Open-access and free articles in Acta Crystallographica Section A: Foundations and Advances
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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.en-gbCopyright (c) 2023 International Union of CrystallographyInternational Union of CrystallographyInternational Union of Crystallographytext/htmlOpen-access and free articles in Acta Crystallographica Section A Foundations and Advanceshttps://journals.iucr.orgtexturn:issn:0108-7673Acta 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.2002-01-01T00:00+00:006yearlymed@iucr.orgActa Crystallographica Section A Foundations and Advancesurn:issn:0108-7673Copyright (c) 2023 International Union of CrystallographyOpen-access and free articles in Acta Crystallographica Section A: Foundations and Advanceshttp://journals.iucr.org/logos/rss10a.gif
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Still imageCrystal diffraction prediction and partiality estimation using Gaussian basis functions
http://scripts.iucr.org/cgi-bin/paper?ik5005
The recent diversification of macromolecular crystallographic experiments including the use of pink beams, convergent electron diffraction and serial snapshot crystallography has shown the limitations of using the Laue equations for diffraction prediction. This article gives a computationally efficient way of calculating approximate crystal diffraction patterns given varying distributions of the incoming beam, crystal shapes and other potentially hidden parameters. This approach models each pixel of a diffraction pattern and improves data processing of integrated peak intensities by enabling the correction of partially recorded reflections. The fundamental idea is to express the distributions as weighted sums of Gaussian functions. The approach is demonstrated on serial femtosecond crystallography data sets, showing a significant decrease in the required number of patterns to refine a structure to a given error.Reflection position, size and shape prediction and partiality estimation of crystal diffraction by integrating using a Gaussian basis are described.International Union of CrystallographyBrehm, W.White, T.Chapman, H.N.PARTIALITY ESTIMATION; DIFFRACTION PREDICTION; MERGING; SERIAL SNAPSHOT CRYSTALLOGRAPHYtext/htmlCrystal diffraction prediction and partiality estimation using Gaussian basis functionsenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273323000682The recent diversification of macromolecular crystallographic experiments including the use of pink beams, convergent electron diffraction and serial snapshot crystallography has shown the limitations of using the Laue equations for diffraction prediction. This article gives a computationally efficient way of calculating approximate crystal diffraction patterns given varying distributions of the incoming beam, crystal shapes and other potentially hidden parameters. This approach models each pixel of a diffraction pattern and improves data processing of integrated peak intensities by enabling the correction of partially recorded reflections. The fundamental idea is to express the distributions as weighted sums of Gaussian functions. The approach is demonstrated on serial femtosecond crystallography data sets, showing a significant decrease in the required number of patterns to refine a structure to a given error.2023-02-17research papers2053-2733March 2023162Acta Crystallographica Section A: Foundations and Advances792053-2733med@iucr.org21452023-02-17https://creativecommons.org/licenses/by/4.0/Dynamic tilting in perovskites
http://scripts.iucr.org/cgi-bin/paper?lu5021
A new computational analysis of tilt behaviour in perovskites is presented. This includes the development of a computational program – PALAMEDES – to extract tilt angles and the tilt phase from molecular dynamics simulations. The results are used to generate simulated selected-area electron and neutron diffraction patterns which are compared with experimental patterns for CaTiO3. The simulations not only reproduced all symmetrically allowed superlattice reflections associated with tilt but also showed local correlations that give rise to symmetrically forbidden reflections and the kinematic origin of diffuse scattering.A new computational program to analyse and extract tilt data from molecular dynamics simulations of perovskites is presented and results compared with experimental data.International Union of CrystallographyHandley, C.M.Ward, R.E.Freeman, C.L.Reaney, I.M.Sinclair, D.C.Harding, J.H.PEROVSKITES; TILT; DIFFRACTION; MOLECULAR DYNAMICS; SUPERLATTICEtext/htmlDynamic tilting in perovskitesenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322011949A new computational analysis of tilt behaviour in perovskites is presented. This includes the development of a computational program – PALAMEDES – to extract tilt angles and the tilt phase from molecular dynamics simulations. The results are used to generate simulated selected-area electron and neutron diffraction patterns which are compared with experimental patterns for CaTiO3. The simulations not only reproduced all symmetrically allowed superlattice reflections associated with tilt but also showed local correlations that give rise to symmetrically forbidden reflections and the kinematic origin of diffuse scattering.2023-01-23research papers79Acta Crystallographica Section A: Foundations and AdvancesMarch 20232053-2733170med@iucr.org2053-2733https://creativecommons.org/licenses/by/4.0/2023-01-231632Dynamical diffraction of high-energy electrons by light-atom structures: a multiple forward scattering interpretation
http://scripts.iucr.org/cgi-bin/paper?lu5020
Because of the strong electron–atom interaction, the kinematic theory of diffraction cannot be used to describe the scattering of electrons by an assembly of atoms due to the strong dynamical diffraction that needs to be taken into account. In this paper, the scattering of high-energy electrons by a regular array of light atoms is solved exactly by applying the T-matrix formalism to the corresponding Schrödinger's equation in spherical coordinates. The independent atom model is used, where each atom is represented by a sphere with an effective constant potential. The validity of the forward scattering approximation and the phase grating approximation, assumed by the popular multislice method, is discussed, and an alternative interpretation of multiple scattering is proposed and compared with existing interpretations.The T-matrix is used to compute the scattering of fast electrons by a regular array of effective spherical potential wells. An assessment of the forward scattering approximation and a real-space multiple scattering interpretation are provided.International Union of CrystallographyDrevon, T.R.Waterman, D.G.Krissinel, E.HIGH-ENERGY ELECTRON DIFFRACTION; T-MATRIX; MULTIPLE SCATTERING; INDEPENDENT ATOM APPROXIMATIONtext/htmlDynamical diffraction of high-energy electrons by light-atom structures: a multiple forward scattering interpretationenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322011779Because of the strong electron–atom interaction, the kinematic theory of diffraction cannot be used to describe the scattering of electrons by an assembly of atoms due to the strong dynamical diffraction that needs to be taken into account. In this paper, the scattering of high-energy electrons by a regular array of light atoms is solved exactly by applying the T-matrix formalism to the corresponding Schrödinger's equation in spherical coordinates. The independent atom model is used, where each atom is represented by a sphere with an effective constant potential. The validity of the forward scattering approximation and the phase grating approximation, assumed by the popular multislice method, is discussed, and an alternative interpretation of multiple scattering is proposed and compared with existing interpretations.2023-02-09research papers2053-2733191March 202379Acta Crystallographica Section A: Foundations and Advancesmed@iucr.org2053-2733https://creativecommons.org/licenses/by/4.0/2023-02-091802A phase retrieval algorithm for triply periodic minimal surface like structures
http://scripts.iucr.org/cgi-bin/paper?ik5006
A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported. In triply periodic minimal surface like structures, the difference between the maximum and minimum electron densities tends to be the smallest for the true phase combination among the possible combinations [Oka (2022). Acta Cryst. A78, 430–436]. Using this feature, a new iterative phase retrieval algorithm for structure determination was developed. The algorithm modifies electron densities outside the upper and lower thresholds in the iterative Fourier transformation process with fixed amplitudes for the structure factors, and efficiently searches for the structure with the smallest difference between the maximum and minimum electron densities. The proper structure was determined by this algorithm for all tested data for lyotropic liquid crystal bicontinuous cubic phases and mesoporous silicas. Although some cases required constraints such as the volume fraction for structure determination, more than half could be determined without any constraints, including space groups.A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported.International Union of CrystallographyOka, T.CRYSTALLOGRAPHIC PHASE RETRIEVAL; LYOTROPIC LIQUID CRYSTALS; MESOPOROUS SILICA; TRIPLY PERIODIC MINIMAL SURFACEStext/htmlA phase retrieval algorithm for triply periodic minimal surface like structuresenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322010786A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported. In triply periodic minimal surface like structures, the difference between the maximum and minimum electron densities tends to be the smallest for the true phase combination among the possible combinations [Oka (2022). Acta Cryst. A78, 430–436]. Using this feature, a new iterative phase retrieval algorithm for structure determination was developed. The algorithm modifies electron densities outside the upper and lower thresholds in the iterative Fourier transformation process with fixed amplitudes for the structure factors, and efficiently searches for the structure with the smallest difference between the maximum and minimum electron densities. The proper structure was determined by this algorithm for all tested data for lyotropic liquid crystal bicontinuous cubic phases and mesoporous silicas. Although some cases required constraints such as the volume fraction for structure determination, more than half could be determined without any constraints, including space groups.2023-01-01research papersJanuary 20232053-273358Acta Crystallographica Section A: Foundations and Advances792053-2733med@iucr.org1512023-01-01https://creativecommons.org/licenses/by/4.0/Electron density and thermal motion of diamond at elevated temperatures
http://scripts.iucr.org/cgi-bin/paper?pl5020
The electron density and thermal motion of diamond are determined at nine temperatures between 100 K and 1000 K via synchrotron powder X-ray diffraction (PXRD) data collected on a high-accuracy detector system. Decoupling of the thermal motion from the thermally smeared electron density is performed via an iterative Wilson–Hansen–Coppens–Rietveld procedure using theoretical static structure factors from density functional theory (DFT) calculations. The thermal motion is found to be harmonic and isotropic in the explored temperature range, and excellent agreement is observed between experimental atomic displacement parameters (ADPs) and those obtained via theoretical harmonic phonon calculations (HPC), even at 1000 K. The Debye temperature of diamond is determined experimentally to be ΘD = 1883 (35) K. A topological analysis of the electron density explores the temperature dependency of the electron density at the bond critical point. The properties are found to be constant throughout the temperature range. The robustness of the electron density confirms the validity of the crystallographic convolution approximation for diamond in the explored temperature range.The electron densities and atomic displacement parameters of diamond are determined from 100 K to 1000 K using synchrotron powder X-ray diffraction.International Union of CrystallographyBeyer, J.Grønbech, T.B.E.Zhang, J.Kato, K.Brummerstedt Iversen, B.X-RAY ELECTRON DENSITY; SYNCHROTRON POWDER X-RAY DIFFRACTION; DIAMOND; CONVOLUTION APPROXIMATIONtext/htmlElectron density and thermal motion of diamond at elevated temperaturesenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322010154The electron density and thermal motion of diamond are determined at nine temperatures between 100 K and 1000 K via synchrotron powder X-ray diffraction (PXRD) data collected on a high-accuracy detector system. Decoupling of the thermal motion from the thermally smeared electron density is performed via an iterative Wilson–Hansen–Coppens–Rietveld procedure using theoretical static structure factors from density functional theory (DFT) calculations. The thermal motion is found to be harmonic and isotropic in the explored temperature range, and excellent agreement is observed between experimental atomic displacement parameters (ADPs) and those obtained via theoretical harmonic phonon calculations (HPC), even at 1000 K. The Debye temperature of diamond is determined experimentally to be ΘD = 1883 (35) K. A topological analysis of the electron density explores the temperature dependency of the electron density at the bond critical point. The properties are found to be constant throughout the temperature range. The robustness of the electron density confirms the validity of the crystallographic convolution approximation for diamond in the explored temperature range.2023-01-012023-01-01https://creativecommons.org/licenses/by/4.0/4112053-2733med@iucr.orgJanuary 2023502053-273379Acta Crystallographica Section A: Foundations and Advancesresearch papersGeographic style maps for two-dimensional lattices
http://scripts.iucr.org/cgi-bin/paper?uv5012
This paper develops geographic style maps containing two-dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two-dimensional lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualize hundreds of thousands of lattices of real crystal structures from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.Continuous invariant-based maps visualize for the first time all two-dimensional lattices extracted from hundreds of thousands of known crystal structures in the Cambridge Structural Database.International Union of CrystallographyBright, M.Cooper, A.I.Kurlin, V.TWO-DIMENSIONAL LATTICES; REDUCED BASIS; OBTUSE SUPERBASE; ISOMETRY; COMPLETE INVARIANTS; METRIC TENSOR; CONTINUITYtext/htmlGeographic style maps for two-dimensional latticesenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322010075This paper develops geographic style maps containing two-dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two-dimensional lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualize hundreds of thousands of lattices of real crystal structures from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.2023-01-01112023-01-01https://creativecommons.org/licenses/by/4.0/2053-2733med@iucr.orgActa Crystallographica Section A: Foundations and Advances79132053-2733January 2023research papersSimulating dark-field X-ray microscopy images with wavefront propagation techniques
http://scripts.iucr.org/cgi-bin/paper?iv5021
Dark-field X-ray microscopy is a diffraction-based synchrotron imaging technique capable of imaging defects in the bulk of extended crystalline samples. Numerical simulations are presented of image formation in such a microscope using numerical integration of the dynamical Takagi–Taupin equations and wavefront propagation. The approach is validated by comparing simulated images with experimental data from a near-perfect single crystal of diamond containing a single stacking-fault defect in the illuminated volume.The simulation of a dark-field X-ray microscopy experiment using wavefront propagation techniques and numerical integration of the Takagi–Taupin equations is shown. The approach is validated by comparing with measurements of a near-perfect diamond crystal containing a single stacking-fault defect.International Union of CrystallographyCarlsen, M.Detlefs, C.Yildirim, C.Ræder, T.Simons, H.DIFFRACTION IMAGING; DYNAMICAL DIFFRACTION; SIMULATIONtext/htmlSimulating dark-field X-ray microscopy images with wavefront propagation techniquesenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S205327332200866XDark-field X-ray microscopy is a diffraction-based synchrotron imaging technique capable of imaging defects in the bulk of extended crystalline samples. Numerical simulations are presented of image formation in such a microscope using numerical integration of the dynamical Takagi–Taupin equations and wavefront propagation. The approach is validated by comparing simulated images with experimental data from a near-perfect single crystal of diamond containing a single stacking-fault defect in the illuminated volume.2022-10-1048262022-10-10https://creativecommons.org/licenses/by/4.0/2053-2733med@iucr.orgActa Crystallographica Section A: Foundations and Advances78November 20222053-2733490research papersExtending the novel |ρ|-based phasing algorithm to the solution of anomalous scattering substructures from SAD data of protein crystals
http://scripts.iucr.org/cgi-bin/paper?ae5119
Owing to the importance of the single-wavelength anomalous diffraction (SAD) technique, the recently developed |ρ|-based phasing algorithm (SM,|ρ|) incorporating the inner-pixel preservation (ipp) procedure [Rius & Torrelles (2021). Acta Cryst A77, 339–347] has been adapted to the determination of anomalous scattering substructures and its applicability tested on a series of 12 representative experimental data sets, mostly retrieved from the Protein Data Bank. To give an idea of the suitability of the data sets, the main indicators measuring their quality are also given. The dominant anomalous scatterers are either SeMet or S atoms, or metals/clusters incorporated by soaking. The resulting SAD-adapted algorithm solves the substructures of the test protein crystals quite efficiently.The novel SM,|ρ| phasing algorithm has been adapted to the determination of anomalous scattering substructures from single-wavelength anomalous diffraction (SAD) data of protein crystals and successfully tested on data sets mostly retrieved from the Protein Data Bank.International Union of CrystallographyRius, J.Torrelles, X.SM,|[RHO]| PHASING ALGORITHM; SMAR PHASING; IPP DENSITY MODIFICATION; SAD-SMAR; |[RHO]|-BASED DIRECT METHODS; STRUCTURE SOLUTIONtext/htmlExtending the novel |ρ|-based phasing algorithm to the solution of anomalous scattering substructures from SAD data of protein crystalsenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322008622Owing to the importance of the single-wavelength anomalous diffraction (SAD) technique, the recently developed |ρ|-based phasing algorithm (SM,|ρ|) incorporating the inner-pixel preservation (ipp) procedure [Rius & Torrelles (2021). Acta Cryst A77, 339–347] has been adapted to the determination of anomalous scattering substructures and its applicability tested on a series of 12 representative experimental data sets, mostly retrieved from the Protein Data Bank. To give an idea of the suitability of the data sets, the main indicators measuring their quality are also given. The dominant anomalous scatterers are either SeMet or S atoms, or metals/clusters incorporated by soaking. The resulting SAD-adapted algorithm solves the substructures of the test protein crystals quite efficiently.2022-10-102053-2733med@iucr.org2022-10-10https://creativecommons.org/licenses/by/4.0/6473research papers2053-2733481November 202278Acta Crystallographica Section A: Foundations and AdvancesOn single-crystal total scattering data reduction and correction protocols for analysis in direct space. Corrigendum
http://scripts.iucr.org/cgi-bin/paper?me6196
The name of the third author of the article by Koch et al. [Acta Cryst. (2021). A77, 611–636] is corrected.Corrigendum to the article by Koch et al. [Acta Cryst. (2021). A77, 611–636].International Union of CrystallographyKoch, R.J.Roth, N.Liu, Y.Ivashko, O.Dippel, A.-C.Petrovic, C.Iversen, B.B.Zimmermann, M. v.Bozin, E.S.PAIR DISTRIBUTION FUNCTION ANALYSIS; PDF ANALYSIS; SINGLE-CRYSTAL 3D DIFFERENTIAL PDF; TOTAL SCATTERING; DATA REDUCTION; CUIR2S4text/htmlOn single-crystal total scattering data reduction and correction protocols for analysis in direct space. Corrigendumenurn:issn:2053-2733textdoi:10.1107/S2053273322009081The name of the third author of the article by Koch et al. [Acta Cryst. (2021). A77, 611–636] is corrected.2022-11-01Acta Crystallographica Section A: Foundations and Advances782053-2733November 2022515addenda and errata51562022-11-01med@iucr.org2053-2733Towards a machine-readable literature: finding relevant papers based on an uploaded powder diffraction pattern
http://scripts.iucr.org/cgi-bin/paper?ae5116
A prototype application for machine-readable literature is investigated. The program is called pyDataRecognition and serves as an example of a data-driven literature search, where the literature search query is an experimental data set provided by the user. The user uploads a powder pattern together with the radiation wavelength. The program compares the user data to a database of existing powder patterns associated with published papers and produces a rank ordered according to their similarity score. The program returns the digital object identifier and full reference of top-ranked papers together with a stack plot of the user data alongside the top-five database entries. The paper describes the approach and explores successes and challenges.A prototype application, pyDataRecognition, is described and tested. It has the goal that, given a measured powder diffraction pattern, it will return a list of publications from the IUCr Journals database that might be related based on the similarity to powder diffraction data deposited for those publications. This explores the possibility of a machine-readable literature where, for example, relevant studies may be found automatically through data similarity matches of online databases.International Union of CrystallographyÖzer, B.Karlsen, M.A.Thatcher, Z.Lan, L.McMahon, B.Strickland, P.R.Westrip, S.P.Sang, K.S.Billing, D.G.Ravnsbaek, D.B.Billinge, S.J.L.MACHINE-READABLE SCIENTIFIC LITERATURE; DATA-DRIVEN LITERATURE SEARCH; POWDER DIFFRACTION; DATA SIMILARITY; CIFtext/htmlTowards a machine-readable literature: finding relevant papers based on an uploaded powder diffraction patternenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322007483A prototype application for machine-readable literature is investigated. The program is called pyDataRecognition and serves as an example of a data-driven literature search, where the literature search query is an experimental data set provided by the user. The user uploads a powder pattern together with the radiation wavelength. The program compares the user data to a database of existing powder patterns associated with published papers and produces a rank ordered according to their similarity score. The program returns the digital object identifier and full reference of top-ranked papers together with a stack plot of the user data alongside the top-five database entries. The paper describes the approach and explores successes and challenges.2022-08-19research papersActa Crystallographica Section A: Foundations and Advances78September 20222053-2733394med@iucr.org2053-27335386https://creativecommons.org/licenses/by/4.0/2022-08-19Crystallographic phase retrieval method for liquid crystal bicontinuous phases: indicator-based method
http://scripts.iucr.org/cgi-bin/paper?ik5003
An indicator-based crystallographic phase retrieval method has been developed for diffraction data of bicontinuous cubic phases of lyotropic liquid crystals. Such liquid crystals have large structural disorder; the number of independent Bragg reflections that can be observed is limited. This paper proposes two indicators to identify plausible combination(s) of crystallographic phases, i.e. electron-density distribution. The indicators are based on the characteristics of the liquid crystals: amphiphilic molecules diffuse mainly in the direction parallel to polar–nonpolar interfaces and the electron density in the direction parallel to the interfaces is averaged. One indicator is the difference between the maximum and minimum electron density, and the other is calculated from the Hessian matrix of the electron density. Using test data, the electron densities were calculated for all possible phase combinations and indicators were obtained. The results indicated that the electron densities with the minimum indicators were close to the true electron density. Therefore, this method is effective for phase retrieval. The accuracy of the phase retrieval decreased when the volume fraction of the region including the triply periodic minimal surface increased.An indicator-based crystallographic phase retrieval method has been developed for diffraction data of bicontinuous cubic phases of lyotropic liquid crystals. The electron densities with the minimum indicators are close to the true electron density.International Union of CrystallographyOka, T.LYOTROPIC LIQUID CRYSTALS; TRIPLY PERIODIC MINIMAL SURFACES; CRYSTALLOGRAPHIC PHASE RETRIEVALtext/htmlCrystallographic phase retrieval method for liquid crystal bicontinuous phases: indicator-based methodenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322006970An indicator-based crystallographic phase retrieval method has been developed for diffraction data of bicontinuous cubic phases of lyotropic liquid crystals. Such liquid crystals have large structural disorder; the number of independent Bragg reflections that can be observed is limited. This paper proposes two indicators to identify plausible combination(s) of crystallographic phases, i.e. electron-density distribution. The indicators are based on the characteristics of the liquid crystals: amphiphilic molecules diffuse mainly in the direction parallel to polar–nonpolar interfaces and the electron density in the direction parallel to the interfaces is averaged. One indicator is the difference between the maximum and minimum electron density, and the other is calculated from the Hessian matrix of the electron density. Using test data, the electron densities were calculated for all possible phase combinations and indicators were obtained. The results indicated that the electron densities with the minimum indicators were close to the true electron density. Therefore, this method is effective for phase retrieval. The accuracy of the phase retrieval decreased when the volume fraction of the region including the triply periodic minimal surface increased.2022-07-282053-2733med@iucr.org43052022-07-28https://creativecommons.org/licenses/by/4.0/research papersSeptember 20224362053-2733Acta Crystallographica Section A: Foundations and Advances78Pure discrete spectrum and regular model sets on some non-unimodular substitution tilings
http://scripts.iucr.org/cgi-bin/paper?uv5005
Substitution tilings with pure discrete spectrum are characterized as regular model sets whose cut-and-project scheme has an internal space that is a product of a Euclidean space and a profinite group. Assumptions made here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with the same multiplicity. A difference from the result of Lee et al. [Acta Cryst. (2020), A76, 600–610] is that unimodularity is no longer assumed in this paper.The equivalence between pure discrete spectrum and regular model sets on some non-unimodular substitution tilings is established. This will help to provide useful information about the cut-and-project scheme used in the description of quasiperiodic structures.International Union of CrystallographyLee, J.-Y.PURE DISCRETE SPECTRUM; REGULAR MODEL SETS; NON-UNIMODULAR SUBSTITUTION; PISOT FAMILY SUBSTITUTION; MEYER SETStext/htmlPure discrete spectrum and regular model sets on some non-unimodular substitution tilingsenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322006714Substitution tilings with pure discrete spectrum are characterized as regular model sets whose cut-and-project scheme has an internal space that is a product of a Euclidean space and a profinite group. Assumptions made here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with the same multiplicity. A difference from the result of Lee et al. [Acta Cryst. (2020), A76, 600–610] is that unimodularity is no longer assumed in this paper.2022-08-12research papers2053-2733451September 2022Acta Crystallographica Section A: Foundations and Advances78med@iucr.org2053-27335437https://creativecommons.org/licenses/by/4.0/2022-08-12Origin of irregular X-ray mirage fringes from a bent, thin crystal
http://scripts.iucr.org/cgi-bin/paper?iv5025
The dynamical theory of diffraction is used to analyse irregular X-ray mirage interference fringes observed in Si220 X-ray reflection topography from a weakly bent, thin crystal due to gravity. The origin of the irregular fringes is attributed to the interference between mirage diffracted beams and a reflected beam from the back surface, which is a new type of interference fringe. The irregular fringes are reproduced by calculating the reflected intensities numerically. The effects of absorption and thermal vibration are quite important for the reproduction. The result shows that the interference fringes depend on the strain as well as the thickness of the crystal, which indicates that the fringes should be useful for analysing weak strain in a crystal as an application.Irregular X-ray mirage interference fringes observed in Si220 X-ray reflection topography from a weakly bent, thin crystal are analysed using the dynamical theory of diffraction. The origin is the interference between two or more mirage diffracted beams and one reflected beam from the back surface.International Union of CrystallographyFukamachi, T.Kawamura, T.INTERFERENCE FRINGES; MIRAGE FRINGES; BENT CRYSTAL; DYNAMICAL THEORY OF X-RAY DIFFRACTIONtext/htmlOrigin of irregular X-ray mirage fringes from a bent, thin crystalenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322006143The dynamical theory of diffraction is used to analyse irregular X-ray mirage interference fringes observed in Si220 X-ray reflection topography from a weakly bent, thin crystal due to gravity. The origin of the irregular fringes is attributed to the interference between mirage diffracted beams and a reflected beam from the back surface, which is a new type of interference fringe. The irregular fringes are reproduced by calculating the reflected intensities numerically. The effects of absorption and thermal vibration are quite important for the reproduction. The result shows that the interference fringes depend on the strain as well as the thickness of the crystal, which indicates that the fringes should be useful for analysing weak strain in a crystal as an application.2022-07-28med@iucr.org2053-2733https://creativecommons.org/licenses/by/4.0/2022-07-285422research papersSeptember 20222053-273342978Acta Crystallographica Section A: Foundations and AdvancesA finite difference scheme for integrating the Takagi–Taupin equations on an arbitrary orthogonal grid
http://scripts.iucr.org/cgi-bin/paper?iv5022
Calculating dynamical diffraction patterns for X-ray diffraction imaging techniques requires numerical integration of the Takagi–Taupin equations. This is usually performed with a simple, second-order finite difference scheme on a sheared computational grid in which two of the axes are aligned with the wavevectors of the incident and scattered beams. This dictates, especially at low scattering angles, an oblique grid of uneven step sizes. Here a finite difference scheme is presented that carries out this integration in slab-shaped samples on an arbitrary orthogonal grid by implicitly utilizing Fourier interpolation. The scheme achieves the expected second-order convergence and a similar error to the traditional approach for similarly dense grids.A finite difference scheme for integrating the Takagi–Taupin equations inside a slab-shaped crystal is demonstrated and tested.International Union of CrystallographyCarlsen, M.Simons, H.DYNAMICAL DIFFRACTION; X-RAY TOPOGRAPHYtext/htmlA finite difference scheme for integrating the Takagi–Taupin equations on an arbitrary orthogonal gridenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322004934Calculating dynamical diffraction patterns for X-ray diffraction imaging techniques requires numerical integration of the Takagi–Taupin equations. This is usually performed with a simple, second-order finite difference scheme on a sheared computational grid in which two of the axes are aligned with the wavevectors of the incident and scattered beams. This dictates, especially at low scattering angles, an oblique grid of uneven step sizes. Here a finite difference scheme is presented that carries out this integration in slab-shaped samples on an arbitrary orthogonal grid by implicitly utilizing Fourier interpolation. The scheme achieves the expected second-order convergence and a similar error to the traditional approach for similarly dense grids.2022-07-0853952022-07-08https://creativecommons.org/licenses/by/4.0/2053-2733med@iucr.orgActa Crystallographica Section A: Foundations and Advances78September 20222053-2733401research papersMultidimensional color codes for chair tilings
http://scripts.iucr.org/cgi-bin/paper?ae5109
Ordered aperiodic structures have been of interest to the crystallographic community for several decades, and study of them has in turn led to the study of lattice substitution systems, model sets and chair tilings. In this work a color code for chair tilings in arbitrary dimensions is presented. In two and three dimensions, it is expedient to translate the digital codes into colors. An explicit example of a three-dimensional color coding covering one octant is constructed. The tiling is then extended to the whole three-dimensional space and an indication is given of how to do this in arbitrary dimensions. Illustrations of some four-dimensional objects are also shown. The principle of color coding can be applied to other complex tilings such as brick tiling.A color code for chair tilings in arbitrary dimensions is presented. The code can be applied also to other lattice substitution tilings.International Union of CrystallographyBen-Abraham, S.I.Flom, D.CHAIR TILINGS; LETTER CODES; DIGITAL CODES; COLOR CODES; APERIODIC STRUCTURES; QUASICRYSTALStext/htmlMultidimensional color codes for chair tilingsenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322004065Ordered aperiodic structures have been of interest to the crystallographic community for several decades, and study of them has in turn led to the study of lattice substitution systems, model sets and chair tilings. In this work a color code for chair tilings in arbitrary dimensions is presented. In two and three dimensions, it is expedient to translate the digital codes into colors. An explicit example of a three-dimensional color coding covering one octant is constructed. The tiling is then extended to the whole three-dimensional space and an indication is given of how to do this in arbitrary dimensions. Illustrations of some four-dimensional objects are also shown. The principle of color coding can be applied to other complex tilings such as brick tiling.2022-06-17research papers2053-2733363July 2022Acta Crystallographica Section A: Foundations and Advances78med@iucr.org2053-27334359https://creativecommons.org/licenses/by/4.0/2022-06-17Identification of a coherent twin relationship from high-resolution reciprocal-space maps
http://scripts.iucr.org/cgi-bin/paper?lu5017
Twinning is a common crystallographic phenomenon which is related to the formation and coexistence of several orientation variants of the same crystal structure. It may occur during symmetry-lowering phase transitions or during the crystal growth itself. Once formed, twin domains play an important role in defining physical properties: for example, they underpin the giant piezoelectric effect in ferroelectrics, superelasticity in ferroelastics and the shape-memory effect in martensitic alloys. Regrettably, there is still a lack of experimental methods for the characterization of twin domain patterns. Here, a theoretical framework and algorithm are presented for the recognition of ferroelastic domains, as well as the identification of the coherent twin relationship using high-resolution reciprocal-space mapping of X-ray diffraction intensity around split Bragg peaks. Specifically, the geometrical theory of twinned ferroelastic crystals [Fousek & Janovec (1969). J. Appl. Phys. 40, 135–142] is adapted for the analysis of the X-ray diffraction patterns. The necessary equations are derived and an algorithm is outlined for the calculation of the separation between the Bragg peaks, diffracted from possible coherent twin domains, connected to one another via a mismatch-free interface. It is demonstrated that such separation is always perpendicular to the planar interface between mechanically matched domains. For illustration purposes, the analysis is presented of the separation between the peaks diffracted from tetragonal and rhombohedral domains in the high-resolution reciprocal-space maps of BaTiO3 and PbZr1−xTixO3 crystals. The demonstrated method can be used to analyse the response of multi-domain patterns to external perturbations such as electric field, change of temperature or pressure.The theory and algorithm are presented for the assignment of ferroelastic domains to the individual components of split Bragg peaks in high-resolution reciprocal-space maps. The formalism of mechanical compatibility of ferroelastic domains is further developed for the analysis of the geometry of the reciprocal space. The application of the algorithm to the reciprocal-space maps of tetragonal BaTiO3 and rhombohedral PbZr1−xTixO3 crystals is demonstrated.International Union of CrystallographyGorfman, S.Spirito, D.Zhang, G.Detlefs, C.Zhang, N.FERROELASTIC DOMAINS; DOMAIN WALLS; HIGH-RESOLUTION X-RAY DIFFRACTIONtext/htmlIdentification of a coherent twin relationship from high-resolution reciprocal-space mapsenurn:issn:2053-2733https://creativecommons.org/licenses/by/4.0/textdoi:10.1107/S2053273322002534Twinning is a common crystallographic phenomenon which is related to the formation and coexistence of several orientation variants of the same crystal structure. It may occur during symmetry-lowering phase transitions or during the crystal growth itself. Once formed, twin domains play an important role in defining physical properties: for example, they underpin the giant piezoelectric effect in ferroelectrics, superelasticity in ferroelastics and the shape-memory effect in martensitic alloys. Regrettably, there is still a lack of experimental methods for the characterization of twin domain patterns. Here, a theoretical framework and algorithm are presented for the recognition of ferroelastic domains, as well as the identification of the coherent twin relationship using high-resolution reciprocal-space mapping of X-ray diffraction intensity around split Bragg peaks. Specifically, the geometrical theory of twinned ferroelastic crystals [Fousek & Janovec (1969). J. Appl. Phys. 40, 135–142] is adapted for the analysis of the X-ray diffraction patterns. The necessary equations are derived and an algorithm is outlined for the calculation of the separation between the Bragg peaks, diffracted from possible coherent twin domains, connected to one another via a mismatch-free interface. It is demonstrated that such separation is always perpendicular to the planar interface between mechanically matched domains. For illustration purposes, the analysis is presented of the separation between the peaks diffracted from tetragonal and rhombohedral domains in the high-resolution reciprocal-space maps of BaTiO3 and PbZr1−xTixO3 crystals. The demonstrated method can be used to analyse the response of multi-domain patterns to external perturbations such as electric field, change of temperature or pressure.2022-04-2878Acta Crystallographica Section A: Foundations and Advances2053-2733May 2022171research papers2022-04-28https://creativecommons.org/licenses/by/4.0/15832053-2733med@iucr.org