it produces a characteristic peak.

High temperature ( $T ≫ Θ_{D}$ ). The heat capacity saturates at the Dulong–Petit value, so $C$ is constant. The number of phonons available for Umklapp scattering grows linearly with $T$ , so the scattering rate $\propto T$ and $ℓ \propto 1/ T$ . Hence $κ \propto 1/ T$ : hotter crystals conduct heat worse, a signature that Umklapp phonon–phonon scattering dominates.
Low temperature ( $T ≪ Θ_{D}$ ). Umklapp processes freeze out exponentially, $ℓ \propto e^{Θ_{D} / b T}$ , so the mean free path grows enormous and is eventually cut off by the sample boundaries or by defects, becoming constant. The heat capacity then controls $κ$ , and with $C \propto T^{3}$ the conductivity rises as $κ \propto T^{3}$ .

Between the rising $T^{3}$ regime and the falling $1/ T$ regime the conductivity reaches a maximum, typically at $T \sim Θ_{D} /20$ . The peak height and position depend on sample size and purity, because those set where boundary scattering takes over.

Lattice thermal conductivity versus temperature (log-log schematic). At low T the conductivity rises as T cubed, tracking the heat capacity while the mean free path is limited by boundaries. At high T it falls as one over T, limited by Umklapp phonon-phonon scattering. The two regimes meet at a peak.

The mean free path and its limits

The mean free path is set by whichever scattering mechanism is shortest, through a sum of rates (Matthiessen's rule):

\frac{1}{ℓ} = \frac{1}{ℓ _{U}} + \frac{1}{ℓ _{def}} + \frac{1}{ℓ _{bdy}} .

Each term dominates a different temperature range:

Umklapp scattering $ℓ_{U} \propto e^{Θ_{D} / b T}$ : dominant at high temperature, where it gives the $1/ T$ falloff. It grows without bound as $T \to 0$ and stops limiting $ℓ$ at low temperature.
Point-defect and isotope scattering $ℓ_{def}$ : roughly temperature-independent, set by the concentration of impurities and isotopic disorder. Rayleigh-like ( $\propto ω^{- 4}$ ) scattering makes it weak for long-wavelength phonons, so it caps $ℓ$ at intermediate temperatures.
Boundary scattering $ℓ_{bdy} \approx d$ , the sample dimension: temperature-independent and the ultimate limit at the lowest temperatures, where the wavelength-long phonons scatter only off the crystal surfaces. In this regime $κ \propto C \propto T^{3}$ with a coefficient proportional to the sample size, one of the few transport quantities that depends on how large the crystal is.

Phonon mean free path versus temperature (log-log schematic). The Umklapp-limited path (accent) diverges exponentially as T falls; the defect-limited and boundary-limited paths (light, dashed) are flat. The observed mean free path follows the lowest curve at each temperature.

The two anharmonic phenomena share one cause. The cubic term in the potential both shifts the equilibrium separation, producing thermal expansion, and couples the phonon modes, producing the scattering that makes thermal conductivity finite. A crystal with no anharmonicity would be a perfect, non-expanding, infinite-conductivity idealization that no material realizes. With lattice dynamics complete, the next module turns from the ions to the conduction electrons and the free-electron Fermi gas.