Symmetry-adapted Basis-vectors Research Articles

Defining an aristotype crystal structure and crystallographic distortions in leucite/pollucite-structured phases with space group $$Ia\overline {3} d$$ I a 3 ¯ d

A universal aristotype crystal structure for leucite/pollucite-structured phases (A2B2IIIC4IVO12, A2BIIC5IVO12) with space group $$Ia\overline {3} d$$ has been determined by a non-linear optimisation of the fractional coordinates. The resultant T-site tetrahedron exhibits close to perfect $$\overline {4} 3m$$ point group symmetry (quadratic elongation 1.0000124, tetrahedral angle variance 1.53 × 10−5 degrees2). For any distorted leucite/pollucite crystal structure, an associated aristotype phase can be derived from the optimised fractional coordinates, with the aristotype phase lattice parameter calculated from the volume of the hettotype T-site tetrahedron. Assuming the aristotype T-site tetrahedron to be ideal, the distortion of the hettotype tetrahedron from that of the aristotype is described as the linear sum of the amplitudes of an appropriate set of symmetry-adapted basis-vectors of the effective regular TO4 molecule consistent with the observed point group symmetry within the crystal structure. The methodology is illustrated by reference to the disordered T-site, leucite-structured phase K2MgSi5O12.

Parameterization of the crystal structure of garnet in terms of symmetry-adapted basis-vectors of the ideal tetrahedron and octahedron: Application to the pressure-dependence of the crystal structure of Y3Al5O12 between 0 and 126 GPa

Abstract Non-linear optimisation of the anion coordinates in garnet-structured compounds (A3B2C3O12) with space group I a 3 ¯ d has been used to define a universal prototypic crystal structure in which the tetrahedral site (CO4) and the octahedral site (BO6) exhibit close to the ideal point group symmetries of 4 ¯ 3 m and m 3 ¯ m respectively. Distortions in actual garnet-structured compounds are then described in terms of condensed normal modes of the two isolated, ideal polyhedra that are consistent with the site point group symmetry within the space group I a 3 ¯ d . The methodology is applied to the pressure-dependence of the crystal structure of Y3Al5O12 from 0 GPa to 126 GPa that have been determined from first-principles calculations. The structural evolution of Y3Al5O12 with pressure is dominated by changes in the AlivO4 site which rotates and becomes significantly more regular as pressure increases. The coupled response of the octahedral site is more complex as it responds to the changes in the tetrahedral site. The evolution of the crystal structure of YAG with pressure is briefly compared to the experimentally determined pressure-dependence of the crystal structure of pyrope, Mg3Al2Si3O12.

A high-resolution powder neutron diffraction study of the crystal structure of neighborite (NaMgF3) between 9 and 440 K

The temperature dependence of the unit-cell dimensions and the crystal structures of the fluorperovskite neighborite, NaMgF 3 (analyzed in the Pbnm setting of the space group), have been determined at 88 temperatures between 9 and 440 K from high-resolution, time-of-flight, powder neutron diffraction data. Lattice parameters exhibit saturation at low temperatures, before developing linear thermal expansion coefficients at temperatures above ~ 350 K. The temperature dependence of each axis has been analyzed, and fitted, using a two-term expression related to an Einstein internal energy function. The unit-cell parameters a and c behave in a conventional manner, however, an unexpected, and previously unobserved, region of negative linear thermal expansion has been found for the b axis in the temperature interval 20 ≤ T ≤ 90 K. Estimated, high-temperature axial thermal expansion coefficients derived from the lattice parameter fitting are in good agreement with those experimentally determined from an earlier synchrotron study, and indicate that high-temperature saturation has been achieved in neighborite by 440 K. The unit-cell volume varies smoothly and monotonically over the whole temperature interval, and the two-term Debye model of Barron has been successfully used to fit these data with characteristic temperatures of 369(2) and 1055(14) K. For the temperature interval 313.15-443.15 K, the thermodynamic Grüneisen constant has been determined using the Debye parameterization of the unit-cell volume, coupled with literature values of the isobaric molar heat capacity. The evolution of the crystal structure as a function of temperature is presented, and explained, in terms of the temperature dependence of the amplitudes of the seven symmetry-adapted basis-vectors of the aristotype phase that are consistent with the orthorhombic space group. The calculated temperature variations of the bond lengths are in excellent agreement with those experimentally determined. The primary order parameters for centrosymmetric, zone-boundary phase transitions in perovskite-structured compounds, i.e., the amplitudes of the basis vectors that transform as the irreducible representations R + 4 (anti-phase tilt) and M + 3 (in-phase tilt), have been fitted to a Landau free energy expansion that incorporates low-temperature saturation. Within the temperature range studied, the temperature dependence of the displacement corresponding to the anti-phase tilt is consistent with tricritical behavior. Experimental evidence is presented for a quadratic coupling of the in-phase tilt to the anti-phase tilt for temperatures greater than ~ 135 K, suggesting critical behavior at the orthorhombic-cubic transition is purely related to an instability at the R point of the pseudocubic Brillouin zone.

Thermoelastic and structural properties of ionically conducting cerate perovskites: (I) BaCeO3 at low temperature in the Pbnm phase

The thermoelastic and structural properties of BaCeO3 perovskite in the Pbnm phase field have been studied using high resolution neutron diffractometry at 37 temperatures between 2K and 350K. From a simultaneous fit of the isochoric heat capacity and unit cell volume, the vibrational density of states in the Pbnm phase is shown to be approximated by a two-term modified Debye model with characteristic temperatures 176(1)K and 543(2)K. Vibrational Debye temperatures, determined from the temperature-variation of the atomic displacement parameters, suggest the cations to be more associated with the lower characteristic temperature, whilst those for the anions, are closer to the higher characteristic temperature. The weighted average of the Grüneisen constants associated with the two characteristic temperatures is in good agreement with the experimentally determined thermodynamic Grüneisen parameter. Structural parameters are presented as the amplitudes of the seven symmetry-adapted basis-vectors of the aristotype phase, and a structural basis for the temperature-dependence of the bond lengths is outlined. The critical exponent associated with the temperature variation of the primary order parameter in the phase transition from Pbnm to Ibnn at 563K is consistent with this transition being tricritical in nature.

Low temperature thermoelastic and structural properties of LaGaO3 perovskite in the Pbnm phase

The thermoelastic and structural properties of LaGaO3 perovskite have been studied using high resolution neutron diffractometry at 158 temperatures between 11K and 548K. Data collected in 2K intervals between 410K and 430K show no evidence for an intermediate phase between the low temperature phase in space group Pbnm and the high temperature phase in space group F3¯2/n. From a simultaneous fit of the unit cell volume and the isochoric heat capacity, the phonon density of states in the Pbnm phase is shown to be approximated by a two-term Debye model, with characteristic temperatures 294(1)K and 831(1)K. Vibrational Debye temperatures, determined from fitting the temperature variation of the atomic displacement parameters, show the cations to be more associated with the lower characteristic temperature, whilst that for the anions, is closer to the higher characteristic temperature. Structural parameters are presented as the amplitudes of the seven symmetry-adapted basis-vectors of the aristotype phase, and a structural basis for the temperature-dependence of the bond lengths is outlined. The phase transition in both temperature and pressure arises when a non-bonded La–O distance approaches the La coordination sphere.

Structural and thermoelastic study of the protonic conducting perovskite SrCe0.95Yb0.05Oξ (ξ ∼ 3) between 373 K and 1273 K

The temperature dependence of the crystal structure of the protonic conducting perovskite SrCe0.95Yb0.05Oξ (ξ ∼ 3) has been determined from Rietveld refinement of neutron time-of-flight powder diffraction data in 25 K steps from 373 K to 1273 K. In contrast to most SrBIVO3 perovskites which show a sequence of structural phase transitions from Pmcn – Incn – I4/mcm - \( Pm\overline 3 m \) with increasing temperature, SrCe0.95Yb0.05Oξ remains orthorhombic with space group Pmcn from 4.2 K to the highest temperature measured. Crystallographic results are presented graphically in the form of the magnitude of the seven unique symmetry adapted basis vectors of a hypothetical aristotype perovskite phase. Thermoelastic properties of SrCe0.95Yb0.05Oξ deduced from the temperature variation of the unit cell volume and Debye-Waller factors are compared with literature values derived from independent calorimetric, Raman and elastic measurements. Thermodynamically, SrCe0.95Yb0.05Oξ behaves as a Debye-like solid with an average Gruneisen parameter in the range 1.4–1.5, but with two characteristic temperatures, ∼200 K and ∼600 K, the lower temperature being more related to the cation vibrations, the higher to the anion vibrations.

Structural and thermoelastic properties of CaTiO 3 perovskite between 7 K and 400 K

The structural and thermoelastic properties of CaTiO 3 perovskite have been studied using high-resolution powder neutron diffractometry at eighty temperatures in the range 7–400 K. The temperature variation of the unit cell volume, the thermodynamic Grüneisen parameter and the isobaric heat capacity are analysed using a two-term Debye model. Structural parameters are presented as the magnitudes of symmetry-adapted basis-vectors of seven normal modes with wavevectors that lie on the surface of the Brillouin zone of the primitive cubic aristotype phase, and a structural basis for the temperature-dependence of the bond lengths is proposed. A consistency between the vibrational Debye temperatures derived from the atomic displacement parameters and the mean values of the vibrational energies of the three atomic species calculated from the partial phonon densities of states has been found.

PARAMETERIZATION OF THE CRYSTAL STRUCTURES OF CENTROSYMMETRIC ZONE-BOUNDARY-TILTED PEROVSKITES: AN ANALYSIS IN TERMS OF SYMMETRY-ADAPTED BASIS-VECTORS OF THE CUBIC ARISTOTYPE PHASE

The methods of group theory have been used to decompose the crystal structures of centrosymmetric perovskites, ABX 3 , that exhibit zone-boundary tilting of the BX 6 octahedra. For the fourteen space-groups consistent with these phenomena, the associated structures are decomposed in terms of the magnitudes of an appropriate set of symmetry-adapted basis-vectors of the primitive cubic aristotype phase of perovskite at high-symmetry points on the surface of the Brillouin zone. The advantage of this parameterization is twofold; firstly, octahedron tilt angles can be determined precisely and independently of the effects of octahedron distortion, and secondly, the degrees of freedom required by the perovskite structure can be rigorously derived. The method is outlined using the results of a neutron-diffraction investigation of CaTiO 3 in space group Pbnm , an example where the structural degrees of freedom are found to be one less than that required by the space group. Full results that can be very simply utilised in the decomposition of the other thirteen space-groups are tabulated. The advantages of decomposing perovskite-structured phases in this way are further illustrated using the temperature dependence of the crystal structure of KCaF 3 between 4.2 and 542 K.

PARAMETERIZATION OF CENTROSYMMETRIC ELPASOLITE-TYPE CRYSTAL STRUCTURES IN TERMS OF SYMMETRY-ADAPTED BASIS-VECTORS OF THE PRIMITIVE CUBIC ARISTOTYPE PHASE

Group-theory methods are used to decompose the crystal structures of centrosymmetric double perovskites, A 2 BB’X 6 , that exhibit zone-boundary tilting of octahedra and the rock-salt pattern of order of the B-site cations, typified by the mineral elpasolite, K 2 NaAlF 6 . For the twelve space-groups consistent with these phenomena, the associated structures are decomposed in terms of the magnitudes of an appropriate set of symmetry-adapted basis-vectors of the primitive cubic aristotype phase of perovskite at high-symmetry points of the Brillouin zone. The method is outlined using the example of Sr 2 YNbO 6 in space group P 2 1 /n ; full results that can be utilized in the decomposition of the other space-groups are tabulated. The advantages of decomposing elpasolite-structured phases in this way are further illustrated using the temperature dependence of the crystal structure of Ba 2 SbBiO 6 between 4.2 and 625 K.

Defect Green's function andTmatrix for an hcp lattice with a substitutional impurity

Starting with a two-neighbor defect space model for a substitutional impurity in an hcp lattice, we apply matrix-partitioning and Green's-function techniques for the study of atomic vibrations in the defect space. Using the fact that the defect site has the point-group symmetry of the Group ${D}_{3h}$, we set up and decompose the total representation $\ensuremath{\Gamma}({D}_{3h})$ of the group as $\ensuremath{\Gamma}({D}_{3h})=4{A}_{1}^{\ensuremath{'}}\ensuremath{\bigoplus}2{A}_{1}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\bigoplus}3{A}_{2}^{\ensuremath{'}}\ensuremath{\bigoplus}4{A}_{2}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\bigoplus}8{E}^{\ensuremath{'}}\ensuremath{\bigoplus}5{E}^{\ensuremath{'}\ensuremath{'}}$. The symmetry-adapted basis vectors of the 39-dimensional defect space are calculated using the projection operator technique and the projected forms of the Green's function and the perturbation matrices are derived. The formal results obtained here are applicable to a variety of problems in hcp crystals.

Damped free vibrations of symmetrical networks by group representation theory

Structures, freely moving in space and having some geometric symmetry, are analyzed for damped free linear vibrations by employing group representation theory. The symmetry adapted basis vectors are determined corresponding to various invariant subspaces for the irreducible representations present in the reducible representation of the configuration space. The kinetic, the damping and the strain energy are written in terms of the symmetry adapted co-ordinates. The Lagrangian equations of motion for damped free systems are expressed in the symmetry adapted co-ordinates for each subspace. These equations of motion are solved for natural frequencies and the normal mode vectors in terms of the symmetry adapted basis vectors. The constant coefficients for the general solution are determined for some initial displacements.

The normal modes of non-linear symmetric systems by group representation theory

Group representation theory is applied to analyze a symmetric spring mass system of finite degree of freedom for non-linear normal modes of vibration. A set of symmetry operations constituting a group characterises the symmetry of a system. The symmetry adapted basis vectors, corresponding to the irreducible representations present in the reducible representation, are determined. The equations of motion in the symmetry adapted coordinates have a lower order of coupling. An equipotential surface in each invariant subspace is determined from which the normal mode vectors are evaluated in terms of the symmetry adapted basis vectors.

Group theory and normal coordinate calculations

A method is proposed to obtain a set of computer generated symmetry adapted basis vectors which completely factorize a matrix associated with any operator which commutes with all the symmetry operations of a molecule. The cartesian coordinates of all atoms are used to derive the irreducible representations of the appropriate molecular point group and the transformation properties of any starting set of coordinates. Symmetry coordinates are then projected out each irreducible representation.

Symmetric network analysis by group representation theory

Symmetric spring mass networks of finite degree of freedom have been analysed for modes of vibration by the group representation theory. A working outline of the theory has been presented. By associating the unit base vectors with each mass point of the network and by operating on them by the symmetry operators of the system, a reducible representation is generated. The number of irreducible representations contained in the reducible representation are then calculated on the basis of group representation theory. By the use of projection operators the symmetry adapted basis vectors are determined. The energy matrices are obtained in a quasi-diagonal form by Wigner's theorem. The Lagrangian is expressed in the symmetry adapted coordinate system and the frequencies along with their corresponding modes are evaluated from the resulting equations of motion.