Application of a Variational Principle to the Calculation of Low-Energy Electron Diffraction Intensities. I. One-Dimensional Problems

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Abstract
A variational principle for the reflectance is derived for elastic scattering from one-dimensional potentials. Using this principle, we show that the reflection coefficient is given by the ratio of two determinants without any subsidiary calculation of the wave field in the crystal and without any need to perform a matching on the boundary. The results are valid for crystals having variable lattice constants, including the possibility of impurity layers. For scattering from periodic potentials, the results are most conveniently obtained by employing Bloch's theorem with the wave number inside the crystal obtained from evaluating a Hill's determinant. The variational principle is also employed to obtain a modified Born approximation for the reflectance. We also compare the reflectance given by approximate wave functions with the exact reflectance for the Kronig-Penney model, the latter also having been obtained by the variational principle.