Magnetic, electric and toroidal polarization modes describing the physical properties of crystals. NdFeO3 case.

The structure and the physical phenomena that occur in a crystal can be described by using a suitable set of symmetry-adapted modes. The classification of magnetic modes in crystals presented in Fabrykiewicz et al. [Acta Cryst. (2021), A77, 327-338] is extended to a classification of electric and toroidal (anapole) modes in crystals. These three classifications are based on magnetic point groups, which are used in two contexts: (i) the magnetic point group of the magnetic crystal class and (ii) the magnetic site-symmetry point group of the Wyckoff position of interest. The classifications for magnetic, electric and toroidal modes are based on the properties of the three generalized inversions: space inversion 1, time inversion 1' and the space-and-time inversion 1'. It is emphasized that none of these three inversions is more important than the other two. A new notation for symmetry operation symbols and magnetic point group symbols is proposed; each operation is presented as a product of one proper rotation and one generalized inversion. For magnetic, electric and toroidal orderings there are 64 modes: three pure ferro(magnetic/electric/toroidal) modes, 13 mixed ferro(magnetic/electric/toroidal) and antiferro(magnetic/electric/toroidal) modes, and 48 pure antiferro(magnetic/electric/toroidal) modes. The proposed classification of modes leads to useful observations: the electric and toroidal modes have many symmetry limitations similar to those already known for the magnetic modes, e.g. a continuous reorientation of the magnetic or electric or toroidal moments is possible only in triclinic or monoclinic symmetry. An antiferro(magnetic/electric/toroidal) ordering with a weak perpendicular ferro(magnetic/electric/toroidal) component is possible only in monoclinic or orthorhombic symmetry. The general classifications of magnetic, electric and toroidal modes are presented for the case of NdFeO3.

Magnetic modes compatible with the symmetry of crystals.

A classification of magnetic point groups is presented which gives an answer to the question: which magnetic groups can describe a given magnetic mode? There are 32 categories of magnetic point groups which describe 64 unique magnetic modes: 16 with a ferromagnetic component and 48 without. This classification focused on magnetic modes is helpful for finding the magnetic space group which can describe the magnetic symmetry of the material.

Crystal symmetry for incommensurate helical and cycloidal modulations.

A classification of magnetic superspace groups compatible with the helical and cycloidal magnetic modulations is presented. Helical modulations are compatible with groups from crystal classes 1, 2, 222, 4, 422, 3, 32, 6 and 622, while cycloidal modulations are compatible with groups from crystal classes 1, 2, m and mm2. For each magnetic crystal class, the directions of the symmetry-allowed (non-modulated) net ferromagnetic moment and electric polarization are given. The proposed classification of superspace groups is tested on experimental studies of type-II multiferroics published in the literature.

Neutron Larmor diffraction on powder samples.

A hitherto unrecognized resolution effect in neutron Larmor diffraction (LD) is reported, resulting from small-angle neutron scattering (SANS) in the sample. Small distortions of the neutron trajectories by SANS give rise to a blurring of the Bragg angles of the order of a few hundredths of a degree, leading to a degradation of the momentum resolution. This effect is negligible for single crystals but may be significant for polycrystalline or powder samples. A procedure is presented to correct the LD data for the parasitic SANS. The latter is accurately determined by the SESANS technique (spin-echo small-angle neutron scattering), which is readily available on Larmor diffractometers. The analysis technique is demonstrated on LD and SESANS data from α-Fe2O3 powder samples. The resulting d-spacing range agrees with experimental data from high-resolution synchrotron radiation powder diffraction on the same sample.

Verification of the de Wolff hypothesis concerning the symmetry of ß-MnO2.

The symmetry lowering from tetragonal to orthorhombic is demonstrated using high-resolution diffraction and also justified by using the magnetic superspace groups formalism for the rutile-type compound ß-MnO2. The (lower) orthorhombic symmetry is observed at temperatures both below and above the Néel temperature. The magnetic ordering of ß-MnO2 is of spin density type and not screw-type helical. The results support the de Wolff [Acta Cryst. (1959), 12, 341-345] hypothesis about the orthorhombic symmetry of ß-MnO2.

Crystal symmetry aspects of materials with magnetic spin reorientation.

The symmetry of materials which undergo a continuous spin reorientation has been analysed. It is shown that continuous spin reorientation is possible only in materials with triclinic or monoclinic crystal structure symmetry, i.e. other symmetries - orthorhombic, tetragonal, trigonal, hexagonal and cubic - are forbidden.

Lack of a threefold rotation axis in α-Fe2O3 and α-Cr2O3 crystals.

The crystal structure of α-Fe2O3 and α-Cr2O3 is usually described with the corundum-type trigonal crystal structure based on the space group R3¯c. There are, however, some observations of the magnetic ordering of both α-Fe2O3 and α-Cr2O3 that are incompatible with the trigonal symmetry. We show experimental evidence based on X-ray powder diffraction and supported by transmission electron microscopy that the symmetry of the crystal structure of both α-Fe2O3 and α-Cr2O3 is monoclinic and it is described with the space group C2/c (derived from R3¯c by removing the threefold rotation axis). The magnetic orderings of α-Fe2O3 and α-Cr2O3 are compatible with the magnetic space groups C2/c and C2/c', respectively. These findings are in agreement with the idea from Curie [(1894), J. Phys. 3, 393-415] that the dissymmetry of the magnetic ordering should be associated with a dissymmetry of the crystal structure.

Crystal and magnetic structure in co-substituted BiFeO3.

Ultra-high-resolution neutron diffraction studies of BiFe(0.8)Co(0.2)O3 show a transition from a cycloidal space modulated spin structure at T = 10 K to a collinear G-type antiferromagnetic structure at T = 120 K. The model of antiparallel directions of Fe(3+) and Co(3+) magnetic moments at the shared Wyckoff position describes well the observed neutron diffraction intensities. On heating above RT, the crystal structure of BiFe(0.8)Co(0.2)O3 changes from a rhombohedral R3c to a monoclinic Cm. At 573 K only the Cm phase is present. The collinear C-type antiferromagnetic structure is present in the Cm phase of BiFe(0.8)Co(0.2)O3 at RT after annealing.

Hierarchically structured scleractinian coral biocrystals.

Microscopic (AFM and FESEM) observations show that scleractinian coral biomineral fibers in extant Desmophyllum and Favia, and fossil Jurassic Isastrea are composed of nanocrystalline grains of about 30-100 nm in size. In contrast to these findings, SR diffraction data on the same coral materials exhibit narrow Bragg peaks suggesting much larger crystallite size. These seemingly contradicting results of microscopic and diffraction studies are reconciled within a new, minute-scale model of scleractinian biomineral fibers. In this model, nanocrystalline aragonite units are interconnected by mineral bridges and form aggregates usually larger than 200 nm. Most likely, the size of the aggregates is resulting from physiological biomineralization cycles that control cellular secretion of ions and biopolymeric species. Intercalation of biopolymers into crystal lattice may influence consistently several structural parameters of the scleractinian coral bio-aragonite in all studied samples: (i) the lattice parameters and internal strains of the bio-aragonite are larger than in mineral aragonite, (ii) lattice parameter elongations and internal strains reveal directional anisotropy with respect to crystallographic axes.

A Cretaceous scleractinian coral with a calcitic skeleton.

It has been generally thought that scleractinian corals form purely aragonitic skeletons. We show that a well-preserved fossil coral, Coelosmilia sp. from the Upper Cretaceous (about 70 million years ago), has preserved skeletal structural features identical to those observed in present-day scleractinians. However, the skeleton of Coelosmilia sp. is entirely calcitic. Its fine-scale structure and chemistry indicate that the calcite is primary and did not form from the diagenetic alteration of aragonite. This result implies that corals, like other groups of marine, calcium carbonate-producing organisms, can form skeletons of different carbonate polymorphs.