Anharmonicity and thermal expansion

Let y denote a bond-length change from its zero-temperature minimum and expand the local potential as $U=Ky^2/2+Ay^3/3+Cy^4/4$. Positive K makes the reference minimum locally stable, while higher terms provide a physically bounded potential. The pure quadratic Boltzmann distribution is even in y, so its mean vanishes at every classical temperature. Its variance grows as thermal energy divided by K. Larger excursions alone are therefore insufficient to explain a larger average bond length.

Treat a weak cubic term perturbatively in the classical canonical distribution. Expanding the Boltzmann weight to first order in A gives a mean proportional to minus the Gaussian fourth moment. The fourth moment is three times the variance squared, producing the result below. A negative cubic coefficient makes the potential softer on its positive-extension side and shifts the mean positively. This is a local weak-anharmonic approximation; a cubic polynomial alone is unbounded and cannot define a global equilibrium ensemble without stabilizing higher terms.

A crystal's equilibrium volume minimizes free energy at the applied pressure. The quasiharmonic approximation computes harmonic phonons separately at different volumes and uses their volume-dependent frequencies in the vibrational free energy. It captures an important part of anharmonic thermodynamics even though each fixed-volume calculation remains harmonic. A mode Grüneisen parameter measures minus the logarithmic frequency derivative with respect to volume. Positive values mean that expansion softens the mode, increasing its thermal entropy and tending to favor a larger equilibrium volume.

Differentiating the vibrational pressure with temperature gives a heat-capacity-weighted Grüneisen expression for volume expansion. The denominator contains the isothermal bulk modulus, which penalizes volume change. Modes contribute according to their own heat capacities, so the effective weighting can change with temperature as different frequencies become populated. Negative mode parameters can produce negative expansion when their weighted contribution dominates. Geometry, collective transverse motion, electronic effects, and structural proximity can all matter; a simple picture of every individual bond lengthening is not a universal explanation.

A quasiharmonic frequency changes when the equilibrium volume changes, but the fixed-volume modes in that approximation still have infinite harmonic lifetime. Explicit cubic and quartic interactions between normal coordinates also produce temperature-dependent self-energy shifts and scattering linewidths. Those effects are needed for intrinsic thermal resistance and can matter strongly near structural instabilities. A calculation can therefore predict expansion reasonably while still omitting the interactions necessary to predict a measured phonon lifetime or thermal conductivity.

The local classical bond derivation is inappropriate when quantum freezing controls the populated fluctuations; quantum free energies then supply the correct temperature dependence. Quasiharmonic theory also fails when large excursions, unstable harmonic modes, or strong explicit anharmonicity invalidate its fixed-volume oscillator description. Compare its predictions with structural data and mode behavior rather than assuming that heating always gives positive linear expansion. For small isotropic strain, the linear expansion coefficient is one-third of the volumetric coefficient; confusing those two definitions creates a factor-of-three error even when the underlying thermodynamics is correct.