Effect of Crystalline Fields on Charge Densities and Magnetic Form Factors

The effects of crystalline fields on $3d$ charge densities and magnetic form factors for transition metal ions are discussed on the basis of recent theoretical investigations augmented by an analysis of optical absorption data. It is shown that the crystalline field has two effects on the free ion $3d$ wave functions and hence on their form factors as well: (1) a differentiation or "splitting" of the two types of cubic $3d$ functions by an expansion of the $t_{2g}\left(e_g\right)$ orbitals and a contraction of the $e_g\left(t_{2g}\right)$ orbitals resulting in two different radial charge densities, and (2) a net expansion of the charge distribution from the free ion value. The magnetic form factors due to this "splitting" effect when calculated according to the methods of Weiss and Freeman show measurable deviations from the free atom results. A form factor for ${\mathrm{Mn}}^{+2\mathrm{}}$ based on optical absorption data shows a large expansion of the $3d$ charge density, in agreement with the magnetic form factor measurements of Hastings, Elliott, and Corliss. This agreement, based on the use of theoretical $F^k(3d, 3d)$ integrals, indicates that the well-known discrepancy between theoretical and experimental values of these integrals arises from the fact that the quantities obtained experimentally are not true $F^k(3d, 3d)$ integrals. The crystalline potential due to an array of negative ion charge densities has been employed to discuss these various effects and their meaning with respect to a proper (essentially molecular) treatment.