Bravais Classes Research Articles

Continuous chiral distances for two-dimensional lattices.

Chirality was traditionally considered a binary property of periodic lattices and crystals. However, the classes of two-dimensional lattices modulo rigid motion form a continuous space, which was recently parametrized by three geographic-style coordinates. The four non-oblique Bravais classes of two-dimensional lattices form low-dimensional singular subspaces in the full continuous space. Now, the deviations of a lattice from its higher symmetry neighbors can be continuously quantified by real-valued distances satisfying metric axioms. This article analyzes these and newer G-chiral distances for millions of two-dimensional lattices that are extracted from thousands of available two-dimensional materials and real crystal structures in the Cambridge Structural Database.

Dualized Ammann-Kramer-Neri tiling

Self-similarity is a key characteristic of quasicrystalline tilings. The similarity ratio can be an n-th power of the irrational scaling unit associated with the relevant non-crystallographic Bravais class, where n is a natural number (e.g., 1, 2, …). The Ammann-Kramer-Neri tiling is a famous quasicrystalline tiling with icosahedral symmetry based on acute and obtuse rhombohedra. It has however remained underexplored in view of its self-similarity, since any attempt to find substitution rules has by far been hindered by the large similarity ratio of no less than τ 3, where τ is the golden mean. We hereby illustrate a new approach to tackle this issue by introducing an auxiliary tiling that combines the rhombohedral tiling and its crystallographic dual supported on the body-centre positions. The vertices of this “dualized” tiling arise from the cut through the same kind of acceptance domain as that of the rhombohedral tiling, namely rhombic triacontahedron, attached to every point of the six-dimensional body-centred icosahedral Bravais lattice. Eleven kinds of polyhedra, nine out of which are tetrahedra, are identified as prototiles. This new tiling admits self-similarity with τ being the similarity ratio.

Geographic style maps for two-dimensional lattices.

This paper develops geographic style maps containing two-dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two-dimensional lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualize hundreds of thousands of lattices of real crystal structures from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.

Combined neutron scattering and resonant ultrasound spectroscopy for lattice dynamics studies

Resonant ultrasound spectroscopy (RUS) and inelastic neutron scattering afford insight into solid state structures and excitations at vastly different frequency and length scales. We will discuss the combined use of these techniques to understand phase transitions in functional and structural materials. Structural complexity, such as chirality or incommensuration leads to emerging features in both techniques and modifies energy transport channels. Novel developments in techniques at the Oak Ridge neutron sources, that now enable in situ ultrasound spectrocopy in the neutron beam, and the RUSCal analysis software that now supports all Bravais classes and implements Monte-Carlo methods, will be discussed in example materials. [Work supported by the Deparment of Energy Office of Basic Energy Science and by Laboratory Directed Research at Oak Ridge National Laboratory. My thanks go to J. Torres, V. Fanelli, Y. Shinohara, E. Cakmak, C. Hua, A. Flores-Bettancourt, M. Ruis-Rodriguez for the collaboration and A. Zevalkink, D. Mandrus, V. Keppens, E. Lara-Curzio, and T. Watkins for fruitful discussions.]

Minimum-strain symmetrization of Bravais lattices

Bravais lattices are the most fundamental building blocks of crystallography. They are classified into groups according to their translational, rotational, and inversion symmetries. In computational analysis of Bravais lattices, fulfilment of symmetry conditions is usually determined by analysis of the metric tensor, using either a numerical tolerance to produce a binary (i.e. yes or no) classification, or a distance function which quantifies the deviation from an ideal lattice type. The metric tensor, though, is not scale-invariant, which complicates the choice of threshold and the interpretation of the distance function. Here, we quantify the distance of a lattice from a target Bravais class using strain. For an arbitrary lattice, we find the minimum-strain transformation needed to fulfil the symmetry conditions of a desired Bravais lattice type; the norm of the strain tensor is used to quantify the degree of symmetry breaking. The resulting distance is invariant to scale and rotation, and is a physically intuitive quantity. By symmetrizing to all Bravais classes, each lattice can be placed in a 14 dimensional space, which we use to create a map of the space of Bravais lattices and the transformation paths between them. A software implementation is available online under a permissive license.

Generation of (3 +d)-dimensional superspace groups for describing the symmetry of modulated crystalline structures

A complete table of (3 + 1)D, (3 + 2)D and (3 + 3)D superspace groups (SSGs) has been enumerated that corrects omissions and duplicate entries in previous tables of superspace groups and Bravais classes. The theoretical methods employed are not new, though the implementation is both novel and robust. The paper also describes conventions for assigning a unique one-line symbol for each group in the table. Finally, a new online data repository is introduced that delivers more complete information about each SSG than has been presented previously.

Symmetry of crystals composed of two Bravais sublattices in 6D crystal spaces

It is shown that the symmetry of crystals composed of two Bravais sublattices can be determined in 6D spaces, which are two orthogonal 3D subspaces. Twenty one sets in 6D space satisfy this requirement. The Bravais classes in these sets and the conditions of translation compatibility of orthogonal 3D subspaces are discussed. A consideration of the subordination scheme for systems leaves only 19 significant sets. Crystals with a fluorite lattice, in which the cation sublattice has the translational symmetry of the crystal, and crystals with a cuprite lattice, in which the anion and cation sublattices have different translational symmetries, are considered as an illustration.

Symmetries of coincidence site lattices of cubic lattices

Abstract We consider the symmetries of coincidence site lattices of 3-dimensional cubic lattices. This includes the discussion of the symmetry groups and the Bravais classes of the CSLs. We derive various criteria and necessary conditions for symmetry operations of CSLs. They are used to obtain a complete list of the symmetry groups and the Bravais classes of those CSLs that are generated by a rotation through the angle π.

Twinning by reticular pseudo-merohedry in trigonal, tetragonal and hexagonal crystals.

Twin laws for trigonal, tetragonal and hexagonal crystals describing twins with principal axes inclined by an angle Phi > 0 are analysed. Twins by reticular merohedry (i.e. obliquity delta = 0) are possible only for certain values s of the axial ratio c/a. For any other axial ratio r, the laws describe twinning by reticular pseudo-merohedry, i.e. with obliquity delta > 0. It is shown that (a) tandelta is a product of two factors, one of which is sinPhi, the other depends only on the relative deviation of r from s; (b) tandelta approximately epsilon, where epsilon denotes the deformation parameter introduced by Bonnet & Durand [Philos. Mag. (1975), 32, 997-1006]. The angle Phi is listed for all cases of reticular merohedry of trigonal, tetragonal and hexagonal (i.e. optically uniaxial) crystals with twin index Sigma </= 5. Mallard's criterion requires that twin laws by (reticular) pseudo-merohedry have Sigma </= 5 and delta </= 6 degrees. Le Page [J. Appl. Cryst. (2002), 35, 175-181] has written a program determining laws with twin index Sigma </= Sigma(max) and obliquity delta </= delta(max) for any given lattice geometry. Here those solutions are analysed and completed for optically uniaxial crystals. Their lattices are characterized by the Bravais class (tP, tI, hP or hR) and the axial ratio c/a = r. For small delta(max), most solutions are related to (reticular) merohedry for an appropriate value s approximately r of the axial ratio. It is argued that other solutions, which are not related to (reticular) merohedry, are not needed to explain observed laws of growth twinning but may be important to interpret observed laws of deformation twinning.

TEM study of superstructure in a perovskite lead lanthanum zirconate stannate titanate ceramic

The superstructure of an antiferroelectric Pb 0.97La 0.02(Zr 0.66Sn 0.25Ti 0.09)O 3 phase, whose composition is near the morphotropic-phase boundary, was characterized. Systematic selected-area diffraction revealed that there were two kinds of superlattice reflections in the pseudocubic reciprocal lattice, i.e. 1 2 hk l superlattice reflections ( h, k, l all odd), and g± 1 7.24 a ∗ +b ∗ one-dimensional incommensurate superlattice reflections, where g denotes the vectors of fundamental or 1 2 hk l superlattice reflections. Convergent-beam electron diffraction disclosed that the average structure of the phase was rhombohedral with space group of R 3m . Based on the rhombohedral reciprocal lattice, the 1 2 hk l reflections were no longer superlattice but fundamental reflections, and the reciprocal vector of the one-dimensional incommensurate reflections was re-expressed as H=ha ∗ +kb ∗ +lc ∗ ± 1 7.24 a ∗ +b ∗ , where h, k, l are integers and (− h + k + l) = 3 n. In the light of the average structure and the reflection condition, the superspace Bravais class of the phase with one-dimensional incommensurate structure was determined to be P 11 R 3m in a (3 + 1)-dimensional space. In addition, the origins of the superlattice reflections were also examined and discussed.

Report of a subcommittee on the nomenclature of n-dimensional crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups.

The Second Report of the Subcommittee on the Nomenclature of n-Dimensional Crystallography recommends specific symbols for R-irreducible groups in 4 and higher dimensions (nD), for centrings, for Bravais classes, for arithmetic crystal classes and for space groups (space-group types). The relation with higher-dimensional crystallographic groups used for the description of aperiodic crystals is briefly discussed. The Introduction discusses the general definitions used in the Report.

Screw rotations and glide mirrors: crystallography in Fourier space.

The traditional crystallographic symmetry elements of screw axes and glide planes are subdivided into those that are removable and those that are essential. A simple real-space criterion, depending only on Bravais class, determines which types can be present in any space group. This terminological refinement is useful in expressing the complementary relation between the real-space and Fourier-space formulations of crystal symmetry, particularly in the case of the two nonsymmorphic space groups that have no systematic extinctions (I212121 and I213). A simple analysis in Fourier space demonstrates the nonsymmorphicity of these two space groups, which finds its physical expression not in a characteristic absence of Bragg peaks, but in a characteristic presence of electronic level degeneracies.

Incommensurate modulated structure study of a Cu–Zn–Al–Zr phase

Zr was added to Cu–Zn–Al shape memory alloy as a grain refinement element. There are two new phases, Cu 50.2Zr 24.6Al 17.3Zn 7.9 (at%) ( Z 1 phase) and Cu 57.4Zr 20.4Zn 10.3Al 11.9 (at%) ( Z 2 phase), in the Cu–19.0Zn–13.1Al–1.1Mn–0.3Zr (at%) alloy. The structure of the Cu 57.4Zr 20.4Zn 10.3Al 11.9 (at%) ( Z 2) phase is studied in detail in the present paper. The results of electron diffraction and high-resolution electron microscopy (HREM) investigations indicated a one-dimensional incommensurate modulated structure in the Z 2 phase. The average structure of the phase is base-centred orthorhombic. The systematic reflection conditions associated with the main and satellite reflections demonstrate that the Bravais class of the Z 2 phase is P Cmmm 11 1 type in a (3+1) dimensional space for the incommensurate modulated structure. The substitution modulation effects of the Al and Zr elements are revealed by energy-filtered transmission electron microscopy (EFTEM). { 1 3 ̄ 2 ̄ } compound-type twins are observed in the Z 2 phase. The relationship between the Z 1 and Z 2 phases in the alloy is discussed.

Voronoi Polyhedra: A Useful Tool to Determine the Symmetry and Bravais Class of Crystal Lattices

AbstractA handy PC‐supported method will be described to deduce the Bravais lattice (Bravais class) of a crystal lattice from an arbitrary set of lattice parameters. The method involves the construction of the related Voronoi polyhedron (Dirichlet domain, Wirkugsbereich, Fedorov parallelohedron, Wigner‐Seitz cell) whose metric relations between its edge lengths can be used to unequivocally identify the Bravais lattice. This is particularly useful if an unconventional setting of the unit cell, or inaccurately determined lattice parameters may result in misleading conclusions as to identify the correct Bravais lattice.

Bravais classes and space groups for trigonal and hexagonal quasiperiodic crystals of arbitrary finite rank

To demonstrate the power of the Fourier-space approach to crystallography, the Bravais classes and space groups of hexagonal and trigonal quasiperiodic crystals are derived for lattices of arbitrary finite rank. The specification of the space groups for each Bravais class is given by an elementary extension of the rank-4 case. The conventional classification of incommensurately modulated hexagonal and trigonal crystals, previously derived using the superspace approach for Bravais classes up to rank (3 + 3) [Janner, Janssen & de Wolff (1993). Acta Cryst. A39, 658–666] and for superspace groups of rank (3 + 1) [de Wolff, Janssen & Janner (1981). Acta Cryst. A37, 625–636], is easily extracted from the general classification for modulations of any finite rank.

Bravais classes for the simplest incommensurate crystal phases

Through the reformulation of crystallography that treats periodic and quasiperiodic structures on an equal footing in three-dimensional Fourier space, a novel computation is given of the Bravais classes for the simplest kinds of incommensurately modulated crystals: (3 + 3) Bravais classes in the cubic system and (3 + 1) Bravais classes in any of the other six crystal systems. The contents of a Bravais class are taken to be sets of ordinary three-dimensional wave vectors inferred from a diffraction pattern. Because no finer distinctions are made based on the intensities of the associated Bragg peaks, a significantly simpler set of Bravais classes is found than Janner, Janssen & de Wolff [Acta Cryst. (1983). A39, 658–666] find by defining their Bravais classes in higher-dimensional superspace. In our scheme, the Janner, Janssen & de Wolff categories appear as different ways to describe identical sets of three-dimensional wave vectors when those sets contain crystallographic (3 + 0) sublattices belonging to more than a single crystallographic Bravais class. While such further discriminations are important to make when the diffraction pattern is well described by a strong lattice of main reflections and weaker satellite peaks, by not making them at the fundamental level of the Bravais class, the crystallographic description of all quasiperiodic materials is placed on a single unified foundation.

The space groups of orthorhombic approximants to the icosahedral quasilattice

The space groups of the orthorhombic approximant lattices to the primitive icosahedral quasilattice are classified. There exist three Bravais classes: Pmmm, Cmmm and Immm. The basis vectors of the Bravais lattice of an approximant are parallel to two-, three- and/or fivefold axes of the quasilattice. It is found that there exist many nonsymmorphic space groups with a common Bravais lattice in addition to symmorphic ones. This is because glides commonly appear.

Copernican crystallography.

Redundancies are pointed out in the widely used extension of the crystallographic concept of Bravais class to quasiperiodic materials. Such pitfalls can be avoided by abandoning the obsolete paradigm that bases ordinary crystallography on microscopic periodicity. The broadening of crystallography to include quasiperiodic materials is accomplished by defining the point group in terms of indistinguishable (as opposed to identical) densities.

Special lines of quasilattices. I. The case of irreducible quasilattices in two and three dimensions

A general theory for special lines in quasilattices is developed. Using the theory, the author classifies completely the special lines of the three Bravais classes of icosahedral quasilattices in three dimensions and another three of n-gonal quasilattices in two dimensions, where n=8, 10 and 12. He establishes compatibility relationships between the classes of special lines and those of special points for each of the six Bravais classes of quasilattices.

Three-dimensional black-and-white Bravais quasilattices with (2+1)-reducible point groups

The author has investigated three-dimensional black-and-white Bravais quasilattices (BWBQL) with point groups D8h (8/mmm), D10h (10/mmm), D12h (12/mm4im) and D5d (5 m), which are (2+1)-reducible. There exists seven Bravais classes of BWBQLs, P2c8/mmm, Pp8/mmm, PF8/mmm, Fp8/mmm, P2c10/mmm, P2c12/mmm and Pp5m.