Pressure Dependence of the Knight Shift in Polycrystallineβ-Tin, Lead, and Platinum

The pressure dependence of the Knight shift in polycrystalline samples of $\beta$-tin, lead, and platinum has been measured at room temperature for hydrostatic pressures up to 12 kbar. In tin both the isotropic and anisotropic contributions to the Knight shift are determined from the asymmetric line shape. The volume dependences for all three metals can be satisfactorily expressed in the form $K\left(V\right)\mathrm{}\propto V^{\alpha }$, where the parameters $\alpha$ are $\alpha \left(\mathrm{Pb}\right)=-0.08\mathrm{}\pm 0.06$, $\alpha \left(\mathrm{Pb}\right)=+0.99\mathrm{}\pm 0.1$, $\alpha (\mathrm{S}\mathrm{n},\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{c})=+0.96\mathrm{}\pm 0.06$, $\alpha (\mathrm{S}\mathrm{n},\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{c})=-4.25\mathrm{}\pm 1.8$. The measured volume dependences are used to separate the temperature dependence at constant pressure into the contribution from thermal expansion and the explicit temperature dependence at constant volume. For lead and tin, the experimental explicit temperature dependences are compared with theoretical predictions based on the effects of lattice vibrations. The theoretical explicit temperature dependence is calculated in two ways: in one, by taking the volume dependence of the Pauli susceptibility given by free-electron theory, and in the other, by using the volume dependence of the density of states determined by pressure studies of the superconducting critical field. Better agreement with experiment is obtained in the latter case. The volume dependence of the anisotropic Knight shift in tin is shown to be more closely determined by a volume dependence of the anisotropy in charge distribution than by a change in the relative $s-p$ character of the wave function.