Parity-odd multipoles, magnetic charges, and chirality in hematite $\alpha$-Fe${}_2$O${}_3$

Collinear and canted magnetic motifs in hematite were investigated by J. Kokubun et al. [Phys. Rev. B 78, 115112 (2008)] using x-ray Bragg diffraction magnified at the iron $K$-edge, and analyses of observations led to various potentially interesting conclusions. We demonstrate that the reported analyses for both nonresonant and resonant magnetic diffraction at low energies near the absorption $K$-edge are not appropriate. In its place, we apply a radically different formulation, thoroughly tried and tested, that incorporates all magnetic contributions to resonant x-ray diffraction allowed by the established chemical and magnetic structures. Essential to a correct formulation of diffraction by a magnetic crystal with resonant ions at sites that are not centers of inversion symmetry are parity-odd atomic multipoles, time-even (polar) and time-odd (magneto-electric), that arise from enhancement by the electric-dipole ($E1$)–electric-quadrupole ($E2$) event. Analyses of azimuthal-angle scans on two space-group forbidden reflections, hexagonal $(0,0,3){}_h$ and $(0,0,9){}_h$, collected by Kokubun $\mathrm{et}\hspace{0.5em}\mathrm{al}.$ [Phys. Rev. B 78, 115112 (2008)] above and below the Morin temperature $(T_M=250$ K), allow us to obtain good estimates of contributing polar and magnetoelectric multipoles, including the iron anapole. We show, beyond reasonable doubt, that available data are inconsistent with parity-even events only ($E1$-$E1$ and $E2$-$E2$). For future experiments, we show that chiral states of hematite couple to circular polarization and differentiate $E1$-$E2$ and $E2$-$E2$ events, while the collinear motif supports magnetic charges.

Figure 1

Magnetic and chemical structure of hematite, space group $R\overline{3}c$. The red (large) and the yellow (small) dots represent oxygen and iron sites, respectively. The left line denotes the magnetic motif along the $c$ axis below the Morin temperature (phase I). The right line denotes the motif above the Morin temperature, where iron moments are contained in the $a$-$b$ plane (phase II).

Figure 2

Azimuthal-angle dependence of intensity of Bragg reflections $(0,0,l){}_h$ with $l=3$ and $l=9$ for phase I (150 K). Continuous curves are fits to structure factors for $E1$-$E2$ and $E2$-$E2$ events with magnetic (time-odd) parity-even multipoles set to zero. Inferred relative atomic multipoles are listed in Table . Experimental data are taken from Kokubun et al.[7]

Figure 3

Azimuthal-angle dependence of intensity of the Bragg reflection $(0,0,9){}_h$ for phases I (150 K) and II (room temperature). Continuous curves are fits to parity-even structure factors $E1$-$E1$ and $E2$-$E2$ including all magnetic multipoles. Experimental data taken from Kokubun et al.[7] appear also in Figs. 2 and 4.

Figure 4

Azimuthal-angle dependence of the intensity of Bragg reflections $(0,0,l){}_h$ with $l=3$ and $l=9$ for phase II (room temperature). Continuous curves are fits to structure factors for $E1$-$E2$ and $E2$-$E2$ events with magnetic (time-odd) parity-even multipoles set to zero. Inferred relative atomic multipoles are listed in Table . Experimental data are taken from Kokubun $\mathrm{et}\hspace{0.5em}\mathrm{al}$.[7]

Figure 5

Simulation of the azimuthal-angle dependence from Eq. (6) for a circular polarized light of Bragg reflections $(0,0,l){}_h$ with $l=3$ and $l=9$ for phase I. Continuous curves are simulations made with the values of the multipoles from the $E1$-$E2$ event gathered in Table . For the $E2$-$E2$ event $I_c$ is zero because our magnetic (time-odd) parity-even multipoles are zero for a ferric ion. Zero $I_c$ does not mean zero intensity because $I_c$ is only the circular polarization contribution to intensity.[17]

Figure 6

Simulation of the azimuthal-angle dependence from Eq. (8) for a circular polarized light of Bragg reflections $(0,0,l){}_h$ with $l=3$ and $l=9$ for phase II (room temperature). Continuous curves are simulations made with the values of the multipoles from the $E1$-$E2$ event gathered in Table . For the $E2$-$E2$ event the $I_c$ is equal to zero because our magnetic (time-odd) parity-even multipoles are zero. Zero $I_c$ does not mean zero intensity since $I_c$ is only the circular polarization contribution.[17]