Showing papers in "Acta Crystallographica Section A in 1975"
TL;DR: In this article, a simple technique is described for ascertaining trial models for the structures of perovskites, which relies on an understanding of the fundamental components of the structure and rules are given for determining trial models rapidly.
Abstract: A simple technique is described for ascertaining trial models for the structures of perovskites. The method relies on an understanding of the fundamental components of the structure. Rules are given for determining trial models rapidly.
1,409 citations
TL;DR: In this paper, a correlation between the tilt angle ω and a flattening of the octahedron is found, and some tentative suggestions are made as to its cause.
Abstract: The rhombohedral perovskites are of interest in lattice dynamics (e.g. LaAlO3, PrAlO3) and for their ferroelectric properties (e.g. LiNbO3, PbZr/TiO3). In this paper, data scattered through the literature are correlated, with correction of some misleading mistakes of calculation. Geometrical descriptions are put in a form allowing comparisons. The structures, classified by their space groups, are described in terms of four structural parameters, the octahedron tilt ω, octahedron distortion d, and A- and B-cation displacements s and t, together with an elongation or flattening ζ of the octahedron, deducible with the help of ω from the interaxial angle αrh. Attention is drawn to the variety of physical causes underlying these departures from ideal perovskite. In the R{\bar 3}c structures a correlation not previously noted in the literature is found between the tilt angle ω and a flattening of the octahedron, and some tentative suggestions are made as to its cause. In the R3m and R3c structures, a lack of general correlation between B-cation displacement and the other parameters is noted, in contradiction to earlier reports.
303 citations
TL;DR: In this article, a pseudo-spherical approximation for polyhedral crystals is described which is accurate to within 2% of the extinction factor y for crystals whose ratio of maximum and minimum dimensions is less than two.
Abstract: Previously derived formalisms for extinction are extended to include crystals of non-spherical shape and anisotropy of mosaic spread and particle size. Expressions derived for extinction in an ellipsoidal crystal are compared with numerical calculations on a polyhedral specimen. A pseudo-spherical approximation for polyhedral crystals is described which is accurate to within 2% of the extinction factor y for crystals whose ratio of maximum and minimum dimensions is less than two. Anisotropy of mosaic spread is introduced in both the Coppens-Hamilton (C.H.) and Thornley-Nelmes (T.N.) descriptions, with both a Lorentzian or a Gaussian distribution function. The formalisms are applied to neutron data sets on LiTbF4 (100°K and 300°K), tetracyanoethylene and LiOH. H2O, and an X-ray data set on α-deutero oxalic acid dihydrate. The distinction between type I and type II crystals is quite clear on the basis of a comparison of R values. Only for LiF, which was studied earlier, was extinction dominated by particle size. In all other cases the best fit corresponds to mosaic-spread-dominated extinction, with a Lorentzian shape of the distribution function. This is especially clear when partial R values summed over the severely extinction-affected reflections are compared. The new formalisms are further supported by the consistency of the final parameters among various refinements in which the most severely extinction-affected reflections are eliminated. Simultaneous refinement on both the particle size and the mosaic spread was only successful in the earlier studied case of SrF2. The T.N. description of anisotropy leads to lower partial R values, in agreement with physical arguments supporting the validity of this distribution.
238 citations
TL;DR: In this article, phase transitions in WO3 crystals were studied by Raman and absorption spectroscopy, and two stable (or metastable) modifications at room temperature, differing slightly but definitely in their physical properties.
Abstract: A study of the phase transitions in WO3 crystals by Raman and absorption spectroscopy reveals that WO3 possesses two stable (or metastable) modifications at room temperature, differing slightly but definitely in their physical properties. Measurements of electrical conductivity show that both are insulators at low temperatures and semiconductors at higher temperatures. One is monoclinic and the other triclinic. Both exhibit at lower temperatures discontinuities in their physical properties without accompanying changes in cell parameters.
194 citations
TL;DR: In this paper, the variance of diffractometer-collected diffraction intensities is discussed in terms of instrumental instability and uncertainties in the parameter used to bring the intensities to a common scale.
Abstract: The variance of diffractometer-collected diffraction intensities is discussed in terms of instrumental instability and uncertainties in the parameter used to bring the intensities to a common scale. It is shown that the inconsistent and/or divergent behavior of the reflections used as standards contributes in a major way to the uncertainty in the scaling parameter and can often account for the largest portion of the variance in excess of the Poisson contribution for reflections with large intensities.
143 citations
TL;DR: In this article, the normal coordinates of tetragonal WO3 have been calculated and a correlation of the phase transitions with the critical behaviour of some of the normal modes has been attempted.
Abstract: The Raman spectrum of WO3 has been investigated from -180 to 290°C. An interpretation of the phonon spectra of the phases with higher symmetries is carried out by group theory. The normal coordinates of tetragonal WO3 have been calculated and a correlation of the phase transitions with the critical behaviour of some of the normal modes has been attempted.
142 citations
TL;DR: In this paper, the simple infinite-aperture charge-density approximation to lattice imaging is extended to include the effects of finite aperture and objective-lens spherical aberration.
Abstract: The simple infinite-aperture charge-density approximation to lattice imaging is extended to include the effects of finite aperture and objective-lens spherical aberration. Images computed using these approximations are compared with images computed by the accurate N-beam technique and with experiment in order to establish the bounds of applicability of the extended-charge-density approximations.
127 citations
TL;DR: In this article, a simple formulae for the interaction energy between two molecules have been used for writing a program which evaluates the total interaction energy of the molecules in a crystal.
Abstract: Simple formulae for the interaction energy between two molecules have been used for writing a program which evaluates the total interaction energy of the molecules in a crystal. These formulae appear as sums of atom-atom and, eventually, atom-bond and bond-bond contributions. The non-additivity of the polarization energy is taken into account, and a rough estimate of the third-order non-additive terms ('triple dipole') is introduced. A suitable modification of the formulae for short interatomic distances allows us to treat hydrogen-bond interactions as well. We present results for the crystals of CH4, CO2, C6H6, and C6H5NO2. The energies calculated for the experimental geometry are in good agreement with experiment. For CO2 and C6H5NO2 minimizations of the computed energy (with respect to unit cell parameters and orientation and position of one molecule in the cell) were performed and it was found that the experimental configuration actually was very close to a minimum. The configurations of neighbour molecules in the crystal are compared with the optimal configuration of a binary complex, and it appears that, for non-hydrogen-bonded molecules, significant differences between these configurations may occur. Finally, for nitrobenzene several local minima seem to exist on the energy hypersurface; the minimum corresponding to the known experimental geometry appears to be the lowest, but only by a small amount.
118 citations
TL;DR: In this article, the negative quartet figure of merit, NQEST, is defined in conjunction with fast multi-solution tangent formula techniques and applied to both known and unknown crystal structures.
Abstract: Recent theoretical advances in the identification of those cosine invariants cos (ϕh + ϕk + ϕ1 + ϕm) which are probably negative suggest algorithms for the calculation of a figure of merit which is sensitive to the integrity of a phase set The negative quartet figure of merit, NQEST, defined here is of particular utility in conjunction with fast multi-solution tangent formula techniques Development of the methods and applications to both known and unknown crystal structures are presented
82 citations
TL;DR: In this article, a technique of local scaling is proposed to reduce systematic errors between sets of isomorphous replacement and anomalous scattering data, which can be used to analyse a data set for significant systematic errors, and to determine whether or not local scaling will be worth applying in a given situation.
Abstract: Systematic errors between sets of isomorphous replacement and anomalous scattering data may in some cases be reduced by a technique of 'local scaling'. Methods are proposed which may be used to analyse a data set for significant systematic errors, and to determine whether or not 'local scaling' will be worth applying in a given situation. The proposed method also predicts the optimum number of reflections to be included in the determination of the local scale factors. The technique is illustrated by several examples including 'good' and 'bad' isomorphous-replacement data for α-chymotrypsin, and anomalous scattering data for the lysozyme from bacteriophage T4, and for cytochrome b562 from Escherichia coli.
79 citations
TL;DR: In this paper, the procedure for computing the resolution function of triple-axis spectrometers is reformulated to make allowance for the spatial configuration of the experimental set-up and for the curvature of the monochromator and analyser crystals.
Abstract: The procedure for computing the resolution function of triple-axis spectrometers is reformulated to make allowance for the spatial configuration of the experimental set-up and for the curvature of the monochromator and analyser crystals. Simple formulae are given for computing the resolution function of conventional instruments as well as of spectrometers with focusing crystals.
TL;DR: The theoretically calculated incoherently scattered X-ray intensities of Cromer & Mann [J. Phys. as mentioned in this paper ] for the elements 2 to 95 and 102 have been fitted to an analytic function.
Abstract: The theoretically calculated incoherently scattered X-ray intensities of Cromer & Mann [J. Chem. Phys. (1967). 47, 1892-1893] and Cromer [J. Chem. Phys. (1969). 50, 4857-4859] for the elements 2 to 95 and 102 have been fitted to an analytic function. The coefficients of fit are tabulated and the accuracy of the fitting is discussed. The accuracy is very good for sin θ/λ ≤ 1.5 A-1.
TL;DR: In this paper, the ratio of the two lattice parameters without the need for a precise knowledge of the X-ray wavelength was given by using pseudo non-dispersive MIMO diffractometry with Mo Ka 1 radiation.
Abstract: The d spacing of the 355 reflexion in silicon has been compared with the d spacing of the 800 reflexion in germanium using pseudo non-dispersive multiple-beam X-ray diffractometry with Mo Ka1 radiation. This technique gives the ratio of the two lattice parameters without the need for a precise knowledge of the X-ray wavelength. Symmetric transmission geometry was used to eliminate the refractive index correction. The results were: {{d(800 Ge)}\over{d(355 Si)}} = 1.0002348 (± 0.0000006) at 22.5 °C and {{d(800 Ge)}\over{d(355 Si}} = 1.0002458 ( ± 0-0000016) at 25 °C. By using the known lattice parameter of silicon obtained by X-ray and optical interferometry it was found that the lattice parameter of germanium at 25 °C is 5.6579060 ± 0.0000092 A.
TL;DR: In this paper, symmetry rules are derived for relating convergent-beam electron-diffraction pattern symmetries to the three-dimensional symmetry of the structure of 2M biotite.
Abstract: Symmetry rules are derived for relating the convergent-beam electron-diffraction pattern symmetries to the three-dimensional symmetry of the structure. These rules are applied to a study of 2M biotite. As a result the space group is determined as C2, or number 5 in International Tables for X-ray Crystallography, whereas it was previously held, in the absence of accurate data, to belong to C2/c or number 15. In addition, a method is described for uniquely determining the centrosymmetry of a crystal independently of space group, hence resolving the ambiguity between a centre of symmetry and a twofold axis present when approximations to dynamic electron scattering are used which take into accoufit only the symmetry of the projection.
TL;DR: In this article, a table of values of A* was prepared, and simple equations were obtained for interpolation between values in the table to an accuracy of 0.1% or better.
Abstract: Accurate X-ray absorption correction factors A* for cylindrical samples were calculated in the range of the product of linear absorption coefficient μand cylinder radius R of 0 ≤ μR ≤ 2.5 and in the range of Bragg angle θ of 0° ≤ 0 ≤ 90°. A method of double numerical integration that is different from the method usually used resulted in a maximum error of 0.1% with the typical error being much less. A table of values of A* was prepared, and simple equations were obtained for interpolation between values in the table to an accuracy of 0.1% or better. A full-range curve-fitting procedure that is accurate to 0.1% also was developed for routine use when many values of A* must be calculated for a single sample.
TL;DR: In this article, high-resolution single-crystal electron diffraction intensity data from fully hydrated, flat, rectangular catalase microcrystals are subjected to various tests which support the thesis that diffraction from crystals with thicknesses up to at least 1500 A may be treated by the kinematical theory.
Abstract: High-resolution single-crystal electron diffraction intensity data from fully hydrated, flat, rectangular catalase microcrystals are subjected to various tests which support the thesis that diffraction from crystals with thicknesses up to at least 1500 A may be treated by the kinematical theory. The ratio of the total diffracted intensity to the incident beam intensity is found to be 0.06 for a 530 A thick crystal and 0.16 for a 1500 A thick crystal. A plot of averaged electron-diffraction intensities vs. sin θ/λ also closely resembles molecular-transform-modulated plots of X-ray diffraction intensity data from protein crystals.
TL;DR: In this paper, the crystal structures of benzene I (orthorhombic) and high-pressure benzene II (monoclinic) were calculated at atmospheric pressure and 25 kbar pressure by minimization of the intermolecular lattice energy subject to the pressure constraint.
Abstract: The crystal structures of benzene I (orthorhombic) and high-pressure benzene II (monoclinic) were calculated at atmospheric pressure and 25 kbar pressure by minimization of the intermolecular lattice energy subject to the pressure constraint. The first set of calculations used an atom-atom (exp-6) potential with no net coulombic charges on the atoms. The second set of calculations used an (exp-6-1) potential which placed charges of ± 0.18 electron on each atom of the benzene molecule. The no-charge calculation incorrectly predicted that benzene should be monoclinic at both atmospheric and 25 kbar pressure. The coulombic charge calculation correctly predicted the observed crystal transaction, favoring the orthorhombic structure at atmospheric pressure but favoring the monoclinic structure at 25 kbar. In the latter calculations there were substantial coulombic contributions to the lattice energy.
TL;DR: In this article, the authors computed lattice images with increasing incident-beam divergence with spherical aberration taken into account, where the outer diffracted beams transmitted by the objective aperture do not contribute to the image, so that the resolution is effectively less than expected from the aperture size.
Abstract: Lattice images were computed with increasing incident-beam divergence with spherical aberration taken into account. The effect is as if the outer diffracted beams transmitted by the objective aperture do not contribute to the image, so that the resolution is effectively less than that expected from the aperture size. Images of Nb12O29, calculated with the inclusion of this effect, show improved agreement with experimental images.
TL;DR: In this paper, a new analytic approximation to atomic incoherent X-ray scattering intensities is proposed, which has the correct asymptotic behaviour at both large and small values of s. Fits to the incoherent intensities calculated by Cromer are presented for all atoms from He through Am.
Abstract: A new analytic approximation to atomic incoherent X-ray scattering intensities is proposed. Unlike other approximations in the literature, the present function has the correct asymptotic behaviour at both large and small values of s. Fits to the incoherent intensities calculated by Cromer are presented for all atoms from He through Am.
TL;DR: In this article, the pairwise potential method was applied to the study of molecular reorientations in organic crystals, and several benzene and naphthalene derivatives were considered, along with some simple heterocycles and a few more complex compounds.
Abstract: The pairwise potential method was applied to the study of molecular reorientations in organic crystals. Several benzene and naphthalene derivatives were considered, along with some simple heterocycles and a few more complex compounds. The results of the calculations have been matched, where possible, with the results of X-ray analysis and other experimental data. The good performance of the method is considered encouraging in view of its application to the study of more complex solid-state processes.
TL;DR: In this article, the structure analysis of the type 2H with space group P{bar 3}m1 has revealed that the z coordinate of the I ion is 0.268 and the thermal vibration of the Pb ion in its own layer is considerably smaller as compared with that perpendicular to the layer.
Abstract: PbI2 single crystals grown in gel and grown from vapour were studied by X-ray diffraction. The structure analysis of the type 2H with space group P{\bar 3}m1 has revealed that the z coordinate of the I ion is 0.268 and the thermal vibration of the Pb ion in its own layer is considerably smaller as compared with that perpendicular to the layer. From the dependence of the frequencies of types 2H and 12R[13]3 on temperature of crystal growth, it has been shown that type 2H is the most common type at room temperature and type 12R[13]3 the most common type at high temperatures. Observations on the 2H-12R transformation by annealing have revealed that this transformation is reversible.
TL;DR: In this article, the presence of Wadsley defects of the type suggested by Chisholm has been confirmed in tremolite, amosite and anthophyllite, and these defects have been shown to correspond to intercalation of lamellae of triple-chain structure parallel to (010).
Abstract: Fibrous specimens of tremolite, amosite, crocidolite and anthophyllite have been studied by electron diffraction and high-resolution microscopy. Amosite (and to a lesser extent crocidolite) are shown to be subject to frequent twinning on (100) leading to stacking disorder, and an explanation is provided of previously supposed anomalies in their diffraction patterns [Chisholm (1973), J. Mater. Sci. 8, 475-483]. Tremolite is sensibly free from these defects and this is explained in terms of crystal chemistry. The presence of Wadsley defects of the type suggested by Chisholm has been confirmed in tremolite, amosite and anthophyllite, and these defects have been shown to correspond to intercalation of lamellae of triple-chain structure parallel to (010). In anthophyllite these lamellae are observed to segregate into domains.
TL;DR: The phase-Q structures of Na0.98K0.02NbO3 and Na 0.10Nb O3 have been determined at room temperature by the neutron powder profile refinement technique as mentioned in this paper.
Abstract: The phase-Q structures of Na0.98K0.02NbO3 and Na0.90K0.10NbO3 have been determined at room temperature by the neutron powder profile refinement technique. Two different models were refined: orthorhombic, space group P21ma (No. 26, in cab setting) and monoclinic, space group Pm (No. 6, in the second setting). The latter is more correct as it takes into account the splitting of the monoclinic ap and cp, but the structures can be described as very nearly orthorhombic. In the 2 mole % structure Nb atoms are all displaced by 0.18 (1) A in the same direction towards the midpoint of the edge of the surrounding octahedron. The octahedra themselves appear to be regular and are tilted by 6.8 (1)° around ap and cp and by 7.8 (1)° around bp. Na, K displacements [0.23 (4) A] are almost parallel to those of Nb. For the 10 mole % structure Na, K and Nb displacements are the same but the tilts (5.2° and 7.3°) are somewhat smaller.
TL;DR: In this article, general equations for the diffuse scattering due to local ionic arrangements and displacements in systems with more than one ion per lattice point are presented, up to fourth-order terms are included.
Abstract: General equations are presented for the diffuse scattering due to local ionic arrangements and displacements in systems with more than one ion per lattice point. Up to fourth-order terms are included. These are placed in a form easy to evaluate in specific cases; examples are given of a solid-solution oxide, and an oxide with vacant sites. Methods for employing these equations for the separation of the various contributions are discussed.
TL;DR: In addition to the classical experimental methods of crystal diffraction, a number of new methods have become available using the time-of-flight technique for neutrons and energy-dispersive detectors for X-rays as discussed by the authors.
Abstract: In addition to the classical experimental methods of crystal diffraction a number of new methods have become available using the time-of-flight technique for neutrons and energy-dispersive detectors for X- rays It is shown that there are simple relations between the formulae for the integrated intensities of the different methods, and that the intensity formulae for all the methods can be simply generated, provided that one of them is derived in the usual way Formulae are given for the powdered crystal and the ideally imperfect crystal in the kinematical approximation as well as for the large perfect crystal in the framework of the dynamical theory
TL;DR: In this article, a least square refinement procedure is developed in which different groups of parameters are defined with respect to different axial systems, and the parameters are refined by describing the reciprocal-lattice vector for a given reflexion for each refinable parameter.
Abstract: A least-squares refinement procedure is developed in which different groups of parameters are defined with respect to different axial systems. The parameters are refined by describing the reciprocal-lattice vector for a given reflexion with respect to the relevant axial system for each refinable parameter. With this procedure features of the structure can be refined in the space most suitable for their description and correlations between parameters are either reduced or at least more readily understood. Constrained refinement is much simpler algebraically and can be made to be extremely flexible.
TL;DR: FeSiF6 6H2O shows extra reflexions inconsistent with the space group R{bar 3}m for T > 300K, and MgSi-F6.6H 2O whose structure is similar to that of MnSiF2O as mentioned in this paper.
Abstract: FeSiF6 6H2O shows extra reflexions inconsistent with the space group R{\bar 3}m. The same conclusion applies, for T> 300K, to MgSiF6.6H2O whose structure is similar to that of MnSiF6.6H2O. These three compounds undergo similar phase transformations with decreasing temperature.
TL;DR: In this article, the possible polytypic structures of a mica monocrystal are generated, using a single screw dislocation emerging on both (001) faces, taking into account: (1) each basic structure the crystallite may have during its previous layer-by-layer growth, (2) the number and position of the elementary layers of the crystal when the spiral becomes active, and (3) the Burgers vector strength of the dislocation.
Abstract: The possible polytypic structures of a mica monocrystal are generated, using a single screw dislocation emerging on both (001) faces. The discussion takes into account: (1) each basic structure the crystallite may have during its previous layer-by-layer growth, (2) the number and position of the elementary layers of the crystal when the spiral becomes active, (3) the Burgers vector strength of the dislocation. Deduced new polytypic structures, belonging to different structural series, are described and their positions with respect to the different basic structures are discussed. Some well-known growth features in micas such as 'epitaxic overgrowths' and 'macroscopic' twinning may, in some cases, be connected with a spiral growth mechanism. The layer-stacking sequences of complex polytypic structures of natural and synthetic micas reported thus far in the literature can be compared with the predictions of the screw dislocation theory of polytypism on perfect or more or less faulted basic structures.
TL;DR: In this article, potential function parameters for CN, HN, and NN non-bonded atom-atom pair interactions were derived from the heats of sublimation of cyanogen and dicyanoacetylene and from the crystal structures of the latter two isomers.
Abstract: Potential function parameters for CN, HN, and NN non-bonded atom-atom pair interactions were derived from the heats of sublimation of cyanogen and dicyanoacetylene and from the crystal structures of cyanogen, dicyanoacetylene, tetracyanoethylene, s-tetrazine, pteridine and pyridazino[4,5-d]pyridazine. Procedure and CC, CH, and HH parameters used were those of Williams [J. Chem. Phys. (1967). 47, 4680-4684]. The exponents of CN, HN, and NN repulsion terms were taken equal to those of CC, CH, and CC repulsion terms, respectively. The lattice summation was cut off at 6, 5.5 and 6 A for the CN, HN, and NN terms, respectively. The coefficients of the attractive and repulsive terms were fitted by weighted least-squares calculations to 45 observational equations. It proved to be necessary to estimate externally the parameters of the HN attractive term. The derived parameters show rough agreement with the values found by other investigators. The experimental values of the heats of sublimation of cyanogen and dicyanoacetylene are reproduced within 4% with the calculated parameters. The parameters predict reasonable values for the heats of sublimation of tetracyanoethylene, s-tetrazine, pteridine and pyridazino[4,5-d]pyridazine. A difference of about 6 kcal mole-1 between the heats of sublimation of the latter two isomers was calculated.
TL;DR: In this paper, simple electron-microscopy techniques are described which allow one to detect the presence of two enantiomorphous forms of a structure within an apparent single crystal.
Abstract: Simple electron-microscopy techniques are described which allow one to detect the presence of two enantiomorphous forms of a structure within an apparent single crystal. The first method consists of a characterization of the interface between the two enantiomorphs. In the second method advantage is taken of violations of Friedel's law which can occur in non-centrosymmetric crystals. These techniques have been illustrated by an analysis of the domain structure in ordered LiFe5O8, which has a space group P4132 or P4332. Consistent results were obtained with both methods. The first method yields a more complete description of the domain structure. Methods which can be used to determine the absolute configuration of the structure in a part of the crystal are discussed.