Occhialini et al. (arXiv:2510.13767; DOI: 10.1103/yr5q-1v1s) add results to several recent experimental studies of bulk magnetism in the rutile compound RuO_2. It is of interest as a candidate altermagnet. The cited publication contains several serious errors. Notably, scattering amplitudes used to interpret measurements accomplished with resonant x-ray Bragg diffraction, which appear in the main text and SM (Eqs. 14 16), are wrong. The authors tender erroneous amplitudes (Phys. Rev. Lett. 122, 017202) and thereby ignore previously published correct results (Phys. Rev. B 105, 014403). As a result, the concluding statement and Footnote SM [54] are misleading.

Symmetry informed diffraction patterns for magnetically ordered MnF2 illuminated by x-rays tuned in energy to a Mn atomic resonance depend on circular polarization in the primary beam. The change in intensity of a Bragg spot with a change in handedness establishes a chiral signature for the fully compensated collinear antiferromagnet. The signature is calculated for electric dipole (E1) and electric quadrupole (E2) axial absorption events. Polar E1-E2 absorption events are forbidden by inversion symmetry in Wyckoff positions assigned to Mn ions in the established MnF2 magnetic symmetry. It is compatible with atomic altermagnetism, and the corresponding order parameter contributes to E2-E2 and magnetic neutron Bragg diffraction patterns. Moreover, spin-flip patterns from polarized neutron diffraction depend on electronic quadrupole and octupole moments that are zero for the nominal 3d5 configuration of Mn2+, which make them good tests of the actual electronic structure.

The magnetic properties of materials hosting Eu2+ (J = 7/2, 4f7) ions have attracted much attention in the science of strongly correlated electrons. In part because crystal electric field effects are impoverished for an S-state ion, as with Gd3+ intermetallics, and Eu2+ substitution in biological and optically active materials is resourceful. The magnetic structure of EuPdSn2 is not wholly resolved. Ferromagnetic and antiferromagnetic structures coexist in powder neutron diffraction patterns, and compete in the ground state. Moreover, the specific heat as a function of temperature is enigmatic and indicative of J = 5/2. We present symmetry-informed analytic magnetic structure factors for single crystal resonant X-ray Bragg diffraction using Eu atomic resonances that reveal significant potential for the technique. Europium ions use Wyckoff positions that are not centres of inversion symmetry in magnetic space groups inferred from neutron diffraction. In consequence, axial and polar Eu multipoles are compulsory components of both magnetic neutron and resonant X-ray Bragg diffraction patterns. The proposed antiferromagnetic phase of EuPdSn2 supports anapoles (magnetic polar dipoles) already observed in magnetic neutron diffraction patterns presented by Gd-doped SmAl2, and several resonant X-ray diffraction patterns.

Magnetic axial and polar (Dirac) nickel multipoles contribute to resonant X-ray Bragg amplitudes in a symmetry-informed analysis of monoclinic Li 2 Ni 3 P 4 O 14 presented for future diffraction experiments. Magnetic long-range order below a temperature of ≃ 14.5 K can be viewed as a two-dimensional trimerized antiferromagnet with Ni ions in two Wyckoff positions in the centrosymmetric ( 1 ) magnetic space group P 2 1 / c . It permits the coupling to circular polarization in the primary X-ray beam, unlike the corresponding diffraction by an antiferromagnet characterized by anti-inversion ( 1 ′) and a linear magnetoelectric effect, e.g. historically significant chromium sesquioxide (Cr 2 O 3 ) and Cu 2 (MoO 4 )(SeO 3 ) [Lovesey & van der Laan (2024). Phys. Rev. B 110 , 174442]. The space group is inferred from neutron Bragg diffraction patterns, without an allowance for permitted Dirac dipoles (anapoles) and quadrupoles [Chikara et al. (2025). Phys. Rev. B 112 , 014438].

Published magnetic data for LaCoO 3 are successfully analyzed with coexisting 5 D and low-spin (LS) cobalt states. Energy levels of the two states are derived in analytic forms. To this end, fictitious orbital angular momentum l of magnitude one defines the Γ 5 ( 5 D ) state. Our Hamiltonian includes the spin–orbit interaction, and a cubic crystal field embellished by a trigonal distortion 9 B 2 0 ( l z 2 − 2 / 3 ) − 80 B 4 0 ( l z 2 − 9 / 10 ) . A singlet ground state with an energy gap to the first excited doublet is realized for certain values of the parameters. The temperature-independent paramagnetic susceptibility (TIPS) of the 5 D state has a finite value, which accords with the observation. Whereas, TIPS is symmetry forbidden in the LS state. A rigorous calculation is made of the excitation spectrum in the LS state. The elementary excitation is modeled as a creation of an electron–hole pair that results in an energy level scheme in which the first excited quartet lies above the singlet ground state. The electron spin resonance data are successfully equated with transitions within the excited quartet. Available magnetization data delineate parameters in the 5 D Hamiltonian. The temperature dependence of the susceptibility of our coexisting model is qualitatively reasonable. To improve on a quantitative outcome, we are led to introduce a temperature dependent concentration for the 5 D and LS states. Calculated Bragg diffraction patterns gathered with x-rays tuned to the Co K -edge reveal potential to refine the current crystal structure and to shed light on the origin of the coexisting states.

A recent experimental study of TbB4 at a low temperature using resonant x-ray Bragg diffraction implies a magnetic symmetry not found in any other rare-earth tetraboride. The evidence for this assertion is a change in the intensity of a TbB4 Bragg spot on reversing the handedness (chirality) of the primary x-ray beam [R. Misawa, K. Arakawa, T. Yoshioka, H. Ueda, F. Iga, K. Tamasaku, Y. Tanaka, and T. Kimura, ]. It reveals a magnetic chiral signature in TbB4 that is forbidden in phases of rare-earth tetraborides known to date, as the previous magnetic symmetries are parity-time (PT) symmetric with anti-inversion present in the magnetic crystal class. Misawa appeal to a PT-symmetric diffraction pattern to interpret their interesting diffraction patterns. In addition to the use of symmetry that does not permit a chiral signature, calculated patterns impose cylindrical symmetry on Tb sites with no justification. We review magnetic symmetries for TbB4 consistent with a published neutron powder diffraction pattern and susceptibility measurements. On the basis of this information, noncollinear antiferromagnetic order exists below a temperature ≈44 K with no ferromagnetic component. Our symmetry-informed patterns encapsulate Tb electronic degrees of freedom in terms of multipoles consistent with established sum rules for dichroic signals. The investigated symmetry templates are noncentrosymmetric, noncollinear antiferromagnetic constructions with propagation vector k=(0,0,0). An inferred chiral signature for a parity-even absorption event has an interesting composition. There is the anticipated product of Tb axial dipoles and chargelike quadrupoles (from Templeton-Templeton scattering). Beyond this contribution, though, symmetry allows a product of dipoles in the chiral signature. A predicted change in the intensity of a Bragg spot with rotation of the crystal about the reflection vector (an azimuthal angle scan) can be tested in future experiments. Likewise contributions to Bragg diffraction patterns from Tb anapoles and higher-order Dirac multipoles. Published by the American Physical Society 2024

The enantiomorphous (chiral) crystal class of the Sohncke-type insulator Pb(TiO)Cu4(PO4)4 permits the rotation of the plane of polarization of light (optical activity). Copper ions participate in antiferromagnetic order below a temperature 7 K, with magnetoelectric and piezomagnetic effects permitted. Lattice and magnetic symmetries of Pb(TiO)Cu4(PO4)4 are fully incorporated in calculated resonant x-ray Bragg diffraction patterns that are successfully compared with existing limited measurements on paramagnetic and magnetically ordered Pb(TiO)Cu4(PO4)4 [Misawa et al. Phys. Rev. B 103, 174409 (2021)]. Specifically, there is additional intensity in the ordered phase to a Bragg spot (a chiral signature) from circular polarization in the primary beam of x-rays. It is created by Cu axial magnetic dipoles, with the prospect of future experiments revealing interference between magnetic dipoles and charge-like (time-even, Templeton-Templeton) quadrupoles. Polar and magnetic (parity- and time-odd) Dirac quadrupoles and octupoles are potentially strong sources of diffraction when the reflection vector is parallel to the unique direction in the tetragonal lattice.

Antiferromagnetic compounds chromium sesquioxide (Cr2O3) and dioxomolybdenum selenite present linear magnetoelectric effects. Anti-inversion symmetry in the corresponding magnetic crystal classes dictate the makeup of magnetic Bragg diffraction patterns. Copper axial and polar magnetic multipoles contribute to resonant x-ray and magnetic neutron amplitudes in a symmetry informed analysis of monoclinic Cu2(MoO4)(SeO3) presented with a view to steering future diffraction experiments. The compound might be viewed as a low- dimensional quantum magnet on account of its crystal structure and the quantum nature of the Cu spin (S= 1/2).

The magnetic structure of RuO2 and the Ru atomic configuration are unknown. A magnetic structure is inferred by confronting measured and calculated Bragg diffraction patterns and adjusting the latter to achieve satisfactory agreement. An accepted pattern, a magnetic symmetry, includes symmetry of sites occupied by the magnetic ions. As a realistic starting point, we provide diffraction patterns for a magnetic symmetry of RuO2, a descendent of the tetragonal parent structure, which accommodates a departure of Ru axial dipoles from the crystal c axis. A chiral signal and piezomagnetic effect are permitted, and a linear magnetoelectric effect forbidden. Features of the neutron diffraction pattern test the nonrelativistic requirement of altermagnetism, and we scrutinize published room-temperature data. Specifically, one Bragg point is consistent with Ru orbital angular momentum and magnetic quadrupole both zero, and the latter result is not expected from nonrelativistic altermagnetism. Azimuthal angle scans in resonant x-ray diffraction are sensitive to the Ru site symmetry and the atomic configuration. Acid tests of the studied magnetic symmetry include a chiral signature and null intensity for unrotated photon polarization.

A magnetic resonance phenomenon is proposed, in which only the orbital degrees of freedom of electrons participate. This previously unexploited magnetic resonance may be called electron orbital resonance (EOR). To this end, a nonmagnetic, singlet ground state is necessary. In addition, the system needs to exhibit the Van Vleck paramagnetism. Conditions for EOR are met by the rare earth perovskite LaCoO3 (LCO). The 5D state of LCO is shown to exhibit the van Vleck paramagnetism that result in EOR. The frequency versus magnetic field relation for EOR in LCO is shown to be readily accessible with present day instrumentation. We predict that conventional electron spin resonance signals will be observed in the 5D state of LCO in addition to EOR.

Scrutiny of an established monoclinic magnetic space group for NdFeO3 reveals hitherto unknown properties of the orthoferrite. Future experiments using neutron and x-ray diffraction techniques can verify them. Neodymium ions possess Dirac multipoles, both time-odd (magnetic) and parity-odd (polar), that come with unique diffraction conditions. Non-magnetic polar Nd multipoles are permitted even though the monoclinic space group is centrosymmetric. Dirac multipoles are forbidden by symmetry at sites occupied by ferric ions. Available diffraction patterns have not been analysed for Dirac multipoles. Nor all permitted components of the axial dipoles and quadrupoles. In the case of neutron diffraction, magnetic quadrupoles are correlations between anapole and orbital degrees of freedom. We give conditions for the observation of Templeton-Templeton scattering of x-rays, created by angular anisotropy in the electronic charge distribution. Axial multipoles are the sole providers of dichroic signals.

Newly published diffraction data on hexagonal YMnO3 at a temperature of 10 K are shown to be consistent with a trusted expression of the magnetic symmetry, although the data alone are not definitive [M. Ramakrishnan et al., Phys. Rev. Research 5, 013203 (2023)]. Howard et al. conclude from an exhaustive review of experimental data that the symmetry of the antiferromagnetic motif of Mn ions is most likely P63'cm' [C. J. Howard et al., Acta Crystallogr. B 69, 534 (2013)]. The data reported by Ramakrishnan et al. does not eliminate symmetry P63' from our calculated diffraction patterns, because the studied reflection vector is parallel to the common chiral vector associated with each triangle of Mn axial dipole moments. Proposed diffraction patterns will give decisive statements about the magnetic symmetry in future investigations using resonant x-ray and magnetic neutron diffraction. To this end, both axial and polar magnetism in the multiferroic are essential in the analysis of diffraction patterns. The account by Ramakrishnan et al. of polar magnetism in resonant x-ray diffraction uses a magnetic symmetry of hexagonal YMnO3 not yet justified, e.g., ferromagnetism and a linear magnetoelectric effect allowed by their chosen symmetry have not been observed. We study polar magnetism in symmetry P63'cm' using Dirac multipoles, including Mn anapoles. They also feature in amplitudes for magnetic neutron diffraction together with Dirac quadrupoles, previously shown to account for diffraction by pseudo-gap phases of cuprate superconductors.

The ferrimagnet Mn3Si2Te6 attracts attention because of a recently discovered colossal magnetoresistance (CMR) with unique magnetic field properties. An improved magnetic structure for the material has emerged from a neutron diffraction study linked to understanding the CMR. A deeper theoretical investigation of the magnetic structure has now revealed anapole, chiral, and orbital states of manganese ions not previously mentioned. Moreover, it is shown that existence of these states in the low temperature form of Mn3Si2Te6, with a magnetic field applied, can be tested by neutron and resonant x-ray diffraction.

The magnetic structure of RuO2 and the Ru atomic configuration are unknown. A magnetic structure is inferred by confronting measured and calculated Bragg diffraction patterns and adjusting the latter to achieve satisfactory agreement. An accepted pattern, a magnetic symmetry, includes symmetry of sites occupied by the magnetic ions. As a realistic starting point, we provide diffraction patterns for a magnetic symmetry of RuO2, a descendent of the tetragonal parent structure, which accommodates a departure of Ru axial dipoles from the crystal c axis. A chiral signal and piezomagnetic effect are permitted, and a linear magnetoelectric effect forbidden. Features of the neutron diffraction pattern test the non-relativistic requirement of altermagnetism, and we scrutinize published room-temperature data. Specifically, one Bragg point is consistent with Ru orbital angular momentum and magnetic quadrupole both zero, and the latter result is not expected from non-relativistic altermagnetism. Azimuthal angle scans in resonant x-ray diffraction are sensitive to the Ru site symmetry and the atomic configuration. Acid tests of the studied magnetic symmetry include a chiral signature and null intensity for unrotated photon polarization.

Uranium ions in sesquinitride alpha-U2N3 occupy independent acentric and centrosymmetric sites according to conventional x-ray diffraction patterns [R. Tro\'c, J. Solid State Chem. 13, 14 (1975)]. We submit that polar uranium multipoles in acentric sites are revealed in resonant x-ray diffraction data recently published by Lawrence Bright et al. [Phys. Rev. B 100, 134426 (2019)]. To this end, their diffraction data gathered with a primary x-ray energy in the vicinity of the uranium M4 absorption edge are compared to symmetry-informed diffraction amplitudes calculated for the bixbyite alpha-Mn2O3 lattice structure. Bragg spots forbidden in this lattice diffraction pattern appear to provide clear-cut evidence for high-order polar uranium multipoles.

The Landau free-energy of a compound that benefits from a linear coupling of an electric field and a magnetic field includes a product of the two fields, one polar and time-even and one axial and time-odd. In ME compounds, expectation values of some atomic magnetic tensors are invariant with respect to anti-inversion. An invariance shared by the Dirac monopole (an element of charge allowed in Maxwell's equations that has not been observed) and a Zeldovich anapole, also known as a Dirac dipole. From the science of materials perspective, it has been established that Dirac multipoles contribute to the diffraction of x-rays and neutrons. We identify Dirac monopoles in bulk magnetic properties of iron tellurate (Fe2TeO6) and a spin ladder (SrFe2S2O). Both cited compounds present a simple antiferromagnetic configuration of axial dipoles, and their different magnetic crystal classes allow a linear ME effect. However, the Kerr effect is symmetry allowed in the spin ladder and forbidden in iron tellurate. Anapoles are forbidden in iron tellurate and allowed in the spin ladder compound, a difference evident in diffraction patterns fully informed by symmetry. More generally, we identify a raft of Dirac multipoles, and axial multipoles beyond dipoles, visible in future experiments using standard techniques with beams of neutrons or x-rays tuned in energy to an iron atomic resonance. ME invariance imposes a phase relationship between nuclear (charge) and magnetic contributions to neutron (x-ray) diffraction amplitudes. In consequence, intensities of Bragg spots in an x-ray pattern do not change when helicity in the primary beam is reversed. A like effect occurs in the magnetic diffraction of polarized neutrons.

The symmetry of long-range magnetic order in manganese telluride (alpha-MnTe) is unknown. Likewise, its standing as an altermagnet. To improve the situation, we present symmetry informed Bragg diffraction patterns based on a primary magnetic order parameter for antiferromagnetic alignment between Mn dipoles. It does not break translation symmetry in a centrosymmetric structure, in keeping with an accepted definition of altermagnetism. Our templates serve x-ray diffraction that benefits from signal enhancement using a Mn atomic resonance, and neutron scattering. Even rank multipoles in magnetic neutron diffraction reflect a core requirement of altermagnetism, because they are zero for strong spin-orbit coupling. Symmetry in the templates demands that nuclear and magnetic contributions possess the same phase, which enables standard neutron polarization analysis on Bragg spots with overlapping contributions. However, three of the four templates generate Bragg spots that do not appear in the lattice (nuclear) diffraction pattern, i.e., Bragg spots that are basis-forbidden and purely magnetic in origin. On the other hand, identical symmetry demands a 90 deg phase shift between magnetic (time-odd) and charge-like (time-even, Templeton-Templeton) contributions to x-ray scattering amplitudes. Consequently, circular polarization in the primary beam of x-rays is rotated. The difference in the intensities of a Bragg spot measured with right- and left-handed circular primary polarization defines a chiral signature. Further tests include predictions in three out four templates of zero intensity in a specified channel of x-ray polarization. Diffraction properties of a template are radically different from those of a parity-time (PT)-symmetric antiferromagnet, for its symmetry allows a linear ME effect and prohibits both a PM effect and a chiral signature.

The micaceous black allotrope of ruthenium trichloride is the subject of many recent experimental and theoretical studies. Even so, its structural and magnetic properties remain undecided; monoclinic, trigonal and rhombohedral space groups for the crystal structure have been proposed on the basis of various types of experiments. The magnetic structure is often discussed in the context of the Kitaev state, but inevitably they are inconclusive discussions in the absence of structural and magnetic space groups. Johnsonet alinfer a candidate for the magnetic structure (Cc2/m) from results gathered in an extensive set of experiments on an untwined sample ofα-RuCl3(Johnsonet al2015Phys. Rev.B92235119). The proposed zigzag antiferromagnetic ground state of Ru ions does not respond to bulk magnetic probes, with optical rotation and all forms of dichroism prohibited by symmetry. Experimental techniques exploited by Johnsonet alincluded x-ray and magnetic neutron diffraction. Properties of the candidate magnetic structure not previously explored include polar magnetism that supports Ru Dirac multipoles, e.g. a ruthenium anapole that is also known as a toroidal dipole. In a general case, Dirac dipoles are capable of generating interactions between magnetic ions, as in an electrical Dzyaloshinskii-Moryia interaction (Kaplan and Mahanti 2011Phys. Rev.B83174432; Zhaoet al2021Nat. Mater.20341). Notably, the existence of Dirac quadrupoles in the pseudo-gap phases of cuprate superconductors YBCO and Hg1201 account for observed magnetic Bragg diffraction patterns. Dirac multipoles contribute to the diffraction of both x-rays and neutrons, and a stringent test of the magnetic structure Cc2/m awaits future experiments. From symmetry-informed calculations we show that, the magnetic candidate permits Bragg spots that arise solely from Dirac multipoles. Stringent tests of Cc2/m can also be accomplished by performing resonant x-ray diffraction with signal enhancement from the chlorineK-edge. X-ray absorption spectra published forα-RuCl3possess a significant low-energy feature (Plumbet al2014Phys. Rev.B90041112(R)). Many experimental studies of other Cl-metal compounds concluded that identical features hallmark the chemical bond. Using a monoclinic Cc2/m structure, we predict the contribution to Bragg diffraction at the ClK-edge absorption. Specifically, the variation of intensity of Bragg spots with rotation of the sample about the reflection vector. The two principal topics of our studies, polar magnetism and the chemical bond in the black allotrope of ruthenium trichloride, are brought together in a minimal model of magnetic Ru ions in Cc2/m.

Polar magnetism is present when ions occupy sites that are not centres of inversion symmetry. Fortunately, such magnetization contributes to neutron scattering that is the bedrock of magnetic structure determinations. Experiments in which the scattered neutron polarization is analysed are not a novelty. Simulations of polarized neutron scattering amplitudes for room temperature haematite (α-Fe2O3) demonstrate the wealth of information on offer. Two magnetic motifs distinguished by the orientation of their bulk ferromagnetism are considered. Additionally, the symmetry-inspired simulations challenge a recent claim to have determined the absolute direction of the Dzyaloshinskii-Moriya (D-M) interaction.

An escalating challenge in condensed-matter research is the characterization of emergent order-parameter nanostructures such as ferroelectric and ferromagnetic skyrmions. Their small length scales coupled with complex, three-dimensional polarization or spin structures makes them demanding to trace out fully. Resonant elastic x-ray scattering (REXS) has emerged as a technique to study chirality in spin textures such as skyrmions and domain walls. It has, however, been used to a considerably lesser extent to study analogous features in ferroelectrics. Here, we present a framework for modeling REXS from an arbitrary arrangement of charge quadrupole moments, which can be applied to nanostructures in materials such as ferroelectrics. With this, we demonstrate how extended reciprocal space scans using REXS with circularly polarized x rays can probe the three-dimensional structure and chirality of polar skyrmions. Measurements, bolstered by quantitative scattering calculations, show that polar skyrmions of mixed chirality coexist, and that REXS allows valuation of relative fractions of right- and left-handed skyrmions. Our quantitative analysis of the structure and chirality of polar skyrmions highlights the capability of REXS for establishing complex topological structures toward future application exploits.