Limitations on the s-state approach to the interpretation of sub-angstrom resolution electron microscope images and microanalysis

The s-state approach is useful for analysing transmission electron microscope images of a thin crystalline foil consisting of well-separated atomic columns. It assumes that the signal collected (e.g., the annular dark-field image, the EELS spectrum) can be attributed to only the lowest-energy bound eigenstate of the two-dimensional projected potential of a single column. When, however, columns are close, the form of the bound states depends on more than one column, which implies that interpretation of the signal may not be so simple. For closely spaced columns we show that the simple s-state approach fails for the case of a sub-Angstrom probe initially centred on one column of a pair, because two bound states are excited. The energy in the probe is almost completely transferred to the neighbouring column after it has propagated some tens of nanometres through the foil and then is transferred back. Signals which relate directly to the local probe intensity (e.g. annular dark-field formed by thermal diffuse scattering, EELS) must be analysed in terms of the two bound states. Accurate calculations of bound states of pairs of columns are more demanding than for a single column but sufficient accuracy can be achieved from knowledge of the 1s-states of isolated columns. We provide formulae for the bound states of a column pair. These can be used to determine if image analysis requires the extension to the s-state approach described in this paper.