Combinatorial aspects of the L ö wenstein avoidance rule. Part I: the independence polynomial
This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. Some properties of this polynomial are discussed; the independence polynomials of some simple ...
Source: Acta Crystallographica Section A - September 24, 2021 Category: Chemistry Authors: Moreira de Oliveira Jr, M. Eon, J.-G. Tags: nets quotient graphs independence quotient research papers Source Type: research
The role of an objective function in the mathematical modelling of wide-angle X-ray diffraction curves of semi-crystalline polymers
To decompose a wide-angle X-ray diffraction (WAXD) curve of a semi-crystalline polymer into crystalline peaks and amorphous halos, a theoretical best-fitted curve, i.e. a mathematical model, is constructed. In fitting the theoretical curve to the experimental one, various functions can be used to quantify and minimize the deviations between the curves. The analyses and calculations performed in this work have proved that the quality of the model, its parameters and consequently the information on the structure of the investigated polymer are considerably dependent on the shape of an objective function. It is shown that the...
Source: Acta Crystallographica Section A - September 24, 2021 Category: Chemistry Authors: Rabiej, M. Rabiej, S. Tags: polymers diffraction curves absolute error relative error objective function mathematical modelling research papers Source Type: research
Teaching Edition of International Tables for Crystallography: Crystallographic symmetry. Edited by Mois I. Aroyo. IUCr/Wiley, 2021. Softcover, pp. xii + 236. ISBN 978-0-470-97422-3. Price GBP 29.99.
(Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - August 27, 2021 Category: Chemistry Authors: Nespolo, M. Tags: book review crystallographic symmetry International Tables for Crystallography teaching book reviews Source Type: research
On the resolution function for powder diffraction with area detectors
In a powder diffraction experiment the resolution function defines the instrumental contribution to the peak widths as a function of the Bragg angle. The Caglioti formula is frequently applied to model the instrumental broadening and used in structural refinement. The parameters in the Caglioti formula are linked to physically meaningful parameters for most diffraction geometries. However, this link is lost for the now very popular powder diffraction geometry using large 2D area detectors. Here we suggest a new physical model for the instrumental broadening specifically developed for powder diffraction data measured with l...
Source: Acta Crystallographica Section A - August 27, 2021 Category: Chemistry Authors: Chernyshov, D. Dyadkin, V. Emerich, H. Valkovskiy, G. McMonagle, C.J. van Beek, W. Tags: powder diffraction resolution function 2D detectors instrumental resolution Caglioti formula research papers Source Type: research
On incoherent diffractive imaging
Incoherent diffractive imaging (IDI) promises structural analysis with atomic resolution based on intensity interferometry of pulsed X-ray fluorescence emission. However, its experimental realization is still pending and a comprehensive theory of contrast formation has not been established to date. Explicit expressions are derived for the equal-pulse two-point intensity correlations, as the principal measured quantity of IDI, with full control of the prefactors, based on a simple model of stochastic fluorescence emission. The model considers the photon detection statistics, the finite temporal coherence of the individual e...
Source: Acta Crystallographica Section A - August 27, 2021 Category: Chemistry Authors: Lohse, L.M. Vassholz, M. Salditt, T. Tags: femtosecond studies free-electron laser correlated fluctuations diffract-then-destroy single particles XFEL research papers Source Type: research
Wilson statistics: derivation, generalization and applications to electron cryomicroscopy
The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in X-ray crystallography and more recently in electron cryomicroscopy (cryo-EM). Specifically, modern computational methods for three-dimensional map sharpening and atomic modelling of macromolecules by single-particle cryo-EM are based on Wilson statistics. Here the first rigorous mathematical derivation of Wilson statistics is provided. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the...
Source: Acta Crystallographica Section A - August 20, 2021 Category: Chemistry Authors: Singer, A. Tags: power spectrum cryo-EM Wilson statistics Fourier analysis Guinier plot research papers Source Type: research
Moir é , Euler and self-similarity – the lattice parameters of twisted hexagonal crystals
A real-space approach for the calculation of the moir é lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moir é lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moir é crystals, i.e. moir é lattices decorated with a basis, require more crucial assessment of the commensurabilities and lead to discrete solutions and a non-continuous angle dependence of the moir é -crystal lattice parameter. In particular, this la...
Source: Acta Crystallographica Section A - August 19, 2021 Category: Chemistry Authors: Feuerbacher, M. Tags: 2D materials moir é pattern twisted bilayers twistronics graphene research papers Source Type: research
A fast algorithm to find reduced hyperplane unit cells and solve N-dimensional B é zout's identities
Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180 ° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N-dimensional B é zout's identity associated with the Miller indices of the hyperplane. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - August 13, 2021 Category: Chemistry Authors: Cayron, C. Tags: N-dimensional B é zout's identity hyperplane unit cell integer relation twinning research papers Source Type: research
On the centennials of the discoveries of the hydrogen bond and the structure of the water molecule: the short life and work of Eustace Jean Cuy (1897 – 1925)
The bent structure of the water molecule, and its hydrogen-bonding properties, arguably rank among the most impactful discoveries in the history of chemistry. Although the fact that the H — O — H angle must deviate from linearity was inferred early in the 20th century, notably from the existence of the electric dipole moment, it was not clear what that angle should be and why. One hundred years ago, a young PhD student at the University of California, Berkeley, Eustace J. Cuy, rationalized the V-shape structure of a water molecule using the Lewis theory of a chemical bond, i.e. a shared electron pair, and its tetrahedr...
Source: Acta Crystallographica Section A - August 13, 2021 Category: Chemistry Authors: Derewenda, Z.S. Tags: hydrogen bond water molecule structure history of structural chemistry feature articles Source Type: research
Experimental observation of carousel-like phason flips in the decagonal quasicrystal Al60Cr20Fe10Si10
Quasicrystals have special crystal structures with long-range order, but without translational symmetry. Unexpectedly, carousel-like successive flippings of groups of atoms inside the ∼ 2 nm decagonal structural subunits of the decagonal quasicrystal Al60Cr20Fe10Si10 were directly observed using in situ high-temperature high-resolution transmission electron microscopy imaging. The observed directionally successive phason flips occur mainly clockwise and occasionally anticlockwise. The origin of these directional phason flips is analyzed and discussed. (Source: Acta Crystallographica Section A)
Source: Acta Crystallographica Section A - August 13, 2021 Category: Chemistry Authors: He, Z. Maurice, J.-L. Ma, H. Wang, Y. Li, H. Zhang, T. Ma, X. Steurer, W. Tags: decagonal quasicrystals TEM transmission electron microscopy atomic clusters clockwise phason flips research papers Source Type: research
Dynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. II. Dynamical theory
The analytical expressions for coherent and diffuse components of the integrated reflection coefficient are considered in the case of Bragg diffraction geometry for single crystals containing randomly distributed microdefects. These expressions are analyzed numerically for the cases when the instrumental integration of the diffracted X-ray intensity is performed on one, two or three dimensions in the reciprocal-lattice space. The influence of dynamical effects, i.e. primary extinction and anomalously weak and strong absorption, on the integrated intensities of X-ray scattering is investigated in relation to the crystal str...
Source: Acta Crystallographica Section A - August 6, 2021 Category: Chemistry Authors: Molodkin, V.B. Olikhovskii, S.I. Dmitriev, S.V. Lizunov, V.V. Tags: dynamical theory of X-ray diffraction diffuse scattering microdefects primary extinction Bragg diffraction geometry research papers Source Type: research
On Borromean links and related structures
The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical `Borromean rings', in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed `n-Borromean' [Liang& Mislow (1994). J. Math. Chem. 16, 27 – 35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number o...
Source: Acta Crystallographica Section A - August 6, 2021 Category: Chemistry Authors: O'Keeffe, M. Treacy, M.M.J. Tags: Borromean rings n-Borromean isonemal piecewise-linear embedding research papers Source Type: research
Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited
A phenomenological model of anisotropic lattice vibrations is proposed, using a temperature-dependent spectral cutoff and varying Debye temperatures for the vibrational normal components. The internal lattice energy, entropy and Debye – Waller B factors of non-cubic elemental crystals are derived. The formalism developed is non-perturbative, based on temperature-dependent linear dispersion relations for the normal modes. The Debye temperatures of the vibrational normal components differ in anisotropic crystals; their temperature dependence and the varying spectral cutoff can be inferred from the experimental lattice heat...
Source: Acta Crystallographica Section A - July 29, 2021 Category: Chemistry Authors: Tomaschitz, R. Tags: anisotropic lattice vibrations thermodynamic functions Debye – Waller factors non-cubic crystals temperature-dependent spectral cutoff directional Debye temperatures effective phonon speed oscillator mass heat capacity zero-point e Source Type: research
Thermodynamics of lattice vibrations in non-cubic crystals: the zinc lattice revisited
A phenomenological model of anisotropic lattice vibrations is proposed, using a temperature-dependent spectral cutoff and varying Debye temperatures for the vibrational normal components. The internal lattice energy, entropy and Debye – Waller B factors of non-cubic elemental crystals are derived. The formalism developed is non-perturbative, based on temperature-dependent linear dispersion relations for the normal modes. The Debye temperatures of the vibrational normal components differ in anisotropic crystals; their temperature dependence and the varying spectral cutoff can be inferred from the experimental lattice heat...
Source: Acta Crystallographica Section A - July 29, 2021 Category: Chemistry Authors: Tomaschitz, R. Tags: anisotropic lattice vibrations thermodynamic functions Debye – Waller factors non-cubic crystals temperature-dependent spectral cutoff directional Debye temperatures effective phonon speed oscillator mass heat capacity zero-point e Source Type: research
On the calculation of the electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model. II. Evaluation of the properties in an infinite crystal
The previously reported exact potential and multipole moment (EP/MM) method for fast and precise evaluation of the intermolecular electrostatic interaction energies in molecular crystals using the pseudoatom representation of the electron density [Nguyen, Macchi& Volkov (2020), Acta Cryst. A76, 630 – 651] has been extended to the calculation of the electrostatic potential (ESP), electric field (EF) and electric field gradient (EFG) in an infinite crystal. The presented approach combines an efficient Ewald-type summation (ES) of atomic multipoles up to the hexadecapolar level in direct and reciprocal spaces with correctio...
Source: Acta Crystallographica Section A - July 29, 2021 Category: Chemistry Authors: Weatherly, J. Macchi, P. Volkov, A. Tags: electrostatic potential electric field electric field gradient charge density pseudoatom model multipole expansion Ewald summation lattice sums research papers Source Type: research
