The theory of X-ray crystal diffraction for finite polyhedral crystals. III. The Bragg-(Bragg)^m cases

The plane-wave and spherical-wave theories are described for the Bragg-(Bragg)^m cases. The treatment is similar to that of Parts I and II [Saka, Katagawa & Kato (1972). Acta Cryst. A28, 102-113, 113-120] for the Laue-(Bragg)^m cases. In the plane-wave theory of the Bragg case, a few aspects which up to now have not been well understood, are described to clarify the mathematical structures of the wave field. In particular, emphasis is put on a method for specifying the plane-wave solution by using a Riemann sheet instead of the dispersion surface. In the spherical-wave theory, the reflected vacuum wave and the transmitted crystal wave at the entrance surface can be represented by two Bessel functions. The crystal wave of the Bragg-(Bragg)^m case reflected at the exit surface is represented by a combination of two Bessel functions. The transmitted vacuum wave, however, is given by a combination of four Bessel functions. It is shown that the solutions are compatible with those of the Laue-(Bragg)^m cases. The solution for finite polyhedral crystals can be constructed by superposing the solutions for individual cases such as of Laue, Laue-(Bragg)^m [Kato (1968). J. Appl. Phys. 39, 2225-2230; Parts I and II] and Bragg-(Bragg)^m obtained in the present paper. A comparison is made with Uragami's results obtained by another approach [J. Phys. Soc. Japan (1969), 27, 147-154; (1970), 28, 1508-1527].