High-resolution electron-microscope images of crystals with correlated atomic displacements

A coordinate-space multislice description of the scattering of high-energy electrons is constructed from consecutions of differential operators acting upon atomic potentials. It is used to find expressions for the intensity distribution in high-resolution electron-microscope images of crystals whose atoms are periodically displaced relative to a reference lattice according to a modulation wave. Both static correlated displacements, such as occur in modulated structures, and time-dependent correlated displacements, as are generated by phonons, are considered. Two aspects of the image are examined in detail; its translational symmetry and its dependence upon the correlations between the atomic displacements. It is shown that the intensity distribution due to scattering from static correlated displacements has the translational symmetry of the modulated structure in that projection, as determined by the component of the modulation wavevector perpendicular to the incident beam, whereas that due to scattering from phonons has the translational symmetry of the reference lattice in that projection. The former is a consequence of higher-order Laue-zone interactions. The intensity distribution due to scattering from static displacements depends upon the absolute phase of the displacement at each scattering atomic site whereas that due to scattering from phonons depends only upon the relative phase of the displacements between different scattering sites, both within the same atomic column parallel to the beam and in adjacent columns. In both cases, the influence of the component of the correlation wavevector parallel to the incident beam is different to that perpendicular to the beam; the former affects the intensity mostly at the atomic sites whilst the latter affects the intensity mostly between the atomic sites. It is also observed that, as a consequence of the periodic nature of the polarization-vector function, the interference terms are small, both relative to the noninterference term and in an absolute sense, particularly for phonon scattering. For this reason, the contribution to the image due to scattering from correlated atomic displacements will have greater and sharper atomic contrast than that due to scattering from the reference structure without displacements. In addition, this component of the intensity distribution will not exhibit strong contrast reversal when the objective-lens defocus is changed.