https://journals.iucr.org/a/issues/2026/04/00/index.html 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 Copyright (c) 2026 International Union of Crystallography 2026-06-12 International Union of Crystallography International Union of Crystallography http://journals.iucr.org urn:issn:2053-2733 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. text/html Acta Crystallographica Section A: Foundations and Advances, Volume 82, Part 4, 2026 text weekly 6 2002-01-01T00:00+00:00 4 82 2026-06-12 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A: Foundations and Advances 229 urn:issn:2053-2733 med@iucr.org June 2026 2026-06-12 http://journals.iucr.org/logos/rss10a.gif https://journals.iucr.org/a/issues/2026/04/00/index.html Still image https://journals.iucr.org/paper?ae5176 The inversion of large-scale diffraction datasets from modern synchrotron sources presents a fundamental challenge in computational crystallography. This paper presents a unified algorithmic framework for the analysis of both near-field (morphological) and far-field (orientational and strain) high-energy diffraction microscopy (HEDM) data. We detail the mathematical formalisms and physical models that form the foundation of this methodology. Key aspects include a generalized model for detector distortion correction, robust algorithms for peak identification in noisy and overlapping patterns, a computationally efficient indexing formalism based on Friedel pair symmetry, and a decoupled iterative refinement scheme that exploits the differing sensitivities of position, orientation and lattice parameters to diffraction observables. We also describe the synergistic integration of near-field and far-field data streams, a critical feature of a truly comprehensive approach. The framework is validated in Part II of this series [Sharma et al. (2026). Acta Cryst. A82, https://doi.org/10.1107/S2053273326004018] using both experimental Ti-7 Al datasets and synthetic reconstructions with known ground truth, achieving orientation accuracy of ∼0.05° and position accuracy of ∼10 µm on experimental data, and a 190× improvement in lattice parameter precision over conventional simultaneous parameter refinement on synthetic data. This integrated framework provides a powerful and extensible solution for turning raw diffraction images into actionable microstructural and micromechanical information. Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 Sharma, H. Park, J.-S. Kenesei, P. 2026-05-28 doi:10.1107/S2053273326004006 International Union of Crystallography This paper details the complete methodological framework implemented in the MIDAS software for processing high-energy diffraction microscopy (HEDM) data. We describe the specific algorithms, coordinate systems and physical models used for both far-field and near-field HEDM analysis. The framework is designed to handle the large and complex datasets from modern synchrotrons, enabling efficient and accurate extraction of microstructural information. EN HEDM high-energy diffraction microscopy 3DXRD 3D X-ray diffraction MIDAS software data reduction methodology diffraction analysis synchrotron radiation The inversion of large-scale diffraction datasets from modern synchrotron sources presents a fundamental challenge in computational crystallography. This paper presents a unified algorithmic framework for the analysis of both near-field (morphological) and far-field (orientational and strain) high-energy diffraction microscopy (HEDM) data. We detail the mathematical formalisms and physical models that form the foundation of this methodology. Key aspects include a generalized model for detector distortion correction, robust algorithms for peak identification in noisy and overlapping patterns, a computationally efficient indexing formalism based on Friedel pair symmetry, and a decoupled iterative refinement scheme that exploits the differing sensitivities of position, orientation and lattice parameters to diffraction observables. We also describe the synergistic integration of near-field and far-field data streams, a critical feature of a truly comprehensive approach. The framework is validated in Part II of this series [Sharma et al. (2026). Acta Cryst. A82, https://doi.org/10.1107/S2053273326004018] using both experimental Ti-7 Al datasets and synthetic reconstructions with known ground truth, achieving orientation accuracy of ∼0.05° and position accuracy of ∼10 µm on experimental data, and a 190× improvement in lattice parameter precision over conventional simultaneous parameter refinement on synthetic data. This integrated framework provides a powerful and extensible solution for turning raw diffraction images into actionable microstructural and micromechanical information. text/html MIDAS: a methodological framework for high-speed high-energy diffraction microscopy data reduction. Part I: methodology text 4 82 2026-05-28 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A research papers 0 0

https://journals.iucr.org/paper?tw5016 The multipole model provides a significantly improved description of electron density compared with the spherical atom approximation and is widely applied in the crystallographic refinement of X-ray diffraction data. The multipolar atom types from theory and statistical clustering (MATTS) data bank collects multipole model parameters for atom types. These parameters are derived from quantum chemical calculations performed on experimental geometries of model molecules and rely on the concept of transferability between chemically similar atoms. An essential component of each atom type is the definition of the local coordinate system (LCS) and the symmetry of the electron density, which so far have been selected individually using expert judgment during data bank construction. In this work, we focus on the electron densities of atoms from model molecules and atom types from the MATTS data bank. We introduce a systematic procedure in which symmetry constraints are removed during multipole model refinement of model molecules and multiple types of LCS are tested using chemically meaningful directions. We examine how different LCS choices influence the pseudosymmetry, the apparent symmetry of electron density, identified empirically based on the refined multipole model parameters and their statistical significance. Our results show that refinement without symmetry constraints improves the representation of pseudosymmetry and, in some cases, leads to changes in both the values of the multipole model parameters and the assigned symmetry of the electron density. We propose an optimal LCS for each topological kind of atom type and provide clear criteria for symmetry assignment. This work contributes to the future development of the MATTS data bank with improved descriptions of LCSs and transferability of electron-density symmetry. Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 Rybicka, P.M. Kulik, M. Ignat'ev, V. Dominiak, P.M. 2026-06-12 doi:10.1107/S2053273326004651 International Union of Crystallography This work focuses on the electron density of atoms modeled with a multipolar approach. We investigate how the refinement of the multipole models with/without symmetry constraints and optimization of the orientation of the local coordinate system and pseudosymmetry assignment improves the accuracy of electron-density representations for atoms and atom types in a pseudoatom data bank. EN quantum crystallography electron density multipolar atom types from theory and statistical clustering MATTS transferable aspherical atom model TAAM The multipole model provides a significantly improved description of electron density compared with the spherical atom approximation and is widely applied in the crystallographic refinement of X-ray diffraction data. The multipolar atom types from theory and statistical clustering (MATTS) data bank collects multipole model parameters for atom types. These parameters are derived from quantum chemical calculations performed on experimental geometries of model molecules and rely on the concept of transferability between chemically similar atoms. An essential component of each atom type is the definition of the local coordinate system (LCS) and the symmetry of the electron density, which so far have been selected individually using expert judgment during data bank construction. In this work, we focus on the electron densities of atoms from model molecules and atom types from the MATTS data bank. We introduce a systematic procedure in which symmetry constraints are removed during multipole model refinement of model molecules and multiple types of LCS are tested using chemically meaningful directions. We examine how different LCS choices influence the pseudosymmetry, the apparent symmetry of electron density, identified empirically based on the refined multipole model parameters and their statistical significance. Our results show that refinement without symmetry constraints improves the representation of pseudosymmetry and, in some cases, leads to changes in both the values of the multipole model parameters and the assigned symmetry of the electron density. We propose an optimal LCS for each topological kind of atom type and provide clear criteria for symmetry assignment. This work contributes to the future development of the MATTS data bank with improved descriptions of LCSs and transferability of electron-density symmetry. text/html From atoms to a data bank: optimizing transferability of electron-density symmetry text 4 82 2026-06-12 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A research papers 0 0

https://journals.iucr.org/paper?cam5015 Coincidence-site lattice (CSL) theory provides a fundamental framework for classifying commensurate crystal orientations and plays a central role in describing grain boundaries. In particular, Ranganathan's formula determines all CSL orientation relationships. However, it is not possible to avoid duplication and also to determine the equivalence of the duplicates, and number theoretical analysis can be required to address these issues. In this study, we provide a simple necessary condition to find all CSL orientation relationships in the cubic system by only using linear algebra. The necessary condition is given by calculating whether the (hkl) plane is commensurate to the (001) plane. We also found that the number of orientation relationships without symmetric tilt grain boundaries (GBs) increases as the Σ value increases by comparing the number of solutions of the equation for constructing symmetric tilt GBs with that of the necessary condition. Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 Kawahara, K. Inoue, K. Saito, M. Chen, Q. Ikuhara, Y. 2026-05-15 doi:10.1107/S2053273326003645 International Union of Crystallography We present a simple linear-algebra-based condition for identifying all coincidence-site lattice orientation relationships in cubic crystals by testing the commensurability of (hkl) planes with (001). EN coincidence-site lattices grain boundaries cubic crystal system Coincidence-site lattice (CSL) theory provides a fundamental framework for classifying commensurate crystal orientations and plays a central role in describing grain boundaries. In particular, Ranganathan's formula determines all CSL orientation relationships. However, it is not possible to avoid duplication and also to determine the equivalence of the duplicates, and number theoretical analysis can be required to address these issues. In this study, we provide a simple necessary condition to find all CSL orientation relationships in the cubic system by only using linear algebra. The necessary condition is given by calculating whether the (hkl) plane is commensurate to the (001) plane. We also found that the number of orientation relationships without symmetric tilt grain boundaries (GBs) increases as the Σ value increases by comparing the number of solutions of the equation for constructing symmetric tilt GBs with that of the necessary condition. text/html New algorithm for generating all coincidence-site lattices of the cubic crystal system text 4 82 2026-05-15 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A research papers 0 0

https://journals.iucr.org/paper?gau5004 We investigate the symmetries of a symbolic dynamical system (Xk, ΓK) of number-theoretic origin. Specifically, we analyze the shift space Xk, defined as the closure of the set Vk of k-free points within the ring of integers {\cal O}_{K} of the biquadratic number field K = {\bb Q}(\sqrt{2},i). The group of shift maps S, which acts on the Minkowski embedding \Gamma_{K}\cong{\bb Z}^{4} by translations, serves as the fundamental action of the system. Our focus is on the homeomorphisms of Xk that interact with the shift action: the automorphism group Aut(Xk, ΓK), consisting of homeomorphisms that commute with every element of S, and the extended symmetry group Sym(Xk, ΓK), which includes homeomorphisms that map the shift action to itself via an automorphism of S. While Aut(Xk, ΓK) is known to be trivial (consisting solely of the shifts themselves), we demonstrate that the extended symmetry group possesses a much richer structure. By leveraging the divisibility and growth properties of {\cal O}_{K}, we prove that Sym(Xk, ΓK) is isomorphic to the semi-direct product {\bb Z}^{4}\times\!\!\hbox{\vrule height 4.6pt depth -0.1pt}\ {\rm Stab}(V_{k}), where the stabilizer is explicitly determined by the unit group {\cal O}_{K}^{\times} and the Galois group {\rm Gal}(K/{\bb Q}). Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 de los Santos, K.A.C. Loyola, M.L. Miro, E.D.P. 2026-05-19 doi:10.1107/S2053273326003967 International Union of Crystallography The study characterizes the symmetries of a symbolic dynamical system arising from k-free integers in the biquadratic field K= {\bb Q}(\sqrt{2}, i). It proves that the extended symmetry group is a semi-direct product {\bb Z}^4 \times\!\!\hbox{\vrule height 4.6pt depth -0.1pt}\ {\rm Stab}(V_k), with the stabilizer explicitly linked to the field's unit and Galois groups. The analysis demonstrates how these number-theoretic shifts mirror the complex symmetry structures typically seen in aperiodic order. EN k-free lattice system symmetry and extended symmetry groups biquadratic number field shift space symbolic dynamical system We investigate the symmetries of a symbolic dynamical system (Xk, ΓK) of number-theoretic origin. Specifically, we analyze the shift space Xk, defined as the closure of the set Vk of k-free points within the ring of integers {\cal O}_{K} of the biquadratic number field K = {\bb Q}(\sqrt{2},i). The group of shift maps S, which acts on the Minkowski embedding \Gamma_{K}\cong{\bb Z}^{4} by translations, serves as the fundamental action of the system. Our focus is on the homeomorphisms of Xk that interact with the shift action: the automorphism group Aut(Xk, ΓK), consisting of homeomorphisms that commute with every element of S, and the extended symmetry group Sym(Xk, ΓK), which includes homeomorphisms that map the shift action to itself via an automorphism of S. While Aut(Xk, ΓK) is known to be trivial (consisting solely of the shifts themselves), we demonstrate that the extended symmetry group possesses a much richer structure. By leveraging the divisibility and growth properties of {\cal O}_{K}, we prove that Sym(Xk, ΓK) is isomorphic to the semi-direct product {\bb Z}^{4}\times\!\!\hbox{\vrule height 4.6pt depth -0.1pt}\ {\rm Stab}(V_{k}), where the stabilizer is explicitly determined by the unit group {\cal O}_{K}^{\times} and the Galois group {\rm Gal}(K/{\bb Q}). text/html Symmetries of a dynamical system arising from a biquadratic number field text 4 82 2026-05-19 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A research papers 0 0

https://journals.iucr.org/paper?gau5007 Exact renormalization techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalization principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for a system with pure-point spectrum, the orbit separation dimension, and its relation to renormalization. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalization and Lyapunov exponents can be used to exclude the presence of absolutely continuous parts of the spectra. Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 Baake, M. Gähler, F. Klick, A. Mañibo, N. Mazáč, J. 2026-05-26 doi:10.1107/S2053273326003918 International Union of Crystallography In this work, renormalization methods for quantities related to the diffraction of inflation systems are surveyed. EN renormalization inflation tilings Exact renormalization techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalization principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for a system with pure-point spectrum, the orbit separation dimension, and its relation to renormalization. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalization and Lyapunov exponents can be used to exclude the presence of absolutely continuous parts of the spectra. text/html Renormalization techniques for inflation systems and some of their applications text 4 82 2026-05-26 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A research papers 0 0

https://journals.iucr.org/paper?ae5177 The increasing complexity of in situ high-energy diffraction microscopy (HEDM) experiments demands a quantitative understanding of the data analysis pipeline to ensure reproducible science. However, the influence of key analysis parameters on the accuracy and precision of microstructural reconstructions is often not well quantified, creating a barrier to progress. This paper addresses this critical gap by presenting a rigorous, systematic validation of the HEDM data reduction methodology as implemented in the MIDAS software suite. Using a new, dedicated Ti-7 Al dataset, we investigate both far-field (FF) and near-field (NF) HEDM. Our results reveal critical sensitivities, demonstrating that grain position accuracy in FF-HEDM is highly dependent on the diversity of sampled diffraction vectors, while orientation precision in NF-HEDM improves dramatically with increased detector separation. We demonstrate the methodology's robustness against common experimental challenges, such as severe diffraction peak overlap, which is effectively filtered by requiring crystallographic consistency. Based on these quantitative findings, we establish a framework of best practices for HEDM data acquisition and analysis to guide the community towards more accurate and reliable results. Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 Sharma, H. Park, J.-S. Shastri, S. Kenesei, P. 2026-05-28 doi:10.1107/S2053273326004018 International Union of Crystallography This paper experimentally establishes the accuracy, robustness and performance limits of the high-energy diffraction microscopy data reduction methodology. Using dedicated far-field and near-field datasets, it quantifies the influence of key analysis parameters, demonstrates computational efficiency, and establishes a framework of best practices to guide the community towards more reproducible results. EN HEDM high-energy diffraction microscopy 3DXRD 3D X-ray diffraction MIDAS experimental validation performance analysis The increasing complexity of in situ high-energy diffraction microscopy (HEDM) experiments demands a quantitative understanding of the data analysis pipeline to ensure reproducible science. However, the influence of key analysis parameters on the accuracy and precision of microstructural reconstructions is often not well quantified, creating a barrier to progress. This paper addresses this critical gap by presenting a rigorous, systematic validation of the HEDM data reduction methodology as implemented in the MIDAS software suite. Using a new, dedicated Ti-7 Al dataset, we investigate both far-field (FF) and near-field (NF) HEDM. Our results reveal critical sensitivities, demonstrating that grain position accuracy in FF-HEDM is highly dependent on the diversity of sampled diffraction vectors, while orientation precision in NF-HEDM improves dramatically with increased detector separation. We demonstrate the methodology's robustness against common experimental challenges, such as severe diffraction peak overlap, which is effectively filtered by requiring crystallographic consistency. Based on these quantitative findings, we establish a framework of best practices for HEDM data acquisition and analysis to guide the community towards more accurate and reliable results. text/html MIDAS: a quantitative framework for high-energy diffraction microscopy. Part II: accuracy, robustness and best practices text 4 82 2026-05-28 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A research papers 0 0

https://journals.iucr.org/paper?nv5024 We propose a novel geometric characterization method for flat-faceted crystals bounded by a single simple crystallographic form – the rhombic dodecahedron. The approach utilizes distances between six pairs of parallel facets and the lengths of three false edges (out of six possible) resulting from anisotropic crystal growth. Our study has established the existence of exactly 34 combinatorially distinct, full-faceted real rhombic dodecahedra. For each type, we determined the range of crystal shape variations using the semi-axial lengths of the minimal-volume circumscribed ellipsoid. The results demonstrate that all real rhombic dodecahedra, except for eight special forms, can exhibit the complete spectrum of possible shape variations. These eight exceptional forms cannot develop tabular morphologies with bidirectional isometry. The obtained results provide valuable tools for precise analysis of rhombic dodecahedral crystal morphology and for establishing morphogenetic correlations between the degree of crystal distortions and the dissymmetry of the crystal-forming environment. This information is relevant for assessing the typomorphic features of natural diamonds, garnets, sodalites and magnetite crystals, as well as their synthetic analogs and other chemical compounds. Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 Stepenshchikov, D.G. Pavlushin, A.D. 2026-05-15 doi:10.1107/S2053273326003979 International Union of Crystallography We discuss a method for describing the geometry and habit of distorted rhombic dodecahedral crystals. The proposed method serves as the basis for analyzing the relationship between the asymmetry of the environment and the mineral individuals of crystals of a wide variety of shapes in any distortion range. EN real crystal form crystal shape rhombic dodecahedron morphological analysis combinatorial crystallography We propose a novel geometric characterization method for flat-faceted crystals bounded by a single simple crystallographic form – the rhombic dodecahedron. The approach utilizes distances between six pairs of parallel facets and the lengths of three false edges (out of six possible) resulting from anisotropic crystal growth. Our study has established the existence of exactly 34 combinatorially distinct, full-faceted real rhombic dodecahedra. For each type, we determined the range of crystal shape variations using the semi-axial lengths of the minimal-volume circumscribed ellipsoid. The results demonstrate that all real rhombic dodecahedra, except for eight special forms, can exhibit the complete spectrum of possible shape variations. These eight exceptional forms cannot develop tabular morphologies with bidirectional isometry. The obtained results provide valuable tools for precise analysis of rhombic dodecahedral crystal morphology and for establishing morphogenetic correlations between the degree of crystal distortions and the dissymmetry of the crystal-forming environment. This information is relevant for assessing the typomorphic features of natural diamonds, garnets, sodalites and magnetite crystals, as well as their synthetic analogs and other chemical compounds. text/html A real rhombic dodecahedron: morphology and crystal shape variations text 4 82 2026-05-15 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A short communications 0 0

https://journals.iucr.org/paper?es5056 Copyright (c) 2026 International Union of Crystallography urn:issn:2053-2733 2026-06-02 doi:10.1107/S2053273325001329 International Union of Crystallography A report of the Twenty-Sixth General Assembly and International Congress of Crystallography is given. EN International Union of Crystallography General Assembly International Congress of Crystallography IUCr 2023 text/html Twenty-Sixth General Assembly and International Congress of Crystallography, Melbourne, Australia, 22–29 August 2023 text 4 82 2026-06-02 Copyright (c) 2026 International Union of Crystallography Acta Crystallographica Section A international union of crystallography 0 0