A theory of the spin-crossover phenomena in the rare earth perovskite LaCoO3induced by temperature or a magnetic field

Published magnetic data for LaCoO3are successfully analyzed with coexisting5Dand low-spin (LS) cobalt states. Energy levels of the two states are derived in analytic forms. To this end, fictitious orbital angular momentumlof magnitude one defines the Γ5(5D) state. Our Hamiltonian includes the spin-orbit interaction, and a cubic crystal field embellished by a trigonal distortion9B20(lz2-2/3)-80B40(lz2-9/10). 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 the5Dstate 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 the5DHamiltonian. 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 the5Dand LS states. Calculated Bragg diffraction patterns gathered with x-rays tuned to the CoK-edge reveal potential to refine the current crystal structure and to shed light on the origin of the coexisting states.

Polar magnetism and chemical bond inα-RuCl3

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 (C_c2/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 C_c2/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 C_c2/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 C_c2/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 C_c2/m.

Ligand-metal bonding effects in resonance enhanced x-ray Bragg diffraction

Chlorine covalently bonded to an open shell metal is present in many materials with desirable or intriguing physical properties. Materials include highly luminescent nontoxic alternatives to lead halide perovskites for optoelectronic applications K₂CuCl₃and Rb₂CuCl₃, enantiomorphic CsCuCl₃that presents magneto-chiral dichroism at a low temperature, and cubic K₂RuCl₆that possesses a singlet ground state generated by antiparallel spin and orbital angular momenta. Structural chirality of CsCuCl₃has been confirmed by resonant x-ray Bragg diffraction. We explore likely benefits of the technique at the chlorine K-edge using a symmetry informed method of calculation applied to chlorine multipoles. Already, a low energy feature in corresponding x-ray absorption spectra of many compounds has been related to the chlorine-metal bond. Bragg diffraction from chlorine in cubic K₂RuCl₆is treated in detail. Diffraction patterns for rhombohedral compounds that present space-group forbidden Bragg spots are found to be relatively simple.

Direct Observation of Anapoles by Neutron Diffraction

The scope of magnetic neutron scattering has been expanded by the observation of electronic Dirac dipoles (anapoles) that are polar (parity odd) and magnetic (time odd). A zero-magnetization ferromagnet Sm_{0.976}Gd_{0.024}Al_{2} with a diamond-type structure presents Dirac multipoles at basis-forbidden reflections that include the standard (2, 2, 2) reflection. Magnetic amplitudes measured at four such reflections are in full accord with a structure factor calculated from the appropriate magnetic space group.

Electronic and magnetic properties of multiferroic ScFeO3 available from diffraction experiments

Electronic and magnetic properties of ferric ions (3d ⁵) in multiferroic ScFeO₃ are puzzling, in part because they are different from the only other multiferroic known to possess the same polar chemical structure, BiFeO₃. Open questions about ScFeO₃ can be addressed by confronting observations with results for G-type antiferromagnetism allowed by the lithium niobate (LiNbO₃)-like parent R3c structure. Calculated structure factors for resonant x-ray diffraction include all charge-like quadrupoles allowed by symmetry, and if experimental results for ScFeO₃ subsequently imply they are different from zero then ferric ions cannot be in the high-spin ⁶S state. The same type of experiment can reveal the moment direction in the G-type antiferromagnetism, according to our calculations, and thereby contribute to understanding magnetic anisotropy. Furthermore, structure factors for magnetic neutron diffraction by ScFeO₃ include Dirac multipoles that are time-odd and parity-odd, e.g. dipoles that are often called anapoles or toroidal moments. Apart from Dirac multipoles, the conventional approach to the interpretation of neutron Bragg diffraction data will be inadequate if ferric ions (Fe³⁺) are not in the high-spin ⁶S state, because the scattering amplitude includes more than simple dipole moments in the general case.

Neutron scattering by Dirac multipoles

Scattering by magnetic charge formed by Dirac multipoles that are magnetic and polar is examined in the context of materials with properties that challenge conventional concepts. An order parameter composed of Dirac quadrupoles has been revealed in the pseudo-gap phase of ceramic, high-T _c superconductors on the basis of Kerr effect and magnetic neutron Bragg diffraction measurements. Construction of Dirac quadrupoles that emerge from centrosymmetric sites used by Cu ions in the ceramic superconductor Hg1201 is illustrated, together with selection rules for excitations that will feature in neutron inelastic scattering, and RIXS experiments. We report magnetic scattering amplitudes for diffraction by polar multipoles that have universal value, because they are not specific to ceramic superconductors. To illustrate this attribute, we consider neutron Bragg diffraction from a magnetically ordered iridate (Sr₂IrO₄) and discuss shortcomings in published interpretations of diffraction data.

Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa2CuO(4+δ))

A symmetry-based interpretation of published experimental results demonstrates that the pseudo-gap phase of underdoped HgBa2CuO(4+δ) (Hg1201) possesses an ordered state of magnetic charge epitomized by Cu magnetic monopoles. Magnetic properties of one-layer Hg1201 and two-layer YBa2Cu3O(6+x) (YBCO) cuprates have much in common, because their pseudo-gap phases possess the same magnetic space-group, e.g. both underdoped cuprates allow the magneto-electric (Kerr) effect. Differences in their properties stem from different Cu site symmetries, leaving Cu magnetic monopoles forbidden in YBCO. Resonant x-ray Bragg diffraction experiments can complement the wealth of information available from neutron diffraction experiments on five Hg1201 samples on which our findings are based. In the case of Hg1201 emergence of the pseudo-gap phase, with time-reversal violation, is accompanied by a reduction of Cu site symmetry that includes loss of a centre of inversion symmetry. In consequence, parity-odd x-ray absorption events herald the onset of the enigmatic phase, and we predict dependence of corresponding Bragg spots on magneto-electric multipoles, including the monopole, and the azimuthal angle (crystal rotation about the Bragg wavevector).

Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor

There is general agreement within the community of researchers that investigate high-Tc materials that it is most important to understand the pseudo-gap phase. To this end, many experiments on various cuprates have been reported. Two prominent investigations-Kerr effect and neutron Bragg diffraction-imply that underdoped YBCO samples possess long-range magnetic order of an unusual kind. However, other measurements do not support the existence of magnetic order. Here we show that the Kerr effect and magnetic Bragg diffraction data are individual manifestations of ordered magneto-electric quadrupoles at Cu sites. While the use of magneto-electric multipoles is new in studies of the electronic properties of cuprates, they are not unknown in other materials, including an investigation with x-rays of the parent compound CuO. We exploit the recent prediction that neutrons are deflected by magneto-electric multipoles. The outcome of our study is a theory for the order-parameter of the pseudo-gap phase without the aforementioned conflict with other measurements, and the first experimental evidence that neutrons interact with multipoles belonging to a state of magnetic charge.

A model of magneto-electric multipoles

A long-known Hamiltonian of electrons with entangled spin and orbital degrees of freedom is re-examined as a model of magneto-electric multipoles (MEs). In the model, a magnetic charge and simple quantum rotator are tightly locked in action, some might say they are enslaved entities. It is shown that MEs almost perfectly accord with those inferred from an analysis of magnetic neutron diffraction data on a ceramic superconductor (YBCO) in the pseudo-gap phase. Nigh on perfection between Stone's model and inferred MEs is achieved by addition to the original model of a crystal-field potential appropriate for the magnetic space group used in the published data analysis. An impression of thermal properties of multipoles is sought from a molecular-field model.

Magneto-electric operators in neutron scattering from electrons

We succeed in deriving an exact expression for the magnetic interaction of neutrons and electrons including magneto-electric operators, allowed in the absence of a centre of inversion symmetry. Central characters are a spin anapole and an orbital (toroidal) analogue, in addition to familiar parity-even operators like the magnetic moment. A simulation of neutron diffraction by antiferromagnetic copper oxide makes full use of information inferred from a thorough investigation with resonant x-ray Bragg diffraction.

Strange magnetic multipoles and neutron diffraction by an iridate perovskite (Sr2IrO4)

A theoretical investigation of a plausible construct for electronic structure in iridate perovskites demonstrates the existence of magnetic multipoles hitherto not identified. The strange multipoles, which are parity-even, time-odd and even rank tensors, are absent from the so-called j(eff) = 1/2 model. We prove that the strange multipoles contribute to magnetic neutron diffraction, and we estimate their contribution to intensities of Bragg spots for Sr(2)IrO(4). The construct encompasses the j(eff) = 1/2 model, and it is consistent with the known magnetic structure, ordered magnetic moment, and published resonant x-ray Bragg diffraction data. Over and above time-odd quadrupoles and hexadecapoles, whose contribution changes neutron Bragg intensities by an order of magnitude, according to our estimates, are relatively small triakontadipoles recently proposed as the primary magnetic order-parameter of Sr(2)IrO(4).

Effects of dispersion and absorption in resonant Bragg diffraction of x-rays

Resonant diffraction of x-rays by crystals with anisotropic optical properties is investigated theoretically, to assess how the intensity of a Bragg spot is influenced by effects related to dispersion (birefringence) and absorption (dichroism). Starting from an exact but opaque expression, simple analytic results are found to expose how intensity depends on dispersion and absorption in the primary and secondary beams and, also, the azimuthal angle (rotation of the crystal about the Bragg wavevector). If not the full story for a given application, our results are more than adequate to explore consequences of dispersion and absorption in the intensity of a Bragg spot. Results are evaluated for antiferromagnetic copper oxide, and low quartz. For CuO, one of our results reproduces all salient features of a previously published simulation of the azimuthal-angle dependence of a magnetic Bragg peak. It is transparent in our analytic result that dispersion and absorption effects alone cannot reproduce published experimental data. Available data for the azimuthal-angle dependence of space-group forbidden reflections (0,0, l), with l ≠ 3n, of low quartz depart from symmetry imposed by the triad axis of rotation symmetry. The observed asymmetry can be induced by dispersion and absorption even though absorption coefficients are constant, independent of the azimuthal angle, in this class of reflections.

Symmetry-protected hidden order and magnetic neutron Bragg diffraction by URu2Si2

We investigate how the order parameter of a continuous phase transition can be protected from view by symmetry in a magnetic crystal. The symmetry in question forbids atomic displacements and formation of magnetic dipoles, rendering the order parameter invisible in standard x-ray and magnetic neutron Bragg diffraction. Analysis of the allowed magnetic space-groups reveals exact properties of the hidden order parameter. We demonstrate that Bragg spots forbidden by the chemical structure can unveil magnetic hidden order. The method is applied to URu2Si2, which has been thoroughly investigated in the past few decades using all manner of experimental techniques. Starting from the established chemical structure of URu2Si2, we have performed a critical analysis of available data for magnetic neutron Bragg diffraction.

Melting of chiral order in terbium manganate (TbMnO3) observed with resonant x-ray Bragg diffraction

Resonant Bragg diffraction of soft, circularly polarized x-rays has been used to observe directly the temperature dependence of chiral-order melting in a motif of Mn ions in terbium manganate. The underlying mechanism uses the b-axis component of a cycloid, which vanishes outside the polar phase. Melting is witnessed by the first and second harmonics of a cycloid, and we explain why the observed temperature dependence differs in the two harmonics. Conclusions follow from an exact treatment of diffraction by using atomic multipoles in a circular cycloid, since a standard treatment of the diffraction, based on a single material-vector identified with the magnetic dipole, does not reproduce correctly observations at the second harmonic.

Magnetic symmetries in neutron and resonant x-ray Bragg diffraction patterns of four iridium oxides

The magnetic properties of Sr(2)IrO(4), Na(2)IrO(3), Sr(3)Ir(2)O(7) and CaIrO(3) are discussed, principally in the light of experimental data in recent literature for Bragg intensities measured in x-ray diffraction with enhancement at iridium L-absorption edges. The electronic structure factors we report, which incorporate parity-even and acentric entities, serve the immediate purpose of making full use of crystal and magnetic symmetry to refine our knowledge of the magnetic properties of the four iridates from resonant x-ray diffraction data. They also offer a platform on which to interpret future investigations, using dichroic signals, resonant x-ray diffraction and neutron diffraction, for example, as well as ab initio calculations of electronic structure. Unit-cell structure factors, suitable for x-ray Bragg diffraction enhanced by an electric dipole-electric dipole (E1-E1) event, reveal exactly which iridium multipoles are visible, e.g., a magnetic dipole parallel to the crystal c-axis (z-axis) and an electric quadrupole with yz-like symmetry in the specific case of CaIrO(3). Magnetic space-groups are assigned to Sr(2)IrO(4), Sr(3)Ir(2)O(7) and CaIrO(3), namely, P(I)cca, P(A)ban and Cm'cm', respectively, in the Belov-Neronova-Smirnova notation. The assignment for Sr(2)IrO(4) is possible because of our new high-resolution neutron diffraction data, gathered on a powder sample. In addition, the new data are used to show that the ordered magnetic moment of an Ir(4+) ion in Sr(2)IrO(4) does not exceed 0.29(4) μ(B). Na(2)IrO(3) has two candidate magnetic space-groups that are not resolved with currently available resonant x-ray data.

Magnetic correlations in a layered iridate, Na2IrO3

Analysis of published data gathered on a sample of Na(2)IrO(3), held deep inside the antiferromagnetic phase at 1.58 K, shows that iridium magnetic dipole moments, measured in resonant x-ray Bragg diffraction, lie in the a-c plane of the monoclinic crystal and enclose an angle ≈118° with the c-axis. These findings, together with bulk measurements, are united in a plausible magnetic ground state for an iridium ion constructed from a Kramers doublet. A magnetic space group, derived from the chemical space group C2/m (unique axis b), possesses an anti-translation, to accommodate antiferromagnetic order, and an odd, two-fold axis of rotation symmetry on the b-axis, C'(2b), placing Ir magnetic dipoles perpendicular to the b-axis. Anapoles (toroidal dipoles) are predicted to be likewise confined to the a-c plane, and magnetic charges forbidden.

Neptunium multipoles and resonant x-ray Bragg diffraction by neptunium dioxide (NpO2)

The low-temperature ordered state of neptunium dioxide (NpO(2)) remains enigmatic. After decades of experimental and theoretical efforts, long-range order of a time-odd (magnetic) high-order atomic multipole moment is now generally considered to be the fundamental order parameter, the most likely candidate being a magnetic triakontadipole (rank 5). To date, however, direct experimental observation of the primary order parameter remains outstanding. In the light of new experimental findings, we re-examine the effect of crystal symmetry on the atomic multipoles and the resulting x-ray resonant scattering signature. Our simulations use the crystallographic point group ̅3m (D(3d)), because corresponding magnetic groups ̅3m', ̅3'm', and ̅3'm are shown by us to be at odds with a wealth of experimental results. In addition to the previously observed (secondary) quadrupole order, we derive expressions for higher-order multipoles that might be observed in future experiments. In particular, magnetic octupole moments are predicted to contribute to Np M(2,3) and L(2,3) resonant scattering via E2–E2 events. The Lorentzian-squared lineshape observed at the M(4) resonance is shown to be the result of the anisotropy of the 3p(3/2) core levels. Quantitative comparison of our calculations to the measured data yields a core–hole width Γ = 2.60(7) eV and a core-state exchange energy [absolute value]ε(1/2)[absolute value] = 0.76(2) eV.

Acentric magnetic and optical properties of chalcopyrite (CuFeS2)

The absence of spatial inversion symmetry at both local (point group 4) and global (crystal class (4)2m) levels greatly influences the electronic properties of chalcopyrite (CuFeS(2)). The predicted dichroic signals (natural circular, non-reciprocal and magneto-chiral) and resonant, parity-odd Bragg diffraction patterns at space-group forbidden reflections portray the uncommon, acentric symmetry. Despite extensive experimental investigations over several decades, by mineralogists, chemists and physicists, there is no consensus view about the electrical and magnetic properties of chalcopyrite. New spectroscopic and diffraction data, gathered at various temperatures in the vicinity of the copper and iron L(2,3) edges, provide necessary confidence in the magnetic motif used in our analytic simulations of x-ray scattering. With the sample held at 10 and 65 K, our data establish beyond reasonable doubt that there is no valence transition, and ordering of the copper moments as the origin of the low-temperature phase (T(c) ≈ 53 K) is ruled out.

Triakontadipole and high-order dysprosium multipoles in the antiferromagnetic phase of DyB2C2

Resonant soft x-ray Bragg diffraction at the Dy M(4,5) edges has been used to study Dy multipoles in the combined magnetic and orbitally ordered phase of DyB(2)C(2). The analysis incorporates both the intra-atomic magnetic and quadrupolar interactions between the 3d core and 4f valence shells. Additionally, we introduce to the formalism the interference of magnetic and nonmagnetic oscillators. This allows a determination of the higher-order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The strength of the Dy 4f multipole moments have been estimated as being up to 80% of the quadrupolar moment.

Determination of the absolute chirality of tellurium using resonant diffraction with circularly polarized x-rays

Many proteins, sugars and pharmaceuticals crystallize into two forms that are mirror images of each other (enantiomers) like our right and left hands. Tellurium is one enantiomer having a space group pair, P3(1)21 (right-handed screw) and P3(2)21 (left-handed screw). X-ray diffraction with dispersion correction terms has been playing an important role in determining the handedness of enantiomers for a long time. However, this approach is not applicable for an elemental crystal such as tellurium or selenium. We have demonstrated that positive and negative circularly polarized x-rays at the resonant energy of tellurium can be used to absolutely distinguish right from left tellurium. This method is applicable to chiral motifs that occur in biomolecules, liquid crystals, ferroelectrics and antiferroelectrics, multiferroics, etc.

Orbital order at Mn and O sites and absence of Zener polaron formation in manganites

We report the doping dependence of the ground state of A-site ordered manganites below and above half doping. Energy and polarization dependence of the orbital reflection, taken by resonant soft-x-ray powder diffraction, at both Mn L(2,3) and O K edges, provides direct evidence for orbital order at Mn(3+) and oxygen sites and absence of Zener polaron formation. For x > or = 0.2 anomalous melting of the orbital order is observed, which is coupled neither to magnetic ordering nor to a structural transition, indicating a two-dimensional character of the interactions.

Orbital dynamics of the 4f shell in DyB2C2

For the first time, fluctuations in the Dy quadrupole (orbital) moment have been observed in DyB(2)C(2) using inelastic neutron scattering. The observed quasielastic response is decomposed into two components, one reflecting transitions within the doublets (narrow) and the other transitions between the doublets (broad) of the effective Dy quartet ground state. The widths of the narrow and broad components are shown to arise from fluctuations in the magnetic dipole and the electric quadrupole moments, respectively.

X-ray birefringence and dichroism obtained from magnetic materials

In the past decade, synchrotron radiation has triggered a surge in studies of the polarization dependence of X-ray beams passing through non-isotropic materials. A vast range of experimental results concerning polarization-dependent absorption (dichroism) and dispersion (birefringence, for example) are available from materials which are either magnetic or exhibit preferred directions due to the local atomic environment. This article aims to bring together the diversity of modern experiments in this field with established methods of optical calculus, in a way that highlights the simplicity of the underlying physics. A useful framework is formed when observable quantities, in the X-ray case, are related to atomic variables of the sample material. Atomic descriptions of absorption spectra with various levels of complexity are considered, and some well documented sum-rules are encountered. The framework is the most general allowed within the electric dipole approximation. By way of illustration, dichroic X-ray absorption by two materials with highly anisotropic properties and magnetic ions with different valence shells are considered; namely, a 3d-transition ion in ferrous niobate, and a lanthanide ion in dysprosium borocarbide. Both materials display interesting magnetic properties that are challenging to interpret at an atomic level of detail, and it is shown how absorption experiments can contribute to resolving some issues.