Acta Crystallographica Section A
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Acta Crystallographica Section A: Foundations of Crystallography 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) 2012 International Union of Crystallography2012-05-01International Union of CrystallographyInternational Union of Crystallographyhttp://journals.iucr.orgurn:issn:0108-7673Acta Crystallographica Section A: Foundations of Crystallography 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 of Crystallography, Volume 68, Part 3, 2012textyearly62002-01-01T00:00+00:003682012-05-01Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section A: Foundations of Crystallography313urn:issn:0108-7673med@iucr.orgMay 20122012-05-01Acta Crystallographica Section Ahttp://journals.iucr.org/logos/rss10a.gif
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Still imageEnumeration of one-dimensional crystal structures obtained from a minimum of diffraction intensities
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A central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small (N ≤ 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N − 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (∼4N/N3/2 for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Al-Asadi, A.Chudin, E.Tsodikov, O.V.2012-03-06doi:10.1107/S0108767312002231International Union of CrystallographyThe number of one-dimensional crystal structures that can be determined from a minimum of diffraction intensities is obtained here for the first time. This number increases exponentially with the increasing size of the structure, illustrating quantitatively the vast structure ambiguity caused by the lack of experimental phase information.ENdirect methodsstructure ambiguityphase problemalgebraic geometryA central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small (N ≤ 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N − 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (∼4N/N3/2 for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures.text/htmlEnumeration of one-dimensional crystal structures obtained from a minimum of diffraction intensitiestext3682012-03-06Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers313318Parameters for temperature dependence of mean-square displacements for B-, Bi- and Tl-containing binary III–V compounds
http://scripts.iucr.org/cgi-bin/paper?td5005
Mean-square displacements were computed within the harmonic approximation from ab initio force constants for binary B-, Tl- and Bi-containing III–V compounds in sphalerite crystal structures in the temperature range from 0 to 1000 K in steps of 1 K. An Einstein model with a temperature-dependent characteristic frequency was used to model the temperature dependence of the mean-square displacements. A Gaussian was fitted to the temperature dependence of the characteristic frequency and parameters of the Gaussian are given. Phonon dispersion relations and densities of states derived during the computation of the mean-square displacements are also shown.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Schowalter, M.Rosenauer, A.Volz, K.2012-03-06doi:10.1107/S0108767312002681International Union of CrystallographyMean-square displacements were computed within the harmonic approximation from ab initio force constants for binary B-, Tl- and Bi-containing III–V compounds with the sphalerite crystal structure in the temperature range from 0 to 1000 K. A fitting scheme based on an Einstein model with a temperature-dependent characteristic frequency was applied to describe the temperature dependence of the mean-square displacements. Respective fit parameters, phonon dispersion relations and densities of states are given.ENDebye–Waller factorsmean-square displacementsforce constantsphonon density of statesphonon dispersion relationsdensity functional theoryMean-square displacements were computed within the harmonic approximation from ab initio force constants for binary B-, Tl- and Bi-containing III–V compounds in sphalerite crystal structures in the temperature range from 0 to 1000 K in steps of 1 K. An Einstein model with a temperature-dependent characteristic frequency was used to model the temperature dependence of the mean-square displacements. A Gaussian was fitted to the temperature dependence of the characteristic frequency and parameters of the Gaussian are given. Phonon dispersion relations and densities of states derived during the computation of the mean-square displacements are also shown.text/htmlParameters for temperature dependence of mean-square displacements for B-, Bi- and Tl-containing binary III–V compoundstext3682012-03-06Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers319323Bayesian analysis of the evidence for minor components in crystallographic models: an alternative to the Hamilton {\cal R} test
http://scripts.iucr.org/cgi-bin/paper?sc5046
A simple test based on Bayesian statistics for the presence of minority populations in single-crystal structure refinement is presented. The test is illustrated by analysis of photocrystallographic experiments on single crystals of ruthenium–sulfur-dioxide-based complexes. In data sets collected after irradiation with light, conventional refinements of the populations of different metastable states to values below 4% are shown to be statistically significant. The results also confirm that the photo-induced states are absent from data collected in the dark.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Phillips, A.E.Cole, J.M.2012-04-05doi:10.1107/S0108767312010094International Union of CrystallographyBayesian model comparison affords a reliable technique for distinguishing externally characterized minority populations from background noise.ENBayesian model comparisonphotocrystallographypopulation analysislikelihood ratio testA simple test based on Bayesian statistics for the presence of minority populations in single-crystal structure refinement is presented. The test is illustrated by analysis of photocrystallographic experiments on single crystals of ruthenium–sulfur-dioxide-based complexes. In data sets collected after irradiation with light, conventional refinements of the populations of different metastable states to values below 4% are shown to be statistically significant. The results also confirm that the photo-induced states are absent from data collected in the dark.text/htmlBayesian analysis of the evidence for minor components in crystallographic models: an alternative to the Hamilton {\cal R} testtext3682012-04-05Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers324330High-resolution study of (002, 113, 11−1) four-beam diffraction in Si
http://scripts.iucr.org/cgi-bin/paper?dm5024
The results of a high-resolution study of the (002, 113, 11\overline{1}) four-beam diffraction in Si are presented. The incident synchrotron radiation beam was highly monochromated and collimated with a multi-crystal arrangement in a dispersive setup in both vertical and horizontal planes, in an attempt to experimentally approach plane-wave incident conditions. The Renninger scheme was used with the forbidden reflection reciprocal-lattice vector 002 normal to the crystal surface. The azimuthal and polar rotations were performed in the crystal surface plane and the vertical plane correspondingly. The polar angular curves for various azimuthal angles were measured and found to be very close to theoretical computer simulations, with only a small deviation from the plane monochromatic wave. The effect of the strong two-beam 002 diffraction was observed for the first time with the maximum reflectivity close to 80%. The structure factor of the 002 reflection in Si was experimentally determined as zero.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Kohn, V.G.Kazimirov, A.2012-04-17doi:10.1107/S0108767312012305International Union of CrystallographyA high-resolution study of (002, 113, 11−1) four-beam diffraction in Si was performed both experimentally and theoretically. Excellent coincidence between theory and experiment was achieved. The forbidden two-beam 002 reflection was excited with the maximum reflectivity of 80%.ENX-ray multiple diffractionsiliconhigh resolutionforbidden reflectionsThe results of a high-resolution study of the (002, 113, 11\overline{1}) four-beam diffraction in Si are presented. The incident synchrotron radiation beam was highly monochromated and collimated with a multi-crystal arrangement in a dispersive setup in both vertical and horizontal planes, in an attempt to experimentally approach plane-wave incident conditions. The Renninger scheme was used with the forbidden reflection reciprocal-lattice vector 002 normal to the crystal surface. The azimuthal and polar rotations were performed in the crystal surface plane and the vertical plane correspondingly. The polar angular curves for various azimuthal angles were measured and found to be very close to theoretical computer simulations, with only a small deviation from the plane monochromatic wave. The effect of the strong two-beam 002 diffraction was observed for the first time with the maximum reflectivity close to 80%. The structure factor of the 002 reflection in Si was experimentally determined as zero.text/htmlHigh-resolution study of (002, 113, 11−1) four-beam diffraction in Sitext3682012-04-17Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers331336An improved experimental databank of transferable multipolar atom models – ELMAM2. Construction details and applications
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ELMAM2 is a generalized and improved library of experimentally derived multipolar atom types. The previously published ELMAM database is restricted mostly to protein atoms. The current database is extended to common functional groups encountered in organic molecules and is based on optimized local axes systems taking into account the local pseudosymmetry of the molecular fragment. In this approach, the symmetry-restricted multipoles have zero populations, while others take generally significant values. The various applications of the database are described. The deformation electron densities, electrostatic potentials and interaction energies calculated for several tripeptides and aromatic molecules are calculated using ELMAM2 electron-density parameters and compared with the former ELMAM database and density functional theory calculations.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Domagała, S.Fournier, B.Liebschner, D.Guillot, B.Jelsch, C.2012-03-29doi:10.1107/S0108767312008197International Union of CrystallographyThe construction and application of the generalized ELMAM2 databank of experimentally derived aspherical atom types are described. With this new database, it is possible to transfer multipolar parameters to proteins and various organic molecules within the limitation of the currently available atom types.ENexperimental electron densitydatabasemultipolar modellingELMAM2 is a generalized and improved library of experimentally derived multipolar atom types. The previously published ELMAM database is restricted mostly to protein atoms. The current database is extended to common functional groups encountered in organic molecules and is based on optimized local axes systems taking into account the local pseudosymmetry of the molecular fragment. In this approach, the symmetry-restricted multipoles have zero populations, while others take generally significant values. The various applications of the database are described. The deformation electron densities, electrostatic potentials and interaction energies calculated for several tripeptides and aromatic molecules are calculated using ELMAM2 electron-density parameters and compared with the former ELMAM database and density functional theory calculations.text/htmlAn improved experimental databank of transferable multipolar atom models – ELMAM2. Construction details and applicationstext3682012-03-29Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers337351Refining structures against reflection rank: an alternative metric for electron crystallography
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A new metric is proposed to improve the fidelity of structures refined against precession electron diffraction data. The inherent dynamical nature of electron diffraction ensures that direct refinement of recorded intensities against structure-factor amplitudes can be prone to systematic errors. Here it is shown that the relative intensity of precessed reflections, their rank, can be used as an alternative metric for refinement. Experimental data from erbium pyrogermanate show that applying precession reduces the dynamical transfer of intensity between reflections and hence stabilizes their rank, enabling accurate and reliable structural refinements. This approach is then applied successfully to an unknown structure of an oxygen-deficient bismuth manganite resulting in a refined structural model that is similar to a calcium analogue.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Eggeman, A.S.Midgley, P.A.2012-03-13doi:10.1107/S0108767312007234International Union of CrystallographyThe relative intensity (or rank) of a reflection is considered as an alternative metric when attempting structural refinement using precession electron diffraction data.ENstructure refinementprecession electron diffractionA new metric is proposed to improve the fidelity of structures refined against precession electron diffraction data. The inherent dynamical nature of electron diffraction ensures that direct refinement of recorded intensities against structure-factor amplitudes can be prone to systematic errors. Here it is shown that the relative intensity of precessed reflections, their rank, can be used as an alternative metric for refinement. Experimental data from erbium pyrogermanate show that applying precession reduces the dynamical transfer of intensity between reflections and hence stabilizes their rank, enabling accurate and reliable structural refinements. This approach is then applied successfully to an unknown structure of an oxygen-deficient bismuth manganite resulting in a refined structural model that is similar to a calcium analogue.text/htmlRefining structures against reflection rank: an alternative metric for electron crystallographytext3682012-03-13Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers352358Density functional calculations of polysynthetic Brazil twinning in α-quartz
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Polysynthetic Brazil twinning in α-quartz, which occurs commonly in amethyst, is interpreted in the literature as having its composition planes parallel to one of the faces of the major rhombohedron r. It is shown that, instead, the composition planes are parallel to one of the faces of the minor rhombohedron z. The proposed translation 0.4547a between neighbouring lamellae leads to binding distances and binding angles across the composition plane that differ less from their bulk values than for the translation 0.5a proposed in the literature. Density functional calculations show that the energy of the unrelaxed polysynthetic twin is lower for the proposed translation. They also show that relaxation increases the thickness of the polytwin by 4 pm per composition plane.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Grimmer, H.Delley, B.2012-03-22doi:10.1107/S0108767312008756International Union of CrystallographyA model of polysynthetic Brazil twins in α-quartz with composition planes parallel to a face of the minor rhombohedron z is proposed. Density functional calculations show that the proposed twin boundary structure has very low energy.ENα-quartztwinningdensity functional calculationsPolysynthetic Brazil twinning in α-quartz, which occurs commonly in amethyst, is interpreted in the literature as having its composition planes parallel to one of the faces of the major rhombohedron r. It is shown that, instead, the composition planes are parallel to one of the faces of the minor rhombohedron z. The proposed translation 0.4547a between neighbouring lamellae leads to binding distances and binding angles across the composition plane that differ less from their bulk values than for the translation 0.5a proposed in the literature. Density functional calculations show that the energy of the unrelaxed polysynthetic twin is lower for the proposed translation. They also show that relaxation increases the thickness of the polytwin by 4 pm per composition plane.text/htmlDensity functional calculations of polysynthetic Brazil twinning in α-quartztext3682012-03-22Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers359365Classifying and assembling two-dimensional X-ray laser diffraction patterns of a single particle to reconstruct the three-dimensional diffraction intensity function: resolution limit due to the quantum noise
http://scripts.iucr.org/cgi-bin/paper?cc5011
A new two-step algorithm is developed for reconstructing the three-dimensional diffraction intensity of a globular biological macromolecule from many experimentally measured quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The first step is classification of the two-dimensional patterns into groups according to the similarity of direction of the incident X-rays with respect to the molecule and an averaging within each group to reduce the noise. The second step is detection of common intersecting circles between the signal-enhanced two-dimensional patterns to identify their mutual location in the three-dimensional wavenumber space. The newly developed algorithm enables one to detect a signal for classification in noisy experimental photon-count data with as low as ∼0.1 photons per effective pixel. The wavenumber of such a limiting pixel determines the attainable structural resolution. From this fact, the resolution limit due to the quantum noise attainable by this new method of analysis as well as two important experimental parameters, the number of two-dimensional patterns to be measured (the load for the detector) and the number of pairs of two-dimensional patterns to be analysed (the load for the computer), are derived as a function of the incident X-ray intensity and quantities characterizing the target molecule.Copyright (c) 2012 Atsushi Tokuhisa et al.urn:issn:0108-7673Tokuhisa, A.Taka, J.Kono, H.Go, N.2012-03-22doi:10.1107/S010876731200493XInternational Union of CrystallographyA new algorithm is developed for reconstructing the high-resolution three-dimensional diffraction intensity function of a globular biological macromolecule from many quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The structural resolution is expressed as a function of the incident X-ray intensity and quantities characterizing the target molecule.ENbiological macromoleculesclassification of two-dimensional diffraction patternscommon intersecting circlesattainable structural resolutionA new two-step algorithm is developed for reconstructing the three-dimensional diffraction intensity of a globular biological macromolecule from many experimentally measured quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The first step is classification of the two-dimensional patterns into groups according to the similarity of direction of the incident X-rays with respect to the molecule and an averaging within each group to reduce the noise. The second step is detection of common intersecting circles between the signal-enhanced two-dimensional patterns to identify their mutual location in the three-dimensional wavenumber space. The newly developed algorithm enables one to detect a signal for classification in noisy experimental photon-count data with as low as ∼0.1 photons per effective pixel. The wavenumber of such a limiting pixel determines the attainable structural resolution. From this fact, the resolution limit due to the quantum noise attainable by this new method of analysis as well as two important experimental parameters, the number of two-dimensional patterns to be measured (the load for the detector) and the number of pairs of two-dimensional patterns to be analysed (the load for the computer), are derived as a function of the incident X-ray intensity and quantities characterizing the target molecule.text/htmlClassifying and assembling two-dimensional X-ray laser diffraction patterns of a single particle to reconstruct the three-dimensional diffraction intensity function: resolution limit due to the quantum noisetext3682012-03-22Copyright (c) 2012 Atsushi Tokuhisa et al.Acta Crystallographica Section Aresearch papers366381Temperature diffuse scattering of nanocrystals
http://scripts.iucr.org/cgi-bin/paper?ib5010
The effects of thermal vibrations on X-ray powder diffraction patterns are discussed. Special considerations for extremely small crystallites are described, including the occurrence of surface and edge vibrational modes, and a restriction on the maximum phonon wavelength. In doing so, a complete temperature diffuse scattering (TDS) model is presented, which includes the influence of these features on: the Debye–Waller parameter; first-order TDS; and higher-order TDS terms. The importance of using an accurate TDS representation is studied as a function of temperature and crystallite size. It is found that a misrepresentation of the TDS for small crystallites can lead to an error in the determined Debye–Waller parameter on the order of 20–40% and a slight overestimation of the peak broadening. While the presented theory is primarily developed considering X-ray scattering, the same expressions are expected to describe the TDS in faster-than-sound neutron powder diffraction measurements.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Beyerlein, K.R.Leoni, M.Scardi, P.2012-03-29doi:10.1107/S0108767312009853International Union of CrystallographyThe effects of thermal vibrations on X-ray powder diffraction patterns are discussed. Special considerations for extremely small crystallites are described.ENX-ray powder diffractiontemperature diffuse scatteringline-profile analysisThe effects of thermal vibrations on X-ray powder diffraction patterns are discussed. Special considerations for extremely small crystallites are described, including the occurrence of surface and edge vibrational modes, and a restriction on the maximum phonon wavelength. In doing so, a complete temperature diffuse scattering (TDS) model is presented, which includes the influence of these features on: the Debye–Waller parameter; first-order TDS; and higher-order TDS terms. The importance of using an accurate TDS representation is studied as a function of temperature and crystallite size. It is found that a misrepresentation of the TDS for small crystallites can lead to an error in the determined Debye–Waller parameter on the order of 20–40% and a slight overestimation of the peak broadening. While the presented theory is primarily developed considering X-ray scattering, the same expressions are expected to describe the TDS in faster-than-sound neutron powder diffraction measurements.text/htmlTemperature diffuse scattering of nanocrystalstext3682012-03-29Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers382392Topological complexity of crystal structures: quantitative approach
http://scripts.iucr.org/cgi-bin/paper?dm5025
The topological complexity of a crystal structure can be quantitatively evaluated using complexity measures of its quotient graph, which is defined as a projection of a periodic network of atoms and bonds onto a finite graph. The Shannon information-based measures of complexity such as topological information content, IG, and information content of the vertex-degree distribution of a quotient graph, Ivd, are shown to be efficient for comparison of the topological complexity of polymorphs and chemically related structures. The IG measure is sensitive to the symmetry of the structure, whereas the Ivd measure better describes the complexity of the bonding network.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Krivovichev, S.2012-04-17doi:10.1107/S0108767312012044International Union of CrystallographyThe topological complexity of crystal structures can be characterized using complexity measures of their quotient graphs. Information-based complexity measures (topological information content, IG, and information content of the vertex-degree distribution of a graph, Ivd) are shown to be efficient for comparison of the topological complexity of polymorphs and chemically related structures.ENcrystal structuretopological complexityquotient graphinformation contentpolymorphsThe topological complexity of a crystal structure can be quantitatively evaluated using complexity measures of its quotient graph, which is defined as a projection of a periodic network of atoms and bonds onto a finite graph. The Shannon information-based measures of complexity such as topological information content, IG, and information content of the vertex-degree distribution of a quotient graph, Ivd, are shown to be efficient for comparison of the topological complexity of polymorphs and chemically related structures. The IG measure is sensitive to the symmetry of the structure, whereas the Ivd measure better describes the complexity of the bonding network.text/htmlTopological complexity of crystal structures: quantitative approachtext3682012-04-17Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Aresearch papers393398Patterson function and δ recycling: derivation of the phasing equations
http://scripts.iucr.org/cgi-bin/paper?sc5049
Two phasing equations based on the Fourier syntheses δP = T−1[(E2 − 〈E2〉)exp(iϕ)] and δM = T−1[(E − 〈E〉)exp(iϕ)] were recently described [Rius (2012). Acta Cryst. A68, 77–81] (E is the quasi-normalized structure factor and 〈E〉 is the average over all reflections). These equations were found by comparison with the direct methods origin-free modulus sum function and constitute the core of the `δ recycling' phasing procedure. The derivation of these phasing equations from the minimization of a residual (RP) between two differently calculated density functions (one of them including the positivity constraint) is shown.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Rius, J.2012-03-22doi:10.1107/S0108767312008768International Union of CrystallographyThe phasing equations of δ recycling direct methods are derived from the minimization of a residual between two differently calculated density functions (one of them including the positivity constraint).ENdirect methodsstructure solutionδ recyclingphasing formulaTwo phasing equations based on the Fourier syntheses δP = T−1[(E2 − 〈E2〉)exp(iϕ)] and δM = T−1[(E − 〈E〉)exp(iϕ)] were recently described [Rius (2012). Acta Cryst. A68, 77–81] (E is the quasi-normalized structure factor and 〈E〉 is the average over all reflections). These equations were found by comparison with the direct methods origin-free modulus sum function and constitute the core of the `δ recycling' phasing procedure. The derivation of these phasing equations from the minimization of a residual (RP) between two differently calculated density functions (one of them including the positivity constraint) is shown.text/htmlPatterson function and δ recycling: derivation of the phasing equationstext3682012-03-22Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Ashort communications399400Report of the Executive Committee for 2010
http://scripts.iucr.org/cgi-bin/paper?es0388
The report of the Executive Committee for 2010 is presented.Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Dacombe, M.2012-03-06doi:10.1107/S0108767311031370International Union of CrystallographyThe report of the Executive Committee for 2010 is presented.ENReport of the Executive CommitteeThe report of the Executive Committee for 2010 is presented.text/htmlReport of the Executive Committee for 2010text3682012-03-06Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Ainternational union of crystallography401429Basic Elements of Crystallography. By Nevill Gonzalez Szwacki and Teresa Szwacka. Pan Stanford Publishing, 2010. Pp. xi + 195. Price (paperback) GBP 49.99. ISBN 978-981-4241-59-5.
http://scripts.iucr.org/cgi-bin/paper?pf0093
Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Nespolo, M.2012-03-29doi:10.1107/S0108767312011968International Union of CrystallographyENbook reviewtext/htmlBasic Elements of Crystallography. By Nevill Gonzalez Szwacki and Teresa Szwacka. Pan Stanford Publishing, 2010. Pp. xi + 195. Price (paperback) GBP 49.99. ISBN 978-981-4241-59-5.text3682012-03-29Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Abook reviews430432Symétrie et propriétés physiques des cristaux. Par Cécile Malgrange, Christian Ricolleau et Françoise Lefaucheux. Pp. xxii+494. Les Ulis: EDP Sciences et CNRS Éditions, 2011. Prix (broché) EUR 52. ISBN-13: 978-2-7598-0499-3. (In French.)
http://scripts.iucr.org/cgi-bin/paper?pf0092
Copyright (c) 2012 International Union of Crystallographyurn:issn:0108-7673Paufler, P.2012-03-06doi:10.1107/S0108767312007106International Union of CrystallographyENbook receivedtext/htmlSymétrie et propriétés physiques des cristaux. Par Cécile Malgrange, Christian Ricolleau et Françoise Lefaucheux. Pp. xxii+494. Les Ulis: EDP Sciences et CNRS Éditions, 2011. Prix (broché) EUR 52. ISBN-13: 978-2-7598-0499-3. (In French.)text3682012-03-06Copyright (c) 2012 International Union of CrystallographyActa Crystallographica Section Abooks received433433