The charge density in vanadium and chromium

Careful measurements of the structure factors of vanadium (Ohba et al. 1981) and chromium (Ohba et al. 1982) up to sin 6/2 = 1.72 A / using AgKa radiation and small spherical crystals ( 0.2 mm diameter), have been reported. The bcc structure of these metals leads to pairs of reflections such as (330/441), (431/510), at identical values of sin 6/2, which have the same intensity for a structure with one spherical atom per lattice point. This is no longer true when the t2g and eg orbitals of the cubic site are no longer equally occupied. This is easiest seen as follows. 

The t2g density is zero where two of the direction cosines x, y, and z are zero, that is, in the planes of the coordinate axes, but peaks along the eight cube diagonals. The density functions are Fourier-transform invariant, as discussed in chapter 3, and expressed by the equation 

For reflections at the same sin 0/A, the radial scattering factors j( are equal, but the angular factors dlmp differ for different occupation of the l2g orbitals. According to Eqs. (11.7) and (11.9), the scattering factor /(eff) 3(/i4 + fc4 + /4) — 1, and f(t2g) (hi)2 + (Ac/)2 + (Wc)2. As a result occupancy of the eg orbitals in excess of the spherical population increases the intensity of reflections with larger values of h4 + k4 + /4, the scattering vectors of which are closer to the directions of the eg lobes, while occupancy of the t2g orbitals beyond 3 electrons increases the reflections with larger values of (hi)2 + (Ac/)2 + (hk)2 for which h4 + k4 + l4 is smaller. 

Experimental and theoretical ratios for paired reflections in Cr are listed in Table 11.1. The ratios are listed such that the reflection with the smallest value of h4 + k4 + l4 is in the numerator. It is evident that the reflections with smaller values of h4 + k4 + l4 are more intense by 1-2.5%, indicating a preferential occupancy of the t2g orbitals, which in the bcc structure are directed towards the nearest neighbors. 

This conclusion is confirmed by both experimental and theoretical deformation density maps. The experimental maps show a positive peak of height 

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