Role of Conduction Electrons in Electric-Field Gradients of Ordered Metals

Measurements of the electric-quadrupole interaction of ${\mathrm{V}}^{51}$ nuclei in the ordered $\beta$-wolfram structure ${\mathrm{V}}_{3}X (X=\mathrm{Ga},\mathrm{Si},\mathrm{Au},\mathrm{etc}.\mathrm{})$ intermetallic compounds suggest a correlation between the magnitudes of the electric-field gradients at the vanadium sites and the density of electronic states at the Fermi energy $\eta \left(E_F\right)$. The conduction-electron contributions to an electric-field gradient in a metal have been inspected and the results indicate that the above correlation can indeed be expected and that conduction-electron field-gradient terms which are linearly related to $\eta \left(E_F\right)$ are of experimental significance in many metals, ordered and disordered. In these investigations, the sources of the field gradient have been divided into three terms: (1) the lattice contribution, $q_{\mathrm{latt}}$ arising from the electronic and nuclear charge external to an atomic sphere drawn about the nuclear site in question, (2) a local contribution, $q_{\mathrm{loc}}$, arising from conduction electrons within the sphere, and (3) Sternheimer antishielding contributions arising from the distortions of the ionic core. Attention is focused on $q_{\mathrm{loc}}$ and, in particular, on those contributions coming from electron states in the vicinity of the Fermi surface. This is done by inspecting the change $q^{\prime }$ in $q_{\mathrm{loc}}$ associated with the repopulation of Bloch states of different symmetries at the Fermi surface when $q_{\mathrm{latt}}$ and its associated potential of $Y_2^{}{}_{}{}^0(\theta ,\phi )$ symmetry within the sphere are turned on (or off). Although this effect does not include all Fermi-surface contributions to $q_{\mathrm{loc}}$, a "coherence" due to the common symmetry of the perturbing potential and the gradient operator tends to make this term important. It is linearly related to both $q_{\mathrm{latt}}$ and $\eta \left(E_F\right)$, and tends to strongly shield the lattice gradient. For example, a maximum estimate of its value for ${\mathrm{V}}_3$Ga is in excess of $-100\mathrm{}{q}_{\mathrm{latt}}$. Thus, we are dealing with an "overshielding" which, contrary to traditional expectations, can-cause a field gradient which is linearly related to $q_{\mathrm{latt}}$ to be opposite in sign to it. The investigation suggests that this term will be of experimental significance in $p$ band as well as high $\eta \left(E_F\right)$ transition metals. Self-consistent effects have been included in the calculation and do not destroy the tendency toward strong shielding. The electron-phonon interaction is inspected and found not to play a role in these terms [i.e., a "bare" $\eta \left(E_F\right)$ should be used]. Finally, the effect of thermal repopulation on the temperature dependence of $q_{\mathrm{loc}}$ is considered.