Theory of neutron scattering by electrons in magnetic materials

A theory of neutron scattering by magnetic materials is reviewed with emphasis on the use of electronic multipoles that have universal appeal, because they are amenable to calculation and appear in theories of many other experimental techniques. The conventional theory of magnetic neutron scattering, which dates back to Schwinger (1937) and Trammell (1953), yields an approximation for the scattering amplitude in terms of magnetic dipoles formed with the spin (S) and orbital angular momentum (L) of valence electrons. The so-called dipole-approximation has been widely adopted by researchers during the past few decades that has seen neutron scattering develop to its present status as the method of choice for investigations of magnetic structure and excitations. Looking beyond the dipole-approximation, however, reveals a wealth of additional information about electronic degrees of freedom conveniently encapsulated in magnetic multipoles. In this language, the dipole-approximation retains electronic axial dipoles, S and L. At the same level of approximation are polar dipoles - called anapoles or toroidal dipoles - allowed in the absence of a centre of inversion symmetry. Anapoles are examples of magneto-electric multipoles, time-odd and parity-odd irreducible tensors, that have come to the fore as signatures of electronic complexity in materials

Orthoferrite with a hidden lanthanide magnetic motif: NdFeO3

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

A case for polar uranium octupoles in cubic U2N3

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

Antiferro--quadrupolar structures in UPd3_3 inferred from x-ray resonant Bragg diffraction

A systematic analysis of resonant x-ray Bragg diffraction data for UPd$_3$, with signal enhancement at the U M$_{IV}$ edge, including possible structural phase-transitions leads to a new determination of the space groups of the material in the phases between $T_0 = 7.8$ K and $T_{+1}$ = 6.9 K as P222$_1$ and between $T_{-1}$ = 6.7 K and $T_{2}$ = 4.4 K as space group P2$_1$. In addition, the quadrupolar order-parameters, , inferred from diffraction data in the phase between $T_{-1}$ and $T_{2}$, are and $$at the (103) Bragg reflection, and$$ at the (104) reflection.Comment: 13 pages, 4 figure

Chiral structures of electric polarization vectors quantified by X-ray resonant scattering

Resonant elastic X-ray scattering (REXS) offers a unique tool to investigate solid-state systems providing spatial knowledge from diffraction combined with electronic information through the enhanced absorption process, allowing the probing of magnetic, charge, spin, and orbital degrees of spatial order together with electronic structure. A new promising application of REXS is to elucidate the chiral structure of electrical polarization emergent in a ferroelectric oxide superlattice in which the polarization vectors in the REXS amplitude are implicitly described through an anisotropic tensor corresponding to the quadrupole moment. Here, we present a detailed theoretical framework and analysis to quantitatively analyze the experimental results of Ti L-edge REXS of a polar vortex array formed in a PbTiO3/SrTiO3 superlattice. Based on this theoretical framework, REXS for polar chiral structures can become a useful tool similar to x-ray resonant magnetic scattering (XRMS), enabling a comprehensive study of both electric and magnetic REXS on the chiral structures.K.T.K., S.Y.P., and D.R.L acknowledge financial support by National Research Foundation of Korea (Grant No. NRF-2020R1A2C1009597, NRF-2019K1A3A7A09033387, and NRF-2021R1C1C1009494). M.M. and R.R. were supported by the Quantum Materials program from the Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231). V.A.S., J.W.F., and L.W.M. acknowledge the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC-0012375 for support to study complex-oxide heterostructure with X-ray scattering. L.W.M. and R.R. acknowledge partial support from the Army Research Office under the ETHOS MURI via cooperative agreement W911NF-21-2-0162. J.Í. acknowledges financial support from the Luxembourg National Research Fund through project FNR/C18/MS/12705883/REFOX. M.A.P.G. was supported by the Czech Science Foundation (project no. 19-28594X). Diamond Light Source, UK, is acknowledged for beamtime on beamline I10 under proposal NT24797. Use of the Advanced Light Source, Lawrence Berkeley National Laboratory, was supported by the U.S. Department of Energy (DOE) under contract no. DE-AC02-05CH11231, and use of the Advanced Photon Source was supported by DOE’s Office of Science under contract DE-AC02-06CH11357

Structural chirality of polar skyrmions probed by resonant elastic x-ray scattering

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.M. R. M. and R. R. were supported by the Quantum Materials program from the Office of Basic Energy Sciences, U.S. Department of Energy (DE-AC02-05CH11231). V. A. S., J. W. F., and L. W. M. acknowledge the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC-0012375 for support to study complex-oxide heterostructure with x-ray scattering. L. W. M. and R. R. acknowledge partial support from the Army Research Office under the ETHOS MURI via cooperative agreement W911NF-21-2-0162. J. Í. acknowledge financial support from the Luxembourg National Research Fund through project FNR/C18/MS/12705883/REFOX. Diamond Light Source, UK, is acknowledged for beam time on beam line I10 under proposal NT24797. K. T. K., S. Y. P., and D. R. L. acknowledge support from the National Research Foundation of Korea, under Grant No. NRF-2020R1A2C1009597, NRF-2019K1A3A7A09033387, and NRF-2021R1C1C1009494. M. A. P. G. acknowledges support by the Czech Science Foundation (Project No. 19-28594X). This research used resources of the Advanced Light Source, a U.S. DOE Office of Science User Facility under Contract No. DE-AC02-05CH11231. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility at Argonne National Laboratory and is based on research supported by the U.S. DOE Office of Science-Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. S. D. gratefully acknowledges a start-up grant from Indian Institute of Science, Bangalore, India. F. G.-O., P. G.-F., and J. J. acknowledge financial support from Grant No. PGC2018-096955-B-C41 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe,” by the European Union. F. G.-O. acknowledges financial support from Grant No. FPU18/04661 funded by MCIN/AEI/10.13039/50110001103