Resonant X-ray Bragg Diffraction Research Articles

Magnetic symmetries of terbium tetraboride (TbB4) revealed by resonant x-ray Bragg diffraction

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

Magnetic properties of Sohncke-type Pb(TiO) Cu4(PO4)4 exposed by resonant x-ray Bragg diffraction

The enantiomorphous (chiral) crystal class of Sohncke-type Pb(TiO)Cu4(PO4)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)Cu4(PO4)4 are fully incorporated in calculated resonant x-ray Bragg diffraction patterns that are successfully compared with existing limited experimental data [Misawa , ]. 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 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)Cu4(PO4)4 [Kimura , ]. Published by the American Physical Society 2024

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 K2CuCl3 and Rb2CuCl3, enantiomorphic CsCuCl3 that presents magneto-chiral dichroism at a low temperature, and cubic K2RuCl6 that possesses a singlet ground state generated by antiparallel spin and orbital angular momenta. Structural chirality of CsCuCl3 has 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 K2RuCl6 is treated in detail. Diffraction patterns for rhombohedral compounds that present space-group forbidden Bragg spots are found to be relatively simple.

Neutron diffraction and the electronic properties of BaFe2Se3

It is argued on the basis of previously published experimental data that, the magnetic space-group Cac (#9.41) is the correct description of magnetically ordered BaFe2Se3. The corresponding crystal class m1′ allows axial and polar dipoles and forbids bulk ferromagnetism. Magneto–electric multipoles that are both time-odd and parity-odd are allowed, e.g., a magnetic charge (monopole) and an anapole (magnetic toroidal dipole). The experimental observation of magneto–electric multipoles must shed light on valence electrons involved in bonding, including charge transfer using 3d(Fe) and p-states of ligand ions. We provide the appropriate structure factors for the Bragg diffraction neutrons, together with estimates of atomic form factors. Structure factors for resonant x-ray Bragg diffraction are also considered, because the analysis of successful experiments will yield complementary information about electronic properties. Magneto–electric multipoles, over and above those that contribute to magnetic neutron diffraction, include the magnetic monopole. A time-odd, parity-even monopole created from the magnetic dipole and an electric toroidal dipole, which is a manifestation of a structural rotation, is allowed in BaFe2Se3 but it is not visible in diffraction, nor is the corresponding dipole.

Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa2CuO4+δ)

A symmetry-based interpretation of published experimental results demonstrates that the pseudo-gap phase of underdoped HgBa2CuO4+δ (Hg1201) possesses an ordered state of magnetic charge epitomized by Cu magnetic monopoles. Magnetic properties of one-layer Hg1201 and two-layer YBa2Cu3O6+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).

Symmetry of reentrant tetragonal phase inBa1−xNaxFe2As2: Magnetic versus orbital ordering mechanism

Magneto-structural phase transitions in Ba1-xAxFe2As2 (A = K, Na) materials are discussed for both magnetically and orbitally driven mechanisms, using a symmetry analysis formulated within the Landau theory of phase transitions. Both mechanisms predict identical orthorhombic space-group symmetries for the nematic and magnetic phases observed over much of the phase diagram, but they predict different tetragonal space-group symmetries for the newly discovered re-entrant tetragonal phase in Ba1-xNaxFe2As2 (x ~ 0.24-0.28). In a magnetic scenario, magnetic order with moments along the c-axis, as found experimentally, does not allow any type of orbital order, but in an orbital scenario, we have determined two possible orbital patterns, specified by P4/mnc1' and I4221' space groups, which do not require atomic displacements relative to the parent I4/mmm1' symmetry and, in consequence, are indistinguishable in conventional diffraction experiments. We demonstrate that the three possible space groups are however, distinct in resonant X-ray Bragg diffraction patterns created by Templeton & Templeton scattering. This provides an experimental method of distinguishing between magnetic and orbital models.

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 jeff = 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 Sr2IrO4. The construct encompasses the jeff = 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 Sr2IrO4.

A new chiral phase of BiFeO3 evidenced from resonant x-ray Bragg diffraction

Results that support a new chiral phase in the only multiferroic material known above room temperature have been obtained by a resonant x-ray Bragg diffraction experiment, with the incoming x-ray beam tuned near the Fe K-edge (7.1135 keV), performed in its ferroelectric phase. The R3c forbidden reflection (0,0,9)H was studied as a function of the rotation of the crystal about the Bragg wavector in both phases, paramagnetic (700 K) and antiferromagnetic (300 K). The data gathered is consistent with a chiral structure formed by a circular cycloid propagating along (1,1,0)H. Templeton and Templeton (T&T) scattering at 700 K is attributed in part to charge-like quadrupoles absent in a standard model of a cycloid in which a material vector generates all electronic states of the resonant ion. Extensive sets of azimuthal-angle data are used to infer values of three atomic multipoles in a satisfactory minimal model of the iron electronic structure, with a quadrupole (E1-E1 event) and a hexadecapole (E2-E2 event) contributing T&T scattering, plus a magnetic dipole (E1-E1).

Chiral Properties of BiFeO3 Inferred from Resonant X-ray Bragg Diffraction

A new chiral phase of ferric ions in bismuth ferrite, the only material known to support multiferroic behaviour at room temperature, is inferred from extensive sets of data gathered by resonant x-ray Bragg diffraction. Values of all ferric multipoles participating in a minimal model of Fe electronic structure are deduced from azimuthal-angle scans. Extensive sets of azimuthal-angle data, gathered by resonant x-ray Bragg diffraction, yield values of all ferric multipoles participating in a minimal model of Fe electronic structure. Paramagnetic (700 K) and magnetically ordered (300 K) phases of a single crystal of BiFeO3 have been studied with x-rays tuned near to the iron K-edge (7.1135 keV). At both temperatures, intensities at a Bragg spot forbidden in the nominal space-group, R3c, are consistent with a chiral motif of ferric ions in a circular cycloid propagating along (1, 1, 0)H. Templeton and Templeton scattering at 700 K is attributed in part to charge-like quadrupoles in a cycloid. The contribution is not present in a standard, simplified model of electronic states of the resonant ion with trivial cylindrical symmetry.

Chiral properties of hematiteα-Fe2O3inferred from resonant Bragg diffraction using circularly polarized x rays

Chiral properties of the two phases - collinear motif (below Morin transition temperature, TM=250 K) and canted motif (above TM) - of magnetically ordered hematite ({\alpha}-Fe2O3) have been identified in single crystal resonant x-ray Bragg diffraction, using circular polarized incident x-rays tuned near the iron K-edge. Magneto-electric multipoles, including an anapole, fully characterize the high-temperature canted phase, whereas the low-temperature collinear phase supports both parity-odd and parity-even multipoles that are time-odd. Orbital angular momentum accompanies the collinear motif, while it is conspicuously absent with the canted motif. Intensities have been successfully confronted with analytic expressions derived from an atomic model fully compliant with chemical and magnetic structures. Values of Fe atomic multipoles previously derived from independent experimental data, are shown to be completely trustworthy.

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 Sr2IrO4, Na2IrO3, Sr3Ir2O7 and CaIrO3 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 CaIrO3. Magnetic space-groups are assigned to Sr2IrO4, Sr3Ir2O7 and CaIrO3, namely, PIcca, PAban and Cm′cm′, respectively, in the Belov–Neronova–Smirnova notation. The assignment for Sr2IrO4 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 Ir4+ ion in Sr2IrO4 does not exceed 0.29(4) μB. Na2IrO3 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 Na2IrO3, 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, , 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 (NpO2) 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 (D3d), because corresponding magnetic groups , and 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 M2,3 and L2,3 resonant scattering via E2–E2 events. The Lorentzian-squared lineshape observed at the M4 resonance is shown to be the result of the anisotropy of the 3p3/2 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 eV.

Ferromagnetic-type order of atomic multipoles in the polar ferrimagnetic GaFeO3

Resonant x-ray Bragg diffraction by gallium ferrate (GaFeO${}_{3}$) at Friedel pairs of charge allowed reflections in the vicinity of the Fe $K$ edge are presented. By use of the magnetization difference, the interference of charge diffraction with the magnetic diffraction is extracted. The study of Friedel pairs and concentrating at the pre-edge allows us to extract the effect of inversion symmetry breaking on the magnetic Fe 3$d$ shell. The data are analyzed using a model based on atomic multipole moments which are magnetic and have no space inversion symmetry (magnetoelectric) interfering with the charge scattering. This model successfully describes data as a function of azimuthal angle and for different incident polarization and it shows that data can be directly related to the magnetoelectric dipole (toroidal) and quadrupole moments. Though the model describes most of the observations successfully, how to correctly describe the observed magnetic nonresonant intensity below the edge remains an open question.

High-order Ho multipoles in HoB2C2 observed with soft resonant x-ray diffraction

Soft resonant x-ray Bragg diffraction (SRXD) at the Ho M4,5 edges has been used to study Ho 4f multipoles in the combined magnetic and orbitally ordered phase of HoB2C2. A full description of the energy dependence for both σ and π incident x-rays at two different azimuthal angles, as well as the ratio Iσ/Iπ as a function of azimuthal angle for a selection of energies, allows a determination of the higher order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The Ho 4f multipole moments have been estimated, indicating a dominant hexadecapole (rank 4) order with an almost negligible influence from either the dipole or the octupole magnetic terms. The analysis incorporates both the intra-atomic magnetic and quadrupolar interactions between the 3d core and 4f valence shells as well as the interference of contributions to the scattering that behave differently under time reversal. Comparison of SRXD, neutron diffraction and non-resonant x-ray diffraction shows that the magnetic and quadrupolar order parameters are distinct. The component of the magnetic order exhibits a Brillouin type increase below the orbital ordering temperature TQ, while the quadrupolar order increases more sharply. We conclude that the quadrupolar interaction is strong, but quadrupolar order only occurs when the magnetic order gives rise to a quasi-doublet ground state, which results in a lock-in of the orbitals at TQ.

Triakontadipole and high-order dysprosium multipoles in the antiferromagnetic phase ofDyB2C2

Resonant soft x-ray Bragg diffraction at the DyM4, 5 edges has been used to study Dy multipoles in the combined magnetic and orbitally ordered phaseof DyB2C2. The analysis incorporates both the intra-atomic magnetic and quadrupolar interactionsbetween the 3d core and 4f valence shells. Additionally, we introduce to the formalism theinterference of magnetic and nonmagnetic oscillators. This allows a determination of thehigher-order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The strengthof the Dy 4f multipole moments have been estimated as being up to 80% of thequadrupolar moment.

Magnetoelectric effects studied by resonant x-ray diffraction in ferrimagneticGaFeO3

Resonant x-ray Bragg diffraction experiments on space group forbidden reflections of type $(0k0)$ with $k$ odd taken in the vicinity of the $\text{Fe}\text{ }K$ edge are used to study magnetoelectric effects in ${\text{GaFeO}}_{3}$. The main intensity of the various spectral features is caused by the E1-E1 transition and can be ascribed to the observation of an electric quadrupole moment, giving different intensities within a Friedel pair. Collecting the magnetic field difference intensity by having the magnetization parallel or antiparallel to the $c$ axis in the ferrimagnetic phase is discussed in the framework of magnetoelectric multipole moments, which are both parity and time odd. The azimuthal angle dependence and the sign change in the magnetic signal when going from $(0k0)$ to $(0\overline{k}0)$ reflections can be understood by the interference of the electric quadrupole with that of magnetoelectric quadrupole and octupole moments.

Magnetic order of multiferroicErMn2O5studied by resonant soft x-ray Bragg diffraction

Magnetic order of multiferroic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mtext>ErMn</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mtext>O</mml:mtext><mml:mn>5</mml:mn></mml:msub></mml:mrow></mml:math>studied by resonant soft x-ray Bragg diffraction