On the quantitative determination of triplet phases by X-ray three-beam diffraction

Calculations of the integrated diffracted intensity for Renninger experiments, i.e. calculations of ψ-scan profiles scanning through three-beam positions, are reported. The fundamental equations of the dynamical theory are solved by means of an eigenvalue procedure and boundary conditions consistent with the diffraction geometry. It is shown that for noncentrosymmetric structures the three-beam ψ-scan profiles bear information on both the magnitude, defined in the range 0 ≤ |ϕ| ≤ 180°, and the sign of the triplet phase involved in the three-beam interference. In general, the ψ-scan profiles can be separated into two parts: a phase-dependent part ('ideal' profile) due to the interference effect and a symmetric phase-independent Umweganregung or Aufhellung profile due to the mean energy flow in a three-beam case. Both parts can be calculated by summing up the ψ-scan profiles for + ϕ and -ϕ, one profile being reversed with respect to the three-beam point. As a result, the experimentally best suited three-beam cases for triplet phase determination should involve structure factors of nearly equal magnitude.