Quaternions: what are they, and why do we need to know?
(Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - August 5, 2020 Category: Chemistry Authors: Horn, B.K.P. Tags: quaternions data alignment rotation orientation orthogonal Procrustes problem orientation distribution function ODF scientific commentaries Source Type: research
X-ray scattering study of water confined in bioactive glasses: experimental and simulated pair distribution function
Temperature-dependent total X-ray scattering measurements for water confined in bioactive glass samples with 5.9 nm pore diameter have been performed. Based on these experimental data, simulations were carried out using the Empirical Potential Structure Refinement (EPSR) code, in order to study the structural organization of the confined water in detail. The results indicate a non-homogeneous structure for water inside the pore, with three different structural organizations of water, depending on the distance from the pore surface: (i) a first layer (4 Å ) of interfacial pore water that forms a strong chemical bon...
Source: Acta Crystallographica Section A - July 19, 2020 Category: Chemistry Authors: Khoder, H. Schaniel, D. Pillet, S. Bendeif, E.-E. Tags: confined water bioactive glasses structural analysis pair distribution function research papers Source Type: research
Embedding-theory-based simulations using experimental electron densities for the environment
The basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg – Kohn density functional EHK[ ρ ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where Ψ A is the embedded NA-electron wavefunction and ρ B(r) is a non-negative function in real space integrating to a given number of electrons NB. This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The applica...
Source: Acta Crystallographica Section A - July 19, 2020 Category: Chemistry Authors: Ricardi, N. Ernst, M. Macchi, P. Wesolowski, T.A. Tags: quantum crystallography density embedding multi-scale simulations electronic structure chromophores research papers Source Type: research
On Cayley graphs of {\bb Z}^4
The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components − 1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently ...
Source: Acta Crystallographica Section A - July 15, 2020 Category: Chemistry Authors: Baburin, I.A. Tags: Cayley graphs free abelian groups computational group theory vertex-transitive graphs isotopy research papers Source Type: research
Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction
Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffr...
Source: Acta Crystallographica Section A - July 15, 2020 Category: Chemistry Authors: Baake, M. Grimm, U. Tags: quasicrystals projection method inflation rules diffraction hyperuniformity topical reviews Source Type: research
Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra
The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of t...
Source: Acta Crystallographica Section A - July 8, 2020 Category: Chemistry Authors: Dauter, Z. Jaskolski, M. Tags: asymmetric unit unit cell Euler's formula space-filling polyhedra Dirichlet domains research papers Source Type: research
Atomic and Molecular Physics. A Primer. By Luciano Colombo. IOP Science, 2019. Ebook, pp. 219. ISBN 978-0-7503-2260-7.
(Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - June 29, 2020 Category: Chemistry Authors: Millot, C. Tags: book review atomic physics molecular physics book reviews Source Type: research
Electrons in Solids: Mesoscopics, Photonics, Quantum Computing, Correlations, Topology (Graduate Texts in Condensed Matter). By Hendrik Bluhm, Thomas Br ü ckel, Markus Morgenstern, Gero Plessen and Christoph Stampfer. De Gruyter, 2019. Paperback, pp. 393. Price EUR 59.65. ISBN 978-3-11-043831-4.
(Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - June 29, 2020 Category: Chemistry Authors: Macchi, P. Tags: book review electrons in solids book reviews Source Type: research
On an extension of Krivovichev's complexity measures
An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Usi...
Source: Acta Crystallographica Section A - June 29, 2020 Category: Chemistry Authors: Hornfeck, W. Tags: Shannon entropy Krivovichev complexity strong additivity crystal structure classification structural complexity research papers Source Type: research
Derived crystal structure of martensitic materials by solid – solid phase transformation
A mathematical description of crystal structure is proposed consisting of two parts: the underlying translational periodicity and the distinct atomic positions up to the symmetry operations in the unit cell, consistent with the International Tables for Crystallography. By the Cauchy – Born hypothesis, such a description can be integrated with the theory of continuum mechanics to calculate a derived crystal structure produced by solid – solid phase transformation. In addition, the expressions for the orientation relationship between the parent lattice and the derived lattice are generalized. The derived structure ration...
Source: Acta Crystallographica Section A - June 29, 2020 Category: Chemistry Authors: Karami, M. Tamura, N. Yang, Y. Chen, X. Tags: derived lattice martensitic phase transformation structure determination synchrotron X-ray diffraction research papers Source Type: research
Theoretical study of the properties of X-ray diffraction moir é fringes. III. Theoretical simulation of previous experimental moir é images
As a practical confirmation of a recently published X-ray moir é -fringe theory [Yoshimura (2015). Acta Cryst. A71, 368 – 381], computer simulations using this theory were conducted for previous experimental moir é images of a strained bicrystal specimen [Yoshimura (1996). Acta Cryst. A52, 312 – 325]. Simulated moir é images with a good or fairly good likeness are presented as a result of this simulation, in which the characteristic fringe-and-band and local strain patterns in the experimental images are reproduced well. Experimental moir é images taken when the inclination of the lattice planes was forcedly increa...
Source: Acta Crystallographica Section A - June 29, 2020 Category: Chemistry Authors: Yoshimura, J. Tags: X-ray moir é fringes strained crystals low-contrast band pattern peculiar experimental fringe profiles research papers Source Type: research
The quaternion-based spatial-coordinate and orientation-frame alignment problems
This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D sp...
Source: Acta Crystallographica Section A - June 17, 2020 Category: Chemistry Authors: Hanson, A.J. Tags: data alignment spatial-coordinate alignment orientation-frame alignment quaternions quaternion frames quaternion eigenvalue methods lead articles Source Type: research
Spherical-wave X-ray dynamical diffraction Talbot effect inside a crystal
Two-wave dynamical diffraction of an X-ray spherical wave in a crystal, when the wave passes through an object with a periodic amplitude transmission function, is considered. The behavior of the diffracted wave (spherical-wave Talbot effect) in the crystal is investigated. The Talbot effect inside the crystal is accompanied by the focusing effect and the pendulum effect. Peculiarities of the effect before the focus point, in the focusing plane and in the region after the focus point inside the crystal are revealed. An expression is found for the Talbot depth and the spherical-wave Talbot effect in these three regions is in...
Source: Acta Crystallographica Section A - May 31, 2020 Category: Chemistry Authors: Balyan, M.K. Levonyan, L.V. Trouni, K.G. Tags: spherical-wave Talbot effect dynamical diffraction X-rays crystals research papers Source Type: research
Novel phasing method using the origin-free modulus sum function expressed in terms of the absolute electron density
The origin-free modulus sum function SM refines the set Φ of phases of the structure factors by maximizing the coincidence between the experimental origin-free modulus function and the calculated one in terms of the ρ ( Φ )2 density function [Rius, J. (1993). Acta Cryst. A49, 406 – 409]. Maximization is normally achieved through the recursive application of a Fourier-based algorithm. The purpose of the present study is: (i) to show that ρ ( Φ )2 can be replaced by | ρ ( Φ )| in SM; (ii) to illustrate the viability of the corresponding phasing algorithm with experimental data. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - May 31, 2020 Category: Chemistry Authors: Rius, J. Tags: modulus sum function direct methods structure solution phasing methods Patterson function research papers Source Type: research
The chord-length distribution of a polyhedron
The chord-length distribution function [ γ ′ ′ (r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ ′ ′ (r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ 1], Δ 1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ , one of the two end points of an r subdomain, the derivative of γ ′ ′ (r) can only show singularities of the forms |r − δ | − n and |r − δ | − m+1/2, with n and...
Source: Acta Crystallographica Section A - May 31, 2020 Category: Chemistry Authors: Ciccariello, S. Tags: small-angle scattering stochastic geometry integral geometry chord-length distribution polyhedra asymptotic behaviour research papers Source Type: research
