Is the mean value of the 3d-electron radius $\langle r^{4} \rangle$ in the equation of the crystal-field theory constant?

Abstract

In the equation of the crystal-field theory $\Delta = \frac{{Q \cdot \langle r^{4} \rangle }}{{\overline{R}{^5} }}$, where Δ is the crystal-field splitting, $\overline{R}$ is the mean metal–ligand distance in the 3d^N-ion-centered coordination polyhedron and Q is constant, $\langle r^{4} \rangle$, the mean value of the fourth power of the 3d-electron radius, is assumed to be also constant (Burns 1993). Here, this assumption is proved by means of high-pressure optical absorption spectroscopy on octahedral Fe²⁺ in siderite FeCO₃ in combination with the data of X-ray diffraction structural refinement by Lavina et al. (2010). It is shown that in the pressure range 1·10^–4 to 44.5 GPa or $\overline{R}$ ranging from 2.15 to 1.97 Å, the value of Q $\cdot \langle r^{4} \rangle$ does not deviate more than ~ 4% from its average value.