Diffraction in crystals at high energies

By means of several distinct stages of approximation, the way in which wave propagation in a lattice becomes classical at high energies is analysed. First, the principle that deflection angles (whether caused by 'quantum' or 'classical' processes) are small at high energies is used to derive a simplified wave equation involving the lattice potential averaged along the direction of the incident beam. Next, the many-beam solution of this equation for the case of systematic reflections is presented in a form which emphasises the spatial variation of the potential, rather than its Fourier components. Third, approximate analytical expressions for the Boch eigenvalues and eigenfunctions, and for the amplitudes of the diffracted beams, are derived by means of the WKB method; this leads to easily calculable expressions for the number of diffracted beams expected in a given situation, as well as for the number of Bloch waves contributing to these beams.