Enumeration and tabulation of magnetic (3+d)-dimensional superspace groups.

A magnetic superspace group (MSSG) simultaneously constrains both the magnetic and non-magnetic (e.g. displacive, occupational, rotation and strain) degrees of freedom of an incommensurately modulated magnetic crystal. We present the first enumeration and tabulation of all non-equivalent (3+d)-dimensional magnetic superspace groups for d = 1, 2 and 3 independent incommensurate modulations, along with a number, symbol and reference setting for each group. We explain the process for generating an exhaustive set of inequivalent magnetic superspace groups, describe several examples, and show how the tables can be accessed via the ISO(3+d)D interface within the ISOTROPY Software Suite. We recommend that published incommensurate magnetic structures indicate a magnetic superspace-group number and symbol from these tables, as well as the transformation matrix from the published group setting to the reference setting used in these tables.

Introducing a unified magnetic space-group symbol.

The two commonly used systems of magnetic space-group (MSG) symbols, with accompanying numbers and settings, are those of Belov-Neronova-Smirnova (BNS) and Opechowski-Guccione (OG). The symbols from both systems have been used for several decades now. Both have advantages and disadvantages. Both present challenges of interpretation to novice and expert users alike, which can inhibit understanding and lead to errors in published magnetic structures. To address each of these challenges going forward, a new unified (UNI) MSG symbol is introduced, which combines a modified BNS symbol with essential information from the OG symbol.

The ISOTILT software for discovering cooperative rigid-unit rotations in networks of interconnected rigid units.

A user-friendly web-based software tool called 'ISOTILT' is introduced for detecting cooperative rigid-unit modes (RUMs) in networks of interconnected rigid units (e.g. molecules, clusters or polyhedral units). This tool implements a recently described algorithm in which symmetry-mode patterns of pivot-atom rotation and displacement vectors are used to construct a linear system of equations whose null space consists entirely of RUMs. The symmetry modes are first separated into independent symmetry-mode blocks and the set of equations for each block is solved separately by singular value decomposition. ISOTILT is the newest member of the ISOTROPY Software Suite. Here, it is shown how to prepare structural and symmetry-mode information for use in ISOTILT, how to use each of ISOTILT's input fields and options, and how to use and interpret ISOTILT output.

An algebraic approach to cooperative rotations in networks of interconnected rigid units.

Crystalline solids consisting of three-dimensional networks of interconnected rigid units are ubiquitous amongst functional materials. In many cases, application-critical properties are sensitive to rigid-unit rotations at low temperature, high pressure or specific stoichiometry. The shared atoms that connect rigid units impose severe constraints on any rotational degrees of freedom, which must then be cooperative throughout the entire network. Successful efforts to identify cooperative-rotational rigid-unit modes (RUMs) in crystals have employed split-atom harmonic potentials, exhaustive testing of the rotational symmetry modes allowed by group representation theory, and even simple geometric considerations. This article presents a purely algebraic approach to RUM identification wherein the conditions of connectedness are used to construct a linear system of equations in the rotational symmetry-mode amplitudes.

Understanding the Behavior of the Above-Room-Temperature Molecular Ferroelectric 5,6-Dichloro-2-methylbenzimidazole Using Symmetry Adapted Distortion Mode Analysis.

The exploitable properties of many functional materials are intimately linked with symmetry-changing phase transitions. These include properties such as ferroelectricity, second harmonic generation, conductivity, magnetism and many others. We describe a new symmetry-inspired method for systematic and exhaustive evaluation of the symmetry changes possible in molecular systems using molecular distortion modes, and how different models can be automatically tested against diffraction data. The method produces a quantitative structural landscape from which the most appropriate structural description of a child phase can be chosen. It can be applied to any molecular or molecular-fragment containing material where a (semi) rigid molecule description is appropriate. We exemplify the method on 5,6-dichloro-2-methylbenzimidazole (DC-MBI), an important molecular ferroelectric. We show that DC-MBI undergoes an unusual symmetry-lowering transition on warming from orthorhombic Pca21 ( T ≲ 400 K) to monoclinic Pc. Contrary to expectations, the high temperature phase of DC-MBI remains polar.

A general algorithm for generating isotropy subgroups in superspace.

This paper presents a general algorithm for generating the isotropy subgroups of superspace extensions of crystallographic space groups involving arbitrary superpositions of multi-k order parameters from incommensurate and commensurate k vectors. Several examples are presented in detail in order to illuminate each step of the algorithm. The practical outcome is that one can now start with any commensurate parent crystal structure and generate a structure model for any conceivable incommensurate modulation of that parent, fully parameterized in terms of order parameters of irreducible representations at the relevant wavevectors. The resulting modulated structures have (3 + d)-dimensional superspace-group symmetry. Because incommensurate structures are now commonly encountered in the context of many scientifically and technologically important functional materials, the opportunity to apply the powerful methods of group representation theory to this broader class of structural distortions is very timely.

An Exhaustive Symmetry Approach to Structure Determination: Phase Transitions in Bi2Sn2O7.

The exploitable properties of many materials are intimately linked to symmetry-lowering structural phase transitions. We present an automated and exhaustive symmetry-mode method for systematically exploring and solving such structures which will be widely applicable to a range of functional materials. We exemplify the method with an investigation of the Bi2Sn2O7 pyrochlore, which has been shown to undergo transitions from a parent γ cubic phase to ß and α structures on cooling. The results include the first reliable structural model for ß-Bi2Sn2O7 (orthorhombic Aba2, a = 7.571833(8), b = 21.41262(2), and c = 15.132459(14) Å) and a much simpler description of α-Bi2Sn2O7 (monoclinic Cc, a = 13.15493(6), b = 7.54118(4), and c = 15.07672(7) Å, ß = 125.0120(3)°) than has been presented previously. We use the symmetry-mode basis to describe the phase transition in terms of coupled rotations of the Bi2O' anti-cristobalite framework, which allow Bi atoms to adopt low-symmetry coordination environments favored by lone-pair cations.

Crystal and magnetic structures of hexagonal YMnO3.

The available data on the structural and magnetic transitions in multiferroic hexagonal YMnO3 have been reviewed, first making use of the computer programs from the group theoretical ISOTROPY software suite to list possible crystal and magnetic structures, then taking into account the capability of neutron diffraction and other physical methods to distinguish them. This leads to a clear view of the transformation sequence, as follows. Hexagonal YMnO3 is paraelectric in P63/mmc at elevated temperatures, and undergoes a single structural transition on cooling through 1250 K to a ferrielectric phase in P63cm that is retained through room temperature. At a much lower temperature, 70 K, there is a magnetic transition from paramagnetic to a triangular antiferromagnetic arrangement, most likely with symmetry P63'cm'. Comment is made on the unusual coupling of ferroelectric and magnetic domains reported to occur in this material, as well as on the so-called `giant magneto-elastic' effect.

Tabulation of irreducible representations of the crystallographic space groups and their superspace extensions.

New tables of irreducible representations (IRs) are introduced for the 230 crystallographic space groups (SGs) in three-dimensional space, at both special and non-special k vectors, and for their extensions to (3 + d)-dimensional superspace (`superspace-extended SGs' or SSESGs). Neither a tabulation of SG IR matrices for non-special k vectors nor a tabulation of SSESG IR matrices for d > 1 have been previously published. These tabulations are made possible by a new form in which the IR matrices of SGs are separated as a product of a translation part T and a point-operation part P, and where the IR matrices of SSESGs are separated as a product of a phase-shift part Q and a point-operation part P(s). Both T and Q have a simple prescribed form that does not need to be tabulated. Also, the new IR matrices are in a convenient block form which allows one to see by inspection which parts of the matrices and the associated order parameters belong to which arm of the star of k. In addition to complex IR matrices, real physically irreducible representation (PIR) matrices are tabulated. The new IR and PIR tables are available on the ISO-IR website (http://stokes.byu.edu/iso/irtables.php) in both convenient human-readable and computer-readable forms.

Equivalence of superspace groups.

An algorithm is presented which determines the equivalence of two settings of a (3 + d)-dimensional superspace group (d = 1, 2, 3). The algorithm has been implemented as a web tool findssg on SSG(3+d)D, providing the transformation of any user-given superspace group to the standard setting of this superspace group in SSG(3+d)D. It is shown how the standard setting of a superspace group can be directly obtained by an appropriate transformation of the external-space lattice vectors (the basic structure unit cell) and a transformation of the internal-space lattice vectors (new modulation wavevectors are linear combinations of old modulation wavevectors plus a three-dimensional reciprocal-lattice vector). The need for non-standard settings in some cases and the desirability of employing standard settings of superspace groups in other cases are illustrated by an analysis of the symmetries of a series of compounds, comparing published and standard settings and the transformations between them. A compilation is provided of standard settings of compounds with two- and three-dimensional modulations. The problem of settings of superspace groups is discussed for incommensurate composite crystals and for chiral superspace groups.

Constrained evolutionary algorithm for structure prediction of molecular crystals: methodology and applications.

Evolutionary crystal structure prediction proved to be a powerful approach for studying a wide range of materials. Here we present a specifically designed algorithm for the prediction of the structure of complex crystals consisting of well defined molecular units. The main feature of this new approach is that each unit is treated as a whole body, which drastically reduces the search space and improves the efficiency, but necessitates the introduction of new variation operators described here. To increase the diversity of the population of structures, the initial population and part (~20%) of the new generations are produced using space-group symmetry combined with random cell parameters, and random positions and orientations of molecular units. We illustrate the efficiency and reliability of this approach by a number of tests (ice, ammonia, carbon dioxide, methane, benzene, glycine and butane-1,4-diammonium dibromide). This approach easily predicts the crystal structure of methane A containing 21 methane molecules (105 atoms) per unit cell. We demonstrate that this new approach also has a high potential for the study of complex inorganic crystals as shown on examples of a complex hydrogen storage material Mg(BH(4))(2) and elemental boron.

The superstructure determination of displacive distortions via symmetry-mode analysis.

For any crystal structure that can be viewed as a low-symmetry distortion of some higher-symmetry parent structure, one can represent the details of the distorted structure in terms of symmetry-adapted distortion modes of the parent structure rather than the traditional list of atomic xyz coordinates. Because most symmetry modes tend to be inactive, and only a relatively small number of mode amplitudes are dominant in producing the observed distortion, symmetry-mode analysis can greatly simplify the determination of a displacively distorted structure from powder diffraction data. This is an important capability when peak splittings are small, superlattice intensities are weak or systematic absences fail to distinguish between candidate symmetries. Here, the symmetry-mode basis is treated as a binary (on/off) parameter set that spans the space of all possible P1 symmetry distortions within the experimentally determined supercell. Using the average R(wp) over repeated local minimizations from random starting points as a cost function for a given mode set, global search strategies are employed to identify the active modes of the distortion. This procedure automatically yields the amplitudes of the active modes and the associated atomic coordinates. The active modes are then used to detect the space-group symmetry of the distorted phase (i.e. the type and location of each of the parent symmetry elements that remain within the distorted supercell). Once a handful of active modes are identified, traditional refinement methods readily yield their amplitudes and the resulting atomic coordinates. A final symmetry-mode refinement is then performed in the correct space-group symmetry to improve the sensitivity to any secondary modes present.

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.

Order parameters for phase transitions to structures with one-dimensional incommensurate modulations.

Phase transitions that result in incommensurate structural modulations are widely observed in crystalline solids and are relevant to a broad range of physical phenomena in magnetic, electronic, optical and structural materials. While the (3+1)-dimensional superspace-group symmetries associated with one-dimensional modulations have been tabulated, the order parameters that produce these modulations have not been explored in detail. Here, using group-theoretical methods, we present a unique and exhaustive enumeration of the isotropy subgroups (and their corresponding order-parameter directions) belonging to irreducible representations of the (3+1)-dimensional superspace extensions of the 230 crystallographic space groups at all incommensurate k points. The vast majority of experimentally observed incommensurately modulated structures have order parameters belonging to one of these subgroups.

Octahedral tilting in cation-ordered perovskites--a group-theoretical analysis.

Group-theoretical methods are used to enumerate the structures of ordered perovskites, in which 1:2 and 1:3 ordering of B and B' cations is considered in combination with the ubiquitous BX6 (or B'X6) octahedral tilting. The cation ordering on the B-cation site is described by irreducible representations of the Pm3 m space group of the cubic aristotype: Lambda1 (k = 1/3,1/3,1/3) for the cation ordering pattern in the 1:2 compound A3BB2'X9 and M1+ (k = 1/2,1/2,0) for the cation ordering in the 1:3 compound A4BB3'X12. The octahedral tilting is mediated by the irreducible representations M3+ and R4+. Ten distinct structures have been identified in the 1:2 case and 11 structures for 1:3.

Group-theoretical analysis of octahedral tilting in ferroelectric perovskites.

Group-theoretical methods are used to analyze perovskite structures where both ferroelectric cation displacements and simple tilting of octahedral units are present. This results in a list of 40 different structures, each with a unique space-group symmetry. The list is compared with that of Aleksandrov & Bartolomé [Phase Transit. (2001), 74, 255-335] and a number of differences are found. The group-subgroup relationships between the structures are also determined, along with an indication of those phase transitions that must be first order by Landau theory.