Time-like perturbation method in high-energy electron diffraction

The small-angle approximation usually encountered in dynamical theories of fast electrons essentially leads to a transformation of the propagation-direction variable z into a time-like parameter [Berry (1971). J. Phys. C, 4, 697-722]. The three-dimensional stationary Schrödinger equation is then approximated by a two-dimensional 'time'-dependent equation which may be solved by using the standard time-perturbation techniques encountered in quantum mechanics. The basic idea of the present approach consists in studying the evolution operator U(z,z₀) instead of the wave function. Depending on the choice of bases, the matrix elements of U(z,z₀) represent either the transition probabilities of diffraction or the kernel function of the propagation issued from Feynman-path integral theory [Berry & Mount (1972). Rep. Prog. Phys. 35, 315-397; Van Dyck (1975). Phys. Status Solidi, 72, 321-336; Jap & Glaeser (1978). Acta Cryst. A34, 94-102]. Special attention is devoted to the so-called 'Bloch waves' and 'physical-optics' formulations which both correspond to the same perturbation expansion but with two different unperturbed 'Hamiltonians'.