Site Symmetry Group Research Articles

Irreducible and site-symmetry-induced representations of single/double ordinary/grey layer groups.

Considered are 80 sets of layer groups, each set consisting of four groups: ordinary single and double, and grey single and double layer groups. The structural properties of layer groups (factorization into cyclic subgroups and the existence of grading according to the sequence of halving subgroups) enable efficient symbolic computation (by the POLSym code) of the relevant properties, real and complex irreducible and allowed (half-)integer (co-)representations in particular. This task includes, as the first step, classification of the irreducible domains based on the group action in the Brillouin zone combined with torus topology. Also, the band (co-)representations induced from the irreducible (co-)representations of Wyckoff-position stabilizers (site-symmetry groups) are decomposed into the irreducible components. These, and other layer group symmetry related theoretical data relevant for physics, layered materials in particular, are tabulated and made available through the web site https://nanolab.group/layer/.

Topological transition in a two-dimensional three-vector model: A non-Boltzmann Monte Carlo study.

Two-dimensional three-vector (d=2,n=3) lattice model of a liquid crystal (LC) system with order parameter space (R) described by the fundamental group Π_{1}(R)=Z_{2} was recently investigated based on non-Boltzmann Monte Carlo simulations. Its results indicated that the system did not undergo a topological transition condensing to a low temperature critical state as was reported earlier. Instead, a crossover to a nematic phase was observed, induced by the onset of a competing relevant length scale. This mechanism is further probed here by assigning a more restrictive R symmetry with Π_{1}(R)=Q (the discrete and non-Abelian group of quaternions), thus engaging the three spin degrees in the formation of point topological defects (disclinations). The results reported here indicate that such a choice of symmetry of the Hamiltonian with suitable model parameters leads to a defect-mediated transition to a low-temperature phase with topological order. It is characterized by a line of critical points with quasi-long-range order of its three spin degrees. The associated temperature-dependent power-law exponent decreases progressively and vanishes linearly as temperature tends to zero. The high-temperature disordered phase shows exponential spin correlations and their temperature-dependent lengths exhibit an essential singular divergence as the system approaches the topological transition point. Biaxial LC models have the required R symmetry owing to their tensor orientational orders and are suggested to serve as prototype examples to exhibit topological transition in (d=2,n=3) lattice models.

An improved symmetry-based approach to reciprocal space path selection in band structure calculations

Band structures for electrons, phonons, and other quasiparticles are often an important aspect of describing the physical properties of periodic solids. Most commonly, energy bands are computed along a one-dimensional path of high-symmetry points and line segments in reciprocal space (the “k-path”), which are assumed to pass through important features of the dispersion landscape. However, existing methods for choosing this path rely on tabulated lists of high-symmetry points and line segments in the first Brillouin zone, determined using different symmetry criteria and unit cell conventions. Here we present a new “on-the-fly” symmetry-based approach to obtaining paths in reciprocal space that attempts to address the previous limitations of these conventions. Given a unit cell of a magnetic or nonmagnetic periodic solid, the site symmetry groups of points and line segments in the irreducible Brillouin zone are obtained from the total space group. The elements in these groups are used alongside general and maximally inclusive high-symmetry criteria to choose segments for the final k-path. A smooth path connecting each segment is obtained using graph theory. This new framework not only allows for increased flexibility and user convenience but also identifies notable overlooked features in certain electronic band structures. In addition, a more intelligent and efficient method for analyzing magnetic materials is also enabled through proper accommodation of magnetic symmetry.

Ferromagnetic and antiferromagnetic orderings in wurtzite diluted magnetic nanostructures

Using a new approach based on symmetry analysis, we have determined the magnetic symmetry groups (Shubnikov groups) of ferro- and antiferromagnetic wurtzite nanostructures doped with magnetic atoms periodically distributed at cation sites as well as the direction of the spontaneous magnetic field. Quantum wells, nanorods (nanowires), nanotubes, and quantum dots have been considered. The direction of the spontaneous magnetic field is determined by magnetic atoms with higher C3v (3m) or Cs (m) site symmetry group (the site symmetry group is defined with respect to the doped nanostructure). When the magnetic-atom distribution becomes more or less disordered (Diluted Magnetic Semiconductors), it seems reasonable to assume that, in most of the cases, the symmetry in regions with a size of some wurtzite unit-cells remains close to that of a periodic one, hence possibly inducing local ferro- or antiferromagnetic properties. The regions can induce overall ferro- or antiferromagnetic states when they have common spontaneous magnetic field direction or, possibly, induce easy-magnetization direction or plane (direction or plane of maximal magnetization intensity for a given external magnetic field strength) when disorder imposes a spread of spontaneous magnetic field direction over the various regions. Of course, such an effect adds itself to magnetic properties eventually induced by grain boundaries, other crystalline phases, and defects such as vacancies, dislocations or interstitial atoms.

Spectroscopic determination of site symmetry and space group in lanthanide-doped crystals: Resolving intricate symmetry aspects for β-NaLnF4

There exists a controversy in the hitherto reported studies concerning the actual space group that accounts for spectroscopic results observed for the trivalent lanthanide (Ln3+) ions doped into β-NaLnF4 hosts. The two competing assignments of the space group: P6¯ or P63/m, bear on the associated point symmetry groups, which describe the local site symmetry of the dopant Ln ions. This paper aims at resolving this controversy. For this purpose two approaches are utilized. First, several new samples were synthesized and characterized by photoluminescence (PL) spectroscopy. Both the emission and excitation PL spectra have been measured. The analysis of the present experimental data indicates unequivocally that the dopant Ln3+ ions enter the crystal lattice β-NaGF4 and β-NaYF4 at two crystallographically distinct sites. Hence, the correct space group for these hosts is P6¯ and not P63/m. The intricacies arising from the sample preparation methods, which bear on the interpretation of the results, are also clarified. Second, the theoretical modeling of the crystal-field parameters (CFPs) is carried out based on superposition model to correlate the available crystallographic and spectroscopic data. The crystal-field analysis enables to ascribe unambiguously the spectroscopically determined sites to the crystallographic sites 1a and 1f. In this way an interpretation of photoluminescence spectra is achieved that fully corroborates the available crystallographic data. The influence of structural disorder occurring in β-NaLnF4 family of compounds on the 5D0→7F1 photoluminescence spectra of dopant Eu3+ ion and its role in interpretation of experimental data is also discussed.

Point groups in crystallography

Two dual spaces are extensively used in crystallography: the space E n , hosting the crystal pattern; and the vector space V n , where face normals and reciprocal-lattice vectors are defined. The term point is used in crystallography to indicate four different types of groups in these two spaces. 1) Morphological groups in V n ; they can be obtained by determining subgroups of maximal holohedries (holohedries not in group-subgroup relation): this gives 21 and 136 groups in V 2 and V 3 , respectively, which are classified into 10 and 32 point-group types (on the basis of which geometric crystal classes are defined) falling into 9 and 18 abstract isomorphism classes. 2) Symmetry groups of atomic groups and coordination polyhedra in E n ; they coincide with molecular groups, which are infinite in number because the symmetry operations forming these groups are not subject to the crystallographic restriction. 3) Site-symmetry groups in E n ; they are finite groups but infinite in number due to conjugation by the translation subgroups of the space groups. They are classified in geometric crystal classes exactly like groups in V n . A finer classification of site-symmetry groups into species is however introduced that takes into account their orientation in space: species of site-symmetry groups in E n uniquely correspond to groups in V n . 4) Groups of matrices representing the linear parts of space group operations in E n ; they are isomorphic to the groups in V n and are also isomorphic to the factor groups G/T, where G is a space group and T its translation subgroup.

Automated derivation of structural class symbols and extended Z′ descriptors for molecular crystal structures in the Cambridge Structural Database

Extended Z′ descriptors that distinguish cases in which molecules occupy crystallographically distinct special equivalent positions (for example, Z′ = ½ + ½) are derived for 173 497 single-component molecular crystal structures in the Cambridge Structural Database. The extended Z′ descriptors are closely related to the structural class (SC) symbols developed by Belsky and Zorky, and the SC symbol is also derived for each CSD entry. SC symbols express more information than Z′ descriptors, listing explicitly the crystallographic site symmetry for all crystallographically distinct molecules: for example, P, Z = 2(, ). The Z′ descriptors are obtained from the SC symbols by taking the reciprocal of the order of each distinct crystallographic site-symmetry group. The automated methodology permits convenient identification and quantification of structures in the CSD that have more than one crystallographically distinct molecule, and it is shown that the proportion of these structures is slowly increasing.

Symmetry analysis for localized function generation and chemical bonding in crystals:SrZrO3and MgO as examples

Different criteria of localization of one-electron states in crystals are discussed and relations between them are established. A new localization criterion determined only by the localized orbital (LO) itself is formulated. The importance of the symmetry analysis in the procedure of the generation of maximally LOs in crystals is shown. It is suggested that the numerical parameters of the LOs corresponding to the valence energy bands of the crystal can be used in the theory of chemical bonding only if these functions form bases of irreducible representations of site symmetry groups of atoms and bonds in crystals. General considerations are illustrated on the examples of LOs corresponding to the valence energy bands in $\mathrm{Sr}\mathrm{Zr}{\mathrm{O}}_{3}$ and MgO crystals. There is in $\mathrm{Sr}\mathrm{Zr}{\mathrm{O}}_{3}$ a LO describing a partly covalent bond between oxygen and zirconium atoms with the degree of covalency $K=0.46$. When applied to the MgO crystal the suggested approach confirms a purely ionic nature of chemical bonding in this crystal.

Characterization of bismuth triborate single crystal using Brillouin and Raman spectroscopy

AbstractThe bismuth triborate (BiB3O6; BIBO) single crystal has been studied by Brillouin and Raman spectroscopy. Using Brillouin spectroscopy the propagation of the acoustic phonons in [100], [010], [001], [110] and [011] directions has been analyzed. The influence of birefringent and optical activity of BIBO crystal on some acoustic phonons has been discussed. The Raman spectra of the crystal in different scattering geometries have been recorded in the 50–1500 cm‐1 spectral range at room temperature. The number and symmetry of Raman active modes was determined on the basis of site symmetry group analysis. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Algorithms for deriving crystallographic space-group information. II. Treatment of special positions.

Algorithms for the treatment of special positions in three-dimensional crystallographic space groups are presented. These include an algorithm for the determination of the site-symmetry group given the coordinates of a point, an algorithm for the determination of the exact location of the nearest special position, an algorithm for the assignment of a Wyckoff letter given the site-symmetry group, and an alternative algorithm for the assignment of a Wyckoff letter given the coordinates of a point directly. All algorithms are implemented in ISO C++ and are integrated into the Computational Crystallography Toolbox. The source code is freely available.

Infrared study of some synthetic phases of malachite (Cu 2(OH) 2CO 3)–hydrozincite (Zn 5(OH) 6(CO 3) 2) series

Synthetic malachite, hydrozincite and five monophasic mixed copper–zinc hydroxycarbonates have been studied by Fourier transform infrared (FTIR) spectroscopy at ambient and liquid nitrogen temperature in the region of 4000–400 cm −1. The analysis of the spectra reveals that the samples containing up to 20% zinc retain the malachite lattice, thus forming solid solutions. The inclusion of zinc ions in malachite reflects on the positions and intensity of the bands corresponding to the internal modes of the carbonate ion, to the OH librations and to the MeO interactions. For example, the higher and the lower frequency components of ν 3 shift to higher and lower frequencies, respectively. The intensity of the bands corresponding to ν 2 decreases with the zinc content increase. The spectrum of the sample Cu 1.31Zn 0.69(OH) 2CO 3 become diffuse and ill-resolved in the region of the MeO interactions (region below 600 cm −1) and the corresponding bands are shifted to lower frequencies due to the weaker ZnO interactions as compared with those of the copper ions. The internal modes of the carbonate ions in hydrozincite and aurichalcite are assigned and discussed taking into account the site symmetry and factor group symmetry. The OH and OD stretches (matrix-isolated HDO molecules) and the hydrogen bond strengths are interpreted in terms of MeO interactions (synergetic effect), hydrogen bond angles and different hydrogen bond acceptor strengths of the oxygen atoms from the carbonate ions. It proves that the hydrogen bonds in hydrozincite are stronger as compared with those in malachite, irrespective of both the larger hydrogen bond lengths and the weaker ZnO interactions in hydrozincite due to the higher hydrogen bond acceptor strength of the non-coordinated oxygen atom and the formation of bifurcated hydrogen bonds.

IR Spectroscopic studies of charged–discharged crystalline WO 3 films

The structural changes of charged–discharged crystalline WO 3 films have been studied using ex-situ IR spectro-electrochemical measurements. The films were deposited on transparent conducting glass (F:SnO 2, Flabeg) and Si wafers by dip coating from a peroxopolytungstic acid solution. Annealing was then performed at 150 and 300°C (60 min) and at 500°C (15 min). At 150°C, an amorphous WO 3· nH 2O phase forms while annealing at 500°C produces a monoclinic WO 3 (m-WO 3) modification. XRD and IR spectroscopies were used to assess the film structure and crystallinity. Crystalline films were galvanostatically charged–discharged to a different level of insertion coefficient ( x) in 0.1 M HClO 4(aq), 1 M LiClO 4/propylene carbonate (PC) and 1 M NaClO 4/PC electrolytes. Ex-situ IR spectra of all intercalated films exhibit a pronounced increase in absorption between 8000–1000 cm −1 due to the quasi-free electrons. We observed differences between the IR spectra of WO 3 when charged to varying levels of x with H +, Li + and Na +. The intercalation of m-WO 3 independently of the inserted ion leads to the transformation from monoclinic to tetragonal and further to cubic phase. By using site symmetry group analysis we predicted the number of IR and Raman active WO stretching vibrations for H x WO 3, Li x WO 3 and Na x WO 3 in cubic, tetragonal and monoclinic phases. We observed a single WO stretching for the c-Li x WO 3 ( Pm3 m) and two for the c-H x WO 3 ( Im3) phase, which agrees with the lowering of the site symmetry of WO 6 groups in the latter phase. For the t-H x WO 3 and t-Li x WO 3 films, a similarity between the corresponding IR spectra indicates that the structure of the tetragonal phases does not differ significantly.

Structure of vanadate in calcium phosphate and vanadate apatite solid solutions

Polycrystalline solid solutions of phosphate and vanadate calcium apatites were synthesized and studied by XRD, XPS, 31P and 51V NMR, FTIR and UV spectroscopies. Homogeneous distributions of vanadium within the solid and the surface were obtained. While 51V NMR spectra suggest an axial symmetry of the vanadate group, FTIR spectra indicate distortion from essentially C3v symmetry in solid solutions with low vanadium content to Cs for pure vanadate apatite, in agreement with the site symmetry group of phosphate in hydroxyapatite. Refinements of XRD pattern of vanadate apatite by Rietveld method confirm the Cs point group attribution. Three vanadium oxygen bond lengths were found around 168 pm and one close to 174 pm, suggesting that the Cs point group could be generated by a small distortion from C3v symmetry. Analysis of UV spectra confirms that distortion from Td symmetry increases with vanadium content and suggests some contraction of the vanadate ion in the apatite lattice.

Supercell Model of V-Doped TiO2: Unrestricted Hartree-Fock Calculations

The host crystal space group and the point defect site symmetry group being given the procedure of generation of the imperfect space group in the supercell or periodic defect model is considered. The approach developed is applied to the symmetry analysis of the possible space groups of metal-doped rutile and anatase structures in the supercell model. The results of unrestricted Hartree-Fock calculations of the V-doped rutile electronic structure (an impurity atom charge, valence states, an impurity level position in the energy gap) are given. Supercells of 36, 60, 84 atoms chosen on the base of the symmetry analysis are calculated. The convergence of the results with the supercell increasing is demonstrated.

Supercell Model of V-Doped TiO2: Unrestricted Hartree-Fock Calculations

The host crystal space group and the point defect site symmetry group being given the procedure of generation of the imperfect space group in the supercell or periodic defect model is considered. The approach developed is applied to the symmetry analysis of the possible space groups of metal-doped rutile and anatase structures in the supercell model. The results of unrestricted Hartree-Fock calculations of the V-doped rutile electronic structure (an impurity atom charge, valence states, an impurity level position in the energy gap) are given. Supercells of 36, 60, 84 atoms chosen on the base of the symmetry analysis are calculated. The convergence of the results with the supercell increasing is demonstrated.

Permutation-Inversion Group of Fullerite C70 in the Intermediate Phase

The permutation-inversion symmetry group of fullerite C 70 in the intermediate phase (at temperatures 270 K < T < 340 K) and the permutation-inversion site symmetry group of a rotating molecule C 70 in a nonrigid crystal are generated. Their irreducible representations compatible with the principle of symmetry of wave functions with respect to the permutations of identical nuclei are found. Group theoretical classification of quantum-mechanical states of a rotating molecule and a crystal in the intermediate phase of fullerite C70 is made. The selection rules for electronic and IR spectra in terms of the irreducible representations of the permutation-inversion group are found.

EPR of Gd3+ in single crystal colquiriite and analysis of the spin Hamiltonian tensorsB 4 andB 6

In single crystal colquiriite LiCaAlF6 doped with Gd3+ ions two EPR spectra of the Gd3+ ions with the Laue site-symmetry groups Ci and C3i were observed. The spectrum angular dependence for trigonal Gd3+ centre was investigated in detail and corresponding spin Hamiltonian parameters were fitted. From analysis of the spin Hamiltonian tensorsB4 andB6 it was established that Gd3+ with the Laue group C3i substitutes at Ca2+ site with the excess charge compensation by an ion located along the threefold axis from this site. The transformation formulas for a sixth-rank irreducible Hermitian tensor under coordinate rotation are tabulated in an explicit form. By using the EPR data for Gd3+ substituted in a variety of host crystals, the fourth-rank and sixth-rank tensors of Gd3+ spin Hamiltonians were tabulated and correlated with structures of the coordination polyhedra at substitution sites. The results suppose a predominance of quadratic crystal field contributions into the spin Hamiltonian tensorB4 of Gd3+.

Splitting of Wyckoff positions (orbits)

A crystal structure is composed of one or more sets of atoms where the atoms of each set are symmetrically equivalent under a space group (‘regular system of atoms’ or ‘orbit of atoms’). For a characterization of the crystal structure it is sufficient to specify one atom of each set (each orbit) by the coordinates of its centre. Relations between crystal structures in crystal chemistry or in continuous phase transitions manifest themselves in group (G)-subgroup (H) relations: H < G. Atoms which are equivalent under G may become non-equivalent under the subgroup H, i.e., ‘the orbit may split’. The splitting is the same for all orbits of a Wyckoff position but may be different for orbits of different Wyckoff positions. In this paper, the possible ways of splitting of orbits and the conditions for this splitting are analyzed. Relevant factors for the splitting are: the index of the group-subgroup relation G-H, the order of the point group of G, the order of the site-symmetry groups of the orbit in G and, to a lesser degree, in the subgroup H. Also the kind of group-subgroup relation plays a role. The splitting is most restricted if H is a normal subgroup of G. The splitting conditions are expressed in an equation which is illustrated by examples.

Spectroscopic study and crystal field analysis of Eu 3+ in the YAl 3(BO 3) 4 huntite matrix

This study reports on the absorption, fluorescence and magnetic circular dichroism (MCD) spectra of YAl 3(BO 3) 4 doped with Eu 3+. A crystal field analysis in the D 3 site symmetry group provided information on the energy levels and corresponding wave functions. Special emphasis is put on the power of the MCD technique for assessing the intensity mechanisms responsible for transitions between energy levels.

Microclusters of classical dipoles on symmetrical systems of sites : on the location of energy extrema and saddles. I. non-centered 2-D systems.

Abstract Microclusters of dipoles free to reorient on a fixed symmetrical system of sites are considered. Here the treatment is limited to planar non-centered systems. The following conjectures are proposed : The extrema and saddles of the energy in the angular configuration hyperspace belong either to the site symmetry group, or its subgroups, or their derived black-white groups, Identity being excluded. These conjectures are verified by calculations in the case of two-and three-dipole microclusters.