Effects of the column approximation on weak-beam calculations

In electron microscope studies of crystal defects, high- resolution detail is known to exist in the weak beams. The most popular calculational tools for predicting and interpreting this detail are systems of ordinary differential equations based on the column approximation. We argue from analytical and numerical studies of Takagi's equation and certain other equations that the column approximation is unreliable for weak beams in the following specific sense. (i) The main image detail will not be where the column approximation would predict, but shifted to the left or the right by easily determined amounts. This shifting has the consequence that dislocation images, from two or more dislocations, will not be separated in space by the same lateral distance as the dislocations themselves unless the dislocations are all the same distance from the exit surface of the crystal. (ii) There is a very interesting fringe structure, due to the interference of waves within the characteristic triangle, that the column-approximation equations will never exhibit; these fringes, which grow in number with specimen thickness, are predicted to occur most prominently in images of point defects.