Images obtained in a TEM or a STEM are never projections of a structure. The fast electron interacts strongly with the nuclear charges and the shell electrons of the sample atoms and strong diffraction effects result from this interaction. As such the image of even a single atom is not a simple projection, but a wave-optical interference phenomenon.  The inverse problem to determine a structure, i.e. the three-dimensional coulomb potential of a sample, from an image or a set of images is hitherto possible for a set of private cases only but not generally formulated.  

As a consequence structure refinement in TEM is based usually on a model-based approach where as a starting point a model of the atomic structure is used to calculate images as they would be created by the TEM in use. A refined is then obtained based on the iterative quantitative differences between experimental images and calculated images.  Such a model refinement is helpful to verify for instance the validity of ad-hoc interpretations that may be falsified by instrumental artefacts such as optical imperfections, even for aberration-corrected instruments.

In practice a multislice calculation is most commonly used, the solid is sliced into thin sub-slices to solve the semi-classical Schroedinger equation for the electron wavefunction interacting with the sample potential in a finite-difference approach. The incident electron wave-function is transferred by the first slice (diffraction) and propagated to the next one. The propagation is done within the Fresnel approximation, the distance between the slices being 20-50 times the wavelength.

Alternatively a Bloch-wave ansatz is frequently used to solve the Schroedinger equation, albeit this procedure is restricted to periodic objects.
For a proper image interpretation both STEM and TEM require simulation. The difference between both is that in STEM the incident electron wave is a cone of plane waves instead of a single plane wave in TEM and the multi-slice calculation has to be performed for each scan pixel separately. Therefore STEM image simulations can be computationally quite demanding.

Figure: Propagation of a focussed STEM probe with a sub-Angstrom diameter through a strontium titanate crystal in <110> viewing direction. The orthoslice shows the high-frequency components of the modulus of the electron wavefunction at various depths of the crystal, which is a measure for the signal obtained on a high-angle annular dark-field detector. Most signal is accumulated from the top of the oxygen column on which the probe is located, further down the sample a substantial signal originates from neighbouring titanium-oxygen column. Analytical signals would show titanium where there is no titanium to be expected at the probe position. Simulations by L. Houben, based on Dr. Probe (Barthel, www.er-c.org/barthel/drprobe/).

Further reading

• J. Barthel, Ultramicroscopy, 193 (2018) 1-11.
• E.J. Kirkland, Advanced Computing in Electron Microscopy. Springer International Publishing. DOI: 10.1007/978-3-030-33260-0.
• P.A. Stadelmann, Ultramicroscopy, 21 (1987) 131-145.