Analytical description of the electron microscope diffraction contrast of lattice defects in the two-beam case

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Abstract
Simple analytical equations are derived for the bright- and dark-field intensities in the two-beam case which describe, for a large variety of lattice defects, the dependence of the contrast on the defect depth position z 0, the specimen thickness t, the excitation error w and the anomalous absorption parameter τ. These dependences may be characterized by the modulus q of the interband scattering at the lattice defect, which is responsible for the variation with z 0 and t, and a thickness parameter t w, which describes the combined influence of w and τ. For zero beam divergence the z 0 and t dependence of the contrast can be neglected only for lattice defects with large q. Lattice defects with small q are invisible for certain values of z 0 and t. For finite beam divergence, the z 0 dependence of weak-beam images is suppressed for lattice defects located near the centre of the specimen (distance from the surfaces larger than 0·7 extinction distances). A simple method is proposed for calculating the contrast for large values of the beam divergence. Very small lattice defects are more easily recognizable on dark-field images taken with w ≠ 0 than on kinematical bright-field images. For t ≃ |t w|, several contrast anomalies are derived from theory and verified by experiment. The black-white contrast found under dynamical imaging conditions may be reversed in thin specimens and may vanish altogether in thick specimens. The weak-beam image of a lattice defect located near the top surface may be different from the image of the same defect located near the bottom surface.