Maximum (Phase-)Contrast in (HR)TEM Imaging (with Scherzer Defocus)
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Assuming the TEM optics are well aligned (including beam tilt and astigmatism are zero), then Gaussian defocus provides TEM images the well-defined minimum contrast and Scherzer defocus gives a maximum contrast.

By defocusing in either direction from Gaussian defocus, the TEM image contrast will be increased. The optimum defocus for obtaining a strong contrast over a broadest range of spatial frequencies is called the Scherzer defocus. Under this condition, the atom columns will show a dark contrast. As listed in Table 2779, the dependence of contrast maxima on defocus is a periodic function. The first and second maxima have optimum applications to large and small features, respectively, since different defocus suppresses different (high or low) resolution information.

Table 2779. Dependence of contrast maxima on defocus. Cs is the spherical aberration coefficient (1 mm here) and λ the wavelength of incident electrons (~0.00251 nm for 200 keV here).
Contrast maxima Special name
Defocus (Δf)
Spatial frequencies well-transferred to images Contrast of atom columns Particular, convenient observations
Formula (nm)*
First Max Scherzer defocus Scherzer defocus -59 A broadest range of spatial frequencies Dark Large features & weak-phase objects
Second Max   Scherzer defocus -87 High spatial frequencies White Small features
Third Max   Scherzer defocus -112   Dark  
Fourth Max   Scherzer defocus -133   White  
* Δf for a specific microscope can be obtained with the excel file.

Note that in the case of Scherzer defocus, the effect of spherical aberration can be partially compensated for by applying the Scherzer defocus, which provides a wide bandpass of the same phase contrast.

In practical TEM operation, to achieve your optimum focusing conditions (first and second contrast maxima on your microscope), the condition with Gaussian defocus can first be obtained with minimum contrast and then you will obtain TEM images by defocusing the microscope with the calculated numbers with the excel file (e.g. the numbers, Δf, in Table 2779).

 

 

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