Friedel's Law Research Articles

Electron microscopy study of enantiomorphic ordered structures

Abstract Taking into account the symmetry properties of enantiomorphic crystals, we have deduced from the Howie–Whelan equations the following new results about the relative intensities of the diffracted beams of electrons: (i) In the two-beam and systematic reflection cases, the deviation from Friedel's law may be observed only if g belongs to a non-centric star. In the two-beam case this possible deviation is related to the phase change between the Fourier coefficients of crystal absorption potentials. (ii) Extinction conditions of inversion boundaries have been established in the case of systematic reflection in relation to the location (with respect to the elements of symmetry of the structure) of inversion centres. The absolute determination of LiFe5O8 ordered enantiomorphic structures is performed by comparison between observed and calculated electron diffraction intensities. In particular the exact operating diffraction conditions are carefully examined.

Experimental Investigations of Bonding Charges in Diamond and Zinc Blende Structures

A major source of information concerning bonding charges in diamond structures is provided by the weak non-zero forbidden reflections of the type: h + k + l = 4n + 2. In the early determinations of the (222) a lot of attention was devoted to perturbing effects such as "Umweganregung" and lattice defects. Neutron and X-ray diffraction measurements confirmed the electronic origin of the (222) intensity, and established the existence of anharmonic contributions at temperatures above R.T., particularly in the case of the (442) in Si. A non-zero (222) intensity was also found in grey-tin. "Pendellösung" techniques can also be applied, especially in conjunction with high intensity X-ray sources. In the case of III-V semiconducting compounds, such as InSb, reflections for which h + k + l = 4n + 2 are not completely forbidden. They are still interesting for bonding charge density studies, however. Reflections like (200), for example, when compared with the free-atom value, provide information about charge transfer and, or, interstitial charges placed off center along the bond. Reflections like (222) and (2̄2̄2̄) are found different whereas they should be rigorously equal, even considering the violation of Friedel's law owing to anomalous dispersion. Such a difference is a direct indication of asphericity of the valence electrons. A phenomenological two-parameter model based on a tetrahedral distortion of the valence charge density and a charge transfer from In to Sb provides excellent fitting with experiment. It is also found that the difference (222-2̄2̄2̄) decreases gradually with temperature, between 400 K and 16 K. We have interpreted this result by assuming that the amount of tetrahedral distortion and charge transfer vary with temperature.

Three-dimensional contrast in dark-field images

Abstract It is shown that, under suitable conditions of N-beam interaction, distinctive contrast is obtained from crystal surface detail in the dark-field image which allows edges or steps on the upper (beam-incident) surface to be distinguished from those on the lower (beam-emergent) surface. The ability to distinguish between the orientations of apex up, and apex down, for a crystal of triangular cross-section is shown to derive from a break-down of Friedel's law which results from the imaging process. An interpretation is given in terms of an extended charge-density approximation’ using the theory developed by Moodie for interpretation of aperiodic features in the bright-field image. Illustrations are from [111] orientations of MgO cubes, although this technique is expected to be of more general application.

Absolute Structure Determination using Electron Microscopy

When the symmetry operations, which constitute the space group of a structure, do not include an inversion or a reflection operation, then the structure can exist in two forms, a right-handed and a left-handed one, called enantiomorphs. The presence of the two enantiomorphs coexisting within a sample can be verified in the electron microscope by imaging in dark field in a multi-beam orientation, with the electron beam parallel with a zone axis, along which the crystal does not show a center of symmetry in projection (1). One takes advantage here of violations in Friedel's Law (2), which may cause a difference in background intensity between the two structures. An example is shown in Fig. 1, where an inversion boundary runs through a thick wedge-shaped crystal of lithium ferrite. Ordered lithium ferrite may occur in two enantiomorphic forms P43 32 (right-handed screw axis), and P41 32 (left-handed screw axis). In this paper it will be shown that it is possible to determine uniquely the configuration of the structure using the results of the dynamical theory.

A second harmonic analyzer for the detection of non-centrosymmetry

The existence of optical second harmonic generation has been shown to be a highly reliable and sensitive physical test for the detection of crystalline non-centrosymmetry. A second harmonic analyzer has been constructed which can resolve space group ambiguities arising from Friedel's Law with a confidence level greater than 99%. The system has been optimized for use with powdered crystalline samples so as to obviate the need for large single crystals and thus facilitates rapid determination of crystalline non-centrosymmetry. The present analyzer can routinely detect second harmonic generation at levels 1/1000 of that generated in a quartz standard, this is about an order of magnitude increase over previously reported systems. Data are reported on several materials including dibenzyldisulfide, and [(C6H5)3P]3CuBF4. The detection of structural phase transitions with the second harmonic analyzer is reported for BaTiO3, colemanite and phenanthrene. Second harmonic generation in the `cubic' phase of BaTiO3 promises to be a powerful tool for determining the dynamics of the ferroelectric phase transition. It is the most direct method for establishing the existence or nonexistence of microscopic polar regions well above the Curie point in a nominally centrosymmetric phase.

Identification of enantiomorphism in crystals by electron microscopy

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.

Observation of the failure of Friedel's law in thickness-bend extinction fringe patterns of electron microscopic image from CdS

The failure of Friedel's law is demonstrated by using the thickness-bend extinction fringe patterns of the (00l) reflexions from CdS. The patterns are simulated by an eight-beam computation. The explanation of this failure is given in terms of the dispersion surface and associated amplitudes. An explanation is also given for the characteristic features of the fringe patterns. Die Verletzung des Friedelschen Gesetzes wird unter Benutzung der “thickness-bend extinction fringe”-Bilder der (00l)-Reflexe von CdS demonstriert. Die Bilder werden mittels Achtstrahl-Naherung simuliert. Diese Verletzung wird mit der Dispersionsflache und den Amplitiiden erklart. Eine Erklarung fur die charakteristischen Einzelheiten der Pendellosungsstreifen wird ebenfalls gegeben.

On the non-centrosymmetric structures which produce centrosymmetric diffraction patterns even with anomalous dispersion

The geometrical conditions that a structure gives centrosymmetric diffraction patterns have been investigated. It is shown that Friedel's law may hold for certain non-centrosymmetric structures containing two or more kinds of anomalous scatterer. Therefore, their absolute configurations can never be established by ordinary absorption-edge techniques. The geometrical characteristics of such a structure are discussed in terms of a vector set.

Réduction de moitié du volume mémoire et de la durée de calcul dans l'usage des algorithmes de transformation de Fourier rapide (FFT)

After a brief review of the crystallographic Fourier transform problem, this paper compares three methods for its computer implementation with reference to core-storage capacity and computation time: the classic method, the simple Fast Fourier Transform algorithms and a new method. The first method needs, especially for a macromolecular three-dimensional structure, too many operations and therefore too long a computation time and poor accuracy. The remarkable FFT algorithm, on the other hand, leads to large savings in computation time with a better accuracy, but using no crystallographic symmetry wastes too much storage capacity. The third method, using FFT but taking into account Friedel's law by means of a simple in situ process, approximately halves the storage capacity as well as the computation time.

Friedel's Law in Electron Diffraction as Applied to the Study of Domain Structures in Non‐Centrosymmetrical Crystals

AbstractIt is shown that a strong contrast may arise in dark‐field electron microscope images between two crystal parts related by an inversion operation, as a result of the violation of Friedel's law. Friedel's law is violated only in the diffracted beam and only in cases where a multiple‐beam situation prevails. Moreover the excited reflections must belong to B zone which reveals the non‐centrosymmetrical character of the crystal. The interfaces between two domains related by an inversion operation, which are proposed to call inversion boundaries are imaged by α‐like fringes which nevertheless have properties which differ somewhat from those of pure α‐fringes. These considerations are applied to the domain structure observed in the γ‐phase. In this case the inversion boundaries separate regions which have suffered displacements as well. It is shown that the displacement vector of such boundaries can be given an unambiguous meaning, although strictly speaking the structures in the two parts cannot be derived one from the other by means of a parallel displacement. It is demonstrated in a concrete case that the observed contrast can be accounted for by a four. beam calculation.

Lattice Curvature in Nearly Perfect Silicon Crystals

Deviations from Friedel's law in the case of anomalous transmission of X-rays have been estimated in as-grown silicon crystals. The results of measurements on a double crystal spectrometer agree with the dynamical theory of elastically deformed crystals. Using this theory, we have obtained information about growth deformation in crystals containing a small number of dislocations or in dislocation-free ones. Radii of curvature of crystal lattice planes of the order of km, orientation differences of the order of 0.1" and relative changes of lattice constants of the order of 10-3% have been estimated. Dislocation loops were observed at the boundary between the dislocation-free and the distorted part of the crystal.

On the diffraction enhancement of symmetry

Diffraction symmetry is in general specified by the point group of a crystal. However, there are some exceptional cases in which the diffraction symmetry becomes, other than as a result of Friedel's law, higher than the point-group symmetry (diffraction enchancement of symmetry). Using a general expression for the square of the structure amplitude, the necessary conditions for the diffraction enhancement have been systematically investigated for the following four kinds of structures: (1) crystals composed of essentially the same substructures, (2) different substructures which have the same symmetry, (3) substructures with the same point group and different space groups, and (4) substructures with isomorphic point groups. It has been shown that symmetry enhancement like 4/m, 4/mmm, {\bar 3}m1 ({\bar 3}1m), 6/m or 6/mmm does occur in addition to 2/m or mmm as suggested by Sadanaga & Takeda [Acta Cryst. (1968) B24, 144] and by Marumo & Saito [Acta Cryst. (1972) B28, 867].

Accurate Measurement of Bijvoet X-ray Intensity Ratios

IN noncentrosymmetric crystals the anomalous dispersion of X-rays leads, in general, to a breakdown of Friedel's law, that is to unequal diffracted intensities from reflexions of the form . A measure of this effect, which for atoms with high atomic number can be appreciable throughout the commonly used range of X-ray wavelengths, is the Bijvoet intensity ratio . Accurate measurements of the Bijvoet ratios are of considerable importance for the determination of phases in structure analysis1, in tests of the theoretically calculated values of the dispersion corrections to the atomic scattering factors2 and in the determination of the absolute configuration of molecules3. Recent determinations of absolute configurations have made use of the anomalous dispersion of CuKα radiation by atoms as light as oxygen4,5.

X-ray diffraction phenomena in elastically distorted crystals

Diffraction phenomena in diametrically compressed single-crystal plates of silicon were systematically studied by topographic methods of the traverse and section types with Cu, Mo and Ag radiations. Black and white contrasts in the traverse topographs depend on the magnitude of the strain gradient, absorption, the sign and the order of the reflection vector, and the crystallographic orientations of the specimens. The results were interpreted in terms of the diffraction theory for a constant strain gradient. The distribution of the strain gradient was calculated under the assumptions of plane stress and isotropic elastic constants. Qualitative agreement was obtained between the theory and experiments. The behaviour of the contrast, including the departure from Friedel's law, is due to Borrmann absorption in distorted crystals. The quantitative discrepancy is explained by anisotropy of the elastic constants. The departure from a constant strain gradient is detectable by the asymmetry of the intensity distribution in section topographs.

Symmetries in Scattering of Slow Neutrons

Reciprocity, detailed balance, and other symmetry laws for slow-neutron scattering are discussed for the case of anisotropic media. While Friedel's law, stating that $\ensuremath{\Sigma}(\mathbf{v}\ensuremath{\rightarrow}{\mathbf{v}}^{\ensuremath{'}})=\ensuremath{\Sigma}(\ensuremath{-}\mathbf{v}\ensuremath{\rightarrow}\ensuremath{-}{\mathbf{v}}^{\ensuremath{'}})$, may fail for polar absorbing media, reciprocity as a consequence of time-reversal invariance, and of thermal equilibrium of the medium, remains valid. The optical theorem helps to derive a reciprocity relation for the total cross section $\ensuremath{\Sigma}(\mathbf{v})=\ensuremath{\Sigma}(\ensuremath{-}\mathbf{v})$. Similar conclusions are derived for the matrices used to describe scattering of polarized neutrons. The two reciprocity relations lead to a macroscopic reciprocity theorem, expressing a symmetry property of Green's functions of the linearized Boltzmann equation.

Observation of the breakdown of Friedel's law in electron diffraction and symmetry determination from zero-layer interactions

Detailed observations are made of the breakdown of Friedel's law in electron diffraction, using the single-crystal intensity distribution from cadmium sulfide in the [21{\bar 3}0] orientation. It has been found by experiment that no breakdown of Friedel's law occurs in the zero-order beam distribution. An analysis of the problem due to Moodie, using multiple-scattering diagrams, has also led to this result. Multiple-scattering diagrams are used here to illustrate the symmetry properties of the zero-beam distribution. Rules are given for the deduction of the other symmetry elements of the projected structure from the diffraction-pattern symmetry, and from the occurrence of dynamic extinction bands in the kinematically forbidden reflections. These are illustrated by further pictures from cadmium sulfide. This analysis is simplified in the present work by neglecting non-zero-layer interactions. Results obtained by n-beam calculation using the multi-slice method show the possibility of quantitative interpretation, and of absolute orientation determination. The limitations of the systematic dynamic approximation and also of a systematic approximation using corrected scattering potentials are examined for this substance.

Die `Diffraction Pattern' des Idealkristalls für Röntgenstrahlinterferenzen im Bragg-Fall

An interpretation of a formula given by Cole and Stemple for the diffraction pattern of non-centrosymmetric perfect crystals in the Bragg case is suggested. Interesting features are the agreement of the shape of the diffraction pattern with Friedel's law whereas the peak height shows no such agreement. The latter is different for h and li reflexions. It depends on whether the planes of the centres of the absorbing matter (the matter providing the imaginary part of the scattering amplitude) lie in that half space between neighbouring net planes which is reached by the nodal planes of the electric field. If this condition is met by the h reflexion it is not met by the-h reflexion and vice versa. The velocity of nodal plane migration (with 0) is coordinated to the relationship between the abscissa of the diffraction pattern maximum and the positional parameter of the above mentioned plane of absorbing centres. This rate is infinite at the limits of the Darwin-Ewald-range of total reflexion and is a minimum in the middle of that range. Tables of the diffraction pattern and its self-convolution have been compiled with the aid of an IBM computer for a selection of parameter combinations for both the polar and non-polar cases. These tables or their computing programs (Fortran IV) are available to those interested.

Extension of the Howie‐Whelan Equations for Electron Diffraction to Non‐Centro Symmetrical Crystals

AbstractThe Howie‐Whelan system of equations [2] is extended to non‐centro symmetrical crystals. This system of equations, which is valid for the Laue case, describes the scattering of electrons by a perfect crystal in the two‐beam approximation and takes anomalous absorption into account. The theory predicts that Friedel's law no longer holds for the dark field image but is still valid in the bright field. Observations on 180° walls in barium titanate support the theory. In addition the theory shows that 180° walls in inclined planes should produce α‐type fringes. This phenomenon is also observed experimentally.

Electron Optical Studies of Barium Titanate Single Crystal Films

Thin single crystal films of barium titanate prepared by chemical thinning of single crystal plates are studied by transmission electron microscopy and electron diffraction. The films as thin as 1000 A are found to have the same lattice parameters as the bulk crystals, giving no evidence for the existence of the anomalous surface layer proposed by Kanzig et al. The surface of barium titanate is found to change substantially when heated above 500°C in air and vacuo, invalidating their evidence for an unusually high Curie temperature of their anomalous surface layer. Ferroelectric 90° and 180° domains are observed by electron microscopy. The contrast of 90° domains is simple diffraction contrast. The contrast of 180° domains is due to the failure of Friedel's law, which is known to be possible to take place in electron diffraction by dynamical diffraction effect. Dislocations, tracks of fission fragments from uranium and their interaction with domains are observed.

Anomalous Neutron Diffraction inαCadmium Sulfide

Anomalous neutron scattering (failure of Friedel's law) was observed in experiments with alpha Cd containing neutron-absorbing Cd/sup 113/ and a space group Pomc, in which the integrated intensities of (hkl) and (hkl)-bar mates of a series of reflections were measured at several wavelengths. The I/sub 002//I/sub 002/ ratio showed a strong energy dependence, due t o the variation of the complex amplitude of cadmium with the wave length, which agrees with the Breit- Wigner theory. The intensity inequalities were compared for x rays and neutrons on one CdS crystal; the direction of the phase shift was found to be the same for both. Possible applications of anomalous neutron scattering are discussed. (D.L.C.)