Propagation of X-ray beams in distorted crystals (Bragg case). I. The case of weak deformations

It has been commonly admitted that the theories of X-ray propagation in distorted crystals based on the principles of geometrical optics [Penning & Polder (1961). Philips Res. Rep. 16, 419-440; Kato (1963). J. Phys. Soc. Jpn, 18, 1785-1791; Kato (1964). J. Phys. Soc. Jpn, 19, 67-71, 971-985] were applicable only in the transmission (Laue) case. It is demonstrated in this paper that they can be applied more generally in all cases where beams can be defined, i.e. also in the Bragg case outside the total reflection range. Simple formulae for the case of a constant strain gradient in symmetric Bragg geometry are derived from a general formulation of the basic equation of geometrical theory using a new universal parameter a. They are verified by solving Takagi's equations numerically. The results are visualized by means of an original method of ray tracing directly from Takagi's equations.