Showing papers in "Acta Crystallographica Section A in 1976"
TL;DR: The effective ionic radii of Shannon & Prewitt [Acta Cryst. (1969), B25, 925-945] are revised to include more unusual oxidation states and coordinations as mentioned in this paper.
Abstract: The effective ionic radii of Shannon & Prewitt [Acta Cryst. (1969), B25, 925-945] are revised to include more unusual oxidation states and coordinations. Revisions are based on new structural data, empirical bond strength-bond length relationships, and plots of (1) radii vs volume, (2) radii vs coordination number, and (3) radii vs oxidation state. Factors which affect radii additivity are polyhedral distortion, partial occupancy of cation sites, covalence, and metallic character. Mean Nb5+-O and Mo6+-O octahedral distances are linearly dependent on distortion. A decrease in cation occupancy increases mean Li+-O, Na+-O, and Ag+-O distances in a predictable manner. Covalence strongly shortens Fe2+-X, Co2+-X, Ni2+-X, Mn2+-X, Cu+-X, Ag+-X, and M-H- bonds as the electronegativity of X or M decreases. Smaller effects are seen for Zn2+-X, Cd2+-X, In2+-X, pb2+-X, and TI+-X. Bonds with delocalized electrons and therefore metallic character, e.g. Sm-S, V-S, and Re-O, are significantly shorter than similar bonds with localized electrons.
51,997 citations
TL;DR: In this paper, a simple procedure is derived which determines a best rotation of a given vector set into a second vector set by minimizing the weighted sum of squared deviations, which is generalized for any given metric constraint on the transformation.
Abstract: A simple procedure is derived which determines a best rotation of a given vector set into a second vector set by minimizing the weighted sum of squared deviations. The method is generalized for any given metric constraint on the transformation.
2,843 citations
TL;DR: In this article, the success of this separation can be tested by comparison of the vibration ellipsoids of bonded atom pairs, which should have equal amplitudes in the bond direction.
Abstract: The frequent practice of combining X-ray and neutron diffraction for distinguishing vibrational smearing from charge deformation due to chemical bonding is neither necessary nor completely satisfactory. The two effects occur principally in distinct regions of real and reciprocal space. They can be distinguished by X-ray data alone provided these extend to reciprocal radii d* > 2 A- 1 and provided the refinement program allows explicitly for the bonding deformation. The success of this separation can be tested by comparison of the vibration ellipsoids of bonded atom pairs, which should have equal amplitudes in the bond direction. Application of this 'rigid-bond' test to four recent refinements using the charge deformation model shows the vibration parameters to be largely unbiased, as those from parallel spherical-atom refinements are not. Hydrogen vibration parameters cannot be derived from X-ray data because of large deformation densities at the nuclei; nor do they satisfy the rigid-bond postulate.
675 citations
TL;DR: In this article, a bond-valence analysis of the repulsion leads to the conclusion that strong and weak hydrogen bonds are different in kind, the stronger ones involve strain and are linear while the weaker ones (O-O greater than 2.7 A) have an extra degree of freedom and are generally bent.
Abstract: The asymmetry of hydrogen bonds arises from the repulsion between the O atoms forming the bonds. A bond-valence analysis of the repulsion leads to the conclusion that strong and weak hydrogen bonds are different in kind, the stronger ones (O-O less than 2.7 A) involve strain and are linear while the weaker ones (O-O greater than 2.7 A) have an extra degree of freedom and are generally bent. The strength of the hydrogen bond is determined by a number of factors such as the requirement that the bond valences around each atom add up to the atomic valence, a tendency for the O-O distance to be close to 2.7 A, and by crystal-packing considerations which often lead to the formation of bent, and hence weaker, hydrogen bonds. The bond-valence analysis correctly predicts the observed correlations between H⋯O distance and O-H-O angle. The frequency with which various hydrogen-bond configurations are observed in crystals is used to propose a method for determining hydrogen-bond energies. This analysis of hydrogen bonding leads to an understanding of the lengthening of hydrogen bonds in high-pressure ices and to proposals for hydrated ion structures which can be used, for example, to predict the acid strengths of anions and to show that in neutral aqueous solutions the oxygen atoms of complex anions each hydrogen-bond to two or three water molecules.
462 citations
TL;DR: In this paper, a real-space averaging of electron density maps is proposed to estimate phase information from the resulting structure, and combine it with that given by isomorphous replacement to produce the phases to be used in the next iteration.
Abstract: Phases can be determined for geometrically redundant amplitudes by iteration of the following procedure: compute an electron density map from the currently available phases; average the electron densities of all the crystallographically independent molecules; rebuild the crystal(s) from this averaged subunit, setting the density outside the molecular boundaries to its average value; obtain phase information from the resulting structure, and combine it with that given by isomorphous replacement to produce the phases to be used in the next iteration. This algorithm converges very rapidly, and has proved to be a powerful tool in the solution of two large unknown protein structures. This paper describes the computational techniques developed to implement it, which include a swift and general method for real-space averaging of electron density maps.
370 citations
TL;DR: In this article, the authors proposed a valence density multipole model for pseudoatoms restricted to single exponential radial functions, which can be used for determination of static charge physical properties and aspects of chemical bonding.
Abstract: The one-electron density function for a group of atoms within the asymmetric region of a unit cell is represented by a finite multipole expansion of the charge density about each atomic center. Each atomic expansion is called a pseudoatom. If the pseudoatom charge density is effectively rigid with nuclear motion, then the model may be used for a static charge density analysis of X-ray diffraction data. A valence density multipole model for pseudoatoms is restricted to single exponential radial functions. The representation is rotationally invariant. The model may be used for determination of static charge physical properties as well as aspects of chemical bonding. These results can be a critical test of the X-ray diffraction experiment for the determination of electron density distributions. The pseudoatoms discussed are primarily intended for crystals comprised of first and second-row atoms. The valence scattering model demands extensive data sets (probably at low temperatures) or an independent determination of atomic positions and mean square amplitudes of vibration.
255 citations
TL;DR: In this article, measurements of lattice parameters and intensities of difference reflexions have been made for all compositions of Nal - xKxNbO3, both at room temperature and above.
Abstract: Measurements of lattice parameters and intensities of difference reflexions have been made for all compositions of Nal - xKxNbO3, both at room temperature and above. Structural models are proposed for each phase found.
168 citations
TL;DR: In this paper, the transition state in binary f.c. based systems is characterized by the repetition of interpenetrating polyhedral clusters of atoms occupying lattice sites and having the macroscopic composition.
Abstract: The 'transition' state in binary f.c.c. based systems is assumed to be characterized by the repetition of interpenetrating polyhedral clusters of atoms occupying lattice sites and having the macroscopic composition. Its mathematical formulation leads to relations between the short-range order (SRO) coefficients, which in turn impose conditions on the distribution in reciprocal space of the diffuse intensity as observed in electron diffraction patterns. It follows that for a given cluster type the diffuse intensity can only differ from zero along certain surfaces or curves. The effect of deviations from the ideal composition of the clusters is briefly discussed. The theory allows an explanation of available diffraction data on certain binary alloys in the initial stages of ordering i.e. in the 'transition' state between the SRO state and the long-range order (LRO) state.
162 citations
TL;DR: In this article, a rapidly converging method for refining approximate atomic models is presented, which combines the conditional structure-factor least-squares procedure described by Waser [Acta Cryst. (1963) with the conjugate gradient method for solving linear systems.
Abstract: A rapidly converging method for refining approximate atomic models is presented. It combines the conditional structure-factor least-squares procedure described by Waser [Acta Cryst. (1963). 16, 1091-1094] with the conjugate gradient method for solving linear systems [Hestenes & Stiefel, J. Res. Natl. Bur. Stand. (1952). 49, 409-436]. The method allows simultaneous variation of all of the structural parameters, although less than 1% of the derivative matrix need be calculated and stored for large systems and less than {{1}\over{64}}th of the diffraction data accessible with Cu radiation need be used. Applications involving a 240 atom mineral and an 812 atom protein are mentioned.
152 citations
TL;DR: In this article, the nature of the bonding in pyrite is considered in relation to the origin of the quadrupole splitting observed by means of the Mossbauer effect and the angular variation of the recoil-free fraction.
Abstract: The nature of the bonding in pyrite is considered in relation to the origin of the quadrupole splitting observed by means of the Mossbauer effect and the angular variation of the recoil-free fraction. It is found that neither crystal-field effects nor non-stoichiometry are enough to account for the observed splitting. Consideration of molecular orbital effects, however, shows that a very small amount of electron delocalization is enough to cause the observed splitting because of the strong effects the valence electrons have on the iron nucleus. It is also found that, contrary to previously published data, the recoil-free fraction is practically independent of angle. This result is derived from both Mossbauer resonance measurements and an X-ray structure determination. The structure determination confirms earlier reports of interatomic distances and angles and also determines the anisotropic thermal parameters for both Fe and S atoms in naturally occurring pyrite. The temperature variation of the isomer shift is measured and interpreted to mean that vibrational modes corresponding to Raman modes are not excited in the pyrite lattice between 10 and 300 K. This complements published IR spectra which show no such modes between 190 and 660 cm-1 (equivalent to 274 and 950 K). The isomer shift of pyrite is found to be 0.074 ± 0.001 mm/s relative to Fe in Cu (source and absorber at room temperature) and the quadrupole splitting is 0.634 ± 0.006 mm/s at room temperature. The ratio of intensities of the 3/2 → ½ and ½ → ½ transitions of the Mossbauer spectra was typically 1.004 ± 0.011.
127 citations
TL;DR: In this paper, it was shown that two-phase domains are possible whenever the derived structure has lost translational symmetry operations, i.e., it has lost translations as well as other symmetry operations with respect to its higher-symmetry (parent) space group.
Abstract: Displacive phase transitions and phase transitions of the type disorder-order usually result in symmetry changes such that the space groups of both structures are related as group and subgroup. The subgroups of a space group always belong to one of the following three types: they have either the same primitive cell as the original space group (they are translationengleich), or they belong to the same crystal class (klassengleich), or they are general (allgemein), i.e. they do not belong to either one of the previous categories exclusively and thus have lost translations as well as other symmetry operations with respect to their higher-symmetry (parent) space group. Twin domains are expected whenever the derived structure belongs to a lower-symmetry crystal class. Antiphase domains are possible whenever the derived structure has lost translational symmetry operations. The usefulness of these considerations is demonstrated and discussed for several materials with derivative structures.
TL;DR: In this paper, an algorithm is proposed which enables one, starting from an arbitrary primitive cell of a three-dimensional Bravais lattice, to reach the Niggli form required for the lattice type determination.
Abstract: An algorithm is proposed which enables one, starting from an arbitrary primitive cell of a three-dimensional Bravais lattice, to reach the Niggli form requisite for the lattice type determination.
TL;DR: In this article, the difference in lattice parameter between the GaAs substrate and the aluminum-substituted epitaxic layer AlxGa1−xAs has been measured accurately on a double-crystal spectrometer for a series of compositions.
Abstract: Heterojunctions GaAs–AlxGa1−xAs involved in the elaboration of IR laser diodes have been studied. The difference in lattice parameter between the GaAs substrate and the aluminum-substituted epitaxic layer AlxGa1−xAs has been measured accurately on a double-crystal spectrometer for a series of compositions. These data coupled with radius of curvature determination have permitted calculation of the stress in the layer and the bulk lattice parameter of AlxGa1−xAs. characterization of the defects introduced during the liquid-phase epitaxy has been performed by X-ray topography.
TL;DR: In this paper, it was shown that the least square refinement of R2 is unbiased if the intensity used in calculating the usual weights is not Io but ⅓ (Io + 2Ic).
Abstract: Statistical fluctuations in counting rates etc., as well as defects in the structural model, can introduce bias in the estimation of parameters by least-squares refinements. Of the residuaIs in common use, only unweighted R2 = Σ(Io - Ic)2 is free from statistical bias. Order-of-magnitude estimates of the bias can be derived, but it seems better to avoid it by adjusting the weights. To the second order, refinement of R2 is unbiased if the intensity used in calculating the usual weights is not Io but ⅓ (Io + 2Ic). There seems to be no simple method of avoiding bias in R1.
TL;DR: In this article, the Ewald method is used to derive an expression for the Lorentz-factor dipole tensor sum in a form convenient for computation, and comparisons are made with the direct and plane-wise summation methods.
Abstract: General principles of the Ewald method for evaluating multipole lattice sums are reviewed The method is used to derive an expression for the Lorentz-factor dipole tensor sum in a form convenient for computation, and comparisons are made with the direct and plane-wise summation methods Expressions are also given for computing quadrupole and octopole sums by the Ewald method The effect of crystal symmetry on lattice sums is outlined; the number of independent sums relating different pairs of equivalent sublattices does not exceed the total number of such sublattices Numerical results are given for the dipole lattice sums of hydrogen cyanide, benzene, durene, anthracene and pyrene Quadrupole sums are given for cuprous chloride and pyrene, and octupole sums are given for hydrogen cyanide, benzene and anthracene For dipole lattice sums, the Ewald method converges much faster than direct summation; for higher multipole sums, the Ewald method has no special advantage in speed, but may prove convenient, especially when sums are required for strained lattices
TL;DR: In this paper, the variance and covariance in the deformation of a centrosymmetric crystal structure were derived for the estimation of the variance of electron density qo/k - qc.
Abstract: Expressions are derived for the estimation of the variance and the covariance in the deformation electron density qo/k - qc, for a centrosymmetric crystal structure. The uncertainty in the calculated density qc depends strongly on the distance to the nuclei. The uncertainty in the observed density qo is fairly constant, except in the vicinity of a symmetry element, where it depends on the same correlation function as the covariance between observed densities. This correlation function is itself dependent on the resolution λ/(2 sin θ)max, the choice of which is of basic importance in any electron density study. The variance-covariance matrix may be used in an averaging procedure of electron densities at chemically, but not crystallographically, equivalent points. The method is applied to the results of X-ray and neutron diffraction studies of chromium hexacarbonyl Cr(CO)6.
TL;DR: In this article, the short-range order parameters were obtained after correcting for the intensity due to atomic displacements in a more complete manner than in earlier studies, and the long-range oscillations in the interatomic potentials determined from the diffuse scattering data fit the Friedel potential only approximately.
Abstract: Absolute measurements of the diffuse intensity in a volume in reciprocal space have been made at six temperatures ranging from 2°C above Tc to 930°C. The Warren short-range order parameters were obtained after correcting for the intensity due to atomic displacements in a more complete manner than in earlier studies. As a result the short-range order parameters are considerably smaller than in these earlier investigations. The (previously disputed) specific heat anomalies in Cu3Au above Tc and the L12 phase have been shown to be associated with unusual changes in diffuse X-ray scattering vs temperature. From studies of the scattering distributions, computer simulation and pair potentials obtained with the Warren short-range order parameters, the anomaly at 600°C appears to be due to the disappearance of DO22-like fluctuations, whereas above 850°C, CuPt-like fluctuations develop. There are premonitory effects just above Tc; there is a large increase in the Debye-Waller factor, in the total intensity due to quadratic terms in atomic displacements, and an apparent change in the sign of average first-neighbor displacement. The long-range oscillations in the interatomic potentials determined from the diffuse-scattering data fit the Friedel potential only approximately. The electron-to-atom ratio in Cu3Au was found to be ~ 0.97, in agreement with results on Cu alloys by other methods.
TL;DR: A series of phases has been found in the tungsten bronze systems MxWO3 for x ≤ 0.10 when M = K,Rb, Cs, Tl.
Abstract: A series of phases has been found in the tungsten bronze systems MxWO3 for x ≤ 0.10 when M = K,Rb, Cs, Tl. The crystal structure of one of these was derived from a high-resolution electron micrograph (lattice image) and confirmed by X-ray diffraction analysis. It can be considered as an (ordered) intergrowth of slices of hexagonal tungsten bronze type with slabs of WO3 structure type. Different members of the series characterized by different thicknesses of the WO3 slabs have been observed. Members have been obtained only intermixed with others, never as single phases. Disorder in the form of varying widths of the two structure elements has been observed quite frequently.
TL;DR: In this article, the solution and refinement of a structure from measurements performed on a twin of class I are discussed, where the twin operation is contained in the Laue symmetry of the crystal and twin-related reflexions have equal intensities; in this case the set of intensities collected from a twin is indistinguishable from that collected on a single crystal.
Abstract: Twins by merohedry can be usefully divided in two classes. Class I: the twin operation is contained in the Laue symmetry of the crystal and twin-related reflexions have equal intensities; in this case the set of intensities collected from a twin is indistinguishable from that collected on a single crystal. Class II: at least one of the independent twin operations is contained in the lattice symmetry but not in the Laue symmetry of the crystal (twin-related reflexions differ in intensity); a knowledge of fractional volumes of crystals is necessary to solve the structure. The solution and refinement of a structure from measurements performed on a twin of class I are discussed.
TL;DR: In this paper, a method of adjusting the Cartesian coordinates to obtain a properly weighted fit to both the positions from the refinement and the molecular parameters is described, and the results of applications of the method to insulin are summarized.
Abstract: The formal refinement methods of least-squares adjustment or difference-map analysis give atomic positions in protein structures with standard deviations which are large compared with the standard deviations of accepted molecular dimensions. This paper describes a method of adjusting the Cartesian coordinates to obtain a properly weighted fit to both the positions from the refinement and the molecular parameters. The equations which have to be solved have many unknowns but few coefficients, and an effective iterative method can be used. The results of applications of the method to insulin are summarized.
TL;DR: A priori estimates of the suitability of a particular crystal for charge density analysis are discussed in terms of a simple criterion based on core electron concentration and a more detailed evaluation of the anticipated errors in the final deformation density maps as discussed by the authors.
Abstract: A priori estimates of the suitability of a particular crystal for charge density analysis are discussed in terms of a simple criterion based on core electron concentration and a more detailed evaluation of the anticipated errors in the final deformation density maps. Predictions are compared with the results obtained in a series of studies and the implications for the design of a charge density analysis are discussed. It is found that counting statistics has often been a major contributor to the standard deviation of the charge density.
TL;DR: The possibility that two arbitrary lattices, 1 and 2, have a coincidence-site lattice (CSL) in common is examined in this article, and the main result is that the density ratio, Σ2, of coincidence points to points of lattice 2 is equal to the least positive integer n such that nT and n||T||T-1 are integral matrices.
Abstract: The possibility that two arbitrary lattices, 1 and 2, have a coincidence-site lattice (CSL) in common is examined. Let T be the 3 × 3 matrix that maps a basis of lattice 1 onto a basis of lattice 2 and let ||T|| be the absolute value of its determinant. It may be assumed that ||T|| ≥ 1. There is a CSL if, and only if, T is rational. The main result is that the density ratio, Σ2, of coincidence points to points of lattice 2 is equal to the least positive integer n such that nT and n||T||T-1 are integral matrices. A basis for the CSL can be determined quickly if lattices 1 and 2 are related by a rotation.
TL;DR: In this paper, the neutron small-angle scattering of hen egg-white lysozyme chloride in solution has been determined and the experimental value of the radius of gyration is R = 13.8 A, in good agreement with the same quantity calculated on the basis of the crystal structure model of tetragonal Lysozyme.
Abstract: The neutron small-angle scattering of hen egg-white lysozyme chloride in solution has been determined. The scattering density of the solvent and accordingly its contrast with the dissolved particles has been varied by changing the H2O/D2O ratio of the solvent. The zero-angle scattering was derived and it was shown that the square root of the zero-angle intensity is proportional to the contrast. The experimental value of the radius of gyration is R = 13.8 A, in good agreement with the same quantity calculated on the basis of the crystal structure model of tetragonal lysozyme. The three basic scattering functions were derived from the experimental data and compared with the corresponding scattering pattern calculated from the coordinates of the X-ray structure. The results do not suggest a deviation of the conformation and structure of lysozyme in solution from the crystal structure. The spherical average of the overall dimensions has been separated from the internal structure. The influence of higher multipoles of the shape of the lysozyme molecule on the scattering curve is discussed.
TL;DR: In this paper, the diffuse X-ray scattering from a single crystal of disordered Cu-29.8 at room temperature was measured at the 110 reciprocal lattice point, in good agreement with the result previously obtained by electron diffraction.
Abstract: The diffuse X-ray scattering from a single crystal of disordered Cu-29.8 at. % Pd alloy was measured at room temperature. A fourfold splitting of the diffuse scattering due to short-range order was observed at the 110 reciprocal lattice point, in good agreement with the result previously obtained by electron diffraction. With the use of the method proposed by Boric & Sparks [Acta Cryst. (1971), A27, 198-201] the short-range-order parameters were determined by separating the other components of the diffuse scattering, i.e. size effect, Huang and thermal diffuse scattering, from the observed intensity data. It was revealed from this analysis that the order parameters beyond the 15th neighbour shell play an important role in characterizing this short-range-order diffuse scattering, although their magnitudes are small compared with those of the lower order parameters. Based on the ordering theory valid above the critical temperature the atomic-pair-interaction potential ratios were also obtained. It is shown that the pair-interaction potential is of long range and oscillates with distance. A comparison is made between the present result and the form of long-range oscillatory potential predicted by the conduction-electron screening model, and then the form of the Fermi surface is discussed.
TL;DR: In this article, single-crystal hk0 electron diffraction patterns from thin ( ≤ 240 A) rhomboid n-hexatriacontane (n-C36H74) crystals contain intensity data which are well fit by the commonly observed O⊥ methylene subcell phasing model but not by a true unit cell model which contains two mutually displaced monolayers.
Abstract: Single-crystal hk0 electron diffraction patterns from thin ( ≤ 240 A) rhomboid n-hexatriacontane (n-C36H74) crystals contain intensity data which are well fit by the commonly observed O⊥ methylene subcell phasing model but not by a true unit cell model which contains two mutually displaced monolayers. The apparent diffraction from a monolayer in these lamellar crystals is thought to be due to bend distortions of the crystal plate. Intensity data conform to a kinematical interpretation as a first approximation, thus allowing a priori structural elucidation, but eventually will require an n-beam dynamical correction.
TL;DR: In this article, the lateral arrangement of a proportion of molecules in the collagen fibril displays a short-range two-dimensional liquid-like order, and the theoretical model used is the integral equation formulation of Percus & Yevick [Phys. Rev.
Abstract: The idea that the lateral arrangement of a proportion of molecules in the collagen fibril displays a short-range two-dimensional liquid-like order is investigated and shown to be true. Theoretical calculations of a dense disordered assembly of hard discs yield X-ray scattering curves which possess the features of the near equatorial diffuse scattering from rat-tail tendon collagen. The theoretical model used is the integral equation formulation of Percus & Yevick [Phys. Rev. (1958). 110, 1] known to be almost exact for hard-disc potentials. The molecular diameter and number density of molecules obtained from the comparison with experiment are close to those suggested by Woodhead-Galloway, Hukins & Wray [Biochem. Biophys. Res. Commun. (1975). 64, 1237-1244] to explain the ordered part of the collagen and to those suggested by Katz & Li [J. Mol. Biol. (1973). 73, 351-369] on the basis of an investigation of the density of wet tendon. A brief discussion of elastoidin is included. Discrepancies between the observed and predicted scattering are discussed.
TL;DR: A means is developed whereby coordinate sets for related molecules may be compared in such a way as to express rigorously the relationship between them in terms of position, orientation, homogeneous strain and curvature.
Abstract: A means is developed whereby coordinate sets for related molecules may be compared in such a way as to express rigorously the relationship between them in terms of position, orientation, homogeneous strain and curvature. It is thought that such curvatures may be of value in characterizing hinge regions of allosteric enzymes. The method is illustrated with examples taken from haemoglobin and myoglobin.
TL;DR: In this paper, the effect of multiple scattering on the nature of the electron-microscope images of graphitic specimens has been investigated by theoretical calculations within the systematics multislice approach.
Abstract: The effect of multiple scattering upon the nature of the electron-microscope images of graphitic specimens has been investigated by theoretical calculations within the systematics multislice approach. For perfect, infinite specimens, imaged under axial illumination, the results agreed closely with those predicted by a simple three-beam dynamical treatment. In all cases the weak-phase-object approximation was found to be invalid, even qualitatively, whenever specimen thicknesses exceeded 50 A. Calculations were also performed for finite crystal sizes and various plausible types of structurally imperfect graphites. The calculated images demonstrate the need for great care in interpreting experimental images of structurally defective graphitic solids. In particular, the positions of missing layers in a crystal of graphite can be 'filled' by spurious fringes under certain conditions of defocus, and fringes corresponding to layers of an intercalated species turn out to have ill defined positions. Characteristics such as these are equally prominent in all images, irrespective of the magnitude of multiple scattering, suggesting that these perturbations arise primarily from instrumental aberrations.
TL;DR: In this article, the relevant group-subgroup relationships are Pm3m ~ R3m and Pm 3m ~ Amm2, which are translation-engleiche subgroups of Pm-O~, which is also the space group of BaTiO3 at high temperature.
Abstract: b ~c ~_ 5\"7 A) (Landolt-B6rnstein, 1969). Neither space group is a subgroup of the other, however, both are subgroups of Pm3m-O~, which is also the space group of BaTiO3 at high temperature. Thus the relevant group-subgroup relationships are Pm3m ~ R3m and Pm3m ~ Amm2. Both, R3m and Amm2 are translationengleiche (though not maximal) subgroups of Pm3m and therefore only twin domains and no antiphase domains are allowed in the R3m and Amm2 forms of BaTiOa.
TL;DR: In this paper, a new formalism of the dynamical diffraction theory is proposed in order to take into account the statistical nature of lattice distortions, and the observed intensity is given by an ensemble average of the intensity field derived from the wave equations of Takagi-Taupin type.
Abstract: A new formalism of the dynamical diffraction theory is proposed in order to take into account the statistical nature of lattice distortions. It is postulated that the observed intensity is given by an ensemble average of the intensity field derived from the wave equations of Takagi-Taupin type. The formalism leads to exactly the same results as those of the conventional dynamical theory for ideally perfect crystals and the results of the conventional kinematical theory for ideally imperfect crystals. The general relations are established among the integrated diffraction powers for three types of incident wave; namely, a spherical wave (narrow beam), a plane wave and the wave from an incoherent homogeneous source (wide beam). This paper is a preparation for the following one on secondary extinction [Kato, Acta Cryst. (1976) A32, 458-466].