Zeeman spectrum, magnetic neutron diffraction pattern, and Dirac multipoles for a multiferroic material $\mathrm{Cu}{\mathrm{B}}_2{\mathrm{O}}_4$

Zeeman spectra, dichroic signals, and neutron Bragg diffraction patterns generated by copper ions in magnetically ordered copper metaborate $\left(\mathrm{Cu}{\mathrm{B}}_2{\mathrm{O}}_4\right)$ are investigated within a minimal model of Cu atomic states. A theory platform, common to understanding optical spectra and neutron diffraction patterns, affords the immediate benefit of a unified description of the experimental probes in terms of electronic multipoles. Results for dichroic signals illustrate a nontrivial use of a general, quantum mechanical theory of photon absorption couched in terms of Dirac multipoles that are magnetic and polar. Anapoles (Dirac dipoles) are predicted to generate Bragg spots in magnetic neutron diffraction that are not indexed by the motif of conventional (axial) magnetic-dipole moments. The minimal model of Cu states is informed by magnetic symmetry, derived from an established commensurate antiferromagnetic order, with a sparse number of parameters that comply with available empirical evidence.

Figure 1

Projection of the chemical structure of $\mathrm{Cu}{\mathrm{B}}_2{\mathrm{O}}_4$ onto the basal plane of the tetragonal cell, space group $I\overline{4}2d$ (No. 122) with crystal class $\overline{4}2m$. $\text{Cu}\left(A\right)$ ions are located at sites $4b$ (0, 0, 1/2 and colored dark blue), and $\text{Cu}\left(B\right)$ ions are at sites $8d$ above and below the plane $(x$, 1/4, 1/8 with $x\approx 0.082$ [2], and colored cyan). A dipole moment at a corner $\text{Cu}\left(A\right)$ is in the direction [1,1, 0], say, and oppositely aligned to the dipole at the midpoint $\text{Cu}\left(A\right)$ in the commensurate, zero-field magnetic structure; cf. Fig. 2 [2].

Figure 2

Motif of magnetic-dipole moments depicted in a central projection [2]. $\text{Cu}\left(A\right)$ with blue arrows occupy sites $8a$ in the magnetic space group $F{d}^{\prime }d{2}^{\prime }$ (No. 43.226), and $\text{Cu}\left(B\right)$ with red arrows.