Pressure Dependence of the Knight Shift in Al and Nb Metal

The pressure dependence of the Knight shift $K$ of Al and Nb metals was measured with a digitally slaved signal averager. The maximum hydrostatic pressure utilized was 8000 kg/${\mathrm{cm}}^2$. The observed value of $\frac{d\ln K}{d\ln V}$ is - 1.01±0.02 and - 0.29±0.02 for Al and Nb, respectively. Since it is essential to know the volume dependence of the (electron-electron-enhanced) spin susceptibility $\frac{d\ln {\chi }_{\mathrm{sp}}^{*}}{d\ln V}$ in order to analyze the observed $\frac{d\ln K}{d\ln V}$, the theoretical implication of the previously proposed methods to estimate $\frac{d\ln {\chi }_{\mathrm{sp}}^{*}}{d\ln V}$ was explored. The linearly temperature-dependent thermal expansion at low temperature ${\alpha }_e$ gives rise to the volume dependence of electron-phonon-enhanced density of states at the Fermi surface. ${\alpha }_e$ includes only a temperature-independent part of the enhancement factor $1+\lambda$. The pressure dependence of the superconductor parameters renders the volume dependence of the density of the states clothed with a full electron-phonon interaction, which includes the temperature-independent part as well as a possible temperature-dependent part. A semiempirical scheme to deduce the volume dependence of $1+\lambda$, the density of states for the bare electrons $N{\left(E_F\right)}_{\mathrm{BS}}$, and the band-structure effective mass $m^{*}$ is proposed. These values are derived from the pressure dependence of the superconducting transition temperature and ${\alpha }_e$. The volume dependence of ${\chi }_{\mathrm{sp}}^{{}^{*}}$ is deduced from $\frac{d\ln N{\left(E_F\right)}_{\mathrm{BS}}}{d\ln V}$ by taking into account the effect of the electron-electron enhancement factor. The volume dependence of the density of wave function at the Fermi surface $〈{\left|\psi \mathrm{}\right(0\left)\right|}^2〉$ was deduced for Al as $\frac{d\ln 〈{\left|\psi \mathrm{}\right(0\left)\right|}^2〉}{d\ln V}=-2.12\mathrm{}$. The volume dependence of the orbital Knight shift $K_0$ for Nb is estimated as $\frac{d\ln K_0}{d\ln V}\simeq 0.4 (or 0.1)$. The possible origin of the discrepancy between the density of states derived from ${\alpha }_e$ and from the pressure dependence of the superconductor parameters is discussed. The origin of inconsistency in the previously reported temperature dependence of $K$ for Al is also suggested.