Thermal vacancies in solidHe3

Thermal vacancy concentrations in crystals of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ have been directly determined from measurements of the temperature dependence of the x-ray lattice parameter. Body-centered-cubic ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ was studied for molar volumes ranging from 20.3 to 24.8 ${\mathrm{cm}}^3$/mole for which the melting temperatures range from 2.36 to 0.435 K, respectively. It was found to contain about 0.5% thermal vacancies at melting, essentially independent of molar volume. A single hcp ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ crystal (18.8 ${\mathrm{cm}}^3$/mole) was studied, and was found to have 0.1% thermal vacancies at melting. From the measured vacancy concentrations, free energies of formation are obtained and compared with results from NMR and ultrasonic experiments. These comparisons suggest that in the bcc phase vacancies move by a tunneling process, while in the hcp phase their motion is thermally activated. Free volumes of formation are found from a comparison of the measured vacancy concentrations with existing thermal expansion and compressibility measurements. In the bcc phase the formation free volume is found to vary from $0.4v_a$ at 20 ${\mathrm{cm}}^3$/mole to $0.3v_a$ at 24 ${\mathrm{cm}}^3$/mole, where $v_a$ is an atomic volume. The apparent heat capacity of the vacancies is calculated with use of a localized vacancy model and found to be unreasonably high. Both this result and an analysis of the volume of formation suggest that the vacancies are nonlocalized.