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

Abstract

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.

The Symmetry of Three-Beam Scattering Equations: Inversion of Three-Beam Diffraction Patterns from Centrosymmetric Crystals

“…There are well established methods for determining whether a crystal is centrosymmetric (Goodman & Lehmpfuhl, 1968) and therefore whether the technique described in this paper is applicable. (These methods derive from the fact that Friedel's law is broken in dynamical scattering.…”

Large-angle convergent-beam electron diffraction; a simple technique for the study of modulated structures with application to V2D

“…The convergent-beam electron diffraction (CBED) technique was developed in the thirties (Kossel & M611enstedt, 1939), but its use in crystallographic studies came much later when it became possible to focus an electron beam on small areas (Goodman & Lehmpfuhl, 1968;Buxton, Eades, Steeds & Rackham, 1976;Goodman & Johnson, 1977;Tanaka, Saito & Watanabe, 1980). With the conventional CBED technique, one associates a diffraction pattern from a zone axis with an angular spread of the incident beam smaller than the smallest Bragg angles involved in order to avoid overlap between the diffraction disks originating from the different Bragg reflections being excited.…”

“…However, several other techniques have been developed to determine the absolute structure-factor value and also the phase of low-index reflections, e.g. thickness fringes in bright-field and dark-field electron microscopy (Ando, Ichimiya & Uyeda, 1974) (Goodman & Lehmpfuhl, 1968), critical voltage (Watanabe, Uyeda & Fukuhara, 1969) and the intersecting Kikuchi-line method (Gjonnes & Hoier, 1971). All these techniques rely on the dynamical behavior of the fast electron in the crystal, and many-beam dynamical calculations are necessary.…”

Friedel's law and non-centrosymmetric space groups

“…For instance, where the kinematical description is a useful approximation, only the Laue groups can be determined unequivocally. Effects due to n-beam scattering in general are required to resolve this problem (Goodman & Lehmpfuhl, 1968). The analysis of non-centrosymmetric diffraction patterns, however, still presents particular difficulties.…”