Heat Capacity of hcp and bcc Solid Helium 3

Abstract

Experimental data are presented for the heat capacity of hcp ${\mathrm{He}}^3$ at five molar volumes (19.05 to 11.42 ${\mathrm{cm}}^3$) and for bcc ${\mathrm{He}}^3$ at four molar volumes (20.18 to 23.80 ${\mathrm{cm}}^3$). These data extend from the true $T^3$ region ($\frac{T}{{\Theta }_0}<0.03\mathrm{}$) to the melting line in all cases with sufficient precision (at least 1% in ${\Theta }_D$) so that the volume dependence of both ${\Theta }_0$ and the reduced ${\Theta }_{-\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{u}\mathrm{s}-}T \left(\frac{\frac{\Theta }{{\Theta }_{0-\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{u}\mathrm{s}-}T}}{{\Theta }_0}\right)$ relationship can be determined. In general, when a comparison is made with other ${\mathrm{He}}^4$ data, $\frac{{\theta }_{03}}{{\theta }_{04}}=1.18\mathrm{}$. The quantity ${\gamma }_0=\frac{-d\ln {\Theta }_0}{d\ln V}$ varies from 2.6 to 2.0 for the hcp data with decreasing molar volume, while ${\gamma }_0=2.2\mathrm{}$ for the bcc phase. The changes in the shapes of the reduced ${\Theta }_{-\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{u}\mathrm{s}-}T$ curves for the hcp phase can be understood in terms of a slightly temperature-dependent Grüeneisen constant $\gamma$, the ratio $\frac{\gamma }{{\gamma }_0}$ being independent of volume to a first approximation and increasing to approximately 1.07 at $\frac{T}{{\Theta }_0}=0.12\mathrm{}$. The shapes of these reduced ${\Theta }_{-\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{u}\mathrm{s}-}T$ curves at the smallest molar volumes are almost identical for hcp ${\mathrm{He}}^3$ and for our one hcp ${\mathrm{He}}^4$ run, and agree with comparable previous data at relatively high temperature. These shapes resemble closely the zero-pressure data for argon and krypton and the theoretical calculations of Horton and Leech. The bcc ${\mathrm{He}}^3$ data can be represented quite precisely as the sum of a Debye-like term [involving ${\Theta }_0\mathrm{}\left(V\right)\mathrm{}$] and an exponential Schottky-like term [involving a characteristic temperature $\phi \mathrm{}\left(V\right)\mathrm{}$]. When compared with the hcp data, the bcc data cannot be explained solely in terms of conventional lattice dynamics.