Vibrational Density of States-Weighted Grüneisen Parameters of Graphite

The vibrational spectrum determines, to a large extent, a material’s thermal expansion. However, extracting information on the material’s vibrational spectrum from the measured thermal expansion is an ill-conditioned inverse problem whose solution generally involves regularization techniques. In this work, the procedure for extracting the vibrational density of states-weighted Grüneisen parameter spectra (\(G_{a}(\Theta )\) and \(G_{c}(\Theta )\)) from the temperature-dependent linear thermal expansion coefficients parallel to the crystallographic a- and c-axis of materials with a hexagonal Bravais lattice is described and then applied to graphite. The resulting spectra, obtained by solving an inverse problem consisting of a pair of coupled Fredholm integrals of the first kind, exhibit two main contributions to \(G_{a}(\Theta )\), centered around \(\Theta\) = 100 K and \(\Theta\) = 1200 K, and one narrow peak in \(G_{c}(\Theta )\) centered around \(\Theta\) = 100 K. The low-energy, narrow peak of negative amplitude in \(G_{a}(\Theta )\) is in good agreement with a large and negative Grüneisen parameter previously reported for transverse acoustic modes in graphite and accounts for the negative thermal expansion exhibited by graphite along the a-axis up to about 655 K. A simplified vibrational density of states-weighted Grüneisen parameter spectra of graphite, consisting of the two lowest energy, narrow peaks in \(G_{a}(\Theta )\) and \(G_{c}(\Theta )\), plus an asymmetric Gaussian peak with positive amplitude in \(G_{a}(\Theta )\), is proposed which still captures the main features of graphite’s thermal expansion.