Pendellösung Fringes in Distorted Crystals I. Fermat’s Principle for Bloch Waves

Abstract

A wave-optical foundation is given for a geometrical optics for electrons and X-rays in distorted crystalline media. A “modified Bloch-wave” and the associated Eikonal function are introduced for this purpose. The modified Bloch-wave behaves like the usual Bloch-wave within local small regions. The trajectory of the wave can be given by a variational principle as to the phase integral along a path with fixed end points A and B, namely
δ∫_A^B(k₀·dr)=0
k₀ being the wave-vector associated with the modified Bloch-wave. This is nothing more than the Fermat’s principle. The Euler’s equation of the variational equation gives the change in k₀ along the trajectory.