Acta Crystallographica Section A
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Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.enCopyright (c) 2021 International Union of Crystallography2021-09-01International Union of CrystallographyInternational Union of Crystallographyhttp://journals.iucr.orgurn:issn:2053-2733Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.text/htmlActa Crystallographica Section A: Foundations and Advances, Volume 77, Part 5, 2021textweekly62002-01-01T00:00+00:005772021-09-01Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances355urn:issn:2053-2733med@iucr.orgSeptember 20212021-09-01Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageExperimental observation of carousel-like phason flips in the decagonal quasicrystal Al60Cr20Fe10Si10
http://scripts.iucr.org/cgi-bin/paper?ug5027
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.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733He, Z.Maurice, J.-L.Ma, H.Wang, Y.Li, H.Zhang, T.Ma, X.Steurer, W.2021-08-13doi:10.1107/S2053273321007518International Union of CrystallographyCarousel-like successive flippings of groups of atoms inside the decagonal clusters of decagonal quasicrystals were observed experimentally. The observed directionally successive phason flips occur mainly clockwise and occasionally anticlockwise.ENdecagonal quasicrystalsTEMtransmission electron microscopyatomic clustersclockwise phason flipsQuasicrystals 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.text/htmlExperimental observation of carousel-like phason flips in the decagonal quasicrystal Al60Cr20Fe10Si10text5772021-08-13Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers355361On 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)
http://scripts.iucr.org/cgi-bin/paper?ae5101
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 tetrahedral stereochemistry. He was inspired, in part, by the proposal of a weak (hydrogen) bond in water by two colleagues at Berkeley, Wendell Latimer and Worth Rodebush, who published their classic paper a year earlier. Cuy went on to suggest that other molecules, notably H2S and NH3, have similar structures, and presciently predicted that this architecture has broader consequences for the structure of water as a liquid. This short, but brilliant paper has been completely forgotten, perhaps due to the tragic death of the author at the age of 28; the hydrogen-bond study is also rarely recognized. One of the most impactful publications on the structure of liquid water, a classic treatise published in 1933 by John Bernal and Ralph Fowler, does not mention either of the two pioneering papers. In this essay, the background for the two discoveries is described, including the brief history of Lewis's research on the nature of the chemical bond, and the history of the discovery of the hydrogen bond, which inspired Cuy to look at the structure of the water molecule. This is – to the best of the author's knowledge – the first biographical sketch of Eustace J. Cuy.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Derewenda, Z.S.2021-08-13doi:10.1107/S2053273321006987International Union of CrystallographyEustace J. Cuy (Couyumdjopoulos) was the first scientist to rationalize the bent shape of the water molecule using the Lewis theory of the chemical bond. His work – inspired in part by the discovery of the hydrogen bond by Latimer, Rodebush and Huggins – was published 100 years ago, but had been forgotten after his tragic death at the age of 28.ENhydrogen bondwater molecule structurehistory of structural chemistryThe 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 tetrahedral stereochemistry. He was inspired, in part, by the proposal of a weak (hydrogen) bond in water by two colleagues at Berkeley, Wendell Latimer and Worth Rodebush, who published their classic paper a year earlier. Cuy went on to suggest that other molecules, notably H2S and NH3, have similar structures, and presciently predicted that this architecture has broader consequences for the structure of water as a liquid. This short, but brilliant paper has been completely forgotten, perhaps due to the tragic death of the author at the age of 28; the hydrogen-bond study is also rarely recognized. One of the most impactful publications on the structure of liquid water, a classic treatise published in 1933 by John Bernal and Ralph Fowler, does not mention either of the two pioneering papers. In this essay, the background for the two discoveries is described, including the brief history of Lewis's research on the nature of the chemical bond, and the history of the discovery of the hydrogen bond, which inspired Cuy to look at the structure of the water molecule. This is – to the best of the author's knowledge – the first biographical sketch of Eustace J. Cuy.text/htmlOn 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)text5772021-08-13Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Afeature articles362378On Borromean links and related structures
http://scripts.iucr.org/cgi-bin/paper?ib5101
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 of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20–25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733O'Keeffe, M.Treacy, M.M.J.2021-07-29doi:10.1107/S2053273321005568International Union of CrystallographyDescriptions are given of optimal piecewise-linear embeddings of families of Borromean structures that can be generated by rotoinversion point-group symmetries. These serve as templates for designed synthesis of polycatenanes.ENBorromean ringsn-Borromeanisonemalpiecewise-linear embeddingThe 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 of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20–25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.text/htmlOn Borromean links and related structurestext5772021-07-29Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers379391Piecewise-linear embeddings of knots and links with rotoinversion symmetry
http://scripts.iucr.org/cgi-bin/paper?ib5102
This article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive – the smallest possible for achiral knots.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733O'Keeffe, M.Treacy, M.M.J.2021-07-29doi:10.1107/S2053273321006136International Union of CrystallographyRotoinversion knots and links are described using the minimum number of sticks. Such structures are attractive targets for molecular synthesis.ENknotslinksdecussate structurespiecewise-linear embeddingThis article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive – the smallest possible for achiral knots.text/htmlPiecewise-linear embeddings of knots and links with rotoinversion symmetrytext5772021-07-29Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers392398On 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
http://scripts.iucr.org/cgi-bin/paper?ae5104
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 corrections for (i) the net polarization of the sample (the `surface term') due to a net dipole moment of the crystallographic unit cell (if present) and (ii) the short-range electron-density penetration effects. The rederived and reported closed-form expressions for all terms in the ES algorithm have been augmented by the expressions for the surface term available in the literature [Stenhammar, Trulsson & Linse (2011), J. Chem. Phys. 134, 224104] and the exact potential expressions reported in a previous study [Volkov, King, Coppens & Farrugia (2006), Acta Cryst. A62, 400–408]. The resulting algorithm, coded using Fortran in the XDPROP module of the software package XD, was tested on several small molecular crystal systems (formamide, benzene, l-dopa, paracetamol, amino acids etc.) and compared with a series of EP/MM-based direct-space summations (DS) performed within a certain number of unit cells generated along both the positive and negative crystallographic directions. The EP/MM-based ES technique allows for a noticeably more precise determination of the EF and EFG and significantly better precision of the evaluated ESP when compared with the DS calculations, even when the latter include contributions from an array of symmetry-equivalent atoms generated within four additional unit cells along each crystallographic direction. In terms of computational performance, the ES/EP/MM method is significantly faster than the DS calculations performed within the extended unit-cell limits but trails the DS calculations within the reduced summation ranges. Nonetheless, the described EP/MM-based ES algorithm is superior to the direct-space summations as it does not require the user to monitor continuously the convergence of the evaluated properties as a function of the summation limits and offers a better precision–performance balance.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Weatherly, J.Macchi, P.Volkov, A.2021-07-29doi:10.1107/S2053273321005532International Union of CrystallographyThe presented hybrid approach for a fast and precise determination of the electrostatic potential, electric field and electric field gradient in an infinite crystal, in which the electron-density distribution is represented using a pseudoatom model, combines an efficient Ewald summation technique of the multipoles up to the hexadecapolar level with corrections that account for the nature of the boundary of an infinite periodic lattice at infinity and the short-range electron-density penetration effects.ENelectrostatic potentialelectric fieldelectric field gradientcharge densitypseudoatom modelmultipole expansionEwald summationlattice sumsThe 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 corrections for (i) the net polarization of the sample (the `surface term') due to a net dipole moment of the crystallographic unit cell (if present) and (ii) the short-range electron-density penetration effects. The rederived and reported closed-form expressions for all terms in the ES algorithm have been augmented by the expressions for the surface term available in the literature [Stenhammar, Trulsson & Linse (2011), J. Chem. Phys. 134, 224104] and the exact potential expressions reported in a previous study [Volkov, King, Coppens & Farrugia (2006), Acta Cryst. A62, 400–408]. The resulting algorithm, coded using Fortran in the XDPROP module of the software package XD, was tested on several small molecular crystal systems (formamide, benzene, l-dopa, paracetamol, amino acids etc.) and compared with a series of EP/MM-based direct-space summations (DS) performed within a certain number of unit cells generated along both the positive and negative crystallographic directions. The EP/MM-based ES technique allows for a noticeably more precise determination of the EF and EFG and significantly better precision of the evaluated ESP when compared with the DS calculations, even when the latter include contributions from an array of symmetry-equivalent atoms generated within four additional unit cells along each crystallographic direction. In terms of computational performance, the ES/EP/MM method is significantly faster than the DS calculations performed within the extended unit-cell limits but trails the DS calculations within the reduced summation ranges. Nonetheless, the described EP/MM-based ES algorithm is superior to the direct-space summations as it does not require the user to monitor continuously the convergence of the evaluated properties as a function of the summation limits and offers a better precision–performance balance.text/htmlOn 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 crystaltext5772021-07-29Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers399419Thermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisited
http://scripts.iucr.org/cgi-bin/paper?ae5103
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 capacity and B factors by least-squares regression. The zero-point internal energy of the phonons is related to the low-temperature limits of the mean-squared vibrational amplitudes of the lattice measured by X-ray and γ-ray diffraction. A specific example is discussed, the thermodynamic variables of the hexagonal close-packed zinc structure, including the temperature evolution of the B factors of zinc. In this case, the lattice vibrations are partitioned into axial and basal normal components, which admit largely differing B factors and Debye temperatures. The second-order B factors defining the non-Gaussian contribution to the Debye–Waller damping factors of zinc are obtained as well. Anharmonicity of the oscillator potential and deviations from the uniform phonon frequency distribution of the Debye theory are modeled effectively by the temperature dependence of the spectral cutoff and Debye temperatures.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Tomaschitz, R.2021-07-29doi:10.1107/S2053273321005507International Union of CrystallographyAn effective theory of lattice vibrations in anisotropic elemental crystals is outlined, based on temperature-dependent spectral cutoffs and varying Debye temperatures. The thermodynamic variables and Debye–Waller B factors of the hexagonal close-packed zinc structure are derived.ENanisotropic lattice vibrationsthermodynamic functionsDebye–Waller factorsnon-cubic crystalstemperature-dependent spectral cutoffdirectional Debye temperatureseffective phonon speedoscillator massheat capacityzero-point energyA 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 capacity and B factors by least-squares regression. The zero-point internal energy of the phonons is related to the low-temperature limits of the mean-squared vibrational amplitudes of the lattice measured by X-ray and γ-ray diffraction. A specific example is discussed, the thermodynamic variables of the hexagonal close-packed zinc structure, including the temperature evolution of the B factors of zinc. In this case, the lattice vibrations are partitioned into axial and basal normal components, which admit largely differing B factors and Debye temperatures. The second-order B factors defining the non-Gaussian contribution to the Debye–Waller damping factors of zinc are obtained as well. Anharmonicity of the oscillator potential and deviations from the uniform phonon frequency distribution of the Debye theory are modeled effectively by the temperature dependence of the spectral cutoff and Debye temperatures.text/htmlThermodynamics of lattice vibrations in non-cubic crystals: the zinc structure revisitedtext5772021-07-29Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers420432Dynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. II. Dynamical theory
http://scripts.iucr.org/cgi-bin/paper?iv5012
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 structure imperfections.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Molodkin, V.B.Olikhovskii, S.I.Dmitriev, S.V.Lizunov, V.V.2021-08-06doi:10.1107/S2053273321005775International Union of CrystallographyA theoretical analysis is carried out of the influence of microdefects on the dynamical effects of primary extinction and anomalous absorption in the integrated intensities of coherent and diffuse scattering of X-rays by imperfect crystals in Bragg diffraction geometry.ENdynamical theory of X-ray diffractiondiffuse scatteringmicrodefectsprimary extinctionBragg diffraction geometryThe 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 structure imperfections.text/htmlDynamical effects in the integrated X-ray scattering intensity from imperfect crystals in Bragg diffraction geometry. II. Dynamical theorytext5772021-08-06Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers433452A fast algorithm to find reduced hyperplane unit cells and solve N-dimensional Bézout's identities
http://scripts.iucr.org/cgi-bin/paper?lu5010
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 Bézout's identity associated with the Miller indices of the hyperplane.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Cayron, C.2021-08-13doi:10.1107/S2053273321006835International Union of CrystallographyThe paper describes a method to determine a short unit cell attached to any hyperplane given by its integer vector p. Equivalently, it gives all the solutions of the N-dimensional Bézout's identity associated with the coordinates of p.ENN-dimensional Bézout's identityhyperplane unit cellinteger relationtwinningDeformation 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 Bézout's identity associated with the Miller indices of the hyperplane.text/htmlA fast algorithm to find reduced hyperplane unit cells and solve N-dimensional Bézout's identitiestext5772021-08-13Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers453459Moiré, Euler and self-similarity – the lattice parameters of twisted hexagonal crystals
http://scripts.iucr.org/cgi-bin/paper?ug5017
A real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré 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 moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Feuerbacher, M.2021-08-19doi:10.1107/S2053273321007245International Union of CrystallographyThe moiré lattice parameters are calculated for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides, and the highly complex pattern of solutions is discussed.EN2D materialsmoiré patterntwisted bilayerstwistronicsgrapheneA real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré 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 moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.text/htmlMoiré, Euler and self-similarity – the lattice parameters of twisted hexagonal crystalstext5772021-08-19Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers460471Wilson statistics: derivation, generalization and applications to electron cryomicroscopy
http://scripts.iucr.org/cgi-bin/paper?ib5103
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 analysis naturally leads to generalizations of the statistics to covariance and higher-order spectra. These in turn provide a theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation-based methods in cryo-EM.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Singer, A.2021-08-20doi:10.1107/S205327332100752XInternational Union of CrystallographyThis paper provides a rigorous mathematical derivation of Wilson's prediction that the power spectrum of many molecules of biological interest is approximately flat at high frequencies. The analysis elucidates the precise cutoff frequency above which the flat approximation holds and extends the result to other types of statistics with applications to electron cryomicroscopy (cryo-EM).ENpower spectrumcryo-EMWilson statisticsFourier analysisGuinier plotThe 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 analysis naturally leads to generalizations of the statistics to covariance and higher-order spectra. These in turn provide a theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation-based methods in cryo-EM.text/htmlWilson statistics: derivation, generalization and applications to electron cryomicroscopytext5772021-08-20Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers472479On incoherent diffractive imaging
http://scripts.iucr.org/cgi-bin/paper?iv5016
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 emissions, as well as the geometry of the scattering volume. The implications are interpreted in view of the most relevant quantities, including the fluorescence lifetime, the excitation pulse, as well as the extent of the scattering volume and pixel size. Importantly, the spatiotemporal overlap between any two emissions in the sample can be identified as a crucial factor limiting the contrast and its dependency on the sample size can be derived. The paper gives rigorous estimates for the optimum sample size, the maximum photon yield and the expected signal-to-noise ratio under optimal conditions. Based on these estimates, the feasibility of IDI experiments for plausible experimental parameters is discussed. It is shown in particular that the mean number of photons per detector pixel which can be achieved with X-ray fluorescence is severely limited and as a consequence imposes restrictive constraints on possible applications.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Lohse, L.M.Vassholz, M.Salditt, T.2021-08-27doi:10.1107/S2053273321007300International Union of CrystallographyStarting from a simple model of stochastic fluorescence emission, a theory is derived of contrast formation and signal-to-noise ratio for incoherent diffractive imaging; its feasibility for plausible experimental parameters is discussed.ENfemtosecond studiesfree-electron lasercorrelated fluctuationsdiffract-then-destroysingle particlesXFELIncoherent 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 emissions, as well as the geometry of the scattering volume. The implications are interpreted in view of the most relevant quantities, including the fluorescence lifetime, the excitation pulse, as well as the extent of the scattering volume and pixel size. Importantly, the spatiotemporal overlap between any two emissions in the sample can be identified as a crucial factor limiting the contrast and its dependency on the sample size can be derived. The paper gives rigorous estimates for the optimum sample size, the maximum photon yield and the expected signal-to-noise ratio under optimal conditions. Based on these estimates, the feasibility of IDI experiments for plausible experimental parameters is discussed. It is shown in particular that the mean number of photons per detector pixel which can be achieved with X-ray fluorescence is severely limited and as a consequence imposes restrictive constraints on possible applications.text/htmlOn incoherent diffractive imagingtext5772021-08-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers480496On the resolution function for powder diffraction with area detectors
http://scripts.iucr.org/cgi-bin/paper?ik5002
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 large 2D area detectors. The model is verified using data from two synchrotron diffraction beamlines with the Pilatus2M and MAR345 detectors. Finally, a functional form is proposed to replace the Caglioti formula for this geometry in the Rietveld method and profile refinements.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Chernyshov, D.Dyadkin, V.Emerich, H.Valkovskiy, G.McMonagle, C.J.van Beek, W.2021-08-27doi:10.1107/S2053273321007506International Union of CrystallographyThe Caglioti function stemming from 1959 is not well suited to describing the resolution function of modern powder diffraction equipment with large 2D detectors. A new function is derived and verified, replacing the Caglioti function for this very popular geometry.ENpowder diffractionresolution function2D detectorsinstrumental resolutionCaglioti formulaIn 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 large 2D area detectors. The model is verified using data from two synchrotron diffraction beamlines with the Pilatus2M and MAR345 detectors. Finally, a functional form is proposed to replace the Caglioti formula for this geometry in the Rietveld method and profile refinements.text/htmlOn the resolution function for powder diffraction with area detectorstext5772021-08-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Aresearch papers497505Teaching 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.
http://scripts.iucr.org/cgi-bin/paper?xo0185
Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Nespolo, M.2021-08-27doi:10.1107/S2053273321006811International Union of CrystallographyENbook reviewcrystallographic symmetryInternational Tables for Crystallographyteachingtext/htmlTeaching 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.text5772021-08-27Copyright (c) 2021 International Union of CrystallographyActa Crystallographica Section Abook reviews506508