Thermodynamic properties of bcc crystals at high temperatures: The alkali metals

We have calculated the thermodynamic properties of monatomic bcc crystals at high temperatures from the Helmholtz free energy $F(V,T)\mathrm{}$ for a second-neighbor, central-force model of the bcc lattice. $F(V,T)\mathrm{}$ includes cubic and quartic anharmonic terms in perturbation theory evaluated in the high-temperature limit. The 25 Brillouin-zone sums that enter the calculation of $F(V,T)\mathrm{}$ are expressed as functions of three parameters, ${\kappa }_1,{\kappa }_2$, and ${\kappa }_3$, that depend on first and second derivatives of the pair potential $\phi \mathrm{}\left(r\right)\mathrm{}$, thus, knowledge of $\phi \mathrm{}\left(r\right)\mathrm{}$ and the Brillouin-zone sums completely determines the outcome of the calculation. Numerical results have been obtained for the alkali metals Li, Na, K, Rb, and Cs. The linear thermal expansion $\epsilon$ is well represented by this theory. The specific heat at constant volume, $C_V$, agrees with experiment to within 3% when allowance is made for the vacancy contribution. The results for the specific heat at constant pressure, $C_P$, are less satisfactory than for $C_V$ (they are within 6% of the experimental values) since they depend on the bulk moduli which, in turn, depend very sensitively on the method of treating the electrons. The free-electron theory is quite inadequate for the alkali metals. We have taken electron correlation into account and obtain results for the bulk moduli that are reasonable (within 10% of the experimental values in most cases, and within 25% for the worst case, Rb), considering the wide range in the experimental values. We discuss the limitations of this approach in dealing with systems where the electrons play a major role in stabilizing the structure of the crystal.