A theoretical consideration of some defects in electron optical images. A formulation of the problem for the incoherent case

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Abstract
Expressions are derived for the resolution functions appropriate to chromatic and spherical aberration in electron microscopy. The resolution functions are given for the two components of the scattered electron beam (i.e. the elastic and inelastic components). The analysis given here for the elastic component is valid only in the case of incoherent radiation (i.e. using a large condenser aperture). The treatment of aberrations caused by inelastic scattering in the specimen assumes that the inelastically scattered radiation is incoherent, an assumption which is valid for amorphous specimens.Numerical results are presented for the angular distributions for plural elastic and inelastic electron scattering in `amorphous' carbon films. It is shown that the contribution of the inelastic scattering to the intensity scattered within the objective aperture (semi-angle in the range 10−3-10−2rad) exceeds the elastic scattering by an order of magnitude for film thicknesses in the range 50-2000 Å and for incident electrons of energy 100 keV. The variation of the electron energy loss spectrum of carbon with the angle of scattering is given. The angular and energy distributions are used to calculate the spherical, chromatic and combined aberration functions appropriate to carbon specimens of thickness 50-1000 Å and for 100 keV electrons. In the incoherent approximation, it is concluded that for carbonaceous materials, inelastic electron scattering dominates all resolution considerations; in particular the chromatic aberration effect due to the inelastic scattering is far more significant than the contribution from spherical aberration.The present treatment of aberrations would seem relevant to low voltage electron microscopy and to thick (about 1000 Å) films of carbonaceous materials at conventional voltages; in both these situations inelastic electron scattering determines the resolution attainable.