Acta Crystallographica Section A
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Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.enCopyright (c) 2014 International Union of Crystallography2014-03-01International Union of CrystallographyInternational Union of Crystallographyhttp://journals.iucr.orgurn:issn:2053-2733Acta Crystallographica Section A: Foundations and Advances covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.text/htmlActa Crystallographica Section A: Foundations and Advances, Volume 70, Part 2, 2014textyearly62002-01-01T00:00+00:002702014-03-01Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section A: Foundations and Advances95urn:issn:2053-2733med@iucr.orgMarch 20142014-03-01Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageProspects for mathematical crystallography
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The potential of mathematical crystallography as an emerging field is examined from a sociological point of view. Mathematical crystallography is unusual as an emerging field as it is also an old field, albeit scattered, with evidence of continued substantial activity. But its situation is similar to that of an emerging field, so we analyse it as such. Comparisons with past emergent efforts suggest that a new field can grow if given an economic demand for its product and a receptive environment. Developing a field entails developing a sense of identity, developing infrastructure and recruiting practitioners.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733McColm, G.2014-02-12doi:10.1107/S2053273313033573International Union of CrystallographyConsidered as an emerging field, how mathematical crystallography grows during the 21st century may depend on how it addresses demand and attracts recruits within the chemical, physical and mathematical communities.ENcrystal structure predictioninnovationmathematical crystallographynanoscale studiesnewly emerging science and technologyThe potential of mathematical crystallography as an emerging field is examined from a sociological point of view. Mathematical crystallography is unusual as an emerging field as it is also an old field, albeit scattered, with evidence of continued substantial activity. But its situation is similar to that of an emerging field, so we analyse it as such. Comparisons with past emergent efforts suggest that a new field can grow if given an economic demand for its product and a receptive environment. Developing a field entails developing a sense of identity, developing infrastructure and recruiting practitioners.text/htmlProspects for mathematical crystallographytext2702014-02-12Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers95105Effects of merohedric twinning on the diffraction pattern
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In merohedric twinning, the lattices of the individuals are perfectly overlapped and the presence of twinning is not easily detected from the diffraction pattern, especially in the case of inversion twinning (class I). In general, the investigator has to consider three possible structural models: a crystal with space-group type H and point group P, either untwinned (H model) or twinned through an operation t in vector space (t-H model), and an untwinned crystal with space group G whose point group P′ is obtained as an extension of P through the twin operation t (G model). In 71 cases, consideration of the reflection conditions may directly rule out the G model; in seven other cases the reflection conditions suggest a space group which does not correspond to the extension of H by the twin operation and the structure solution or at least the refinement will fail. When the twin operation belongs to a different crystal family (class IIB twinning: the crystal has a specialized metric), the presence of twinning can often be recognized by the peculiar effect it has on the reflection conditions.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Nespolo, M.Ferraris, G.Souvignier, B.2014-02-12doi:10.1107/S2053273313029082International Union of CrystallographyThe effect of merohedric twinning on the reflection conditions and the symmetry of the diffraction pattern is analysed systematically and criteria to confirm or exclude the presence of twinning are presented.ENmerohedric twinningreflection conditionsdiffraction symmetryIn merohedric twinning, the lattices of the individuals are perfectly overlapped and the presence of twinning is not easily detected from the diffraction pattern, especially in the case of inversion twinning (class I). In general, the investigator has to consider three possible structural models: a crystal with space-group type H and point group P, either untwinned (H model) or twinned through an operation t in vector space (t-H model), and an untwinned crystal with space group G whose point group P′ is obtained as an extension of P through the twin operation t (G model). In 71 cases, consideration of the reflection conditions may directly rule out the G model; in seven other cases the reflection conditions suggest a space group which does not correspond to the extension of H by the twin operation and the structure solution or at least the refinement will fail. When the twin operation belongs to a different crystal family (class IIB twinning: the crystal has a specialized metric), the presence of twinning can often be recognized by the peculiar effect it has on the reflection conditions.text/htmlEffects of merohedric twinning on the diffraction patterntext2702014-02-12Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers106125Brillouin-zone database on the Bilbao Crystallographic Server
http://scripts.iucr.org/cgi-bin/paper?xo5018
The Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Aroyo, M.I.Orobengoa, D.de la Flor, G.Tasci, E.S.Perez-Mato, J.M.Wondratschek, H.2014-02-12doi:10.1107/S205327331303091XInternational Union of CrystallographyThe Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) is presented. Recent improvements and modifications of the database are discussed and illustrated by several examples.ENBilbao Crystallographic ServerBrillouin-zone databasereciprocal-space groupsThe Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.text/htmlBrillouin-zone database on the Bilbao Crystallographic Servertext2702014-02-12Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers126137On physical property tensors invariant under line groups
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The form of physical property tensors of a quasi-one-dimensional material such as a nanotube or a polymer can be determined from the point group of its symmetry group, one of an infinite number of line groups. Such forms are calculated using a method based on the use of trigonometric summations. With this method, it is shown that materials invariant under infinite subsets of line groups have physical property tensors of the same form. For line group types of a family of line groups characterized by an index n and a physical property tensor of rank m, the form of the tensor for all line group types indexed with n > m is the same, leaving only a finite number of tensor forms to be determined.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Litvin, D.B.2014-02-18doi:10.1107/S2053273313033585International Union of CrystallographyMaterials invariant under infinite subsets of line groups are shown to have physical property tensors of the same form.ENphysical propertiesline groupsquasi-one-dimensional materialsThe form of physical property tensors of a quasi-one-dimensional material such as a nanotube or a polymer can be determined from the point group of its symmetry group, one of an infinite number of line groups. Such forms are calculated using a method based on the use of trigonometric summations. With this method, it is shown that materials invariant under infinite subsets of line groups have physical property tensors of the same form. For line group types of a family of line groups characterized by an index n and a physical property tensor of rank m, the form of the tensor for all line group types indexed with n > m is the same, leaving only a finite number of tensor forms to be determined.text/htmlOn physical property tensors invariant under line groupstext2702014-02-18Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers138142Direct phasing in femtosecond nanocrystallography. I. Diffraction characteristics
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X-ray free-electron lasers solve a number of difficulties in protein crystallography by providing intense but ultra-short pulses of X-rays, allowing collection of useful diffraction data from nanocrystals. Whereas the diffraction from large crystals corresponds only to samples of the Fourier amplitude of the molecular transform at the Bragg peaks, diffraction from very small crystals allows measurement of the diffraction amplitudes between the Bragg samples. Although highly attenuated, these additional samples offer the possibility of iterative phase retrieval without the use of ancillary experimental data [Spence et al. (2011). Opt. Express, 19, 2866–2873]. This first of a series of two papers examines in detail the characteristics of diffraction patterns from collections of nanocrystals, estimation of the molecular transform and the noise characteristics of the measurements. The second paper [Chen et al. (2014). Acta Cryst. A70, 154–161] examines iterative phase-retrieval methods for reconstructing molecular structures in the presence of the variable noise levels in such data.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Chen, J.P.J.Spence, J.C.H.Millane, R.P.2014-01-15doi:10.1107/S2053273313032038International Union of CrystallographyThe diffraction of X-ray free-electron laser pulses by protein nanocrystals and the implications for direct phasing are investigated.ENnanocrystalsfemtosecond nanocrystallographydirect phasingX-ray free-electron lasersshape transformX-ray free-electron lasers solve a number of difficulties in protein crystallography by providing intense but ultra-short pulses of X-rays, allowing collection of useful diffraction data from nanocrystals. Whereas the diffraction from large crystals corresponds only to samples of the Fourier amplitude of the molecular transform at the Bragg peaks, diffraction from very small crystals allows measurement of the diffraction amplitudes between the Bragg samples. Although highly attenuated, these additional samples offer the possibility of iterative phase retrieval without the use of ancillary experimental data [Spence et al. (2011). Opt. Express, 19, 2866–2873]. This first of a series of two papers examines in detail the characteristics of diffraction patterns from collections of nanocrystals, estimation of the molecular transform and the noise characteristics of the measurements. The second paper [Chen et al. (2014). Acta Cryst. A70, 154–161] examines iterative phase-retrieval methods for reconstructing molecular structures in the presence of the variable noise levels in such data.text/htmlDirect phasing in femtosecond nanocrystallography. I. Diffraction characteristicstext2702014-01-15Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers143153Direct phasing in femtosecond nanocrystallography. II. Phase retrieval
http://scripts.iucr.org/cgi-bin/paper?mq5017
X-ray free-electron laser diffraction patterns from protein nanocrystals provide information on the diffracted amplitudes between the Bragg reflections, offering the possibility of direct phase retrieval without the use of ancillary experimental diffraction data [Spence et al. (2011). Opt. Express, 19, 2866–2873]. The estimated continuous transform is highly noisy however [Chen et al. (2014). Acta Cryst. A70, 143–153]. This second of a series of two papers describes a data-selection strategy to ameliorate the effects of the high noise levels and the subsequent use of iterative phase-retrieval algorithms to reconstruct the electron density. Simulation results show that employing such a strategy increases the noise levels that can be tolerated.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Chen, J.P.J.Spence, J.C.H.Millane, R.P.2014-01-15doi:10.1107/S2053273313032725International Union of CrystallographyA selective sampling scheme is described to improve the noise tolerance of direct phasing based on shape transform diffraction between Bragg reflections in nanocrystallography using X-ray free-electron lasers.ENnanocrystalsfemtosecond crystallographydirect phasingX-ray free-electron lasersshape transformiterative transform algorithmsX-ray free-electron laser diffraction patterns from protein nanocrystals provide information on the diffracted amplitudes between the Bragg reflections, offering the possibility of direct phase retrieval without the use of ancillary experimental diffraction data [Spence et al. (2011). Opt. Express, 19, 2866–2873]. The estimated continuous transform is highly noisy however [Chen et al. (2014). Acta Cryst. A70, 143–153]. This second of a series of two papers describes a data-selection strategy to ameliorate the effects of the high noise levels and the subsequent use of iterative phase-retrieval algorithms to reconstruct the electron density. Simulation results show that employing such a strategy increases the noise levels that can be tolerated.text/htmlDirect phasing in femtosecond nanocrystallography. II. Phase retrievaltext2702014-01-15Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers154161Viruses and fullerenes – symmetry as a common thread?
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The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Dechant, P.-P.Wardman, J.Keef, T.Twarock, R.2014-02-18doi:10.1107/S2053273313034220International Union of CrystallographyCarbon onions from affine extensions are considered in analogy with virus work.ENsymmetryvirusesfullerenescarbon onionsCoxeter groupsaffine extensionsquasicrystalsThe principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.text/htmlViruses and fullerenes – symmetry as a common thread?text2702014-02-18Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers162167Alternative approaches to onion-like icosahedral fullerenes
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The fullerenes of the C60 series (C60, C240, C540, C960, C1500, C2160 etc.) form onion-like shells with icosahedral Ih symmetry. Up to C2160, their geometry has been optimized by Dunlap & Zope from computations according to the analytic density-functional theory and shown by Wardman to obey structural constraints derived from an affine-extended Ih group. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry-inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine-extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C60 onion molecule is defined, with shells consisting of all successive fullerenes of the C60 series. The structural relations between the C60 onion and graphite lead to a one-parameter model with the same Euclidean symmetry P63mc as graphite and having a c/a = τ2 ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Janner, A.2014-02-20doi:10.1107/S2053273313034219International Union of CrystallographyThe carbon atomic positions obtained by a quantum-chemical computation are compared with model ones having six integral indices. In one approach these indices follow from affine extensions of the icosahedral group and in another one from crystallographic scaling transformations.ENfullerenescarbon onionsmolecular crystallographyaffine extensionscrystallographic scalingsanalytic density-functional theoryThe fullerenes of the C60 series (C60, C240, C540, C960, C1500, C2160 etc.) form onion-like shells with icosahedral Ih symmetry. Up to C2160, their geometry has been optimized by Dunlap & Zope from computations according to the analytic density-functional theory and shown by Wardman to obey structural constraints derived from an affine-extended Ih group. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry-inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine-extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C60 onion molecule is defined, with shells consisting of all successive fullerenes of the C60 series. The structural relations between the C60 onion and graphite lead to a one-parameter model with the same Euclidean symmetry P63mc as graphite and having a c/a = τ2 ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.text/htmlAlternative approaches to onion-like icosahedral fullerenestext2702014-02-20Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers168180What periodicities can be found in diffraction patterns of quasicrystals?
http://scripts.iucr.org/cgi-bin/paper?dm5049
The structure of quasicrystals is aperiodic. Their diffraction patterns, however, can be considered periodic. They are composed solely of series of peaks which exhibit a fully periodic arrangement in reciprocal space. Furthermore, the peak intensities in each series define the so-called `envelope function'. A Fourier transform of the envelope function gives an average unit cell, whose definition is based on the statistical distribution of atomic coordinates in physical space. If such a distribution is lifted to higher-dimensional space, it becomes the so-called atomic surface – the most fundamental feature of higher-dimensional analysis.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Wolny, J.Kozakowski, B.Kuczera, P.Pytlik, L.Strzalka, R.2014-02-20doi:10.1107/S2053273313034384International Union of CrystallographyThe structure of quasicrystals is aperiodic; however, their diffraction patterns comprise periodic series of peaks, which can be used to retrieve essential features of the quasicrystalline structure.ENquasicrystalsdiffraction patternFibonacci sequencePenrose tilingaverage unit cellhigher-dimensional analysisThe structure of quasicrystals is aperiodic. Their diffraction patterns, however, can be considered periodic. They are composed solely of series of peaks which exhibit a fully periodic arrangement in reciprocal space. Furthermore, the peak intensities in each series define the so-called `envelope function'. A Fourier transform of the envelope function gives an average unit cell, whose definition is based on the statistical distribution of atomic coordinates in physical space. If such a distribution is lifted to higher-dimensional space, it becomes the so-called atomic surface – the most fundamental feature of higher-dimensional analysis.text/htmlWhat periodicities can be found in diffraction patterns of quasicrystals?text2702014-02-20Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers181185Symmetry of helicoidal biopolymers in the frameworks of algebraic geometry: α-helix and DNA structures
http://scripts.iucr.org/cgi-bin/paper?dm5047
The chain of algebraic geometry and topology constructions is mapped on a structural level that allows one to single out a special class of discrete helicoidal structures. A structure that belongs to this class is locally periodic, topologically stable in three-dimensional Euclidean space and corresponds to the bifurcation domain. Singular points of its bounding minimal surface are related by transformations determined by symmetries of the second coordination sphere of the eight-dimensional crystallographic lattice E8. These points represent cluster vertices, whose helicoid joining determines the topology and structural parameters of linear biopolymers. In particular, structural parameters of the α-helix are determined by the seven-vertex face-to-face joining of tetrahedra with the E8 non-integer helical axis 40/11 having a rotation angle of 99°, and the development of its surface coincides with the cylindrical development of the α-helix. Also, packing models have been created which determine the topology of the A, B and Z forms of DNA.Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Samoylovich, M.Talis, A.2014-02-26doi:10.1107/S2053273313033822International Union of CrystallographyThe chain of algebraic geometry and topology constructions permits one to single out a special class of discrete helicoidal structures. The symmetry of these structures determines the structural parameters of the α-helix and topology of the A, B and Z forms of DNA.ENalgebraic geometrytopologyeight-dimensional lattice E8polytopeshelical axis 40/11α-helixstructuresThe chain of algebraic geometry and topology constructions is mapped on a structural level that allows one to single out a special class of discrete helicoidal structures. A structure that belongs to this class is locally periodic, topologically stable in three-dimensional Euclidean space and corresponds to the bifurcation domain. Singular points of its bounding minimal surface are related by transformations determined by symmetries of the second coordination sphere of the eight-dimensional crystallographic lattice E8. These points represent cluster vertices, whose helicoid joining determines the topology and structural parameters of linear biopolymers. In particular, structural parameters of the α-helix are determined by the seven-vertex face-to-face joining of tetrahedra with the E8 non-integer helical axis 40/11 having a rotation angle of 99°, and the development of its surface coincides with the cylindrical development of the α-helix. Also, packing models have been created which determine the topology of the A, B and Z forms of DNA.text/htmlSymmetry of helicoidal biopolymers in the frameworks of algebraic geometry: α-helix and DNA structurestext2702014-02-26Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Aresearch papers186198Essentials of Crystallography, second edition. By M. A. Wahab. Narosa Publishing House, 2014. Pp. xix + 335. Price USD 98.00 (North and South America), GBP 49.95 (rest of the World outside the Indian sub-continent). ISBN 978-1842658413 (outside the Indian sub-continent), 978-81-8487-316-0 (in the Indian sub-continent).
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Copyright (c) 2014 International Union of Crystallographyurn:issn:2053-2733Nespolo, M.2014-02-20doi:10.1107/S2053273313032919International Union of CrystallographyENbook reviewtext/htmlEssentials of Crystallography, second edition. By M. A. Wahab. Narosa Publishing House, 2014. Pp. xix + 335. Price USD 98.00 (North and South America), GBP 49.95 (rest of the World outside the Indian sub-continent). ISBN 978-1842658413 (outside the Indian sub-continent), 978-81-8487-316-0 (in the Indian sub-continent).text2702014-02-20Copyright (c) 2014 International Union of CrystallographyActa Crystallographica Section Abook reviews199202