Af. Moodie et al., THE SYMMETRY OF 3-BEAM SCATTERING EQUATIONS - INVERSION OF 3-BEAM DIFFRACTION PATTERNS FROM CENTROSYMMETRIC CRYSTALS, Acta crystallographica. Section A, Foundations of crystallography, 52, 1996, pp. 596-605
It is shown that for a centrosymmetric crystal, and within the range o
f 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 d
iscs of a convergent-beam diffraction pattern. In fact, the solution i
s given in terms of three distances measured on the diffraction patter
n. It is further shown that the inversion is independent of thickness.
Three proofs are given, each illustrating a different aspect of the p
hysical processes involved. In the first, the fundamental symmetry of
the diffraction process is shown to be that of the special unitary gro
up of order three and the Gell-Mann representation is used to construc
t three sub-algebras, in terms of which the explicit solution is writt
en. 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 i
nvolves the projection of the scattering matrix into the spaces of the
eigenvectors. Unlike the first method, this makes use of a basis howe
ver, it is not necessary to calculate explicitly either the eigenvecto
rs or the eigenvalues. The third method, based on the Bloch-wave expan
sion, shows that the system is characterized by three lines, which are
ruled on one of the dispersion surfaces, and that all of the informat
ion in the system is embodied in these lines. Although this theory is
scalar and developed here for electron diffraction, it can apply equal
ly to the right circular component of the wave function of X-rays. Som
e brief remarks are made on the practicability of this method based on
preliminary experiments that indicate that phase is the easiest of th
e parameters to measure.