A Theoretical Investigation of Plural and Multiple Scattering of Electrons by Amorphous Films, with Special Reference to Image Contrast in the Electron Microscope

The integral transformations of Bothe have provided the main basis of the theory of the plural and multiple scattering of electrons by amorphous films. A new method of performing the integrations is developed here, taking both elastic and inelastic scattering into account, which enables equations to be derived in closed form for the angular distribution of scattered electrons. It is assumed that the scattering films are composed of spherically symmetrical, isolated, neutral atoms. The integrations are made possible by fitting the variation with scattering angle of the total mass-scattering cross section S t for single scattering with a summation of Gaussian terms. Since the fitting parameters occur in the final equations, this method is extremely flexible and allows the most accurate data for S t which are available to be incorporated. A mass-scattering cross section S p is defined so that when substituted for S t in the single-scattering theory the effects of plural scattering are incorporated. Under certain conditions S p is found to be identical with S t , and the equation giving the total fraction of the beam scattered by the film reduces to that given by Zeitler and Bahr. As examples representative of the elements of the periodic table, the theory is applied to carbon and gold films. The variations of the angular distribution of scattered electrons and their effect on electron diffraction patterns and image contrast in the electron microscope are determined for various film thicknesses. Electron energies in the range 20-500 keV, and electron-microscope objective apertures of semi-angle 0-2 × 10 -2 radian are considered. Comparisons are made between the calculated results, the Lenz theory, and the experimental data which are available. The maximum thickness to which the single-scattering theory can be applied is examined and compared with definitions based on the electron mean free path in the film. Methods of improving the accuracy of the theoretical results are discussed.