Effective partition function of crystals: Reconstruction from heat capacity data and Debye-Waller factor

Abstract

A semi-empirical method to define a partition function for phonons is proposed, which is capable of accurately reproducing thermodynamic functions, especially in the intermediate temperature range, where the Debye theory occasionally fails to describe the emerging phonon peaks in the heat capacity. The phonon partition function is defined with a temperature-dependent spectral cutoff Λ(T) and Debye temperature θ(T). The varying θ(T) can be reconstructed from heat capacity measurements by least-squares regression, and the spectral cutoff is chosen so that the partition function defines a genuine equilibrium system consistent with the equilibrium condition ∂S/∂U=1/T on the internal-energy derivative of entropy. The zero-point energy of the phonons is not predetermined by the amplitude of the cubic low-temperature slope of the heat capacity but emerges as an integration constant, which can be inferred from X-ray diffraction measurements of the Debye-Waller B-factor. The formalism is put to test with the rutile polymorph of TiO2.