Anharmonic lattice vibrations in simple metals

Entropy Debye temperatures ${\Theta }_S\mathrm{}\left(T\right)\mathrm{}$ are calculated from $T=0\mathrm{}$ K to the melting temperature $T_m$, at constant pressure and (partly) at constant volume, from the recommended heat-capacity data of Hultgren et al. of 1973 for the elements ${}_3{}^{}{}_{}{}^{}\mathrm{Li}$, ${}_{11}{}^{}{}_{}{}^{}\mathrm{Na}$, ${}_{19}{}^{}{}_{}{}^{}\mathrm{K}$, ${}_{37}{}^{}{}_{}{}^{}\mathrm{Rb}$, ${}_{55}{}^{}{}_{}{}^{}\mathrm{Cs}$, ${}_{29}{}^{}{}_{}{}^{}\mathrm{Cu}$, ${}_{47}{}^{}{}_{}{}^{}\mathrm{Ag}$, ${}_{79}{}^{}{}_{}{}^{}\mathrm{Au}$, ${}_4{}^{}{}_{}{}^{}\mathrm{Be}$, ${}_{12}{}^{}{}_{}{}^{}\mathrm{Mg}$, ${}_{20}{}^{}{}_{}{}^{}\mathrm{Ca}$, ${}_{30}{}^{}{}_{}{}^{}\mathrm{Zn}$, ${}_{48}{}^{}{}_{}{}^{}\mathrm{Cd}$, ${}_{80}{}^{}{}_{}{}^{}\mathrm{Hg}$, ${}_{13}{}^{}{}_{}{}^{}\mathrm{Al}$, ${}_{31}{}^{}{}_{}{}^{}\mathrm{Ga}$, ${}_{49}{}^{}{}_{}{}^{}\mathrm{In}$, ${}_{81}{}^{}{}_{}{}^{}\mathrm{Tl}$, ${}_{50}{}^{}{}_{}{}^{}\mathrm{Sn}$, and ${}_{82}{}^{}{}_{}{}^{}\mathrm{Pb}$. The result is discussed in terms of low-order explicitly anharmonic phonon frequency shifts ${\Delta }_3+{\Delta }_4$ and shifts ${\Delta }_2$ due to the thermal expansion. The total shift in ${\Theta }_S$ (${\Delta }_2+{\Delta }_3+{\Delta }_4+\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{r}-\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{s}$) is very similar for all the metals. At $T_m$ it lies between $-0.10\mathrm{}{\Theta }_S$ and $-0.15\mathrm{}{\Theta }_S$ except for Li ($-0.08\mathrm{}{\Theta }_S$). $\frac{({\Delta }_3+{\Delta }_4)}{{\Delta }_2}$ typically ranges from -0.4 to + 0.6. Higher-order anharmonic shifts are found to be small, decreasing ${\Theta }_S$ by 1 to 3% at $T_m$. Accurate values are derived for some frequency moments of the harmonic (i.e., low-temperature) phonon density of states.