Magnetic properties of Sohncke-type Pb(TiO)$C{u}_4{\left(P{O}_4\right)}_4$ exposed by resonant x-ray Bragg diffraction

The enantiomorphous (chiral) crystal class of Sohncke-type Pb(TiO)$C{u}_4{\left(P{O}_4\right)}_4$ permits the rotation of the plane of polarization of light (optical activity). Copper ions participate in noncollinear antiferromagnetic order below a temperature ≈ 7 K, with magnetoelectric and piezomagnetic effects permitted. Crystal and magnetic symmetries of Pb(TiO)$C{u}_4{\left(P{O}_4\right)}_4$ are fully incorporated in calculated resonant x-ray Bragg diffraction patterns that are successfully compared with existing limited experimental data [Misawa et al., Phys. Rev. B 103, 174409 (2021)]. Specifically, there is additional intensity of a Bragg spot (a chiral signature) from circular polarization (helicity) in the primary beam of x rays. The chiral signature is shown to arise from Cu axial magnetic dipoles, with the prospect of future experiments revealing interference between magnetic dipoles and (Templeton-Templeton) chargelike quadrupoles. An expression for the additional intensity used by Misawa et al. does not respect magnetic symmetry, and our symmetry-informed answer overturns their principal conclusions about chirality and magnetic order. Dirac quadrupoles and octupoles are potentially strong sources of diffraction when the reflection vector is parallel to the unique direction in the tetragonal lattice. Notably, a Dirac quadrupole is a parity- and time-odd atomic multipole unlike a multisite spin entity previously mentioned in the context of Pb(TiO)$C{u}_4{\left(P{O}_4\right)}_4$ [Kimura et al., Phys. Rev. B 97, 134418 (2018)].