Showing papers in "Acta Crystallographica Section A in 1995"
TL;DR: A least-squares procedure is described for modeling an empirical transmission surface as sampled by multiple symmetry-equivalent and/or azimuth rotation-equ equivalent intensity measurements.
Abstract: A least-squares procedure is described for modeling an empirical transmission surface as sampled by multiple symmetry-equivalent and/or azimuth rotation-equivalent intensity measurements. The fitting functions are sums of real spherical harmonic functions of even order, ylm(− u0) + ylm(u1), 2 ≤ l = 2n ≤ 8. The arguments of the functions are the components of unit direction vectors, −u0 for the reverse incident beam and u1 for the scattered beam, referred to crystal-fixed Cartesian axes. The procedure has been checked by calculations against standard absorption test data.
7,395 citations
TL;DR: In this paper, the exact analytic method of evaluating the absorption during scattering in multifaceted convex crystals is developed in a way that permits efficient computation, and a fast and accurate algorithm is given for finding the Howells polyhedra whose determination is fundamental to the analytic method.
Abstract: The exact analytic method of evaluating the absorption during scattering in multifaceted convex crystals is developed in a way that permits efficient computation. A fast and accurate algorithm is given for finding the Howells polyhedra whose determination is fundamental to the analytic method. The algorithm allows for the evaluation of cases when the sample is only partly illuminated, can be adapted to more general situations and can be used to generate an estimate of the error in the computation. In most cases this is to 1 part in 1014. Results of standard tests are given to greater accuracy than previously available and results for multifaceted approximations to a cylinder and a sphere are given to illustrate the power of the method.
1,476 citations
TL;DR: In this paper, new analytical X-ray scattering-factor representations valid for the full range of sin Θ/λ from 0.0 to 6.0 were developed from fits of a linear combination of five Gaussians to the values of the scattering factors tabulated in International Tables for Crystallography (1992).
Abstract: New analytical X-ray scattering-factor representations valid for the full range of sin Θ/λ from 0.0 to 6.0 A−1 have been developed from fits of a linear combination of five Gaussians to the values of the scattering factors tabulated in International Tables for Crystallography (1992) [Vol. C. Dordrecht: Kluwer Academic Publishers]. The resulting functions for both neutral atoms and ions are compared with the existing parametrizations, which are applicable for 0.0–2.0 A−1 and 2.0–6.0 A−1, respectively. The quality of the new parametrization involving 11 parameters per atom (ion) compares well with the previous work or is even superior. Examples are discussed, some errors in International Tables for Crystallography, Vol. C are indicated and a warning is given that most of the previously published four Gaussian expansions for ions are inadequate for calculations involving sin Θ/λ ≥ 2.0 A−1.
536 citations
TL;DR: In this paper, the amplitudes of beams reflected from a crystal surface by high-energy electrons are expressed in terms of matrix operators based on Bloch waves, and the solution is derived in the limiting case of an infinite slab and is therefore applicable to cases involving overlayers of different composition and structure.
Abstract: The amplitudes of beams reflected from a crystal surface by high-energy electrons are expressed in terms of matrix operators based on Bloch waves. The solution is derived in terms of the limiting case of an infinite slab and is therefore applicable to cases involving overlayers of different composition and structure. The possibility of using a kinematical theory and its relationship to the full dynamical treatment are investigated. Calculations show that the method is limited by numerical problems but that it can be used to give a qualitative description of features in reflection high-energy electron diffraction.
248 citations
TL;DR: In this article, the topology of the electron density within the theory of atoms in crystals has been used to define primary and secondary interactions in terms of the topological properties of crystal lattices.
Abstract: The heavier halide molecules form layered crystals indicative of the presence of a specific directed intermolecular interaction. It is shown that this interaction within the crystal can be defined and characterized using the topology of the electron density within the theory of atoms in crystals. It is also shown that its presence in the crystal and the resulting geometry of the layered structure can be predicted in terms of the topology of the Laplacian distribution of an isolated Cl2 molecule, as it relates to the definition of Lewis acid and base sites within the valence shell of an atom. The generality of the definition of both primary and secondary interactions in terms of the topology of the electron density is demonstrated for all types of crystal. The electron density of solid molecular chlorine was determined by fitting the experimental X-ray structure factors and by theoretical calculation and its topology determined. Each Cl atom is found to be linked by bond paths, lines of maximum electron density, to twelve other atoms in the crystal: to four atoms in the same layer parallel to the bc plane, one of which defines the intramolecular bond of the Cl2 group, to six atoms in the four neighbouring molecules lying in the same stack parallel to the b axis and to two atoms in molecules situated in a neighbouring stack.
180 citations
TL;DR: In this article, a new atomic electron density function is derived by Fourier transformation of resolution-truncated atomic scattering factors, which forms the basis of a new real-space refinement method, RSthis article, that is a substantial improvement on prior implementations that did not formally consider the absence of high-resolution terms in a typical macromolecular electron density map.
Abstract: A new atomic electron-density function is derived by Fourier transformation of resolution-truncated atomic scattering factors. It forms the basis of a new real-space refinement method, RSREF, that is a substantial improvement on prior implementations that did not formally consider the absence of high-resolution terms in a typical macromolecular electron-density map. Real-space refinement is further improved through the simultaneous refinement of stereochemical restraints analogous to reciprocal-space methods. Parallel refinements of a viral capsid structure show that real-space refinement produces models that are at least as good as those refined in reciprocal space, by either restrained or molecular-dynamics methods, and that refinement cycles are ~50 times faster. Real-space refinement will not replace reciprocal-space methods for proteins, where, without the high noncrystallographic symmetry of viruses, experimental phases and electron-density maps are not of the same high quality. However, applied to local regions, it can be used to speed up and improve the quality of interactive model building before a full refinement is started.
153 citations
TL;DR: In this article, the validity of form-factor, modified-form-factor and anomalous-scattering-factor approximations in predictions of elastic photon-atom scattering is assessed with the aid of the state-of-the-art numerical calculation of Rayleigh scattering obtained using the second-order S-matrix theory, in the photon energy range from 100 eV to 1 eV.
Abstract: The validity of form-factor, modified-form-factor and anomalous-scattering-factor approximations in predictions of elastic photon-atom scattering is assessed with the aid of the state-of-the-art numerical calculation of Rayleigh scattering obtained using the second-order S-matrix theory, in the photon energy range from 100 eV to 1 MeV. A comparison is made with predictions from S-matrix theory in the same atomic model for representative low-Z (carbon, Z = 6) and high-Z (lead, Z = 82) elements to get a general idea of the validity of these simpler more approximate methods. The importance of bound–bound contributions and the angle dependence of the anomalous scattering factors is discussed. A prescription is suggested, with the assumption of angle independence, that uses simpler approaches to obtain the elastic scattering cross sections in the soft-X-ray regime at the level of accuracy of the S-matrix calculation, failing at large momentum transfers for high-Z elements. Predictions from this prescription are compared with experiment. With starting point the many-body elastic scattering amplitude, a detailed discussion is presented of the partition of the elastic scattering amplitude into Rayleigh and Delbruck scattering components. This partition of the optical theorem reveals contributions from bound–bound atomic transitions, bound pair annihilation and bound pair production that are not usually associated with elastic scattering. In the partitioned optical theorem for Rayleigh scattering, as in the many-body optical theorem for scattering from excited states, subtracted cross sections naturally appear. These terms are needed, in addition to the familiar terms for photoionization, to relate the real and imaginary parts of the scattering amplitude.
129 citations
TL;DR: In this article, a method of analysis of crystal space topology by means of Voronoi-Dirichlet tessellation is described, and examples of the application of the proposed method in crystal chemical analysis are given.
Abstract: The method of analysis of crystal space topology by means of Voronoi-Dirichlet tessellation is described. The possibilities of using Voronoi-Dirichlet polyhedra in the investigation of local and global geometrical/topological properties of the crystal lattice in structures of simple and complex substances are discussed. Examples of the application of the proposed method in crystal-chemical analysis are given.
120 citations
TL;DR: In this article, the use of a crenel function, i.e. a difference between two Heaviside functions of amplitude 1, for strong occupation modulation waves and its influence on the refinement of accompanying displacive modulation waves is discussed.
Abstract: The use of a crenel function, i.e. a difference between two Heaviside functions of amplitude 1, for strong occupation modulation waves and its influence on the refinement of accompanying displacive modulation waves is discussed. The basic set of harmonic functions that is usually employed for the modelling of the displacive modulation wave is no longer orthogonal on the interval where the crenel function takes the value 1. This causes severe correlations between different displacive modulation amplitudes during refinement. The best solution to prevent these correlations is to select functions for inclusion in the refinement according to the criterion that their generalized cosine to the subspace of already selected functions has to be smaller than a certain threshold value. A quality-of-selection parameter is used to estimate the completeness of the selected functions. Finally, the selected functions are orthogonalized. One artificial illustration and one real example are given to demonstrate the use and application of the proposed methods.
113 citations
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TL;DR: X-rays, which have played the leading role in imaging crystallizable materials, may also prove to be highly valuable in the imaging of very large non-crystalline structures.
Abstract: The subject of X-ray microscopy (high-resolution X-ray imaging of general nonperiodic structures), an area in which much progress has been made in recent years, is reviewed. The main techniques are briefly described. Achievable performance levels, which for many years were highly speculative, can now be understood with fair accuracy in terms of basic X-ray and specimen properties, and techniques have progressed to the point where actual results are nearing those levels. In terms of specimen size and imaging resolution, X-ray microscopies lie between electron and light microscopy, and are thus suited to imaging extremely large and complex structures; in addition, they demand little or no specimen preparation, and can be used to observe local composition and chemical state as well as structure. Thus X-rays, which have played the leading role in imaging crystallizable materials, may also prove to be highly valuable in the imaging of very large non-crystalline structures. Throughout the treatment, attention is paid to the relationships connecting the subject with X-ray crystallography.
103 citations
TL;DR: In this article, the distribution of structural classes (Structural classes) among organic homomolecular crystals has been investigated and 305 structural classes having very unequal frequencies have been discovered.
Abstract: Structural classes (SC) are very useful for the description and consideration of molecular arrangements in crystals. The distribution of 19642 organic homomolecular crystals among SC has been investigated. 305 SC having very unequal frequencies were discovered. A full list is given.
TL;DR: In this article, the authors describe the theory and results obtained with the correlation search technique to solve the translation problem in the molecular-replacement method, where the correlation function is expressed in terms of intensities of structure factors and is calculated by fast Fourier transforms.
Abstract: This paper describes the theory and results obtained with the correlation-search technique to solve the translation problem in the molecular-replacement method. The correlation function is expressed in terms of intensities of structure factors and is calculated by fast Fourier transforms.
TL;DR: In this article, the authors used convergent-beam electron diffraction (CBED) to refine crystal structure parameters (atom positions and Debye-Waller factors) using a small specimen area down to a few nm in diameter.
Abstract: The method to refine crystal structure parameters (atom positions and Debye–Waller factors) using convergent-beam electron diffraction (CBED), which is applicable to crystal structure analysis of a small specimen area down to a few nm in diameter, is studied. The line profiles of higher-order Laue-zone reflections are recorded using imaging plates. Theoretical intensities are calculated based on the dynamical theory of electron diffraction with the aid of the generalized Bethe approximation to shorten calculation time. The structural parameters are determined by fitting the experimental profiles with the theoretical ones using a nonlinear least-squares method. The present method has been applied to the low-temperature phase of SrTiO3. The structural parameter or the rotation angle of the oxygen octahedron has been determined to be \varphi = 1.12 (4)°, which shows good agreement with that obtained from electron spin resonance experiments.
TL;DR: The R-free methodology of Brunger as discussed by the authors applied to the likelihood-function calculation allows realistic phase-error estimates to be obtained for both unrefined and refined models, and these estimates may be used as an additional indicator in the refinement process.
Abstract: Reasonable assumptions about the statistical properties of errors in an atomic model lead to the probability distributions for the values of structure-factor phases. These distributions contain some generally unknown parameters reflecting how large the model errors are. These parameters must be determined properly to give realistic estimates of phase errors. Maximum-likelihood-based estimates suggested by Lunin & Urzhumtsev [Acta Cryst. (1984), A40, 269–277] are good for models not subjected to refinement but underestimate the errors when being used for refined models. The R-free methodology of Brunger [Nature (London), (1992), 355, 472–474] applied to the likelihood-function calculation allows realistic phase-error estimates to be obtained for both unrefined and refined models. These estimates may be used as an additional indicator in the refinement process.
TL;DR: Schwarzenbach, Abrahams, Flack, Gonschorek, Hahn, Huml, Marsh, Prince, Robertson, Rollett & Wilson as discussed by the authors presented the concepts of standard uncertainty, of combined standard uncertainty and of Type A and Type B evaluations of standard uncertainties.
Abstract: The Working Group has examined recent recommendations for evaluating and expressing uncertainty in measurement [Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization (ISO, 1993)]. The present publication updates an earlier report of the IUCr Subcommittee on Statistical Descriptors [Schwarzenbach, Abrahams, Flack, Gonschorek, Hahn, Huml, Marsh, Prince, Robertson, Rollett & Wilson (1989). Acta Cryst. A45, 63-75]. This new report presents the concepts of standard uncertainty, of combined standard uncertainty, and of Type A and Type B evaluations of standard uncertainties. It expands the earlier dictionary of statistical terms, recommends replacement of the term estimated standard deviation (e.s.d.) by standard uncertainty (s.u.) or by combined standard uncertainty (c.s.u.) in statements of the statistical uncertainties of data and results, and requests a complete description of the experimental and computational procedures used to obtain all results submitted to IUCr publications.
TL;DR: In this paper, a general case of incident radiation partially coherent in time and space is considered, where the time-delay effect of the diffracted radiation is described when the ultrashort time duration incident pulse strikes the crystal surface.
Abstract: The theory of time-dependent X-ray Bragg diffraction by crystals is developed on the basis of the Green-function (point-source) formalism. A general case of incident radiation partially coherent in time and space is considered. The time-delay effect of the diffracted radiation is described when the ultrashort time duration incident pulse strikes the crystal surface. The problem in question is closely connected with the effect of time delay in the resonance scattering of synchrotron radiation by nuclei in a crystal. It is found that, for the case where the incident wave is plane (or is an incoherent superposition of plane waves) and the time-dependent pulse is a pseudo δ function in time, the instantaneous crystal reflectivity is a smooth temporal function and tends to the value corresponding to the integrated reflectivity calculated by means of the conventional dynamical–kinematical X-ray diffraction theory. If the incident X-ray pulse profile is a pseudo 6 function in both time and space, the temporal crystal response has the same functional dependence as the spatial distribution of the diffracted intensity under the condition of conventional Bragg diffraction of the X-ray beam with lateral width < toc, where the time delay to is equal to Λ/2πc and (μoC)−1 in the cases of dynamical and kinematical X-ray scattering within a crystal, respectively (Λ is the X-ray extinction length, μo is the linear absorption coefficient and c is the velocity of light in vacuum).
TL;DR: In this article, the nuclear mean square thermal displacements have been analyzed assuming no coupling between the external (rigid body) and internal vibrations in which zero-point vibrational effects are apparent.
Abstract: Neutron diffraction data have been collected for hexamethylenetetramine (HMT) at 15, 50, 80, 120, 160 and 200K using a single crystal (mass 8.1 mg). The structure refinement at each temperature included two extinction parameters and third-order thermal parameters for the H nuclei. Extinction effects are very severe with extinction factors as small as 0.2Fkin2 for three reflections (800, 110 and 440). Application of the Sabine extinction theory indicates that the crystal domain size decreases from 115 microns at 200 K to 85 microns at 15 K. The half-width in the mosaic spread (7" of arc) is almost independent of temperature. An extinction model without phase correlations between mosaic blocks gives a slightly better fit to the diffraction data. The nuclear mean square thermal displacements have been analysed assuming no coupling between the external (rigid body) and internal vibrations. This gives mean square displacements for rigid-body vibration in which zero-point vibrational effects are apparent. The methylene H nuclei have internal vibrations approximately independent of temperature. At 200 K, the H nuclear vibrations have a small anharmonic component, but at temperatures below 160 K this becomes insignificant in terms of the experimental error.
TL;DR: In this article, a new and widely applicable technique for modeling single-crystal diffuse scattering has been developed, which is most useful for the study of disordered crystalline materials.
Abstract: The scattering from crystals has two components, Bragg and diffuse. In the case of disordered crystalline materials, or those at high temperature, the latter contribution is considerable and contains a great deal of information about any static or thermal disorder in the system. However, interpretation of this diffuse scattering is in general difficult. A new and widely applicable technique for modelling single-crystal diffuse scattering has been developed, which is most useful for the study of disordered crystalline materials. The algorithm, based on the reverse Monte Carlo method, is described in detail, and the information that can be obtained using it is discussed with reference to a study on ice Ih.
TL;DR: In this paper, an ab initio solution of the structure of [9,9′-bianthryl]-10-carbonitrile is presented using a routine application of the maximum entropy method combined with likelihood evaluation employing 150 unique diffraction intensities.
Abstract: Ab initio structure determination and refinement from electron diffraction data is not a widely used technique in structural science because of the inaccuracies inherent in the process of intensity measurement and because the relative sparseness of the data sets collected makes the structures hard to solve; there are also problems of verifying the correctness of the results. In this paper, the techniques of model building from electron diffraction data were employed to solve the structure. In addition, an ab initio solution of the structure of [9,9′-bianthryl]-10-carbonitrile is presented using a routine application of the maximum-entropy method combined with likelihood evaluation employing 150 unique diffraction intensities. The structure thus determined was obtained independently of the model-building studies. The agreement between the two methods is excellent and both agree with a single-crystal X-ray study on the same material. In addition, the high-resolution images agree with the images calculated from the model and with the potential maps after correction for the transfer function and dynamic scattering.
TL;DR: In this paper, the threading dislocation density of the epitaxic layer can be determined quantitatively using a Barrels five-crystal diffactometer by making use of known instrumental effects for this diffractometer.
Abstract: Recently, a general technique for the measurement for the threading dislocation densities in epitaxic semiconductors by high-resolution X-ray diffraction was reported [Ayers (1994). J. Cryst. Growth, 135, 71-77]. Here, this method has been extended to the case of a Barrels five-crystal diffactometer by making use of known instrumental effects for this diffractometer. The usefulness of the method has been demonstrated by application of the technique to epitaxic ZnSe grown on GaAs (001) by photo-assisted metalorganic vapor-phase epitaxy. It is shown that in this case the threading dislocation density of the epitaxic layer can be determined quantitatively. Evidence for the introduction of dislocations in the underlying GaAs substrate is also presented.
TL;DR: In this paper, it was shown that information of order six suffices to determine a structure uniquely, where six is the number of atoms in a set of equal atoms that can occupy positions on a circle subdivided into equally spaced markings.
Abstract: The three-dimensional configuration of crystallized structures is obtained by reading off partial information about the Fourier transform of such structures from diffraction data obtained with an X-ray source. We consider a discrete version of this problem and discuss the extent to which `intensity only' measurements as well as `higher-order invariants' can be used to settle the reconstruction problem. This discrete version is an extension of the study undertaken by Patterson in terms of `cyclotomic sets', corresponding to arrangements of equal atoms that can occupy positions on a circle subdivided into N equally spaced markings. This model comes about when the usual three-dimensional Fourier transform is replaced by a one-dimensional discrete Fourier transform. The model in this paper considers molecules made up of atoms with possibly different (integer-valued) atomic numbers. It is shown that information of order six suffices to determine a structure uniquely.
TL;DR: In this paper, a dynamical theory of atomic number sensitive image (or Z-contrast image) formed by thermal diffusely scattered (TDS) electrons is proposed based on first-principles considerations.
Abstract: Dynamical theories of atomic number sensitive image (or Z-contrast image) formed by thermal diffusely scattered (TDS) electrons are proposed based on first-principles considerations. `Exact' theories are derived for simulating images obtained either in scanning transmission electron microscopy (STEM) using an annular dark-field detector or in transmission electron microscopy (TEM) using an on-axis objective aperture under hollow-cone beam illumination. The atom thermal vibrations are described using lattice dynamics with consideration of phase correlations. The effects that are comprehensively covered in the theory include: dynamical diffraction of the beam before and after TDS, thickness-dependent beam broadening or channelling, Huang scattering from defect regions, coherence of the thermal diffusely scattered electrons generated from the atomic layers packed within the coherent length, multiphonon and multiple phonon excitations, and the detector geometry. Simplified theories have been derived from this unified approach under various approximations. It has been shown that the incoherent imaging theory is a much simplified case of the practical imaging condition, and can be applied only for qualitative image interpretation. The coherent length in the z direction varies with the change of atomic mass in the column. It is thus possible that the z coherence may disappear for heavy elements. Finally, the theory of Huang scattering in high-angle dark-field TEM imaging has been illustrated, and the theoretically expected results have been observed experimentally.
TL;DR: In this paper, Ewald's method of accelerated convergence was generalized to calculate the electrostatic potential of a cubic NaF crystal in which the atoms have overlapping spherical densities, and the algorithm was applied to the cubic Na-F crystal.
Abstract: Ewald's method of accelerated convergence [Ewald (1921). Ann. Phys. (Leipzig), 64, 253–287] is generalized to calculate the electrostatic potential of a crystal in which the atoms have overlapping spherical densities. The algorithm is applied to the cubic NaF crystal. The potentials at the Na and F nuclei are calculated for the free-ion model and for the results from a κ refinement of the experimental data of Howard & Jones [Acta Cryst. (1977), A33, 776–783]. The κ refinement indicates an incomplete charge transfer but gives an electrostatic energy close to that of the point-charge model with full charge transfer and a lattice energy that is in good agreement with the experimental value.
TL;DR: A description of 4-connected nets with one kind of vertex is given in this article, in which at least three of the shortest rings containing each pair of edges are 4-rings.
Abstract: A description is given of 4-connected nets with one kind of vertex in which at least three of the shortest rings containing each pair of edges are 4-rings. 21 such nets are identified and characterized topologically. Some correspond to well known zeolite structures, but most are believed to be new.
TL;DR: A fast algorithm is described that makes it possible to apply the holographic method to sizable crystallographic problems and can incorporate a 'target' electron density, making it similar to solvent flattening.
Abstract: The holographic method makes use of partially modeled electron density and experimentally measured structure-factor amplitudes to recover electron density corresponding to the unmodeled part of a crystal structure. This paper describes a fast algorithm that makes it possible to apply the holographic method to sizable crystallographic problems. The algorithm uses positivity constraints on the electron density and can incorporate a 'target' electron density, making it similar to solvent flattening. The potential for applying the holographic method to macromolecular X-ray crystallography is assessed using both synthetic and experimental data.
TL;DR: In this article, the symmetry properties of many-particle interatomic potentials are investigated within the framework of a generalized model of a multicomponent lattice gas, and the relations of the invadance (with respect to the symmetry transformations) for the potentials and the relationships between their Fourier components are represented in a form convenient to check the symmetry adequacy of microscopic model potentials in real as well as in reciprocal space.
Abstract: Within the framework of a generalized model of a multicomponent lattice gas, the symmetry properties of many-particle interatomic potentials are investigated. The relations of the invadance (with respect to the symmetry transformations) for the potentials and the relationships between their Fourier components are represented in a form convenient to check the symmetry adequacy of microscopic model potentials in real as well as in reciprocal space. The method of statistical-thermodynamic description of multicomponent solid solutions is suggested, taking into consideration the symmetry changes of interatomic potentials due to the structural phase transformations.
TL;DR: In this article, experimental electron diffraction data from copper perchlorophthalocyanine collected at 1200kV were re-assessed to determine the most important deviation of these intensities from the single-scattering approximation.
Abstract: A recent simulation has attempted to evaluate the validity of direct phasing and Fourier techniques in electron crystallography. In response to this study, experimental electron diffraction data from copper perchlorophthalocyanine collected at 1200kV were re-assessed to determine the most important deviation of these intensities from the single-scattering approximation. While n-beam dynamical scattering has indeed been observed for these electron diffraction intensities (in agreement with the simulation) and has been shown to be important for selection of data suitable for ab initio structure analysis, it is, however, not the major perturbation to data obtained at very high voltages. Rather, a simple correction for secondary scattering provides the best fit to the experimental data, an observation consistent with the analyses of other organic structures. Thus, in order to justify the use of electron diffraction intensities from any substance for an ab initio structure determination, it is, first of all, requisite that the actual conditions used for the diffraction experiment be closely modeled.
TL;DR: In this paper, it was shown that the correlation function [γ(r)] of any N-component sample with interfaces made up of planar facets is always a third-degree polynomial in r.
Abstract: It is shown that, close to the origin, the correlation function [γ(r)] of any N-component sample with interfaces made up of planar facets is always a third-degree polynomial in r. Hence, the only monotonically decreasing terms present in the asymptotic expansion of the relevant small-angle scattered intensity are the Porod [−2γ'(0+)/h4] and the Kirste–Porod [4γ(3)(0+)/h6] contributions. The latter contribution is non-zero owing to the contributions arising from each vertex of the interphase surfaces. The general vertex contribution is evaluated in closed form and the γ(3)(0+) values relevant to the regular polyhedra are reported.
TL;DR: In this paper, the maximum entropy method was used to reconstruct a 2D projected density map from 3D diffraction data, where the spatial resolution in one given direction is very small compared to that in perpendicular directions.
Abstract: When density distributions in crystals are reconstructed from 3D diffraction data, a problem sometimes occurs when the spatial resolution in one given direction is very small compared to that in perpendicular directions. In this case, a 2D projected density is usually reconstructed. For this task, the conventional Fourier inversion method only makes use of those structure factors measured in the projection plane. All the other structure factors contribute zero to the reconstruction of a projected density. On the contrary, the maximum-entropy method uses all the 3D data, to yield 3D-enhanced 2D projected density maps. It is even possible to reconstruct a projection in the extreme case when not one structure factor in the plane of projection is known. In the case of poor resolution along one given direction, a Fourier inversion reconstruction gives very low quality 3D densities «smeared» in the third dimension. The application of the maximum-entropy procedure reduces the smearing significantly and reasonably well resolved projections along most directions can now be obtained from the MaxEnt 3D density. To illustrate these two ideas, particular examples based on real polarized neutron diffraction data sets are presented
TL;DR: In this paper, the authors examined the validity of the zeroth-order single-pixel approximation (ZSPA) by using a one-dimensional two-pixel model for which the exact solution can be analytically obtained.
Abstract: By a recent development of the maximum-entropy method (MEM) following Sakata & Sato [Acta Cryst. (1990), A46, 263–270], electron- (or nuclear-) density distributions have been obtained for crystalline materials of simple structures from single-crystal or powder diffraction data. In order to obtain a ME density map, the ME equation is solved iteratively under the zeroth-order single-pixel approximation (ZSPA) starting from the uniform density. The purpose of this paper is to examine the validity of the ZSPA by using a one-dimensional two-pixel model for which the exact solution can be analytically obtained. For this model, it is also possible to solve the ME equation numerically without ZSPA by the same iterative procedure as in the case of ZSPA. By comparison of these three solutions for a one-dimensional two-pixel model, it is found that the solutions obtained iteratively both with and without ZSPA always converge to the exact solution so long as the value of the Lagrange undetermined multiplier, λ, is chosen to be sufficiently small. This means the ZSPA solution does not depend on λ when the convergence is attained. When, λ exceeds a critical value, iteration with ZSPA gives oscillatory divergence but iteration without ZSPA converges to a different value from the exact solution. It is concluded that the introduction of ZSPA does not cause any serious problem in the solution of the ME equation, when a sufficiently small λ value is used in the ME analysis.