Imaging conditions for reliable measurement of displacement and strain in high-resolution electron microscopy

We analyse the degree to which the lattice fringe displacements in an image correspond to displacements of the atomic planes in the specimen using lens transfer theory. Our basic assumption is that the exit wave function faithfully reproduces the displacements of the projected atomic structure. The way this information is imaged by the objective lens is then developed analytically. We observe an interchange of amplitude and phase information between the original and the reconstructed wave function. For symmetry-related reflections, we show that in the absence of beam amplitude variations, the displacements are imaged perfectly by the objective lens. The theoretical results are confirmed using one-dimensional simulations. For the more complicated case of non-centrosymmetric structures, beam tilts and crystal tilts, we study the implications for slowly varying displacement fields. Errors are found to be minimised in areas where the contrast of the lattice fringes is highest. Finally, we deduce from these theoretical results a number of practical rules.