Electron density distribution in vanadium and niobium from $\gamma$-ray diffraction

The electron density distribution in paramagnetic vanadium and niobium is derived via multipole refinement of high-quality single-crystal diffraction data, complete up to sinθ/λ $=$ 1.9 Å${}^{-1}$, collected at room temperature with 316.5 keV $\gamma$ radiation. Four $\gamma$ lines in the energy range 200–600 keV have been used to extrapolate extinction-free low-order structure factors. Both metals display charge asphericity due to preferential occupancy of the $t_{2g}$ orbital. The asphericity in V is in quantitative agreement with ab initio calculations, but not with earlier experimental results, laying at rest a long-standing debate. In both metals, the $d$-electron density is strongly contracted relative to the free atoms. The high-precision thermal parameters are in good agreement with some of the older results. No support is found in the data for anharmonic contributions to the thermal parameters, contrary to claims in previous reports on V. Attention is paid to the 3$d$-4$s$ occupation problem in V, indicating no change in electronic configuration between atom and metal. Relativistic form factor issues are also discussed. It is pointed out that for the analysis of crystallographic data nonrelativistic form factors are to be preferred over their relativistic counterparts.

Figure 1

Observed squared structure factors on absolute scale of the 110 reflection for niobium and vanadium. The error bars are within the size of the data points. The solid curves were calculated from the Zachariasen theory with the following fit parameters: $F^2$(λ $=$ 0) $=$ 3847(25), $g$ $=$ 1397(37) rad${}^{-1}$ in the case of Nb, and $F^2$(λ $=$ 0) $=$ 986(7), $g$ $=$ 5001(120) rad${}^{-1}$ in the case of V.

Figure 2

Aspherical contributions to the static model density of vanadium in the (110) plane. The density range is from −0.38 to 0.25 $e$ Å${}^{-3}$. Solid lines represent regions of excessive density, dashed lines depleted regions in steps of 0.05 $e$ Å${}^{-3}$. The zero contour is omitted. The densities are truncated at ±0.3 $e$ Å${}^{-3}$.

Figure 3

Relative deviation, ( $f$ − $g$)/$g$, of the ordinary nonrelativistic (NFF) and relativistic (RFF) form factors from the modified form factor for niobium.