Showing papers in "Acta Crystallographica Section A in 1978"

Testing aspherical atom refinements on small-molecule data sets

Abstract: X-ray data on silicon, tetracyanoethylene, p-nitropyridine N-oxide and ammonium thiocyanate are refined with a generalized aspherical-atom formalism as introduced by Stewart, but modified to have a spherical valence more similar to the unperturbed HF valence shell. Several types of radial dependences of the multipole functions are tested and criteria are developed for judging the adequacy of the aspherical-atom refinement. The aspherical-atom model leads to a significant decrease in the least-squares error function, a reduction of features in the residual map, and an improvement in thermal parameters when comparison is made with the neutron results or when the rigid-bond postulate proposed by Hirshfeld is applied. Positional parameters are often improved except in the case of terminal atoms for which discrepancies, attributed to correlation between dipole-population and positional parameters, are sometimes observed. Deformation maps based on the aspherical-atom least-squares parameters contain less noise than X -- N maps and benefit from inclusion of calculated values of weak structure amplitudes in the summation. In the cases studied, deformation maps including terms beyond the experimental resolution do not yield additional information.

A discussion of the solution for the best rotation to relate two sets of vectors

Abstract: A method is discussed for obtaining the best proper rotation to relate two sets of vectors.

On the treatment of negative intensity observations

Abstract: A method is described which produces sensible estimates of structure factor moduli from intensity observations, whether the latter are positive or negative. Preliminary applications of the method to data from the protein phosphorylase b are summarized.

Refinement of Large Structures by Simultaneous Minimization of Energy and R Factor

Abstract: An improved method of crystallographic structure refinement, especially suitable for large molecules, is described, it is based on simultaneous minimization of a realistic potential-energy function and a crystallographic residual. The method has already proved its worth in the final stages of refinement of two structures; an application to crude wiremodel coordinates of a small protein is described and evaluated.

LALS: a linked-atom least-squares reciprocal-space refinement system incorporating stereochemical restraints to supplement sparse diffraction data

Abstract: LALS is a computer program for the refinement of molecular structures. It is primarily intended for helical macromolecules but has applicability in other fields. The refinement uses as data, X-ray structure factors, usually from fibre diffraction studies, and/or stereochemical information, including a comprehensive short-contact search. Various other geometrical constraints and restraints may be placed on the molecular conformation. As parameters of the refinement, LALS uses primarily dihedral angles about single bonds, assuming bond lengths and angles to be fixed and known. This greatly reduces the number of parameters from conventional atom-position refinements, as is necessary in systems where data are sparse. LALS has been used successfully to investigate the structures of a large number of more or less ordered polynucleotides, polysaccharides and other fibrous materials.

A new least-squares refinement technique based on the fast Fourier transform algorithm

Abstract: A new atomic-parameters least-squares refinement method is presented which makes use of the fast Fourier transform algorithm at all stages of the computation. For large structures, the amount of computation is almost proportional to the size of the structure making it very attractive for large biological structures such as proteins. In addition the method has a radius of convergence of approximately 0.75 A making it applicable at a very early stage of the structure-determination process. The method has been tested on hypothetical as well as real structures. The method has been used to refine the structure of insulin at 1.5 A resolution, barium beauvuricin complex at 1.2 A resolution, and myoglobin at 2 A resolution. Details of the method and brief summaries of its applications are presented in the paper.

The crystal structure of η-Cu3Si precipitates in silicon

Abstract: The crystal structure of Cu-Si precipitates in silicon has been studied by means of transmission electron diffraction and microscopy. The precipitates are shown to be of the equilibrium Cu3Si phase, probably of the low-temperature η” form. Consistent with earlier X-ray diffraction data for this phase, the crystal structure of the precipitates is determined to be based on a trigonally distorted b.c.c. arrangement. The lattice is orthorhombic (C), and is a two-dimensional long-period superlattice, which it is suggested is based on the intermediate-temperature η' phase; lattice parameters aη” = 76.76, bη” = 7.00, cη” = 21.94 A. On the basis of the diffraction data for the precipitates, it is suggested that the ηand η' phases are both trigonal, space groups R{\bar 3}m and R{\bar 3} respectively, with lattice parameters aη = 2.47 A, αη = 109.74° and aη' = 4.72 A, αη' = 95.72°.

Stable calculation of coordinates from distance information

Abstract: A new method is described for the calculation of Cartesian coordinates for n points given the n × n matrix of interpoint distances. The algorithm is faster than some earlier methods, and it is remarkably stable with respect to both numerical roundoff errors and errors in the given distance matrix. The resultant coordinates have their origin near the center of mass and axes approximately along the three principal rotational axes. The calculation is described of distances to the center of mass directly from the distance matrix. Results of computer trials of the algorithm are given.

Crystal-structure of p-hydroxybenzoate hydroxylase

Abstract: Abstract The structure of the enzyme p-hydroxybenzoate hydroxylase (EC 1.14.13.2) in a complex with its substrate has been determined at a resolution of 2.5 A. The molecular weight is 43,000 and the dimensions of one molecule are approximately 70 A × 50 A × 45 A. The crystal structure contains dimers of these molecules. Approximately 16% of the residues occur in β-sheets and 26% in α-heliees. The molecule can be divided into three domains. The active site, near the isoalloxazine ring, is formed by side-chains of the three domains. The N-5 edge of the isoalloxazine ring points to p-hydroxybenzoate, which is bound in a deep cleft.

Strain-Tensor Components Expressed in Terms of Lattice Parameters

Abstract: Expressions are developed for the components of the linear Lagrangian, linear Eulerian, finite Lagrangian (Green's), and finite Eulerian (Almansi's) strain tensors in terms of a crystal's lattice parameters before and after a deformation. The development has been undertaken with the concepts and notations of linear algebra.

The defect structure of hyper-stoichiometric uranium dioxide

Abstract: Earlier neutron-diffraction measurements [Willis (1964), Proc. Br. Ceram. Soc. 1, 9-19] on a single crystal of UO2.12 at 800°C showed that interstitial oxygen atoms enter the UO2 lattice at two kinds of site, which are displaced by about 1 A along the (110) and (111) directions from the cubic-coordinated interstitial sites. It was also shown that the solution of oxygen is accompanied by the formation of vacancies in the normal oxygen sites, but that the uranium sublattice remains undisturbed. A new analysis of the same data has been carried out, which introduces constraints between some of the parameters and allows a more satisfactory treatment of the anharmonic motion of the oxygen atoms. The new analysis leads to a much more precise determination of the occupation numbers of the (110) and (111) oxygen atoms and of the normal oxygen vacancies. The simplest model to account for these new results is the so-called 2:2:2 configuration of oxygen defects.

The magnetic structure of Bi2Fe4O9 – analysis of neutron diffraction measurements

Abstract: The compound Bi2Fe4O9 belongs to the space group Pbam ( D92h), with two formula units per unit cell. Neutron diffraction measurements showed that it is paramagnetic at room temperature and undergoes a transition to an antiferromagnetic state at TN = (264 ± 3) K in agreement with previous susceptibility and Mossbauer measurements. Analysis of the 80 K neutron diffraction pattern yielded a magnetic structure with the following features: (a) The basic translations ao, bo, co of the chemical lattice change into antitranslations in the magnetic lattice. (b) The spins are perpendicular to co. (c) The magnetic structure belongs to the PC2/m space group and is a basis vector to an irreducible space under the Pbam irreducible representations, in accord with Landau's theory of second-order phase transition. The position parameters of the Fe3+ ions in the unit cell were refined. The magnetic moment of the compound was found to be (4.95 ± 0.08) μB, compared with the value of 5 μB for the Fe3+ free ion. The temperature dependence of the { 131 } magnetic reflection peak intensity was measured and found to be in agreement with the sublattice magnetization predicted by the molecular field approximation.

Structural Phase Transitions in Sodium-Potassium Niobate Solid Solutions by Neutron Powder Diffraction

Abstract: The structures of (Na,K)NbO3 solid solutions have been determined by using the neutron powder-profile refinement technique: phases K and L at room temperature, and G and F at higher temperatures. In each case approximate values have been found for position parameters and temperature factors. The structural models proposed by Ahtee & Glazer [Acta Cryst. (1976), A32, 434-446] based on the lattice parameters and intensities of difference reflexions have been found to be substantially correct. Indications of vibrational modes have been found in the sequential phase transitions of Na0.90K0.10NbO3.

A test for rigid‐body vibrations based on a generalization of Hirshfeld's `rigid‐bond' postulate

Abstract: A simple test for the validity of the rigid-body model for molecular vibrations in crystals is proposed.

Experimental study of disordered mica structures by high‐resolution electron microscopy

Abstract: Disorder in stacking sequences of mica minerals, predominantly 1M muscovite from York, Ontario and biotite from Mitchell Co., North Carolina, was observed using high-resolution electron microscopy. The specimens were prepared by sectioning the flakes of mica in the microtome with a diamond knife, so that crystals were viewed down the direction parallel to the Si-O layers. Disordered sequences of the layers, as well as ordered crystals, are best described by citing the positions of tunnels between alkali ions lying in the interlayers, since these individual sites are resolved in electron micrographs of micas. An evaluation of the usefulness of one-dimensional lattice fringe images for studying disordered states in crystals is also discussed by comparing them with structure images of corresponding crystals. Intimate intergrowths of different mica polytypes on a scale of tens of angstroms raise questions as to the definition of origins of unit cells and therefore polytypes.

The use of molecular‐orbital calculations on model systems for the prediction of bridging‐bond‐angle variations in siloxanes, silicates, silicon nitrides and silicon suffides

Abstract: CNDO/2 molecular-orbital calculations are shown to predict trends in bridging angle, ∠T-Y-T, with reasonable accuracy for gas-phase molecules of the type TX3YTX3, where Y is an anion (= O, OH, S, NH) bridging the tetrahedrally coordinated cations T(= C, Si, Al, Be, B) and where X are the nonbridging anions (= H, F). For example, calculated equilibrium ∠T-O-T are 100, 125 and 150° in CH3OCH3, SiH3OCH3 and SiH3OSiH3, respectively, compared to experimental values of 111, 121 and 144°. The equilibrium ∠Si-O-Si in SiH3OSiH3 is also in close agreement with the average value of this angle (148°) in the silica polymorphs. The SiH3YTH3 molecule is proposed as a model system for studying bridging bond angles in silicates and is found to give bridging-angle trends in reasonable agreement with those observed in Si-containing solids. In particular, calculations on SiH3YSiH3 predict LSi-Y-Si to decrease from 150 to 130 to 109° as Y is changed from O to NH to S, in agreement with observed average angles of 144, 120 and 110°, respectively, in silicates, silicon nitrides and silicon sulfides. For SiH3OSiH3 the equilibrium ∠Si-O-Si is calculated to increase as the Si-O bond distance decreases, at a rate within a factor of two of that observed for the silica polymorphs. Calculations on clusters with Be or B replacing Si invariably lead to predicted equilibrium ∠Si -O-T values from 125 to 105°, in accord with the small angles that tend to be observed for such bridge bonds in minerals. Since Si(3d) orbitals are excluded in these calculations, it is apparent that the observed angular trends can be explained without invoking d-p π bonding. Inspection of one and two-center energy components of the CNDO/2 total energy also shows that the form of the Si-T interaction is not that of a nonbonded repulsion.

The electron stationary picture method for high-speed measurement of reflection intensities from crystals with large unit cells

Abstract: A method for measuring X-ray intensity data from crystals with large unit ceils is presented. The method takes full advantage of the capabilities of the multiwire area detector diffractometer. This diffractometer is a high-speed data collection system utilizing a multiwire proportional chamber that can detect photons from all simultaneously occurring reflections and record the diffraction pattern as a two-dimensional histogram in a computer mass core memory. In the electronic stationary picture method of data collection, reflection intensities are extracted from a sequence of electronic pictures each of which is exposed while the crystal is stationary. Between successive pictures the crystal is rotated about a fixed axis by a small constant angle of approximately 0.05°. Because the integrated intensity of each reflection is extracted from several consecutive pictures, advance prediction of detector coordinates and picture number is required for all reflections and the data-extraction computer program is necessarily complex. Expressions are derived for reflection detector coordinates and setting angles, the system hardware and software are described, the data collection procedure is outlined, and the quality of 7 × 105 reflection intensities measured during six months of operation is analyzed.

Analysis of molecular motion with allowance for intramolecular torsion

Abstract: The one-parameter model of Dunitz & White [Acta Cryst. (1973), A29, 93-94] for the analysis of torsional degrees of freedom within an otherwise rigid body has been applied to data for a number of representative compounds. The model gives results not significantly different from those of more elaborate models for librations about bonds, even when the r.m.s, amplitudes of these librations are as great as 24°. The limitations of this kind of analysis and some of the problems in applying it are discussed.

Electrostatic lattice energy in ionic crystals: optimization of the convergence of Ewald series

Abstract: Ewald's method is reconsidered to express the dependence of Madelung energy on the ionic charges explicitly, also taking into account the space-group symmetry of the structure. Upper bounds for the residues of the two partial series have been calculated by integral approximation; that relative to the direct-lattice series is shown to depend on the cube root of the unit-cell volume. The optimum value of the parameter A, which equalizes the rates of convergence of the two sums and minimizes the total number of terms, has been determined numerically for a given termination error and for a range of unit-cell dimensions. Theoretical results are tested by calculations on some specific crystal structures.

Is there an N3− ion in the crystal structure of the ionic conductor lithium nitride (Li3N)?

Abstract: The structure parameters of Li3N in the temperature range -120 to 20°C were refined both for neutral atoms (Li0 and N0) and for ions (Li+ and N3-). For N3- new scattering curves were used which were calculated by applying stabilizing Watson-sphere potentials for different radii. All structure parameters depend critically on the scattering curves used. However, only on the assumption of ions, and only for N3- scattering curves corresponding to a small range of Watson radii, were physically meaningful structure parameters and R values down to 0.9% obtained. These structure refinements demonstrate that Li3N can be considered as an ionic crystal in which the N3- ion, though unstable as a free ion, is stabilized by the surrounding Li+ ions.

Crystal perfection in a noncentrosymmetric alloy. Refinement and test of twinning of the γ-Cu9Al4 structure

Abstract: The crystal structure of Cu9Al4 has been refined on the basis of 114 independent structure factors, averaged from a set of 2416 intensity measurements with a diffractometer on a spherical crystal of 0.05 mm diameter. Graphite-monochromatized Cu Kα radiation was used. The final R value was 2.4% when corrections for absorption and extinction had been applied. Anisotropic thermal parameters were introduced and shown to be significant. The occurrence of coherently scattering antiphase domains was found to be less than I%. All equivalent reflections and Friedel pairs were recorded with graphite-monochromatized Mo Kα radiation for ten selected index combinations. Observed and calculated anomalous dispersion effects were found to match closely, indicating that incoherent twinning was also practically non-existent.

Approximations for dynamical calculations of microdiffraction patterns and images of defects

Abstract: The calculation of dynamical elastic diffuse electron scattering from a crystal containing a point defect or dislocation may require prohibitively large amounts of computing time. Two approximations are described which greatly reduce the computer time required for these calculations, and so allow the simulation of electron-microscope images of defects at atomic resolution. The diffuse scattering expected from microdiffraction experiments can also be predicted in this way.

The short‐range‐order structure of α‐phase Cu–Al alloys

Abstract: A three-dimensional X-ray diffuse scattering investigation of the short-range order in a-phase Cu-AI alloys, and its dependence on alloy composition, quenching temperature and isothermal annealing at 250°C, has been carried out. The Cowley-Warren order coefficients were determined after separation of the size effects using a procedure based on the Boric-Sparks quadratic approximation of atomic displacements. These short-range-order coefficients were analyzed in terms of the complete spectrum of nearest-neighbor atomic configurations, without the necessity of invoking a specific model for the characteristic feature(s) of the localorder structure. Although in the composition range investigated the equilibrium condition is a short-rangeordered structure, the nearest-neighbor configurations bear a marked resemblance to the long-period antiphase-shift structures which have been found in alloys with more AI. Because essentially all of the most highly ordered configurations were found to exist isolated from each other in a 9.13 at.% A1 alloy, it is concluded that they represent inherently stable, spatial arrangements of the atoms. In alloys containing 13.56 and 14.76 at.% AI, connected ordered configurations, or small ordered domains, were detected, and this effect increases markedly with increasing A1 content. Isothermal annealing at 250°C of a Cu-14.76 at.% A1 alloy quenched from 650°C results in a net disordering process; however, superimposed on this, in the early stages of annealing, is an enhancement of the population density of an atomic configuration confined to planes of the { 111 } type. This is interpreted as evidence for occurrence of the Suzuki mechanism. It is argued that the body of published experimental observations on this alloy system is better understood if one recognizes the existence of stacking faults, and their interaction with the matrix, in addition to that of short-range order.

X‐ray study of the atomic charge densities in MgO, CaO, SrO and BaO

Abstract: Accurate X-ray structure factors of CaO, SrO and BaO were measured at 293 K from single crystals and analysed together with the data of Sanger [Acta Cryst (1969), A25, 694-702] on MgO The oxygen ion is seen to suffer an average compression and a fourth-order cubic harmonic type deformation, as compared to the atomic superposition model, yielding a lower average charge density in the surrounding region and a charge transfer from [111] to the [100] direction These features strengthen with increasing atomic number of the cation Mg and Ca show a weak broadening About nine electrons are localized around the oxygen position The tenth electron of O2- is more widely distributed The electron counts of the cation correspond most closely to Mg+, Ca2+, Sr2+ and Ba2+ The results confirm the dominance of ionic character and increase of ionicity with increasing atomic number of the cation

The vicissitudes of the low‐quartz crystal setting or the pitfalls of enantiomorphism

Abstract: Based on a literature study, the following proposals endeavour to combine the advantages of previous conventions for setting low-quartz crystals: (1) Knowing that symmetry axes are parallel to the edges of the smallest hexagonal cell, place Oz along the 3-axis and Ox, Oy, Ou along the macroscopic 2-axes. (2) Direct the polar axes positively toward the s and x faces, if present, or toward the positive charge developed on extension by piezoelectric test. (3) Match the hand of the coordinate system to that of the structural screw; thus, the right-handed coordinate system (RHCS) for the known right screw (RS) of laevorotatory quartz. (4) Let the coordinate axis that coincides with a 2-axis in the structure be Ox[100]. These rules result in both enantiomorphs obeying Lang's mnemonic rule and being in the r(+) setting, in which {10{\bar 1}1} symbolizes the major rhombohedron r, defined by Iobs(10{\bar 1}1) > Iobs(01{\bar 1}1). Low quartz has only one structure. It should be published only for the RS (P3121) and in the RHCS: anyone can visualize the LS enantiomorph by plotting the atomic coordinates in the LHCS. This policy should apply to any enantiomorphous substance with known absolute configuration. If a crystal structure, determined in the RHCS, turns out to have a LS, transform to LHCS and publish the new coordinates, which also fit the standard RS in RHCS, but add, 'Known only as LS' or 'Observed as LS'.

Computed electron microscope images of atomic defects in f.c.c. metals

Abstract: The bright-field and dark-field electron microscope images expected for [100] split interstitials in thin crystals of gold and aluminum without and with lattice relaxation have been calculated by the method of periodic continuation including full n-beam dynamical interactions of both the Bragg reflections and the diffuse scattering. The advantage of using 1 MeV rather than 100 keV electrons is demonstrated in that, even with the same nominal resolution, the 1 MeV electrons give images in which the defect structure is more readily recognized. The conditions have been determined for which the defect images have high contrast and provide clear representations of the atom configurations.

The dynamical theory of X-ray Bragg diffraction from a crystal with a uniform strain gradient. The Green Riemann functions

Abstract: The dynamical problem of X-ray Bragg diffraction from a thick (semi-infinite) crystal deformed by a uniform strain gradient (USG) is treated on the basis of the Green-Riemann function formalism. The rigorous solution of the problem is formulated by means of the Huygens-Fresnel principle. The exact Green functions are obtained in the form of the Laplace integrals suitable in physical applications. The quasi-classical and the Born (kinematical) asymptotic expansions of the Green functions are constructed as functions of the effective USG parameter B. Special attention is paid to the analysis of the wave-field propagation in a crystal with USG. The spatial harmonics Re(q) of the diffracted Green function, when Re(qB) 0 are damped exponentially in the bulk of the crystal. The Taupin problem of the Bragg dynamical diffraction of the X-ray incident plane wave from a thick crystal, the lattice spacing being a linear function of the coordinate z (along the inward normal to the entrance surface) only is solved exactly in analytical form. In the latter case the waveguide nature of the propagation of the spatial harmonics inside such a crystal, provided that Re(qB) < 0, is clearly revealed.

The electron density and bonding in beryllium metal as studied by Fourier methods

Abstract: The bonding in Be metal is studied by calculation of valence and deformation density maps from data collected by Brown [Philos. Mag. (1972), 26, 1377-1394]. The deformation maps show clearly that the predominant interaction is bonding through the tetrahedral holes of the h.c.p. lattice. The valence-density maps generally agree with least-squares-model maps published by Stewart [Acta Cryst. (1977), A33, 33-38], though differences in detail exist which may result from incomplete space filling of the least-squares model functions. The experimental densities are interpreted in terms of the hybrid-orbital model of Altmann, Coulson & Hume-Rothery [Proc. R. Soc. London Ser. A, (1957), 240, 145-155] as corresponding to (sp2)a (sp)b hybridization of the Be atoms with b > a. Agreement with a theoretical density obtained by the augmented plane wave calculation of lnoue & Yamashita [J. Phys. Soc. Jpn. (1973), 35, 677-683] supports both theoretical and experimental procedures.

Experience with fast Fourier least squares in the refinement of the crystal structure of rhombohedral 2-zinc insulin at 1.5 Å resolution

Abstract: The newly developed fast Fourier least-squares algorithm [Agarwal (1978). Acta Cryst. A34, 791-809] has been used to refine the structure of rhombohedral 2-Zn pig insulin at a resolution of 1.5 A. The CPU time required for each cycle of refinement was about 3 min on an IBM 370/168 computer. After 67 cycles of refinement the conventional R factor was 0.113 for 11 890 terms. The procedure used in the refinement is described and an analysis of the results presented. From the experience learned in this refinement a procedure is suggested which should provide faster convergence and require considerably fewer refinement cycles.

Crystal powder statistics. II. Line profiles in diffraction spectra of identical crystals and of Gaussian samples. Crystal size distributions

Abstract: The average interference function ({\cal J}hkl(ΔS)) of a powder sample containing perfect crystals at a reciprocal distance ΔS from the peak is evaluated both for the case of identical parallelepiped crystals and for a Gaussian sample [probability of thickness d along a given crystal direction = C1 exp (-C2d2)]. In the latter case ({\cal J}hkl(ΔS)) decreases as 1/ΔS2 for large ΔS, by analogy with the Bernoullian model [Ailegra, Bassi & Meille (1978). Acta Cryst. A34, 652-655] although with a smaller amplitude, for a fixed integrated intensity and half-peak width. It is shown that the Gaussian interference function, or line profile, cannot be given by any real sample, at least if its crystals neither contain holes nor have concave surfaces. Number and weight probability distributions are calculated both for the Bernoullian and for the Gaussian crystal-size statistics. As expected from the calculated line profiles, the Bernoullian statistics correspond to a larger weight percentage of crystals smaller than the average.