Pressure Dependence of the Knight Shift in Al and Nb Metal

Abstract

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 $\frac{d\mathrm{ln}K}{d\mathrm{ln}V}$ is - 1.01\ifmmode\pm\else\textpm\fi{}0.02 and - 0.29\ifmmode\pm\else\textpm\fi{}0.02 for Al and Nb, respectively. Since it is essential to know the volume dependence of the (electron-electron-enhanced) spin susceptibility $\frac{d\mathrm{ln}{\ensuremath{\chi}}_{\mathrm{sp}}^{*}}{d\mathrm{ln}V}$ in order to analyze the observed $\frac{d\mathrm{ln}K}{d\mathrm{ln}V}$, the theoretical implication of the previously proposed methods to estimate $\frac{d\mathrm{ln}{\ensuremath{\chi}}_{\mathrm{sp}}^{*}}{d\mathrm{ln}V}$ was explored. The linearly temperature-dependent thermal expansion at low temperature ${\ensuremath{\alpha}}_{e}$ gives rise to the volume dependence of electron-phonon-enhanced density of states at the Fermi surface. ${\ensuremath{\alpha}}_{e}$ includes only a temperature-independent part of the enhancement factor $1+\ensuremath{\lambda}$. 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 $1+\ensuremath{\lambda}$, the density of states for the bare electrons $N{({E}_{F})}_{\mathrm{BS}}$, and the band-structure effective mass ${m}^{*}$ is proposed. These values are derived from the pressure dependence of the superconducting transition temperature and ${\ensuremath{\alpha}}_{e}$. The volume dependence of ${\ensuremath{\chi}}_{\mathrm{sp}}^{{}^{*}}$ is deduced from $\frac{d\mathrm{ln}N{({E}_{F})}_{\mathrm{BS}}}{d\mathrm{ln}V}$ 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 $〈{|\ensuremath{\psi}(0)|}^{2}〉$ was deduced for Al as $\frac{d\mathrm{ln}〈{|\ensuremath{\psi}(0)|}^{2}〉}{d\mathrm{ln}V}=\ensuremath{-}2.12$. The volume dependence of the orbital Knight shift ${K}_{0}$ for Nb is estimated as $\frac{d\mathrm{ln}{K}_{0}}{d\mathrm{ln}V}\ensuremath{\simeq}0.4 (or 0.1)$. The possible origin of the discrepancy between the density of states derived from ${\ensuremath{\alpha}}_{e}$ 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.