Simplified Theory of Anharmonic Contributions to the Thermodynamic Properties of Solid Sodium Metal

Since attempts at the evaluation of anharmonic effects in crystals invariably lead to great difficulties, a prescription is offered by which the estimation of anharmonic effects is greatly simplified. As a first step, the first-order anharmonic contributions to the free energy are looked upon as being pseudoshifts in the frequencies ${\omega }_i$ of the usual normal modes. Thus the anharmonic part of the free energy is considered to affect the total free energy only insofar as it causes ${\omega }_i$ to become ${\omega }_i(1+\frac{\Delta {\omega }_i}{{\omega }_i})$, where $\frac{\Delta {\omega }_i}{{\omega }_i}$ is the fractional shift in frequency. It is then argued, on the basis of a simple calculation, that a simple and plausible relation exists between the quantities $\frac{\Delta {\omega }_i}{{\omega }_i}$ and the thermal fractional frequency shifts of certain acoustic waves which occur even when anharmonic contributions to the free energy are neglected. The relation mentioned is one of simple proportionality, and since the acoustic frequency shifts can be calculated for some materials, it permits an estimation of the anharmonic part of the free energy. Calculations are carried out for sodium metal, and the results are compared with experimental data. In all cases the agreement between theory and experiment is shown to be quite good. Among the calculations is one which evaluates the anharmonic temperature-dependent contribution to the Grüneisen parameter $\gamma$. This is shown to increase $\gamma$ by more than 20% at room temperature and pressure, and cause it to vary almost linearly with volume. At higher pressures, however, anharmonic effects diminish until at about 40% compression they disappear almost completely and $\gamma$ behaves as a constant.