(sodium)= 0.8685; B(fluorine)=0.8671 A2], using force constants derived by fitting a shell model to measured dispersion curves. This shows that the treatment of the extinction-affected reflexions in the least-squares refinement was entirely satisfactory. The author's thanks are due te the X-ray diffraction group of the University of St. Andrews, Scotland, for the use of a Siemens four-circle diffractometer. His thanks are also due to the referee for his constructive criticism. A method for the numerical evaluation of N-beam diffraction amplitudes and intensities which has been successfully employed over the last few years is described. This derives from the multi-slice formulation of Cowley and Moodie. The physical basis of the method and practical approaches to calculation are described. References
The structures of the orthorhombic RuddlesdenPopper (A,+ tBnX3,+ 1) phases Ca3Ti207 (n=2) (CaTiO3) blocks, n TiO6 octahedra thick, with single layers of CaO having a distorted NaC1 configuration. TiO6 octahedra are tilted and distorted in a very similar fashion to those in CaTiO3 (n = ~). This fact was used to determine the space groups of the layered structures. Convergent-beam electron diffraction patterns are best matched by calculations in the above space groups which are thus confirmed. Octahedral tilt angles increase slightly in the sequence n =2, 3, oo. Strontium addition reduces the octahedral tilt angles because of preferential substitution of Sr on the Ca sites within the perovskite blocks of Ca4Ti3OIo. The algorithm used to produce starting models for structure refinements is thought to be generally applicable to Ruddlesden-Popper and possibly other layered perovskite structures. It furnishes the predictions: (a) all n-even compounds in the Can+~TinO3n+l series will have space group[
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.PART 1
An investigation was made of the MgO h00-systematic interactions using the experimental method of fine-focus convergent beam electron diffraction. The experimental data were compared directly with intensity distributions computed from assumed MgO structure factors by multiple-beam diffraction theory.The investigation showed that close agreement could be obtained between calculation and experiment provided specific values were chosen for the scattering potentials and the crystal thickness. Close agreement was found with a value of V200 = 6"92 volt. This value is appreciably lower than the value of 7"29 volt derived from the Hartree-Fock calculations for neutral atoms, but is in agreement with the scattering curve derived by Tokonami using a model for the O2-ion recently developed by Yamashita.A form of distribution was found for the absorbing potential (leading to a Borrmann effect) which i was close to that of 6 functions located at the atom sites. The value obtained for V20o, 0.14 volt, was considerably lower than that estimated in some earlier investigations. Experimental and theoretical reasons for this difference are given. It was apparent that study of a more strongly interacting reflexion set than the MgO h00-systematics would give more precision in the potential determination. In fitting the calculated and experimental curves the crystal thickness was determined to within one unit cell or with an accuracy of 0.5 ~o for the crystals studied. IntroductionMany studies have been made in the past of dynamic effects in diffraction patterns from magnesium oxide. Recently single wedge-crystals have been examined, both in a diffraction camera (Lehmpfuhl & Moli6re, 1961, 1962, and in the electron microscope (Kamiya, 1963;Uyeda & Nonoyama, 1965) to obtain values of Vh and V~,, the Fourier coefficients of the elastic and inelastic scattering potential. Mostly the two-beam approximation of Bethe has been used for these determinations. From the magnitude of the term (V gig z) implicit inBethe's derivation for addition-potentials, the importance of the higher order interferences can be estimated in each case. It is expected that this systematic interference effect will be more important for the 200 reflexion than for example for the 220 and 420 reflexions.Estimates made for 11200 and V~0o from the two-beam ealculati0n will be in error from this effect. Corrections for the weak beams in the form of addition-potentials can be applied from Bethe's theory under ":ertain conditions, namely when the excitation error for the main beam is very small and the excitation errors of all other reflexions appearing in the correction terms are large. However, this correction gives no solution for the weak beams, nor for the main beams far from the Bragg position. On the other hand as has been pointed out by Howie & Whelan (1960), when n-beam theory is used the systematic effect itself can lead to a better determination of the scattering potentials, and in particular of the absorption factors, provided the experimental data are adequat...
Detailed observations are made of the breakdown of Friedel's law in electron diffraction, using the single-crystal intensity distribution from cadmium sulphide in the [2130] 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 reflexions. These are illustrated by further pictures from cadmium sulphide. 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.
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