Nonlinear optics, and particularly second harmonic generation (SHG), is increasingly used in many modern disciplines from material characterization in physical sciences to bio-imaging in medicine and optical signal processing in information technology. We present a theoretical analysis providing a strong estimate for the energy-integrated SHG response. Requirements of symmetry in time and space are fully respected in the calculation, and estimates of natural and magnetic circular dichroic signals are superior to previous ones. Like symmetry requirements are traced in the amplitude for magnetic neutron scattering, which includes all axial and polar (Dirac) contributions. Our method of working, in terms of spherical multipoles and implementation of symmetry, could be of use in a variety of other probes of electronic properties.
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 publi…
Ordered magnetic charge created by Dirac multipoles that are magnetic and polar is examined. It has previously been revealed in the pseudo-gap phase of high-Tc materials by use of the Kerr effect and magnetic neutron Bragg diffraction. There are several forms of the polar operator for magnetic neutron scattering built from spin and electric dipole operators of unpaired electrons. 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 …
A recent polarized neutron diffraction experiment on the 5d2 rhenium double perovskite Ba2YReO6 held at a low temperature uncovered weak magnetic diffraction peaks. Data analysis inferred a significantly reduced Re dipole moment, and long-range order compatible with an antiferromagnet, non-collinear motif. To interpret the experimental findings, we present a model wavefunction for Re ions derived from the crystal field potential, Coulomb interaction, and spin-orbit coupling that fully respects the symmetry of the low-temperature ordered state. It is used to calculate in analytic form all multi…
We submit that the magnetic space-group Cac (#9.41) is consistent with the established magnetic structure of BaFe2Se3, with magnetic dipole moments in a motif that uses two ladders [Caron J M et al 2011 Phys. Rev. B 84 180409(R)]. The corresponding crystal class m1' allows axial and polar dipoles and forbids bulk ferromagnetism. The compound supports magneto-electric multipoles, including a magnetic charge (monopole) and an anapole (magnetic toroidal dipole) visible in the Bragg diffraction of x-rays and neutrons. Our comprehensive simulation of neutron Bragg diffraction by BaFe2Se3 exploits e…
Cerium electronic multipoles in trigonal symmetry viewed in magnetic neutron diffraction are investigated. Analytic expressions for all the magnetic multipoles, and radial integrals calculated with an established atomic code, are used to scrutinize a recent claim to have observed an octupole-ice configuration in a powder sample of pyrochlore Ce2Sn2O7. Three equally plausible suites of multipoles belonging to uncorrelated cerium ions give equally satisfactory accounts of the available diffraction pattern. Our multipoles are suitable for future experiments using Bragg diffraction from a compound…
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 duri…
Fundamental block and staggered orders of magnetic Fe multipoles in Na2FeSe2 are classified by their symmetry and magnetoelectric properties. Our structure model incorporates ferromagnetic or antiferromagnetic coupling between chains. The ferrate salt is valued in studies of highly correlated electrons as the only iron selenide known to possess chain-like structural units hosting ferrous cations. Axial and polar (Dirac) multipoles are compulsory in the electronic structure since Fe ions exhibit enantiomorphic symmetry in the parent K2ZnO2-type compound. Calculated Bragg diffraction patterns fo…
The currently accepted magnetic ground state of Sr2IrO4 (the -++- state) preserves inversion symmetry. This is at odds, though, with recent experiments that indicate a magnetoelectric ground state, leading to the speculation that orbital currents or more exotic magnetic multipoles might exist in this material. Here, we analyze various magnetic configurations and demonstrate that two of them, the magnetoelectric -+-+ state and the non-magnetoelectric ++++ state, can explain these recent second-harmonic generation (SHG) experiments, obviating the need to invoke orbital currents. The SHG-probed m…
Structural and magnetic transitions in a double perovskite hosting 5d1 Re ions are discussed on the basis of recently published high-resolution x-ray diffraction patterns [D. Hirai, et al., Phys. Rev. Res. 2, 022063(R) (2020)]. A reported structural transition below room temperature, from cubic to tetragonal symmetry, appears not to be driven by T2g-type quadrupoles, as suggested. A magnetic motif at lower temperature is shown to be composed of two order parameters, associated with propagation vectors k = (0, 0, 1) and k = (0, 0, 0). Findings from our studies, for structural and magnetic prope…
An ancient Hamiltonian of electrons with entangled spin and orbital degrees of freedom is re-examined as a model of magneto-electric multipoles. 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 magneto-electric multipoles 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 magneto-electric multipoles is achieved by addition to the origina…
The sesquioxide, Ce$_{2}$O$_{3}$, has been a material of intense interest in recent years due to reports of an anomalous giant magnetodielectric effect and emergent mixed crystal field-phonon (vibronic) excitations below a putative antiferromagnetic transition at T$_{N}$ = 6.2 K. The claim of long-range magnetic order in this material is based on heat capacity and temperature-dependent susceptibility measurements; however, multiple neutron diffraction studies have been unable to distinguish any magnetic Bragg peaks. In this article, we present the results of a comprehensive investigation of th…
Compulsory Dirac multipoles in the bilayer perovskite Ca3Ru2O7 are absent in published analyses of experimental data. In a first step at correcting knowledge of the magnetic structure, we have analysed existing Bragg diffraction patterns gathered on samples held well below the Néel temperature at which A-type antiferromagnetic order of axial dipoles spontaneously develops. Patterns were gathered with neutrons, and linearly polarized x-rays tuned in energy to a ruthenium atomic resonance. Neutron diffraction data contains solid evidence of Dirac dipoles (anapoles or toroidal moments). No such c…
A systematic description of multipole degrees of freedom is discussed on the basis of the Stevens' operator-equivalent technique. The generalized Stevens' multiplicative factors are derived for all of the electric and the magnetic multipoles relevant to f-electron systems. With extensive use of the Stevens' factors, we express the spatial dependences of the electric and the magnetic fields, and the electric and the magnetic charge densities of localized f electrons. The latter is utilized to draw wave functions including their magnetic profile in addition to their shape with the charge density…
Polarization-dependent photon spectroscopy (dichroism) of the second-harmonic generation (SHG) response is shown to reveal chiral and magnetic properties of a sample. Two dichroic signals are allowed with electric-dipole (E1) and electric-quadrupole (E2) scattering events, and both require circular polarization in the primary beam. Explicit expressions for electronic multipoles, which benefit from equivalent electronic operators, in NCD and MCD of the SHG response are derived using theoretical techniques from atomic physics.
Real-space manifestation of ferroic multipoles in altermagnetic MnF$_2$
European Spallation Source, ERIC（欧洲散裂源）；Laboratoire Léon Brillouin, CEA, CNRS（勒内·布朗利实验室）；Institut Laue-Langevin（朗之万研究所）；Laboratoire de Physique des Solides, CNRS, Université Paris-Saclay（巴黎综合理工学院固体物理实验室）；Quantum Innovation Centre (Q.InC), (A*STAR)（量子创新中心）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, University of Windsor（温莎大学物理系）
The Bragg diffraction of neutrons and x-rays are well-suited to the task of determining the distribution of magnetization in crystals. Applications of the two techniques proceed by contrasting observed intensities with intensities calculated with a specific model, and changing the model as need be to achieve satisfactory agreement. An all-in-all-out (AIAO) magnetic configuration of magnetic dipoles on a cubic face-centred lattice with networks of corner-sharing tetrahedra is often mentioned in the context of pyrochlore oxides, for example, but the corresponding neutron and x-ray diffraction pa…
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 Sm0.976Gd0.024Al2 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.
