From affine A4 to affine H2: group-theoretical analysis of fivefold symmetric tilings
The projections of lattices may be used as models of quasicrystals, and the particular affine extension of the H2 symmetry as a subgroup of A4, discussed in this work, presents a different perspective on fivefold symmetric quasicrystallography. Affine H2 is obtained as the subgroup of affine A4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral face...
Source: Acta Crystallographica Section A - April 25, 2022 Category: Chemistry Authors: Ozdes Koca, N. Koc, R. Koca, M. Al-Reasi, R. Tags: affine groups quasicrystals aperiodic tilings lattices projections of polytopes research papers Source Type: research
Correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. II. Experimental results
While the crystal structure of the polymorph phase can be studied in three dimensions conveniently by X-ray methods like grazing-incidence X-ray diffraction (GIXD), the first monolayer is only accessible by surface-sensitive methods that allow the determination of a two-dimensional lattice. Here, GIXD measurements with sample rotation are compared with distortion-corrected low-energy electron diffraction (LEED) experiments on conjugated molecules: 3,4;9,10-perylenetetracarboxylic dianhydride (PTCDA), 6,13-pentacenequinone (P2O), 1,2;8,9-dibenzopentacene (trans-DBPen) and dicyanovinyl-quaterthiophene (DCV4T-Et2) grown by ph...
Source: Acta Crystallographica Section A - April 11, 2022 Category: Chemistry Authors: Simbrunner, J. Domke, J. Sojka, F. Jeindl, A. Otto, F. Gruenewald, M. Hofmann, O.T. Fritz, T. Resel, R. Forker, R. Tags: organic epitaxy GIXD LEED thin films indexing polymorphism research papers Source Type: research
Correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. I. Theory
The epitaxial growth of molecular crystals at single-crystalline surfaces is often strongly related to the first monolayer at the substrate surface. The present work presents a theoretical approach to compare three-dimensional lattices of epitaxially grown crystals with two-dimensional lattices of the molecules formed within the first monolayer. Real-space and reciprocal-space representations are considered. Depending on the crystallographic orientation relative to the substrate surface, proper linear combinations of the lattice vectors of the three-dimensional unit cell result in a rhomboid in the xy plane, representing a...
Source: Acta Crystallographica Section A - April 11, 2022 Category: Chemistry Authors: Simbrunner, J. Domke, J. Forker, R. Resel, R. Fritz, T. Tags: crystallographic lattices surface unit cell GIXD mathematical crystallography thin films research papers Source Type: research
Ab initio reconstruction from one-dimensional crystal diffraction data
Filamentary and rod-like assemblies are ubiquitous in biological systems, and single such assemblies can form one-dimensional (1D) crystals. New, intense X-ray sources, such as X-ray free-electron lasers, make it feasible to measure diffraction data from single 1D crystals. Such experiments would present some advantages, since cylindrical averaging of the diffraction data in conventional fiber diffraction analysis is avoided, there is coherent signal amplification relative to single-particle imaging, and the diffraction data are oversampled compared with those from a 3D crystal so that the phase problem is better determine...
Source: Acta Crystallographica Section A - April 5, 2022 Category: Chemistry Authors: Arnal, R.D. Millane, R.P. Tags: 1D crystals phase problem iterative projection algorithms shrink-wrap algorithm filaments research papers Source Type: research
nmfMapping: a cloud-based web application for non-negative matrix factorization of powder diffraction and pair distribution function datasets
A cloud-hosted web-based software application, nmfMapping, for carrying out a non-negative matrix factorization of a set of powder diffraction or atomic pair distribution function datasets is described. This application allows structure scientists to find trends rapidly in sets of related data such as from in situ and operando diffraction experiments. The application is easy to use and does not require any programming expertise. It is available at https://pdfitc.org/. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - April 5, 2022 Category: Chemistry Authors: Thatcher, Z. Liu, C.-H. Yang, L. McBride, B.C. Thinh Tran, G. Wustrow, A. Karlsen, M.A. Neilson, J.R. Ravnsb æ k, D.B. Billinge, S.J.L. Tags: non-negative matrix factorization NMF pair distribution function PDF data analysis web applications research papers Source Type: research
A Metropolis Monte Carlo algorithm for merging single-particle diffraction intensities
Single-particle imaging with X-ray free-electron lasers depends crucially on algorithms that merge large numbers of weak diffraction patterns despite missing measurements of parameters such as particle orientations. The expand – maximize – compress (EMC) algorithm is highly effective at merging single-particle diffraction patterns with missing orientation values, but most implementations exhaustively sample the space of missing parameters and may become computationally prohibitive as the number of degrees of freedom extends beyond orientation angles. This paper describes how the EMC algorithm can be modified to employ ...
Source: Acta Crystallographica Section A - April 5, 2022 Category: Chemistry Authors: Mobley, B.R. Schmidt, K.E. Chen, J.P.J. Kirian, R.A. Tags: coherent X-ray diffractive imaging (CXDI) single particles molecular orientation determination XFEL research papers Source Type: research
Isogonal piecewise-linear embeddings of 1-periodic knots and links, and related 2-periodic chain-link and knitting patterns
Families of 1- and 2-periodic knots and weavings that have isogonal (vertex-transitive) piecewise-linear embeddings are described. In these structures there is just one thread, or multiple threads with parallel or collinear axes. The principal structures are a large family of 1-periodic knots and related multi-thread infinite links, knitting patterns and chain-link weaving. The relevance to synthetic chemistry is described in terms of targets for designed synthesis such as mechanically interlocked polymers. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - April 4, 2022 Category: Chemistry Authors: O'Keeffe, M. Treacy, M.M.J. Tags: knots weaves isogonal structures links knitting patterns mechanically interlocked polymers research papers Source Type: research
Bond topology of chain, ribbon and tube silicates. Part I. Graph-theory generation of infinite one-dimensional arrangements of (TO4)n − tetrahedra
Chain, ribbon and tube silicates are based on one-dimensional polymerizations of (TO4)n − tetrahedra, where T = Si4+ plus P5+, V5+, As5+, Al3+, Fe3+ and B3+. Such polymerizations may be represented by infinite graphs (designated chain graphs) in which vertices represent tetrahedra and edges represent linkages between tetrahedra. The valence-sum rule of bond-valence theory limits the maximum degree of any vertex to 4 and the number of edges linking two vertices to 1 (corner-sharing tetrahedra). The unit cell (or repeat unit) of the chain graph generates the chain graph through action of translational symmetry operators. T...
Source: Acta Crystallographica Section A - April 4, 2022 Category: Chemistry Authors: Day, M.C. Hawthorne, F.C. Tags: bond topology tetrahedra chain graph adjacency matrix vertex set edge set silicates graph generation isomorphism translational symmetry research papers Source Type: research
Coordination sequences of periodic structures are rational via automata theory
The conjecture of Grosse-Kunstleve et al. [(1996), Acta Cryst. A52, 879 – 889], that coordination sequences of periodic structures in n-dimensional Euclidean space are rational, is proved. This has been recently proven by Nakamura et al. [(2021), Acta Cryst. A77, 138 – 148]; however, the proof presented here is a straightforward application of classic techniques from automata theory. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - February 18, 2022 Category: Chemistry Authors: Kopczy ń ski, E. Tags: coordination sequences periodic structures rational generating functions short communications Source Type: research
A simplified relationship between the modified O-lattice and the rotation matrix for generating the coincidence site lattice of an arbitrary Bravais lattice system
The coincidence site lattice (CSL) is important for characterizing the structure and energy state of grain boundaries in polycrystalline materials. A simplified relationship between the modified O-lattice and the corresponding rotation matrix is proposed to establish a general formula for the CSL and the near coincidence site lattice (NCSL) in Bravais lattice systems. The general formula paves the way to computer simulation and crystallographic analysis of grain boundaries. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - February 18, 2022 Category: Chemistry Authors: Liu, H. Tags: grain boundaries phase transitions coincidence site lattice CSL near coincidence site lattice NCSL rotation matrix O-lattice research papers Source Type: research
Tangled piecewise-linear embeddings of trivalent graphs
A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent grap...
Source: Acta Crystallographica Section A - February 18, 2022 Category: Chemistry Authors: O'Keeffe, M. Treacy, M.M.J. Tags: tangled structures trivalent graphs tangled molecular targets research papers Source Type: research
On the combinatorics of crystal structures: number of Wyckoff sequences of given length
A formula for the calculation of the number of Wyckoff sequences of a given length is presented, based on the combinatorics of multisets with finite multiplicities and a generating function approach, assuming a certain space-group type and taking into account the number of non-fixed and fixed Wyckoff positions, respectively. The formula is applied to the 44 distinguishable combinatorial types of the 230 space-group types. A comparison is made between the calculated frequencies of occurrence of Wyckoff sequences of given space-group type and length and the observed ones for actual crystal structures, as retrieved from the P...
Source: Acta Crystallographica Section A - February 4, 2022 Category: Chemistry Authors: Hornfeck, W. Tags: Wyckoff sequences combinatorics short communications Source Type: research
Combinatorial aspects of the L ö wenstein avoidance rule. Part II: local and global constraints
This article deals with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets, with an application to the 200 2-periodic nets listed on the RCSR (Reticular Chemistry Structure Resource) site. A companion paper described a calculation technique of independence ratios of finite graphs based on propositional calculus. This paper focuses on criteria for the choice of the translation subgroup and of the transversal. The translation subgroup should be chosen in such a way as to eliminate every cycle in the quotient graph that is shorter than structural cycles, or rings, of the net. Topolo...
Source: Acta Crystallographica Section A - February 4, 2022 Category: Chemistry Authors: Moreira de Oliveira Jr, M. de Abreu Mendes, F. Eon, J.-G. Tags: nets quotient graphs independence quotient research papers Source Type: research
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 t...
Source: Acta Crystallographica Section A - February 4, 2022 Category: Chemistry Authors: Nikoli ć , B. Milo š evi ć , I. Vukovi ć , T. Lazi ć , N. Dmitrovi ć , S. Popovi ć , Z. Damnjanovi ć , M. Tags: group theory layer groups computer algebra crystallographic database Brillouin-zone topology research papers Source Type: research
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...
Source: Acta Crystallographica Section A - February 4, 2022 Category: Chemistry Authors: Campbell, B.J. Stokes, H.T. Perez-Mato, J.M. Rodr í guez-Carvajal, J. Tags: time reversal magnetic space group Belov – Neronova Smirnova Opechowski Guccione Hermann Mauguin research papers Source Type: research
