Perfect precise colorings of plane semiregular tilings
A coloring of a planar semiregular tiling {\cal T} is an assignment of a unique color to each tile of {\cal T}. If G is the symmetry group of {\cal T}, the coloring is said to be perfect if every element of G induces a permutation on the finite set of colors. If {\cal T} is k-valent, then a coloring of {\cal T} with k colors is said to be precise if no two tiles of {\cal T} sharing the same vertex have the same color. In this work, perfect precise colorings are obtained for some families of k-valent semiregular tilings in the plane, where k ≤ 6. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - August 17, 2023 Category: Chemistry Authors: Loquias, M.J.C. Santos, R.B. Tags: semiregular tilings hyperbolic tilings perfect colorings precise colorings triangle groups research papers Source Type: research
Optimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data
Estimating the error in the merged reflection intensities requires a full understanding of all the possible sources of error arising from the measurements. Most diffraction-spot integration methods focus mainly on errors arising from counting statistics for the estimation of uncertainties associated with the reflection intensities. This treatment may be incomplete and partly inadequate. In an attempt to fully understand and identify all the contributions to these errors, three methods are examined for the correction of estimated errors of reflection intensities in electron diffraction data. For a direct comparison, the thr...
Source: Acta Crystallographica Section A - August 14, 2023 Category: Chemistry Authors: Khouchen, M. Klar, P.B. Chintakindi, H. Suresh, A. Palatinus, L. Tags: error modelling error analysis data reduction electron diffraction research papers Source Type: research
Delone lattice studies in C3, the space of three complex variables
The Delone (Selling) scalars, which are used in unit-cell reduction and in lattice-type determination, are studied in C3, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars. The transformations at boundaries of the Selling-reduced orthant are described as matrices of operators. A graphical representation as the projections onto the three coordinates is described. Note, in his later publications, Boris Delaunay used the Russian version of his surname, Delone. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - August 10, 2023 Category: Chemistry Authors: Andrews, L.C. Bernstein, H.J. Tags: lattices Delone Delaunay cell reduction Selling C3 space research papers Source Type: research
Measuring lattices
Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3. The process is a simplified version of the process needed when working with the more complex six-dimensional real space of Ni...
Source: Acta Crystallographica Section A - August 10, 2023 Category: Chemistry Authors: Andrews, L.C. Bernstein, H.J. Tags: Delaunay Delone Selling lattices unit cell research papers Source Type: research
Background optimization of powder electron diffraction for implementation of the e-PDF technique and study of the local structure of iron oxide nanocrystals
The local structural characterization of iron oxide nanoparticles is explored using a total scattering analysis method known as pair distribution function (PDF) (also known as reduced density function) analysis. The PDF profiles are derived from background-corrected powder electron diffraction patterns (the e-PDF technique). Due to the strong Coulombic interaction between the electron beam and the sample, electron diffraction generally leads to multiple scattering, causing redistribution of intensities towards higher scattering angles and an increased background in the diffraction profile. In addition to this, the electron...
Source: Acta Crystallographica Section A - July 25, 2023 Category: Chemistry Authors: Mogili, N.V.V. Verissimo, N.C. Abeykoon, A.M.M. Bozin, E.S. Bettini, J. Leite, E.R. Souza Junior, J.B. Tags: electron powder diffraction background subtraction electron pair distribution function (e-PDF) electron reduced density function (e-RDF) nanocrystalline maghemite research papers Source Type: research
Patch frequencies in rhombic Penrose tilings
This exposition presents an efficient algorithm for an exact calculation of patch frequencies for rhombic Penrose tilings. A construction of Penrose tilings via dualization is recalled and, by extending the known method for obtaining vertex configurations, the desired algorithm is obtained. It is then used to determine the frequencies of several particularly large patches which appear in the literature. An analogous approach works for a particular class of tilings and this is also explained in detail for the Ammann – Beenker tiling. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - July 25, 2023 Category: Chemistry Authors: Maz á č , J. Tags: patch frequency tiling dualization method research papers Source Type: research
Approximating lattice similarity
A method is proposed for choosing unit cells for a group of crystals so that they all appear as nearly similar as possible to a selected cell. Related unit cells with varying cell parameters or indexed with different lattice centering can be accommodated. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - July 24, 2023 Category: Chemistry Authors: Andrews, L.C. Bernstein, H.J. Sauter, N.K. Tags: lattice matching Delaunay Delone Niggli Selling research papers Source Type: research
Uri Shmueli (1928 – 2023)
(Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - June 27, 2023 Category: Chemistry Authors: Brock, C.P. Tags: International Tables for Crystallography structure-factor statistics crystal packing thermal motion atomic displacement parameters obituaries Source Type: research
Andr é Authier (1932 – 2023)
(Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - June 20, 2023 Category: Chemistry Authors: Epelboin, Y. Tags: European Crystallographic Association International Union of Crystallography dynamical diffraction crystal defects X-ray topography obituaries Source Type: research
An invertible seven-dimensional Dirichlet cell characterization of lattices
Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice vectors) or by Delaunay-reduced cells (based on four non-coplanar vectors summing to zero and all meeting at obtuse or right angles) is commonly performed. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner – Seitz (or Dirichlet, or Voronoi) cell of the lat...
Source: Acta Crystallographica Section A - June 20, 2023 Category: Chemistry Authors: Bernstein, H.J. Andrews, L.C. Xerri, M. Tags: Dirichlet cell Voronoi cell Wigner – Seitz cell Niggli reduction research papers Source Type: research
Machine learning for classifying narrow-beam electron diffraction data
As an alternative approach to X-ray crystallography and single-particle cryo-electron microscopy, single-molecule electron diffraction has a better signal-to-noise ratio and the potential to increase the resolution of protein models. This technology requires collection of numerous diffraction patterns, which can lead to congestion of data collection pipelines. However, only a minority of the diffraction data are useful for structure determination because the chances of hitting a protein of interest with a narrow electron beam may be small. This necessitates novel concepts for quick and accurate data selection. For this pur...
Source: Acta Crystallographica Section A - June 20, 2023 Category: Chemistry Authors: Matinyan, S. Demir, B. Filipcik, P. Abrahams, J.P. van Genderen, E. Tags: diffraction single-molecule electron diffraction TEM transmission electron microscopy machine learning neural networks research papers Source Type: research
Double-slit X-ray dynamical diffraction in curved crystals
The theoretical investigation of double-slit X-ray dynamical diffraction in curved crystals shows that Young's interference fringes are formed. An expression for the period of the fringes has been established which is polarization sensitive. The position of the fringes in the cross section of the beam depends on the deviation from the Bragg exact orientation for a perfect crystal, on the curvature radius and on the thickness of the crystal. This type of diffraction can be used for determination of the curvature radius by measuring the shift of the fringes from the centre of the beam. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - June 20, 2023 Category: Chemistry Authors: Balyan, M.K. Tags: double-slit dynamical diffraction Young's fringes X-ray diffraction interferometer curved crystals research papers Source Type: research
Efficient structure-factor modeling for crystals with multiple components
Diffraction intensities from a crystallographic experiment include contributions from the entire unit cell of the crystal: the macromolecule, the solvent around it and eventually other compounds. These contributions cannot typically be well described by an atomic model alone, i.e. using point scatterers. Indeed, entities such as disordered (bulk) solvent, semi-ordered solvent (e.g. lipid belts in membrane proteins, ligands, ion channels) and disordered polymer loops require other types of modeling than a collection of individual atoms. This results in the model structure factors containing multiple contributions. Most macr...
Source: Acta Crystallographica Section A - June 20, 2023 Category: Chemistry Authors: Afonine, P.V. Adams, P.D. Urzhumtsev, A.G. Tags: structure factors multiple components scattering functions bulk solvent refinement density maps research papers Source Type: research
A note on the wedge reversion antisymmetry operation and 51 types of physical quantities in arbitrary dimensions
The paper by Gopalan [(2020). Acta Cryst. A76, 318 – 327] presented an enumeration of the 41 physical quantity types in non-relativistic physics, in arbitrary dimensions, based on the formalism of Clifford algebra. Gopalan considered three antisymmetries: spatial inversion, 1, time reversal, 1 ′ , and wedge reversion, 1 † . A consideration of the set of all seven antisymmetries (1, 1 ′ , 1 † , 1 ′ † , 1 † , 1 ′ , 1 ′ † ) leads to an extension of the results obtained by Gopalan. It is shown that there are 51 types of physical quantities with distinct symmetry properties in total. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - June 5, 2023 Category: Chemistry Authors: Fabrykiewicz, P. Tags: multivectors wedge reversion antisymmetry Clifford algebra short communications Source Type: research
Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups
This paper presents the basic tools used to describe the global symmetry of so-called bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias& Portier (1982). J. Phys. Colloq. 43, C6-15 – C6-24; Pond& Vlachavas (1983). Proc. R. Soc. Lond. Ser. A, 386, 95 – 143]. The focus here is on the twist rotations such that the s...
Source: Acta Crystallographica Section A - June 5, 2023 Category: Chemistry Authors: Gratias, D. Quiquandon, M. Tags: bicrystallography with complex numbers bilayers coincidence lattices space groups research papers Source Type: research
