Temperature dependence of lattice vibrations and analysis of the specific heat of graphite

Abstract

In the semicontinuum model of lattice vibrations in graphite proposed by Komatsu and Nagamiya, the expressions of the dispersion relation contain the elastic constants ${C}_{11},$ ${C}_{12},$ ${C}_{13},$ ${C}_{33},$ ${C}_{44},$ and \ensuremath{\kappa} and the interlayer spacing c as parameters, where $c\ensuremath{\rho}{\ensuremath{\kappa}}^{2}$ is the bending elastic constant of a graphite layer and \ensuremath{\rho} is the density. We improve this semicontinuum model by taking these parameters as a function of temperature. For the parameters except \ensuremath{\kappa}, we use the experimental data already known and the relations derived from them. \ensuremath{\kappa} is determined by fitting the calculated specific heat to the experimental one. The experimental specific heat for single-crystal graphite is derived by evaluating the data reported in the literature. With the value of \ensuremath{\kappa} thus determined, the improved semicontinuum model can explain the experimental specific heat well in the temperature range below 350 K. Then, \ensuremath{\kappa} decreases more rapidly with increasing temperature than the other elastic constants; this result means that softening of the out-of-plane vibrations occurs. It is suggested that the softening is closely related to the negative thermal expansion parallel to the layer planes. The calculated Gr\uneisen constants are compared with the experimental ones. The second derivative of the specific heat curve with respect to temperature gives information on the frequency distribution function of lattice vibrations. From the analysis of the low-temperature specific heat, the value of ${C}_{44}$ at room temperature is determined to be $0.425\ifmmode\times\else\texttimes\fi{}{10}^{11}{\mathrm{d}\mathrm{y}\mathrm{n}/\mathrm{c}\mathrm{m}}^{2}.$