The path integral formalism as a new description for the diffraction of high‐energy electrons in crystals

Abstract

It is shown that the Feynman path integral formalism provides an elegant way of describing the elastic scattering of electrons in crystals. Transformation of the formulae into momentum space permits the use of the Fourier transform of the periodic crystal potential so that the evaluation of the path integral results in a multiple sum. In the case of high‐energy electron diffraction, the introduction of the boundary conditions can be performed in a “forward scattering approximation”. It can be shown that the result obtained is equivalent with the solution of the system of Howie and Whelan, derived from the Schrödinger equation under a similar approximation. Applying a parabolic approximation to the multiple sum leads to the multi‐slice formulae of Cowley and Moodie. The kinematic and phase grating expressions can also be obtained using appropriate approximations.