A new calculation of electron scattering cross sections and a theoretical discussion of image contrast in the electron microscope

Abstract

The current basis for the theoretical discussion of the contrast in an image in the electron microscope is the 1954 Lenz theory. The basis of this theory is reviewed and the reasons are given for the two interpretations of Lenz's theory in its application to all the elements which have arisen in the literature; sample calculations based on these interpretations and theoretical reasons for favouring one of them are given. Since 1954, theoretical descriptions of the atom have improved and the Hartree-Fock wave functions have replaced the Hartree wave functions and the Thomas-Fermi-Dirac potentials have superseded the Thomas-Fermi potentials. In this article the results from the newer atomic models are incorporated into the theory of single-electron scattering as applied to electron microscopy. The new method of calculation avoids the rigid expressions used by Lenz for the potential distribution within an atom and the atomic radius by using directly the values of the atomic scattering factors for X-rays and for electrons evaluated from the latest atomic models. A feature of the new expressions for the elastic and inelastic atomic scattering cross sections is their flexibility; the results of future developments of the atomic models can be readily incorporated into the calculations. The variations of the elastic, inelastic and total mass-scattering cross sections and differential atomic scattering cross sections with the accelerating voltage of the electron beam, the angular aperture of the objective lens and the atomic number of the amorphous specimen giving rise to the scattering, are discussed in detail for both the newly developed theory and the Lenz theory in its two interpretations. Most of the results are given for accelerating voltages in the range 20-100 kV but a few results for the new theory are given for voltages of 500 kV and 1000 kV. The differences and similarities between the results of the two theories are discussed and a section is included on the agreement between theoretical and experimental parameters describing the image contrast. The theoretical values of the total mass-scattering cross sections are appreciably higher than the experimental values, but the theoretical predictions about the variations of the cross sections with the atomic number of the scattering element and the instrumental parameters are in general agreement with the experimental results. Methods are suggested by which the basis of the new calculations of the cross sections may be improved.