Aspherical Spin Density inS-State Cations

A calculation of the first-order effect of spin-orbit coupling on a ${\mathrm{}\left(3d\right)\mathrm{}}^5$ unperturbed $S$-state ion in a cubic crystal field has led to a new contribution ${\sigma }_1\left(\mathrm{r}\right)$ to the ionic spin density. The integral over space of ${\sigma }_1\left(\mathrm{r}\right)$ is zero and ${\sigma }_1\left(\mathrm{r}\right)$ is perpendicular to the unperturbed spin density ${\sigma }_0\mathrm{}\left(r\right)\mathrm{}$; while ${\sigma }_0\mathrm{}\left(r\right)\mathrm{}$ is spherically symmetric about the nucleus, ${\sigma }_1\left(\mathrm{r}\right)$ is highly aspherical. In $\alpha$-${\mathrm{Fe}}_2$ ${\mathrm{O}}_3$ at room temperature, the spin density consists of the large "antiferromagnetic" component ${\sigma }_0$, plus the new term ${\sigma }_1$, plus the weak ferromagnetic or Dzialoshinsky term ${\sigma }_D$, which is spherical. In neutron scattering, it is found that ${\sigma }_1$ contributes to the same "ferromagnetic" Bragg peaks as does ${\sigma }_D$, and in the same order of magnitude. Hence the new term is probably important for understanding the surprising, highly aspherical ferromagnetic spin-density distribution recently observed in $\alpha$-${\mathrm{Fe}}_2$ ${\mathrm{O}}_3$ by Pickart, Nathans, and Halperin. In general, ${\sigma }_1\left(\mathrm{r}\right)$ will be nonzero under much less restrictive symmetry requirements than those needed for the nonvanishing of ${\sigma }_D$. In the course of discussion, it is pointed out that the Dzialoshinsky-Moriya theory of the weak ferromagnetism in $\alpha$-${\mathrm{Fe}}_2$ ${\mathrm{O}}_3$, for example, implies that the application of a spatially uniform magnetic field should influence the distribution of domains of antiferromagnetic spin components. This effect was observed by Pickart et al.