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 (Ga(Θ)and Gc(Θ)) 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 Ga(Θ), centered around Θ= 100 K and Θ= 1200 K, and one narrow peak in Gc(Θ)centered around Θ= 100 K. The low-energy, narrow peak of negative amplitude in Ga(Θ)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 Ga(Θ)and Gc(Θ), plus an asymmetric Gaussian peak with positive amplitude in Ga(Θ), is proposed which still captures the main features of graphite’s thermal expansion.