Institute of Metals Division - An Analytical Method for Calculating Electron Diffraction Patterns Containing Twin Reflections

An analytical method is presented which enables the electron diffraction patterns from a twinned region to be indexed. The method has only been applied to (112) twins in bcc lattices, but can easily be adapted to other types of twinned lattices. Calculations of all twin spots, including those superimposed on matrix spots, is made possible by this method. SINCE the advent of transmission electron microscopy, the analysis of selected-area electron diffraction patterns has become very important in the interpretation of electron micrographs. Frequently, as in the case of some martensites, twinned areas are observed and give rise to complex diffraction patterns, which so far can be analyzed only for the simplest orientations of the thin foil specimen with respect to the electron beam. In order to simplify the analysis of twin patterns of more complex orientations, an analytical method was devised which is capable of predicting the positions of all reflections from both matrix and twin regions for all possible foil orientations. Although the method has only been applied to the case of bcc metals twinned about a (112) plane, it is equally applicable to other Bravais lattices. This method has been successfully applied to selected-area electron diffraction patterns of Fe-Ni and Fe-Ni-C martensites and drawn tungsten wire. PROCEDURE Imagine the reciprocal lattice corresponding to a bcc material, with each point hkl representing the crystallographic plane (hkl). Since the extinction conditions require that for this lattice, only reflections where h + k + 1 = even integer are allowed, the three dimensional reciprocal lattice is actually fcc. Since in real space twinned regions and matrix regions are mirrored across a (112) plane, the twinned reciprocal lattice is derived from the matrix reciprocal lattice by a rotation of 180 deg about a (112) direction. While it is possible to calculate how a point hkl will shift under a twinning transformation to a new point htktlt using only the reciprocal lattice based on the orthonormal vectors [100], [010], and [001], the calculations are greatly simplified if a new orthonormal coordinate system containing the twin axis is introduced. The twinning operation then becomes a simple rotation of 180 deg about a principal coordinate axis (in this case the twinning axis), and a retransformation back to the original reciprocal space will give the position htktIt of the twinned reflection.