The symmetry of three-beam scattering equations: Inversion of three-beam diffraction patterns from centrosymmetric crystals

It is shown that for a centrosymmetric crystal, and within the range of validity of the three-beam approximation, the phases as well as the magnitudes of the three structure amplitudes can be measured uniquely and directly from the geometry of the intensity distributions in the discs of a convergent-beam diffraction pattern. In fact, the solution is given in terms of three distances measured on the diffraction pattern. It is further shown that the inversion is independent of thickness. Three proofs are given, each illustrating a different aspect of the physical processes involved. In the first, the fundamental symmetry of the diffraction process is shown to be that of the special unitary group of order three and the Gell-Mann representation is used to construct three sub-algebras, in terms of which the explicit solution is written. SU(3) is shown to have particular significance in crystallography, namely, that it is the group of lowest order with symmetries that can be analysed to yield structural phase. The second and longer method involves the projection of the scattering matrix into the spaces of the eigenvectors. Unlike the first method, this makes use of a basis however, it is not necessary to calculate explicitly either the eigenvectors or the eigenvalues. The third method, based on the Bloch-wave expansion, shows that the system is characterized by three lines, which are ruled on one of the dispersion surfaces, and that all of the information in the system is embodied in these lines. Although this theory is scalar and developed here for electron diffraction, it can apply equally to the right circular component of the wave function of X-rays. Some brief remarks are made on the practicability of this method based on preliminary experiments that indicate that phase is the easiest of the parameters to measure.