The effect of lens aberrations on the spatial resolution of an energy-filtered TEM image

Using geometrical optics, we derive analytical expressions for the chromatic- and spherical-aberration point-spread functions (PSF) of an electron lens, taking into account the angular distribution of inelastic scattering and energy distribution of the electrons forming the image. Because the PSF has extended tails, the chromatic limit to point resolution in an energy-selected image is typically a factor of 3-8 times smaller than the diameter of the aberration disk, as given by the usual chromatic-aberration formula. In the case of spherical aberration, the diameter containing 50% of the electrons is usually between 1 and 35% of the total diameter. The effects of electron diffraction and delocalization of the inelastic scattering are also considered, particularly in relation to the possibility of atomic-scale imaging of selected atomic species.