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) 2021 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) 2021 International Union of Crystallographyurn:issn:0108-7673Open-access and free articles in Acta Crystallographica Section A: Foundations and Advanceshttp://journals.iucr.org/logos/rss10a.gif
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Still imageCombining X-rays, neutrons and electrons, and NMR, for precision and accuracy in structure–function studies
http://scripts.iucr.org/cgi-bin/paper?ae5100
The distinctive features of the physics-based probes used in understanding the structure of matter focusing on biological sciences, but not exclusively, are described in the modern context. This is set in a wider scope of holistic biology and the scepticism about `reductionism', what is called the `molecular level', and how to respond constructively. These topics will be set alongside the principles of accuracy and precision, and their boundaries. The combination of probes and their application together is the usual way of realizing accuracy. The distinction between precision and accuracy can be blurred by the predictive force of a precise structure, thereby lending confidence in its potential accuracy. These descriptions will be applied to the comparison of cryo and room-temperature protein crystal structures as well as the solid state of a crystal and the same molecules studied by small-angle X-ray scattering in solution and by electron microscopy on a sample grid. Examples will include: time-resolved X-ray Laue crystallography of an enzyme Michaelis complex formed directly in a crystal equivalent to in vivo; a new iodoplatin for radiation therapy predicted from studies of platin crystal structures; and the field of colouration of carotenoids, as an effective assay of function, i.e. their colouration, when unbound and bound to a protein. The complementarity of probes, as well as their combinatory use, is then at the foundation of real (biologically relevant), probe-artefacts-free, structure–function studies. The foundations of our methodologies are being transformed by colossal improvements in technologies of X-ray and neutron sources and their beamline instruments, as well as improved electron microscopes and NMR spectrometers. The success of protein structure prediction from gene sequence recently reported by CASP14 also opens new doors to change and extend the foundations of the structural sciences.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Helliwell, J.R.2021-05-04doi:10.1107/S205327332100317XInternational Union of CrystallographyThe distinctive features of the probes used in understanding the structure of matter focusing on biological sciences, but not exclusively, are described in the modern context to minimize the consequences of artefactual information in data interpretation. The precision and accuracy of both data and technique are revisited. A variety of structural results are described that reach beyond reductionism to the whole biological organism. All these aspects open new doors to change and extend the foundations of the structural sciences.enX-RAYS; NEUTRONS; ELECTRONS; NMR; STRUCTURE AND FUNCTIONThe distinctive features of the physics-based probes used in understanding the structure of matter focusing on biological sciences, but not exclusively, are described in the modern context. This is set in a wider scope of holistic biology and the scepticism about `reductionism', what is called the `molecular level', and how to respond constructively. These topics will be set alongside the principles of accuracy and precision, and their boundaries. The combination of probes and their application together is the usual way of realizing accuracy. The distinction between precision and accuracy can be blurred by the predictive force of a precise structure, thereby lending confidence in its potential accuracy. These descriptions will be applied to the comparison of cryo and room-temperature protein crystal structures as well as the solid state of a crystal and the same molecules studied by small-angle X-ray scattering in solution and by electron microscopy on a sample grid. Examples will include: time-resolved X-ray Laue crystallography of an enzyme Michaelis complex formed directly in a crystal equivalent to in vivo; a new iodoplatin for radiation therapy predicted from studies of platin crystal structures; and the field of colouration of carotenoids, as an effective assay of function, i.e. their colouration, when unbound and bound to a protein. The complementarity of probes, as well as their combinatory use, is then at the foundation of real (biologically relevant), probe-artefacts-free, structure–function studies. The foundations of our methodologies are being transformed by colossal improvements in technologies of X-ray and neutron sources and their beamline instruments, as well as improved electron microscopes and NMR spectrometers. The success of protein structure prediction from gene sequence recently reported by CASP14 also opens new doors to change and extend the foundations of the structural sciences.text/htmlCombining X-rays, neutrons and electrons, and NMR, for precision and accuracy in structure–function studiestext3772021-05-04Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733lead articles173med@iucr.orgMay 20211852053-2733Diffraction line profiles from polydisperse crystalline systems. Corrigenda
http://scripts.iucr.org/cgi-bin/paper?me6128
Equation (16) and some entries in Table 1 in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.Copyright (c) 2021 International Union of Crystallographyurn:issn:2053-2733Scardi, P.2021-03-17doi:10.1107/S2053273321002813International Union of CrystallographyErrors in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.enLINE PROFILE ANALYSIS; LPA; WHOLE POWDER PATTERN MODELLING; WPPM; CRYSTALLINE DOMAIN SIZE; CORRIGENDAEquation (16) and some entries in Table 1 in the article by Scardi & Leoni [(2001), Acta Cryst. A57, 604–613] are corrected.text/htmlDiffraction line profiles from polydisperse crystalline systems. Corrigendatext3772021-03-17Acta Crystallographica Section A: Foundations and AdvancesCopyright (c) 2021 International Union of Crystallography2053-2733addenda and errata232med@iucr.orgMay 20212322053-2733A new electron diffraction approach for structure refinement applied to Ca3Mn2O7
http://scripts.iucr.org/cgi-bin/paper?lu5005
The digital large-angle convergent-beam electron diffraction (D-LACBED) technique is applied to Ca3Mn2O7 for a range of temperatures. Bloch-wave simulations are used to examine the effects that changes in different parameters have on the intensity in D-LACBED patterns, and atomic coordinates, thermal atomic displacement parameters and apparent occupancy are refined to achieve a good fit between simulation and experiment. The sensitivity of the technique to subtle changes in structure is demonstrated. Refined structures are in good agreement with previous determinations of Ca3Mn2O7 and show the decay of anti-phase oxygen octahedral tilts perpendicular to the c axis of the A21am unit cell with increasing temperature, as well as the robustness of oxygen octahedral tilts about the c axis up to ∼400°C. The technique samples only the zero-order Laue zone and is therefore insensitive to atom displacements along the electron-beam direction. For this reason it is not possible to distinguish between in-phase and anti-phase oxygen octahedral tilting about the c axis using the [110] data collected in this study.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Beanland, R.Smith, K.Vaněk, P.Zhang, H.Hubert, A.Evans, K.Römer, R.A.Kamba, S.2021-03-17doi:10.1107/S2053273321001546International Union of CrystallographyThe `digital' large-angle convergent-beam electron diffraction (D-LACBED) method uses computer control of a transmission electron microscope to collect hundreds of diffraction patterns from a region a few nanometres in size, which are combined into a single data set. The sensitivity of the resulting patterns to crystal structure is explored using the Ruddlesden–Popper oxide Ca3Mn2O7 and it is found that refinement of atomic coordinates can be performed to sub-picometre precision.enDIGITAL DIFFRACTION; ELECTRON DIFFRACTION; CA3MN2O7; CBED; LACBEDThe digital large-angle convergent-beam electron diffraction (D-LACBED) technique is applied to Ca3Mn2O7 for a range of temperatures. Bloch-wave simulations are used to examine the effects that changes in different parameters have on the intensity in D-LACBED patterns, and atomic coordinates, thermal atomic displacement parameters and apparent occupancy are refined to achieve a good fit between simulation and experiment. The sensitivity of the technique to subtle changes in structure is demonstrated. Refined structures are in good agreement with previous determinations of Ca3Mn2O7 and show the decay of anti-phase oxygen octahedral tilts perpendicular to the c axis of the A21am unit cell with increasing temperature, as well as the robustness of oxygen octahedral tilts about the c axis up to ∼400°C. The technique samples only the zero-order Laue zone and is therefore insensitive to atom displacements along the electron-beam direction. For this reason it is not possible to distinguish between in-phase and anti-phase oxygen octahedral tilting about the c axis using the [110] data collected in this study.text/htmlA new electron diffraction approach for structure refinement applied to Ca3Mn2O7text3772021-03-17Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers196med@iucr.orgMay 20212072053-2733A discussion on `Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis'
http://scripts.iucr.org/cgi-bin/paper?ug5021
Heterovalent ternary nitrides are considered one of the promising classes of materials for photovoltaics, combining attractive physical properties with low toxicity and element abundance. One of the front-runner systems under consideration is ZnSnN2. Although it is nominally a ternary compound, no clear crystallographic evidence for cation ordering has been observed so far. An attempt to elucidate this discrepancy [Quayle (2020). Acta Cryst. A76, 410–420] was the trigger for an intensive discussion between the authors, and an agreement was reached to elaborate on some points in order to set things in perspective. Rather than using a conventional comment–answer scheme, this is published in the form of a joint discussion to celebrate constructive criticism and collegiality.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Quayle, P.C.Breternitz, J.2021-03-23doi:10.1107/S2053273321001376International Union of CrystallographyA scientific exchange on an earlier paper [Quayle (2020). Acta Cryst. A76, 410–420] has led to the clarification of some of the points.enGROUP-SUBGROUP RELATIONSHIPS; NITRIDE MATERIALS; WURTZITE TYPEHeterovalent ternary nitrides are considered one of the promising classes of materials for photovoltaics, combining attractive physical properties with low toxicity and element abundance. One of the front-runner systems under consideration is ZnSnN2. Although it is nominally a ternary compound, no clear crystallographic evidence for cation ordering has been observed so far. An attempt to elucidate this discrepancy [Quayle (2020). Acta Cryst. A76, 410–420] was the trigger for an intensive discussion between the authors, and an agreement was reached to elaborate on some points in order to set things in perspective. Rather than using a conventional comment–answer scheme, this is published in the form of a joint discussion to celebrate constructive criticism and collegiality.text/htmlA discussion on `Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis'text3772021-03-23Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers217med@iucr.orgMay 20212212053-2733Nothing trumps good data
http://scripts.iucr.org/cgi-bin/paper?me6117
urn:issn:2053-2733Pinkerton, A.A.2021-01-29doi:10.1107/S2053273321000759International Union of CrystallographyThe advantages of a powerful new tool for determining the electron density of small inorganic systems using high-quality powder diffraction data from the MYTHEN microstrip detector [Svane et al. (2021). Acta Cryst. A77, 85–95] are considered.enELECTRON DENSITY; DATA QUALITY; CHARGE DENSITY; POWDER DIFFRACTION; MYTHEN DETECTORtext/htmlNothing trumps good datatext2772021-01-29Acta Crystallographica Section A: Foundations and Advances2053-2733scientific commentaries83med@iucr.orgMarch 2021842053-2733Coordination sequences of crystals are of quasi-polynomial type
http://scripts.iucr.org/cgi-bin/paper?pl5008
The coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Nakamura, Y.Sakamoto, R.Mase, T.Nakagawa, J.2021-02-18doi:10.1107/S2053273320016769International Union of CrystallographyIt is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889] in their study of coordination sequences of zeolites.enCOORDINATION SEQUENCES; GRAPH THEORY; HILBERT POLYNOMIAL; MONOID THEORYThe coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].text/htmlCoordination sequences of crystals are of quasi-polynomial typetext2772021-02-18Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers138med@iucr.orgMarch 20211482053-2733Multipole electron densities and structural parameters from synchrotron powder X-ray diffraction data obtained with a MYTHEN detector system (OHGI)
http://scripts.iucr.org/cgi-bin/paper?pl5010
Powder X-ray diffraction has some inherent advantages over traditional single-crystal X-ray diffraction in accurately determining electron densities and structural parameters due to the lower requirements for sample crystallinity, simpler corrections and measurement simultaneity. For some simple inorganic materials, it has been shown that these advantages can compensate for disadvantages such as peak overlap and error-prone background subtraction. Although it is challenging to extend powder X-ray diffraction-based electron-density studies to organic materials with significant peak overlap, previous results using a dedicated vacuum diffractometer with a large image-plate camera (AVID) demonstrated that it can be done. However, the vacuum setup with the off-line detector system was found to prohibit a widespread use. Fast microstrip detectors, which have been employed at a number of powder diffraction beamlines, have the potential to facilitate electron-density studies. Nevertheless, no electron-density studies even for materials with slight peak overlap have been performed with microstrip detectors. One of the most critical problems has been a difference in sensitivity between microstrip channels, which substantially defines the dynamic range of a detector. Recently, a robust approach to this problem has been developed and applied to a total scattering measurement system (OHGI) with 15 MYTHEN microstrip modules. In the present study, synchrotron powder X-ray diffraction data obtained with OHGI are evaulated in terms of multipole electron densities and structural parameters (atomic positions and displacement parameters). These results show that, even without a dedicated setup and perfect samples, electron-density modelling can be carried out on high-quality powder X-ray diffraction data. However, it was also found that the required prior information about the sample prohibits widespread use of the method. With the presently obtainable data quality, electron densities of molecular crystals in general are not reliably obtained from powder data, but it is an excellent, possibly superior, alternative to single-crystal measurements for small-unit-cell inorganic solids. If aspherical atomic scattering factors can be obtained from other means (multipole databases, theoretical calculations), then atomic positions (including for hydrogen) and anisotropic atomic displacement parameters (non-hydrogen atoms) of excellent accuracy can be refined from synchrotron powder X-ray diffraction data on organic crystals.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Svane, B.Tolborg, K.Kato, K.Iversen, B.B.2021-01-28doi:10.1107/S2053273320016605International Union of CrystallographyMultipole electron densities and structural parameters of inorganic and organic materials were evaluated on the basis of synchrotron powder X-ray diffraction data obtained with a MYTHEN detector system (OHGI).enELECTRON DENSITY; STRUCTURAL PARAMETERS; POWDER DIFFRACTION; MOLECULAR CRYSTALS; SYNCHROTRON RADIATIONPowder X-ray diffraction has some inherent advantages over traditional single-crystal X-ray diffraction in accurately determining electron densities and structural parameters due to the lower requirements for sample crystallinity, simpler corrections and measurement simultaneity. For some simple inorganic materials, it has been shown that these advantages can compensate for disadvantages such as peak overlap and error-prone background subtraction. Although it is challenging to extend powder X-ray diffraction-based electron-density studies to organic materials with significant peak overlap, previous results using a dedicated vacuum diffractometer with a large image-plate camera (AVID) demonstrated that it can be done. However, the vacuum setup with the off-line detector system was found to prohibit a widespread use. Fast microstrip detectors, which have been employed at a number of powder diffraction beamlines, have the potential to facilitate electron-density studies. Nevertheless, no electron-density studies even for materials with slight peak overlap have been performed with microstrip detectors. One of the most critical problems has been a difference in sensitivity between microstrip channels, which substantially defines the dynamic range of a detector. Recently, a robust approach to this problem has been developed and applied to a total scattering measurement system (OHGI) with 15 MYTHEN microstrip modules. In the present study, synchrotron powder X-ray diffraction data obtained with OHGI are evaulated in terms of multipole electron densities and structural parameters (atomic positions and displacement parameters). These results show that, even without a dedicated setup and perfect samples, electron-density modelling can be carried out on high-quality powder X-ray diffraction data. However, it was also found that the required prior information about the sample prohibits widespread use of the method. With the presently obtainable data quality, electron densities of molecular crystals in general are not reliably obtained from powder data, but it is an excellent, possibly superior, alternative to single-crystal measurements for small-unit-cell inorganic solids. If aspherical atomic scattering factors can be obtained from other means (multipole databases, theoretical calculations), then atomic positions (including for hydrogen) and anisotropic atomic displacement parameters (non-hydrogen atoms) of excellent accuracy can be refined from synchrotron powder X-ray diffraction data on organic crystals.text/htmlMultipole electron densities and structural parameters from synchrotron powder X-ray diffraction data obtained with a MYTHEN detector system (OHGI)text2772021-01-28Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers85med@iucr.orgMarch 2021952053-2733Arithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedra
http://scripts.iucr.org/cgi-bin/paper?pl5009
The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V − E + F = 2 or Euler characteristic χ = V − E + F − I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = Vm − Em + Fm − Im = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Naskręcki, B.Dauter, Z.Jaskolski, M.2021-02-04doi:10.1107/S2053273320016186International Union of CrystallographyA mathematical proof based on arithmetic argument is presented for the modified Euler characteristic \chi_{\rm m}=\sum_{i=0}^N (-1)^i \sum _{j=1}^{n(i)} 1/m(ij)=0 (where the first summation runs from 0-dimensional vertices to the N-dimensional cell or `interior'), applicable to symmetrically arranged space-filling polytopes in N-dimensional space, where the contribution of each jth i-dimensional element of the polytope is weighted by a factor inversely proportional to its multiplicity m(ij).enEULER'S FORMULA; MULTIPLICITY-WEIGHTED EULER CHARACTERISTIC; SPACE-FILLING POLYHEDRA; POLYTOPES; ASYMMETRIC UNIT; DIRICHLET DOMAINSThe puzzling observation that the famous Euler's formula for three-dimensional polyhedra V − E + F = 2 or Euler characteristic χ = V − E + F − I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = Vm − Em + Fm − Im = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.text/htmlArithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedratext2772021-02-04Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers126med@iucr.orgMarch 20211292053-2733Symmetry relations in wurtzite nitrides and oxide nitrides and the curious case of Pmc21
http://scripts.iucr.org/cgi-bin/paper?ug5020
Binary III–V nitrides such as AlN, GaN and InN in the wurtzite-type structure have long been considered as potent semiconducting materials because of their optoelectronic properties, amongst others. With rising concerns over the utilization of scarce elements, a replacement of the trivalent cations by others in ternary and multinary nitrides has led to the development of different variants of nitrides and oxide nitrides crystallizing in lower-symmetry variants of wurtzite. This work presents the symmetry relationships between these structural types specific to nitrides and oxide nitrides and updates some prior work on this matter. The non-existence of compounds crystallizing in Pmc21, formally the highest subgroup of the wurtzite type fulfilling Pauling's rules for 1:1:2 stoichiometries, has been puzzling scientists for a while; a rationalization is given, from a crystallographic basis, of why this space group is unlikely to be adopted.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Breternitz, J.Schorr, S.2021-03-23doi:10.1107/S2053273320015971International Union of CrystallographyBinary and multinary nitrides in a wurtzitic arrangement are very interesting semiconductor materials. The group–subgroup relationship between the different structural types is established.enGROUP-SUBGROUP RELATIONSHIPS; NITRIDE MATERIALS; WURTZITE TYPEBinary III–V nitrides such as AlN, GaN and InN in the wurtzite-type structure have long been considered as potent semiconducting materials because of their optoelectronic properties, amongst others. With rising concerns over the utilization of scarce elements, a replacement of the trivalent cations by others in ternary and multinary nitrides has led to the development of different variants of nitrides and oxide nitrides crystallizing in lower-symmetry variants of wurtzite. This work presents the symmetry relationships between these structural types specific to nitrides and oxide nitrides and updates some prior work on this matter. The non-existence of compounds crystallizing in Pmc21, formally the highest subgroup of the wurtzite type fulfilling Pauling's rules for 1:1:2 stoichiometries, has been puzzling scientists for a while; a rationalization is given, from a crystallographic basis, of why this space group is unlikely to be adopted.text/htmlSymmetry relations in wurtzite nitrides and oxide nitrides and the curious case of Pmc21text3772021-03-23Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers208med@iucr.orgMay 20212162053-2733Small-angle X-ray scattering from GaN nanowires on Si(111): facet truncation rods, facet roughness and Porod's law
http://scripts.iucr.org/cgi-bin/paper?iv5011
Small-angle X-ray scattering from GaN nanowires grown on Si(111) is measured in the grazing-incidence geometry and modelled by means of a Monte Carlo simulation that takes into account the orientational distribution of the faceted nanowires and the roughness of their side facets. It is found that the scattering intensity at large wavevectors does not follow Porod's law I(q) ∝ q−4. The intensity depends on the orientation of the side facets with respect to the incident X-ray beam. It is maximum when the scattering vector is directed along a facet normal, reminiscent of surface truncation rod scattering. At large wavevectors q, the scattering intensity is reduced by surface roughness. A root-mean-square roughness of 0.9 nm, which is the height of just 3–4 atomic steps per micrometre-long facet, already gives rise to a strong intensity reduction.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Kaganer, V.M.Konovalov, O.V.Fernández-Garrido, S.2021-01-05doi:10.1107/S205327332001548XInternational Union of CrystallographyThe intensity of small-angle X-ray scattering from GaN nanowires on Si(111) depends on the orientation of the side facets with respect to the incident beam. This reminiscence of truncation rod scattering gives rise to a deviation from Porod's law. A roughness of just 3–4 atomic steps per micrometre-long side facet notably changes the intensity curves.enNANOWIRES; POROD'S LAW; FACET TRUNCATION RODS; SMALL-ANGLE X-RAY SCATTERING; SAXS; GRAZING-INCIDENCE SMALL-ANGLE X-RAY SCATTERING; GISAXSSmall-angle X-ray scattering from GaN nanowires grown on Si(111) is measured in the grazing-incidence geometry and modelled by means of a Monte Carlo simulation that takes into account the orientational distribution of the faceted nanowires and the roughness of their side facets. It is found that the scattering intensity at large wavevectors does not follow Porod's law I(q) ∝ q−4. The intensity depends on the orientation of the side facets with respect to the incident X-ray beam. It is maximum when the scattering vector is directed along a facet normal, reminiscent of surface truncation rod scattering. At large wavevectors q, the scattering intensity is reduced by surface roughness. A root-mean-square roughness of 0.9 nm, which is the height of just 3–4 atomic steps per micrometre-long facet, already gives rise to a strong intensity reduction.text/htmlSmall-angle X-ray scattering from GaN nanowires on Si(111): facet truncation rods, facet roughness and Porod's lawtext771https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2021-01-0542research papers2053-2733January 2021med@iucr.org532053-2733Macromolecular phasing using diffraction from multiple crystal forms
http://scripts.iucr.org/cgi-bin/paper?sc5137
A phasing algorithm for macromolecular crystallography is proposed that utilizes diffraction data from multiple crystal forms – crystals of the same molecule with different unit-cell packings (different unit-cell parameters or space-group symmetries). The approach is based on the method of iterated projections, starting with no initial phase information. The practicality of the method is demonstrated by simulation using known structures that exist in multiple crystal forms, assuming some information on the molecular envelope and positional relationships between the molecules in the different unit cells. With incorporation of new or existing methods for determination of these parameters, the approach has potential as a method for ab initio phasing.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Metz, M.Arnal, R.D.Brehm, W.Chapman, H.N.Morgan, A.J.Millane, R.P.2021-01-05doi:10.1107/S2053273320013650International Union of CrystallographyA phasing algorithm for protein crystallography using diffraction data from multiple crystal forms is proposed. The algorithm is evaluated by simulation, and practical aspects and potential for ab initio phasing are discussed.enMULTIPLE CRYSTAL FORMS; AB INITIO PHASING; ITERATIVE PROJECTION ALGORITHMS; X-RAY FREE-ELECTRON LASERS; XFELSA phasing algorithm for macromolecular crystallography is proposed that utilizes diffraction data from multiple crystal forms – crystals of the same molecule with different unit-cell packings (different unit-cell parameters or space-group symmetries). The approach is based on the method of iterated projections, starting with no initial phase information. The practicality of the method is demonstrated by simulation using known structures that exist in multiple crystal forms, assuming some information on the molecular envelope and positional relationships between the molecules in the different unit cells. With incorporation of new or existing methods for determination of these parameters, the approach has potential as a method for ab initio phasing.text/htmlMacromolecular phasing using diffraction from multiple crystal formstext771https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2021-01-0519research papers2053-2733January 2021med@iucr.org352053-2733A cloud platform for atomic pair distribution function analysis: PDFitc
http://scripts.iucr.org/cgi-bin/paper?ae5091
A cloud web platform for analysis and interpretation of atomic pair distribution function (PDF) data (PDFitc) is described. The platform is able to host applications for PDF analysis to help researchers study the local and nanoscale structure of nanostructured materials. The applications are designed to be powerful and easy to use and can, and will, be extended over time through community adoption and development. The currently available PDF analysis applications, structureMining, spacegroupMining and similarityMapping, are described. In the first and second the user uploads a single PDF and the application returns a list of best-fit candidate structures, and the most likely space group of the underlying structure, respectively. In the third, the user can upload a set of measured or calculated PDFs and the application returns a matrix of Pearson correlations, allowing assessment of the similarity between different data sets. structureMining is presented here as an example to show the easy-to-use workflow on PDFitc. In the future, as well as using the PDFitc applications for data analysis, it is hoped that the community will contribute their own codes and software to the platform.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Yang, L.Culbertson, E.A.Thomas, N.K.Vuong, H.T.Kjær, E.T.S.Jensen, K.M.Ø.Tucker, M.G.Billinge, S.J.L.2021-01-05doi:10.1107/S2053273320013066International Union of CrystallographyA new web platform is presented for the pair distribution function (PDF) community to use and share advanced PDF analysis software in the cloud.enPAIR DISTRIBUTION FUNCTION; PDF; DATA ANALYSIS; WEB APPLICATIONS; CLOUD COMPUTINGA cloud web platform for analysis and interpretation of atomic pair distribution function (PDF) data (PDFitc) is described. The platform is able to host applications for PDF analysis to help researchers study the local and nanoscale structure of nanostructured materials. The applications are designed to be powerful and easy to use and can, and will, be extended over time through community adoption and development. The currently available PDF analysis applications, structureMining, spacegroupMining and similarityMapping, are described. In the first and second the user uploads a single PDF and the application returns a list of best-fit candidate structures, and the most likely space group of the underlying structure, respectively. In the third, the user can upload a set of measured or calculated PDFs and the application returns a matrix of Pearson correlations, allowing assessment of the similarity between different data sets. structureMining is presented here as an example to show the easy-to-use workflow on PDFitc. In the future, as well as using the PDFitc applications for data analysis, it is hoped that the community will contribute their own codes and software to the platform.text/htmlA cloud platform for atomic pair distribution function analysis: PDFitctext771https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2021-01-052research papers2053-2733January 2021med@iucr.org62053-2733Algorithms for target transformations of lattice basis vectors
http://scripts.iucr.org/cgi-bin/paper?ae5090
Simple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi-dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi-dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Gorfman, S.2020-10-29doi:10.1107/S2053273320012668International Union of CrystallographyPresented here are algorithms for the transformation of lattice basis vectors to a specific target. The algorithms are useful for crystallographic operations in direct and reciprocal spaces alike. The algorithms are demonstrated graphically and numerically.enCRYSTAL LATTICE; TRANSFORMATIONS; LATTICE PLANES; ZONESSimple algorithms are proposed for the transformation of lattice basis vectors to a specific target. In the first case, one of the new basis vectors is aligned to a predefined lattice direction, while in the second case, two of the new basis vectors are brought to a lattice plane with predefined Miller indices. The multi-dimensional generalization of the algorithm is available in the supporting materials. The algorithms are useful for such crystallographic operations as simulation of zone planes (i.e. geometry of electron diffraction patterns) or transformation of a unit cell for surface or cleavage energy calculations. The most general multi-dimensional version of the algorithm may be useful for the analysis of quasiperiodic crystals or as an alternative method of calculating Bézout coefficients. The algorithms are demonstrated both graphically and numerically.text/htmlAlgorithms for target transformations of lattice basis vectorstext766https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-10-29713research papers2053-2733November 2020med@iucr.org7182053-2733A flexible and standalone forward simulation model for laboratory X-ray diffraction contrast tomography
http://scripts.iucr.org/cgi-bin/paper?iv5008
Laboratory X-ray diffraction contrast tomography (LabDCT) has recently been developed as a powerful technique for non-destructive mapping of grain microstructures in bulk materials. As the grain reconstruction relies on segmentation of diffraction spots, it is essential to understand the physics of the diffraction process and resolve all the spot features in detail. To this aim, a flexible and standalone forward simulation model has been developed to compute the diffraction projections from polycrystalline samples with any crystal structure. The accuracy of the forward simulation model is demonstrated by good agreements in grain orientations, boundary positions and shapes between a virtual input structure and that reconstructed based on the forward simulated diffraction projections of the input structure. Further experimental verification is made by comparisons of diffraction spots between simulations and experiments for a partially recrystallized Al sample, where a satisfactory agreement is found for the spot positions, sizes and intensities. Finally, applications of this model to analyze specific spot features are presented.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Fang, H.Juul Jensen, D.Zhang, Y.2020-09-18doi:10.1107/S2053273320010852International Union of CrystallographyA flexible and standalone forward simulation model has been developed to compute the diffraction projections for laboratory diffraction contrast tomography (LabDCT). The outputs are expected to be of great value for all present users of LabDCT as well as interested new users.en3D GRAIN MAPPING; DIFFRACTION CONTRAST TOMOGRAPHY; X-RAY DIFFRACTION; FORWARD SIMULATION; GRAIN RECONSTRUCTIONLaboratory X-ray diffraction contrast tomography (LabDCT) has recently been developed as a powerful technique for non-destructive mapping of grain microstructures in bulk materials. As the grain reconstruction relies on segmentation of diffraction spots, it is essential to understand the physics of the diffraction process and resolve all the spot features in detail. To this aim, a flexible and standalone forward simulation model has been developed to compute the diffraction projections from polycrystalline samples with any crystal structure. The accuracy of the forward simulation model is demonstrated by good agreements in grain orientations, boundary positions and shapes between a virtual input structure and that reconstructed based on the forward simulated diffraction projections of the input structure. Further experimental verification is made by comparisons of diffraction spots between simulations and experiments for a partially recrystallized Al sample, where a satisfactory agreement is found for the spot positions, sizes and intensities. Finally, applications of this model to analyze specific spot features are presented.text/htmlA flexible and standalone forward simulation model for laboratory X-ray diffraction contrast tomographytext766https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-09-18652research papers2053-2733November 2020med@iucr.org6632053-2733Quaternions: what are they, and why do we need to know?
http://scripts.iucr.org/cgi-bin/paper?me6092
Copyright (c) 2020 International Union of Crystallographyurn:issn:2053-2733Horn, B.K.P.2020-08-06doi:10.1107/S2053273320010359International Union of CrystallographyThe significance of the work by A. J. Hanson [Acta Cryst. (2020), A76, 432–457] on finding the optimal alignment of pairs of spatial and/or orientation data sets is discussed.enQUATERNIONS; DATA ALIGNMENT; ROTATION; ORIENTATION; ORTHOGONAL PROCRUSTES PROBLEM; ORIENTATION DISTRIBUTION FUNCTION; ODFtext/htmlQuaternions: what are they, and why do we need to know?text5762020-08-06Acta Crystallographica Section A: Foundations and AdvancesCopyright (c) 2020 International Union of Crystallography2053-2733scientific commentaries556med@iucr.orgSeptember 20205582053-2733Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings
http://scripts.iucr.org/cgi-bin/paper?ug5007
Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Lee, D.-I.Akiyama, S.Lee, J.-Y.2020-08-21doi:10.1107/S2053273320009717International Union of CrystallographyThe equivalence between pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings is discussed.enPISOT FAMILY SUBSTITUTION TILINGS; PURE DISCRETE SPECTRUM; REGULAR MODEL SETS; MEYER SETS; RIGIDITYPrimitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.text/htmlPure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilingstext765https://creativecommons.org/licenses/by/4.0/Acta Crystallographica Section A: Foundations and Advances2020-08-21600research papers2053-2733September 2020med@iucr.org6102053-2733Embedding-theory-based simulations using experimental electron densities for the environment
http://scripts.iucr.org/cgi-bin/paper?ug5015
The basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg–Kohn density functional EHK[ρ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where ΨA is the embedded NA-electron wavefunction and ρB(r) is a non-negative function in real space integrating to a given number of electrons NB. This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρB(r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as ΨA) and the glycylglycine molecule [represented as ρB(r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρB(r) in FDET-based simulations.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Ricardi, N.Ernst, M.Macchi, P.Wesolowski, T.A.2020-07-20doi:10.1107/S2053273320008062International Union of CrystallographyFor the first time, the use of experimentally derived molecular electron densities as ρB(r) in calculations based on frozen-density embedding theory (FDET) of environment-induced shifts of electronic excitations for chromophores in clusters is demonstrated. ρB(r) was derived from X-ray restrained molecular wavefunctions of glycylglycine to obtain environment densities for simulating electronic excitations in clusters.enQUANTUM CRYSTALLOGRAPHY; DENSITY EMBEDDING; MULTI-SCALE SIMULATIONS; ELECTRONIC STRUCTURE; CHROMOPHORESThe basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg–Kohn density functional EHK[ρ] performed using the auxiliary functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B], where ΨA is the embedded NA-electron wavefunction and ρB(r) is a non-negative function in real space integrating to a given number of electrons NB. This choice of independent variables in the total energy functional E_{v_{AB}}^{\rm FDET}[\Psi _A, \rho _B] makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρB(r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as ΨA) and the glycylglycine molecule [represented as ρB(r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρB(r) in FDET-based simulations.text/htmlEmbedding-theory-based simulations using experimental electron densities for the environmenttext5762020-07-20Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers571med@iucr.orgSeptember 20205792053-2733Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction
http://scripts.iucr.org/cgi-bin/paper?ae5086
Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Baake, M.Grimm, U.2020-07-09doi:10.1107/S2053273320007421International Union of CrystallographyAveraged quantities such as mean shelling numbers, scaling behaviour or diffraction for cut-and-project sets can conveniently be computed in internal space, also for systems with fractally bounded windows.enQUASICRYSTALS; PROJECTION METHOD; INFLATION RULES; DIFFRACTION; HYPERUNIFORMITYTilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.text/htmlInflation versus projection sets in aperiodic systems: the role of the window in averaging and diffractiontext5762020-07-09Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733topical reviews559med@iucr.orgSeptember 20205702053-2733On Cayley graphs of {\bb Z}^4
http://scripts.iucr.org/cgi-bin/paper?eo5107
The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Baburin, I.A.2020-07-16doi:10.1107/S2053273320007159International Union of CrystallographyCayley graphs of {\bb Z}^4 with valency 10 have been enumerated which correspond to generating sets of integral vectors with components −1, 0, 1 and which are embedded in a four-dimensional Euclidean space without edge intersections.enCAYLEY GRAPHS; FREE ABELIAN GROUPS; COMPUTATIONAL GROUP THEORY; VERTEX-TRANSITIVE GRAPHS; ISOTOPYThe generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.text/htmlOn Cayley graphs of {\bb Z}^4text5762020-07-16Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers584med@iucr.orgSeptember 20205882053-2733Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra
http://scripts.iucr.org/cgi-bin/paper?sc5138
The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.https://creativecommons.org/licenses/by/4.0/urn:issn:2053-2733Dauter, Z.Jaskolski, M.2020-07-09doi:10.1107/S2053273320007093International Union of CrystallographyFor many tested cases of identical space-filling polyhedra, such as the space-group-specific asymmetric units or Dirichlet domains, the numbers of their faces (Fn), edges (En) and vertices (Vn), in each case normalized by division by the multiplicity of their (potentially special) symmetry position, fulfill a modified Euler's formula Fn − En + Vn = 1.enASYMMETRIC UNIT; UNIT CELL; EULER'S FORMULA; SPACE-FILLING POLYHEDRA; DIRICHLET DOMAINSThe famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.text/htmlMultiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedratext5762020-07-09Acta Crystallographica Section A: Foundations and Advanceshttps://creativecommons.org/licenses/by/4.0/2053-2733research papers580med@iucr.orgSeptember 20205832053-2733