Open-access and free articles in Acta Crystallographica Section A: Foundations of Crystallography
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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) 2020 International Union of CrystallographyInternational Union of CrystallographyInternational Union of Crystallographyhttps://journals.iucr.orgurn: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.text/htmlOpen-access and free articles in Acta Crystallographica Section A Foundations and Advancestextyearly62002-01-01T00:00+00:00med@iucr.orgActa Crystallographica Section A Foundations and AdvancesCopyright (c) 2020 International Union of Crystallographyurn:issn:0108-7673Open-access and free articles in Acta Crystallographica Section A: Foundations of Crystallographyhttp://journals.iucr.org/logos/rss10a.gif
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Still imageIsotopy classification of three-dimensional embedded nets
http://scripts.iucr.org/cgi-bin/paper?me6077
Copyright (c) 2020 International Union of Crystallographyurn:issn:2053-2733Schulte, E.2020-04-29doi:10.1107/S2053273320005616International Union of CrystallographyThe article by Power et al. [Acta Cryst. (2020), A76, 275–301] on the isotopy classification of crystal nets is discussed.enEMBEDDED NETS; ISOTOPY CLASSIFICATION; TOPOLOGYtext/htmlIsotopy classification of three-dimensional embedded netstext763Copyright (c) 2020 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances2020-04-29273scientific commentaries2053-2733May 2020med@iucr.org2742053-2733Theoretical study of the properties of X-ray diffraction moiré fringes. III. Theoretical simulation of previous experimental moiré images
http://scripts.iucr.org/cgi-bin/paper?td5063
As a practical confirmation of a recently published X-ray moiré-fringe theory [Yoshimura (2015). Acta Cryst. A71, 368–381], computer simulations using this theory were conducted for previous experimental moiré images of a strained bicrystal specimen [Yoshimura (1996). Acta Cryst. A52, 312–325]. Simulated moiré images with a good or fairly good likeness are presented as a result of this simulation, in which the characteristic fringe-and-band and local strain patterns in the experimental images are reproduced well. Experimental moiré images taken when the inclination of the lattice planes was forcedly increased in one of the component crystals of the bicrystal specimen were also fairly well simulated in this computation, and their fringe patterns of inclined fringes are shown to be in accordance with the prediction by the theory. This moiré-fringe theory is thus considered to be widely applicable to the study of moiré images. Furthermore, the successful simulation of the previous experimental moiré images means that a satisfactory theoretical explanation was given for the experimental images, with respect to their characteristic global features. However, this study by the theoretical simulation shows explicitly that some significant peculiarities in the fringe profiles of the experimental images still remain unexplained by this moiré-fringe theory.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Yoshimura, J.2020-06-30doi:10.1107/S205327332000532XInternational Union of CrystallographyUsing a recently developed moiré-fringe theory of X-ray diffraction, computer simulations of previous experimental moiré images are presented, for an experimental verification of the moiré-fringe theory and for a theoretical explanation of the peculiar experimental moiré images.enX-RAY MOIRE FRINGES; STRAINED CRYSTALS; LOW-CONTRAST BAND PATTERN; PECULIAR EXPERIMENTAL FRINGE PROFILESAs a practical confirmation of a recently published X-ray moiré-fringe theory [Yoshimura (2015). Acta Cryst. A71, 368–381], computer simulations using this theory were conducted for previous experimental moiré images of a strained bicrystal specimen [Yoshimura (1996). Acta Cryst. A52, 312–325]. Simulated moiré images with a good or fairly good likeness are presented as a result of this simulation, in which the characteristic fringe-and-band and local strain patterns in the experimental images are reproduced well. Experimental moiré images taken when the inclination of the lattice planes was forcedly increased in one of the component crystals of the bicrystal specimen were also fairly well simulated in this computation, and their fringe patterns of inclined fringes are shown to be in accordance with the prediction by the theory. This moiré-fringe theory is thus considered to be widely applicable to the study of moiré images. Furthermore, the successful simulation of the previous experimental moiré images means that a satisfactory theoretical explanation was given for the experimental images, with respect to their characteristic global features. However, this study by the theoretical simulation shows explicitly that some significant peculiarities in the fringe profiles of the experimental images still remain unexplained by this moiré-fringe theory.text/htmlTheoretical study of the properties of X-ray diffraction moiré fringes. III. Theoretical simulation of previous experimental moiré imagestext764https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-06-30503research papers2053-2733July 2020med@iucr.org5202053-2733The quaternion-based spatial-coordinate and orientation-frame alignment problems
http://scripts.iucr.org/cgi-bin/paper?ib5072
The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation–orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Hanson, A.J.2020-06-18doi:10.1107/S2053273320002648International Union of CrystallographyQuaternion methods for obtaining solutions to the problem of finding global rotations that optimally align pairs of corresponding lists of 3D spatial and/or orientation data are critically studied. The existence of multiple literatures and historical contexts is pointed out, and the algebraic solutions of the quaternion approach to the classic 3D spatial problem are emphasized. The treatment is extended to novel quaternion-based solutions to the alignment problems for 4D spatial and orientation data.enDATA ALIGNMENT; SPATIAL-COORDINATE ALIGNMENT; ORIENTATION-FRAME ALIGNMENT; QUATERNIONS; QUATERNION FRAMES; QUATERNION EIGENVALUE METHODSThe general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation–orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.text/htmlThe quaternion-based spatial-coordinate and orientation-frame alignment problemstext764https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-06-18432lead articles2053-2733July 2020med@iucr.org4572053-2733Multiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a grid
http://scripts.iucr.org/cgi-bin/paper?ae5082
The generalized Darwin–Hamilton equations [Wuttke (2014). Acta Cryst. A70, 429–440] describe multiple Bragg reflection from a thick, ideally imperfect crystal. These equations are simplified by making full use of energy conservation, and it is demonstrated that the conventional two-ray Darwin–Hamilton equations are obtained as a first-order approximation. Then an efficient numeric solution method is presented, based on a transfer matrix for discretized directional distribution functions and on spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially if the detector only covers a finite solid angle.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Bornemann, F.Li, Y.Y.Wuttke, J.2020-04-16doi:10.1107/S2053273320002065International Union of CrystallographyTo describe multiple Bragg reflection from a thick, ideally imperfect crystal, the transport equations are reformulated in three-dimensional phase space and solved by spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially for finite detectors.enMOSAIC CRYSTALS; MULTIPLE SCATTERING; DARWIN-HAMILTON EQUATIONS; SPECTRAL COLLOCATIONThe generalized Darwin–Hamilton equations [Wuttke (2014). Acta Cryst. A70, 429–440] describe multiple Bragg reflection from a thick, ideally imperfect crystal. These equations are simplified by making full use of energy conservation, and it is demonstrated that the conventional two-ray Darwin–Hamilton equations are obtained as a first-order approximation. Then an efficient numeric solution method is presented, based on a transfer matrix for discretized directional distribution functions and on spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially if the detector only covers a finite solid angle.text/htmlMultiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a gridtext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-16376research papers2053-2733May 2020med@iucr.org3892053-2733Structure-mining: screening structure models by automated fitting to the atomic pair distribution function over large numbers of models
http://scripts.iucr.org/cgi-bin/paper?vk5039
A new approach is presented to obtain candidate structures from atomic pair distribution function (PDF) data in a highly automated way. It fetches, from web-based structural databases, all the structures meeting the experimenter's search criteria and performs structure refinements on them without human intervention. It supports both X-ray and neutron PDFs. Tests on various material systems show the effectiveness and robustness of the algorithm in finding the correct atomic crystal structure. It works on crystalline and nanocrystalline materials including complex oxide nanoparticles and nanowires, low-symmetry and locally distorted structures, and complicated doped and magnetic materials. This approach could greatly reduce the traditional structure searching work and enable the possibility of high-throughput real-time auto-analysis PDF experiments in the future.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Yang, L.Juhás, P.Terban, M.W.Tucker, M.G.Billinge, S.J.L.2020-04-28doi:10.1107/S2053273320002028International Union of CrystallographyStructure-mining finds and returns the best-fit structures from structural databases given a measured pair distribution function data set. Using databases and heuristics for automation, it has the potential to save experimenters a large amount of time as they explore candidate structures from the literature.enPAIR DISTRIBUTION FUNCTION; PDF; AUTOMATED FITTING; ATOMIC STRUCTURE; STRUCTURE DISCOVERYA new approach is presented to obtain candidate structures from atomic pair distribution function (PDF) data in a highly automated way. It fetches, from web-based structural databases, all the structures meeting the experimenter's search criteria and performs structure refinements on them without human intervention. It supports both X-ray and neutron PDFs. Tests on various material systems show the effectiveness and robustness of the algorithm in finding the correct atomic crystal structure. It works on crystalline and nanocrystalline materials including complex oxide nanoparticles and nanowires, low-symmetry and locally distorted structures, and complicated doped and magnetic materials. This approach could greatly reduce the traditional structure searching work and enable the possibility of high-throughput real-time auto-analysis PDF experiments in the future.text/htmlStructure-mining: screening structure models by automated fitting to the atomic pair distribution function over large numbers of modelstext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-28395research papers2053-2733May 2020med@iucr.org4092053-2733SPIND-TC: an indexing method for two-color X-ray diffraction data
http://scripts.iucr.org/cgi-bin/paper?ib5084
Recent developments of two-color operation modes at X-ray free-electron laser facilities provide new research opportunities, such as X-ray pump/X-ray probe experiments and multiple-wavelength anomalous dispersion phasing methods. However, most existing indexing methods were developed for indexing diffraction data from monochromatic X-ray beams. Here, a new algorithm is presented for indexing two-color diffraction data, as an extension of the sparse-pattern indexing algorithm SPIND, which has been demonstrated to be capable of indexing diffraction patterns with as few as five peaks. The principle and implementation of the two-color indexing method, SPIND-TC, are reported in this paper. The algorithm was tested on both simulated and experimental data of protein crystals. The results show that the diffraction data can be accurately indexed in both cases. Source codes are publicly available at https://github.com/lixx11/SPIND-TC.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Li, X.Li, C.Liu, H.2020-04-02doi:10.1107/S2053273320001916International Union of CrystallographyAn auto-indexing method for two-color X-ray diffraction data is presented, which has been tested on both simulated and experimental protein diffraction data. The indexing yield is increased significantly compared with the previous approach using conventional indexers.enSERIAL CRYSTALLOGRAPHY; TWO-COLOR DIFFRACTION; INDEXING ALGORITHMRecent developments of two-color operation modes at X-ray free-electron laser facilities provide new research opportunities, such as X-ray pump/X-ray probe experiments and multiple-wavelength anomalous dispersion phasing methods. However, most existing indexing methods were developed for indexing diffraction data from monochromatic X-ray beams. Here, a new algorithm is presented for indexing two-color diffraction data, as an extension of the sparse-pattern indexing algorithm SPIND, which has been demonstrated to be capable of indexing diffraction patterns with as few as five peaks. The principle and implementation of the two-color indexing method, SPIND-TC, are reported in this paper. The algorithm was tested on both simulated and experimental data of protein crystals. The results show that the diffraction data can be accurately indexed in both cases. Source codes are publicly available at https://github.com/lixx11/SPIND-TC.text/htmlSPIND-TC: an indexing method for two-color X-ray diffraction datatext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-02369research papers2053-2733May 2020med@iucr.org3752053-2733An efficient method for indexing grazing-incidence X-ray diffraction data of epitaxially grown thin films
http://scripts.iucr.org/cgi-bin/paper?wo5036
Crystal structure identification of thin organic films entails a number of technical and methodological challenges. In particular, if molecular crystals are epitaxially grown on single-crystalline substrates a complex scenario of multiple preferred orientations of the adsorbate, several symmetry-related in-plane alignments and the occurrence of unknown polymorphs is frequently observed. In theory, the parameters of the reduced unit cell and its orientation can simply be obtained from the matrix of three linearly independent reciprocal-space vectors. However, if the sample exhibits unit cells in various orientations and/or with different lattice parameters, it is necessary to assign all experimentally obtained reflections to their associated individual origin. In the present work, an effective algorithm is described to accomplish this task in order to determine the unit-cell parameters of complex systems comprising different orientations and polymorphs. This method is applied to a polycrystalline thin film of the conjugated organic material 6,13-pentacenequinone (PQ) epitaxially grown on an Ag(111) surface. All reciprocal vectors can be allocated to unit cells of the same lattice constants but grown in various orientations [sixfold rotational symmetry for the contact planes (102) and (102)]. The as-determined unit cell is identical to that reported in a previous study determined for a fibre-textured PQ film. Preliminary results further indicate that the algorithm is especially effective in analysing epitaxially grown crystallites not only for various orientations, but also if different polymorphs are present in the film.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Simbrunner, J.Schrode, B.Domke, J.Fritz, T.Salzmann, I.Resel, R.2020-04-02doi:10.1107/S2053273320001266International Union of CrystallographyA method is described for indexing grazing-incidence X-ray diffraction data of epitaxially grown thin films comprising various crystal orientations and/or polymorphs by measuring reciprocal-lattice vectors.enEPITAXY; INDEXING; MATHEMATICAL CRYSTALLOGRAPHYCrystal structure identification of thin organic films entails a number of technical and methodological challenges. In particular, if molecular crystals are epitaxially grown on single-crystalline substrates a complex scenario of multiple preferred orientations of the adsorbate, several symmetry-related in-plane alignments and the occurrence of unknown polymorphs is frequently observed. In theory, the parameters of the reduced unit cell and its orientation can simply be obtained from the matrix of three linearly independent reciprocal-space vectors. However, if the sample exhibits unit cells in various orientations and/or with different lattice parameters, it is necessary to assign all experimentally obtained reflections to their associated individual origin. In the present work, an effective algorithm is described to accomplish this task in order to determine the unit-cell parameters of complex systems comprising different orientations and polymorphs. This method is applied to a polycrystalline thin film of the conjugated organic material 6,13-pentacenequinone (PQ) epitaxially grown on an Ag(111) surface. All reciprocal vectors can be allocated to unit cells of the same lattice constants but grown in various orientations [sixfold rotational symmetry for the contact planes (102) and (102)]. The as-determined unit cell is identical to that reported in a previous study determined for a fibre-textured PQ film. Preliminary results further indicate that the algorithm is especially effective in analysing epitaxially grown crystallites not only for various orientations, but also if different polymorphs are present in the film.text/htmlAn efficient method for indexing grazing-incidence X-ray diffraction data of epitaxially grown thin filmstext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-04-02345research papers2053-2733May 2020med@iucr.org3572053-2733Isotopy classes for 3-periodic net embeddings
http://scripts.iucr.org/cgi-bin/paper?ib5087
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Power, S.C.Baburin, I.A.Proserpio, D.M.2020-03-05doi:10.1107/S2053273320000625International Union of CrystallographyEntangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type.enPERIODIC NETS; EMBEDDED NETS; COORDINATION POLYMERS; ISOTOPY TYPES; CRYSTALLOGRAPHIC FRAMEWORKSEntangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.text/htmlIsotopy classes for 3-periodic net embeddingstext763https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-03-05275lead articles2053-2733May 2020med@iucr.org3012053-2733The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tiling
http://scripts.iucr.org/cgi-bin/paper?ae5079
In this study, the atomic structure of the ternary icosahedral ZnMgTm quasicrystal (QC) is investigated by means of single-crystal X-ray diffraction. The structure is found to be a member of the Bergman QC family, frequently found in Zn–Mg–rare-earth systems. The ab initio structure solution was obtained by the use of the Superflip software. The infinite structure model was founded on the atomic decoration of two golden rhombohedra, with an edge length of 21.7 Å, constituting the Ammann–Kramer–Neri tiling. The refined structure converged well with the experimental diffraction diagram, with the crystallographic R factor equal to 9.8%. The Bergman clusters were found to be bonded by four possible linkages. Only two linkages, b and c, are detected in approximant crystals and are employed to model the icosahedral QCs in the cluster approach known for the CdYb Tsai-type QC. Additional short b and a linkages are found in this study. Short interatomic distances are not generated by those linkages due to the systematic absence of atoms and the formation of split atomic positions. The presence of four linkages allows the structure to be pictured as a complete covering by rhombic triacontahedral clusters and consequently there is no need to define the interstitial part of the structure (i.e. that outside the cluster). The 6D embedding of the solved structure is discussed for the final verification of the model.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Buganski, I.Wolny, J.Takakura, H.2020-02-11doi:10.1107/S2053273319017339International Union of CrystallographyThe article discusses the atomic structure modelling based on the Ammann–Kramer–Neri tiling of the ternary Bergman quasicrystal in the 3D real space.enBERGMAN QUASICRYSTAL; ATOMIC STRUCTURE; AVERAGE UNIT CELLIn this study, the atomic structure of the ternary icosahedral ZnMgTm quasicrystal (QC) is investigated by means of single-crystal X-ray diffraction. The structure is found to be a member of the Bergman QC family, frequently found in Zn–Mg–rare-earth systems. The ab initio structure solution was obtained by the use of the Superflip software. The infinite structure model was founded on the atomic decoration of two golden rhombohedra, with an edge length of 21.7 Å, constituting the Ammann–Kramer–Neri tiling. The refined structure converged well with the experimental diffraction diagram, with the crystallographic R factor equal to 9.8%. The Bergman clusters were found to be bonded by four possible linkages. Only two linkages, b and c, are detected in approximant crystals and are employed to model the icosahedral QCs in the cluster approach known for the CdYb Tsai-type QC. Additional short b and a linkages are found in this study. Short interatomic distances are not generated by those linkages due to the systematic absence of atoms and the formation of split atomic positions. The presence of four linkages allows the structure to be pictured as a complete covering by rhombic triacontahedral clusters and consequently there is no need to define the interstitial part of the structure (i.e. that outside the cluster). The 6D embedding of the solved structure is discussed for the final verification of the model.text/htmlThe atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tilingtext762https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-02-11180research papers2053-2733March 2020med@iucr.org1962053-2733Distinguishing space groups by electron channelling: centrosymmetric full-Heusler or non-centrosymmetric half-Heusler?
http://scripts.iucr.org/cgi-bin/paper?lk5054
X-ray emission under electron-channelling conditions is used to distinguish between a non-centrosymmetric half-Heusler and a centrosymmetric full-Heusler crystal. For TiCo1.5+xSn the space-group determination based on a Rietveld refinement procedure became challenging for increasing Co content (x > 0.2), while electron channelling proved successful for higher Co content (x = 0.35). This technique can be used on crystals as small as (10 nm)3.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Hansen, V.Kosinskiy, A.Taftø, J.2020-02-19doi:10.1107/S2053273319016942International Union of CrystallographyElectron channelling was successfully used to determine the space group of a crystal where conventional diffraction failed to distinguish between half-Heusler and full-Heusler.enSPACE GROUPS; INVERSION SYMMETRY; ELECTRON CHANNELLING; X-RAY EMISSION; HEUSLER CRYSTALSX-ray emission under electron-channelling conditions is used to distinguish between a non-centrosymmetric half-Heusler and a centrosymmetric full-Heusler crystal. For TiCo1.5+xSn the space-group determination based on a Rietveld refinement procedure became challenging for increasing Co content (x > 0.2), while electron channelling proved successful for higher Co content (x = 0.35). This technique can be used on crystals as small as (10 nm)3.text/htmlDistinguishing space groups by electron channelling: centrosymmetric full-Heusler or non-centrosymmetric half-Heusler?text762https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-02-19211short communications2053-2733March 2020med@iucr.org2132053-2733pinkIndexer – a universal indexer for pink-beam X-ray and electron diffraction snapshots
http://scripts.iucr.org/cgi-bin/paper?ae5078
A crystallographic indexing algorithm, pinkIndexer, is presented for the analysis of snapshot diffraction patterns. It can be used in a variety of contexts including measurements made with a monochromatic radiation source, a polychromatic source or with radiation of very short wavelength. As such, the algorithm is particularly suited to automated data processing for two emerging measurement techniques for macromolecular structure determination: serial pink-beam X-ray crystallography and serial electron crystallography, which until now lacked reliable programs for analyzing many individual diffraction patterns from crystals of uncorrelated orientation. The algorithm requires approximate knowledge of the unit-cell parameters of the crystal, but not the wavelengths associated with each Bragg spot. The use of pinkIndexer is demonstrated by obtaining 1005 lattices from a published pink-beam serial crystallography data set that had previously yielded 140 indexed lattices. Additionally, in tests on experimental serial crystallography diffraction data recorded with quasi-monochromatic X-rays and with electrons the algorithm indexed more patterns than other programs tested.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Gevorkov, Y.Barty, A.Brehm, W.White, T.A.Tolstikova, A.Wiedorn, M.O.Meents, A.Grigat, R.-R.Chapman, H.N.Yefanov, O.2020-01-10doi:10.1107/S2053273319015559International Union of CrystallographypinkIndexer, an algorithm developed for indexing of snapshot diffraction patterns recorded with pink-beam X-rays, monochromatic X-rays and electrons, is described and its use evaluated.enINDEXING; PINKINDEXER; CRYSTFEL; PINK X-RAY BEAM; SERIAL ELECTRON DIFFRACTIONA crystallographic indexing algorithm, pinkIndexer, is presented for the analysis of snapshot diffraction patterns. It can be used in a variety of contexts including measurements made with a monochromatic radiation source, a polychromatic source or with radiation of very short wavelength. As such, the algorithm is particularly suited to automated data processing for two emerging measurement techniques for macromolecular structure determination: serial pink-beam X-ray crystallography and serial electron crystallography, which until now lacked reliable programs for analyzing many individual diffraction patterns from crystals of uncorrelated orientation. The algorithm requires approximate knowledge of the unit-cell parameters of the crystal, but not the wavelengths associated with each Bragg spot. The use of pinkIndexer is demonstrated by obtaining 1005 lattices from a published pink-beam serial crystallography data set that had previously yielded 140 indexed lattices. Additionally, in tests on experimental serial crystallography diffraction data recorded with quasi-monochromatic X-rays and with electrons the algorithm indexed more patterns than other programs tested.text/htmlpinkIndexer – a universal indexer for pink-beam X-ray and electron diffraction snapshotstext762https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-01-10121research papers2053-2733March 2020med@iucr.org1312053-2733Converting three-space matrices to equivalent six-space matrices for Delone scalars in S6
http://scripts.iucr.org/cgi-bin/paper?ae5074
The transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices when working in the six-dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three-space representation, apply the three-by-three matrices and then back-transform to the six-space representation, but it is much simpler to have the equivalent six-by-six matrices and apply them directly. The general form of the transformation from the three-space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Andrews, L.C.Bernstein, H.J.Sauter, N.K.2020-01-01doi:10.1107/S2053273319014542International Union of CrystallographyGiven a matrix for transforming vectors in the three-space of unit-cell edge vectors, the corresponding matrix to transform vectors in the six-space of Delone scalars is derived.enDELAUNAY; DELONE; CENTERING TRANSFORMATIONS; CENTERED LATTICES; REDUCED CELLS; LATTICE CENTERING; NIGGLI; SELLING; MATRIX TRANSFORMATIONSThe transformations from the primitive cells of the centered Bravais lattices to the corresponding centered cells have conventionally been listed as three-by-three matrices that transform three-space lattice vectors. Using those three-by-three matrices when working in the six-dimensional space of lattices represented as Selling scalars as used in Delone (Delaunay) reduction, one could transform to the three-space representation, apply the three-by-three matrices and then back-transform to the six-space representation, but it is much simpler to have the equivalent six-by-six matrices and apply them directly. The general form of the transformation from the three-space matrix to the corresponding matrix operating on Selling scalars (expressed in space S6) is derived, and the particular S6matrices for the centered Delone types are listed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)text/htmlConverting three-space matrices to equivalent six-space matrices for Delone scalars in S6text1762020-01-01Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers79med@iucr.orgJanuary 2020832053-2733X-ray diffraction from strongly bent crystals and spectroscopy of X-ray free-electron laser pulses
http://scripts.iucr.org/cgi-bin/paper?iv5002
The use of strongly bent crystals in spectrometers for pulses of a hard X-ray free-electron laser is explored theoretically. Diffraction is calculated in both dynamical and kinematical theories. It is shown that diffraction can be treated kinematically when the bending radius is small compared with the critical radius given by the ratio of the Bragg-case extinction length for the actual reflection to the Darwin width of this reflection. As a result, the spectral resolution is limited by the crystal thickness, rather than the extinction length, and can become better than the resolution of a planar dynamically diffracting crystal. As an example, it is demonstrated that spectra of the 12 keV pulses can be resolved in the 440 reflection from a 20 µm-thick diamond crystal bent to a radius of 10 cm.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Kaganer, V.M.Petrov, I.Samoylova, L.2020-01-01doi:10.1107/S2053273319014347International Union of CrystallographyA strongly bent crystal diffracts kinematically when the bending radius is small compared with the critical radius given by the ratio of the extinction length to the Darwin width of the reflection. Under these conditions, the spectral resolution of the X-ray free-electron laser pulse is limited by the crystal thickness and can be better than under dynamical diffraction conditions.enX-RAY FREE-ELECTRON LASERS; X-RAY SPECTROSCOPY; BENT CRYSTALS; DIAMOND CRYSTAL OPTICS; FEMTOSECOND X-RAY DIFFRACTION; DYNAMICAL DIFFRACTIONThe use of strongly bent crystals in spectrometers for pulses of a hard X-ray free-electron laser is explored theoretically. Diffraction is calculated in both dynamical and kinematical theories. It is shown that diffraction can be treated kinematically when the bending radius is small compared with the critical radius given by the ratio of the Bragg-case extinction length for the actual reflection to the Darwin width of this reflection. As a result, the spectral resolution is limited by the crystal thickness, rather than the extinction length, and can become better than the resolution of a planar dynamically diffracting crystal. As an example, it is demonstrated that spectra of the 12 keV pulses can be resolved in the 440 reflection from a 20 µm-thick diamond crystal bent to a radius of 10 cm.text/htmlX-ray diffraction from strongly bent crystals and spectroscopy of X-ray free-electron laser pulsestext1762020-01-01Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers55med@iucr.orgJanuary 2020692053-2733Cluster-mining: an approach for determining core structures of metallic nanoparticles from atomic pair distribution function data
http://scripts.iucr.org/cgi-bin/paper?lk5048
A novel approach for finding and evaluating structural models of small metallic nanoparticles is presented. Rather than fitting a single model with many degrees of freedom, libraries of clusters from multiple structural motifs are built algorithmically and individually refined against experimental pair distribution functions. Each cluster fit is highly constrained. The approach, called cluster-mining, returns all candidate structure models that are consistent with the data as measured by a goodness of fit. It is highly automated, easy to use, and yields models that are more physically realistic and result in better agreement to the data than models based on cubic close-packed crystallographic cores, often reported in the literature for metallic nanoparticles.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Banerjee, S.Liu, C.-H.Jensen, K.M.ØJuhás, P.Lee, J.D.Tofanelli, M.Ackerson, C.J.Murray, C.B.Billinge, S.J.L.2020-01-01doi:10.1107/S2053273319013214International Union of CrystallographyA novel approach for finding and evaluating structural models of small metallic nanoparticles is presented.enSTRUCTURAL MODELS; NANOPARTICLES; CLUSTERS; PAIR DISTRIBUTION FUNCTIONS; DATA MINING; SCREENINGA novel approach for finding and evaluating structural models of small metallic nanoparticles is presented. Rather than fitting a single model with many degrees of freedom, libraries of clusters from multiple structural motifs are built algorithmically and individually refined against experimental pair distribution functions. Each cluster fit is highly constrained. The approach, called cluster-mining, returns all candidate structure models that are consistent with the data as measured by a goodness of fit. It is highly automated, easy to use, and yields models that are more physically realistic and result in better agreement to the data than models based on cubic close-packed crystallographic cores, often reported in the literature for metallic nanoparticles.text/htmlCluster-mining: an approach for determining core structures of metallic nanoparticles from atomic pair distribution function datatext1762020-01-01Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers24med@iucr.orgJanuary 2020312053-2733Elastic propagation of fast electron vortices through amorphous materials
http://scripts.iucr.org/cgi-bin/paper?lk5051
This work studies the elastic scattering behavior of electron vortices when propagating through amorphous samples. A formulation of the multislice approach in cylindrical coordinates is used to theoretically investigate the redistribution of intensity between different angular momentum components due to scattering. To corroborate and elaborate on our theoretical results, extensive numerical simulations are performed on three model systems (Si3N4, Fe0.8B0.2, Pt) for a wide variety of experimental parameters to quantify the purity of the vortices, the net angular momentum transfer, and the variability of the results with respect to the random relative position between the electron beam and the scattering atoms. These results will help scientists to further improve the creation of electron vortices and enhance applications involving them.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Löffler, S.Sack, S.Schachinger, T.2019-11-04doi:10.1107/S2053273319012889International Union of CrystallographyThis article describes the scattering behavior of electron vortices inside amorphous samples. It focuses on the vortex purity, net angular momentum transfer, and statistical variations due to random beam and atom positions.enELECTRON VORTEX BEAMS; AMORPHOUS MATERIALS; ELASTIC SCATTERINGThis work studies the elastic scattering behavior of electron vortices when propagating through amorphous samples. A formulation of the multislice approach in cylindrical coordinates is used to theoretically investigate the redistribution of intensity between different angular momentum components due to scattering. To corroborate and elaborate on our theoretical results, extensive numerical simulations are performed on three model systems (Si3N4, Fe0.8B0.2, Pt) for a wide variety of experimental parameters to quantify the purity of the vortices, the net angular momentum transfer, and the variability of the results with respect to the random relative position between the electron beam and the scattering atoms. These results will help scientists to further improve the creation of electron vortices and enhance applications involving them.text/htmlElastic propagation of fast electron vortices through amorphous materialstext6752019-11-04Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers902med@iucr.orgNovember 20199102053-2733Relativistic correction of atomic scattering factors for high-energy electron diffraction
http://scripts.iucr.org/cgi-bin/paper?lk5052
Relativistic electron diffraction depends on linear and quadratic terms in the electric potential, the latter being neglected in the frequently used relativistically corrected Schrödinger equation. The quadratic electric potential term modifies atomic scattering amplitudes in particular for large-angle scattering and backscattering. The respective correction increases with increasing scattering angle, increasing atomic number and increasing kinetic energy. Conventional tabulations for electron scattering and its large-angle extrapolations can be amended in closed form by a universal correction based on the screened Coulomb potential squared.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Lentzen, M.2019-10-24doi:10.1107/S2053273319012191International Union of CrystallographyRelativistic electron diffraction depends on linear and quadratic terms in the electric potential, the latter being neglected in the frequently used relativistically corrected Schrödinger equation. Conventional tabulations for electron scattering and its large-angle extrapolations can be amended in closed form by a universal correction based on the screened Coulomb potential squared.enELECTRON DIFFRACTION; ATOMIC SCATTERING FACTORS; RELATIVITY THEORY; SCHRODINGER EQUATIONRelativistic electron diffraction depends on linear and quadratic terms in the electric potential, the latter being neglected in the frequently used relativistically corrected Schrödinger equation. The quadratic electric potential term modifies atomic scattering amplitudes in particular for large-angle scattering and backscattering. The respective correction increases with increasing scattering angle, increasing atomic number and increasing kinetic energy. Conventional tabulations for electron scattering and its large-angle extrapolations can be amended in closed form by a universal correction based on the screened Coulomb potential squared.text/htmlRelativistic correction of atomic scattering factors for high-energy electron diffractiontext6752019-10-24Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers861med@iucr.orgNovember 20198652053-2733X-ray interference fringes from a weakly bent plane-parallel crystal with negative strain gradient
http://scripts.iucr.org/cgi-bin/paper?lk5040
Under the anomalous transmission condition in the Bragg mode, X-ray interference fringes were observed between two beams with different hyperbolic trajectories in a very weakly bent plane-parallel perfect crystal with negative strain gradient. The origin of the fringes was analysed based on the dynamical theory of diffraction for a distorted crystal. In the reflected beam from the entrance surface, the interference fringes were observed between once- and twice-reflected beams from the back surface. In the transmitted beam from the back surface, the interference fringes were observed between the direct beam and once-reflected beam from the entrance surface. In the emitted beam from the lateral surface, the interference fringes were observed between the beams after different numbers of reflections in the crystal. The multiply reflected beams were formed by a combined result of long propagation length along the beam direction with large divergence of the refracted beams when the strain gradient was negative. The period of these interference fringes was sensitive to very weak strain, of the order of 10−7.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Fukamachi, T.Jongsukswat, S.Ju, D.Negishi, R.Hirano, K.Kawamura, T.2019-10-07doi:10.1107/S2053273319011859International Union of CrystallographyIn the waves emitted from the entrance, back and lateral surfaces of a very weakly bent plane-parallel perfect crystal with negative strain gradient, X-ray interference fringes between two refracted beams with different hyperbolic trajectories were observed when the strain was very weak, of the order of 10−7.enINTERFERENCE FRINGES; MIRAGE FRINGES; X-RAY BEAM TRAJECTORY; BENT CRYSTAL; MULTIPLE BRAGG DIFFRACTION; DYNAMICAL THEORY OF X-RAY DIFFRACTIONUnder the anomalous transmission condition in the Bragg mode, X-ray interference fringes were observed between two beams with different hyperbolic trajectories in a very weakly bent plane-parallel perfect crystal with negative strain gradient. The origin of the fringes was analysed based on the dynamical theory of diffraction for a distorted crystal. In the reflected beam from the entrance surface, the interference fringes were observed between once- and twice-reflected beams from the back surface. In the transmitted beam from the back surface, the interference fringes were observed between the direct beam and once-reflected beam from the entrance surface. In the emitted beam from the lateral surface, the interference fringes were observed between the beams after different numbers of reflections in the crystal. The multiply reflected beams were formed by a combined result of long propagation length along the beam direction with large divergence of the refracted beams when the strain gradient was negative. The period of these interference fringes was sensitive to very weak strain, of the order of 10−7.text/htmlX-ray interference fringes from a weakly bent plane-parallel crystal with negative strain gradienttext6752019-10-07Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers842med@iucr.orgNovember 20198502053-2733Bayesian machine learning improves single-wavelength anomalous diffraction phasing
http://scripts.iucr.org/cgi-bin/paper?ae5069
Single-wavelength X-ray anomalous diffraction (SAD) is a frequently employed technique to solve the phase problem in X-ray crystallography. The precision and accuracy of recovered anomalous differences are crucial for determining the correct phases. Continuous rotation (CR) and inverse-beam geometry (IBG) anomalous data collection methods have been performed on tetragonal lysozyme and monoclinic survivin crystals and analysis carried out of how correlated the pairs of Friedel's reflections are after scaling. A multivariate Bayesian model for estimating anomalous differences was tested, which takes into account the correlation between pairs of intensity observations and incorporates the a priori knowledge about the positivity of intensity. The CR and IBG data collection methods resulted in positive correlation between I(+) and I(−) observations, indicating that the anomalous difference dominates between these observations, rather than different levels of radiation damage. An alternative pairing method based on near simultaneously observed Bijvoet's pairs displayed lower correlation and it was unsuccessful for recovering useful anomalous differences when using the multivariate Bayesian model. In contrast, multivariate Bayesian treatment of Friedel's pairs improved the initial phasing of the two tested crystal systems and the two data collection methods.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Garcia-Bonete, M.-J.Katona, G.2019-10-07doi:10.1107/S2053273319011446International Union of CrystallographyThe a posteriori probability densities of anomalous structure-factor amplitude differences were estimated by the Markov chain Monte Carlo machine-learning method. The model incorporated the correlation between the different Bijvoet pairs and the improved estimates were shown to be beneficial for SAD phasing.enSINGLE-WAVELENGTH X-RAY ANOMALOUS DIFFRACTION; SAD; FRIEDEL PAIRS; BIJVOET PAIRS; CONTINUOUS ROTATION DATA COLLECTION; INVERSE-BEAM GEOMETRY; BAYESIAN INFERENCE; SURVIVINSingle-wavelength X-ray anomalous diffraction (SAD) is a frequently employed technique to solve the phase problem in X-ray crystallography. The precision and accuracy of recovered anomalous differences are crucial for determining the correct phases. Continuous rotation (CR) and inverse-beam geometry (IBG) anomalous data collection methods have been performed on tetragonal lysozyme and monoclinic survivin crystals and analysis carried out of how correlated the pairs of Friedel's reflections are after scaling. A multivariate Bayesian model for estimating anomalous differences was tested, which takes into account the correlation between pairs of intensity observations and incorporates the a priori knowledge about the positivity of intensity. The CR and IBG data collection methods resulted in positive correlation between I(+) and I(−) observations, indicating that the anomalous difference dominates between these observations, rather than different levels of radiation damage. An alternative pairing method based on near simultaneously observed Bijvoet's pairs displayed lower correlation and it was unsuccessful for recovering useful anomalous differences when using the multivariate Bayesian model. In contrast, multivariate Bayesian treatment of Friedel's pairs improved the initial phasing of the two tested crystal systems and the two data collection methods.text/htmlBayesian machine learning improves single-wavelength anomalous diffraction phasingtext6752019-10-07Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers851med@iucr.orgNovember 20198602053-2733XGANDALF – extended gradient descent algorithm for lattice finding
http://scripts.iucr.org/cgi-bin/paper?ae5071
Serial crystallography records still diffraction patterns from single, randomly oriented crystals, then merges data from hundreds or thousands of them to form a complete data set. To process the data, the diffraction patterns must first be indexed, equivalent to determining the orientation of each crystal. A novel automatic indexing algorithm is presented, which in tests usually gives significantly higher indexing rates than alternative programs currently available for this task. The algorithm does not require prior knowledge of the lattice parameters but can make use of that information if provided, and also allows indexing of diffraction patterns generated by several crystals in the beam. Cases with a small number of Bragg spots per pattern appear to particularly benefit from the new approach. The algorithm has been implemented and optimized for fast execution, making it suitable for real-time feedback during serial crystallography experiments. It is implemented in an open-source C++ library and distributed under the LGPLv3 licence. An interface to it has been added to the CrystFEL software suite.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Gevorkov, Y.Yefanov, O.Barty, A.White, T.A.Mariani, V.Brehm, W.Tolstikova, A.Grigat, R.-R.Chapman, H.N.2019-08-30doi:10.1107/S2053273319010593International Union of CrystallographyA description and evaluation are given of XGANDALF, extended gradient descent algorithm for lattice finding, an algorithm developed for fast and accurate indexing of snapshot diffraction patterns.enINDEXING; XGANDALF; CRYSTFEL; MULTIPLE LATTICES; SERIAL CRYSTALLOGRAPHYSerial crystallography records still diffraction patterns from single, randomly oriented crystals, then merges data from hundreds or thousands of them to form a complete data set. To process the data, the diffraction patterns must first be indexed, equivalent to determining the orientation of each crystal. A novel automatic indexing algorithm is presented, which in tests usually gives significantly higher indexing rates than alternative programs currently available for this task. The algorithm does not require prior knowledge of the lattice parameters but can make use of that information if provided, and also allows indexing of diffraction patterns generated by several crystals in the beam. Cases with a small number of Bragg spots per pattern appear to particularly benefit from the new approach. The algorithm has been implemented and optimized for fast execution, making it suitable for real-time feedback during serial crystallography experiments. It is implemented in an open-source C++ library and distributed under the LGPLv3 licence. An interface to it has been added to the CrystFEL software suite.text/htmlXGANDALF – extended gradient descent algorithm for lattice findingtext755https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2019-08-30694research papers2053-2733September 2019med@iucr.org7042053-2733