Modified two-beam description of X-ray fields and intensities near a three-beam diffraction point. General formulation and first-order solution

The dynamical three-beam problem in Renninger geometry is cast in a pseudo-two-beam formulation for the primary OH reflection, with the inverse of the excitation error ξ_L with respect to the third reciprocal-lattice point L acting as a perturbation parameter for modifying the true two-beam solutions. This approach introduces a quasi-universal angular scale x for measuring the onset of all three-beam effects, and it leads to a first-order solution that preserves all features of a two-beam case, but around a shifted Lorentz point, and with modified structure factors. The modified structure factors, odd in x, cause pronounced asymmetries in the diffracted intensities on both sides of the three-beam point, for |x| 1. In this range of x, the first-order solution provides a simple analytic expression for the integrated diffracted intensity vs angle, for a sequence of neighboring three-beam or higher-order points. This is exemplified for the Ge 222 primary reflection. The physics of the onset of the three-beam interaction, and the limitations of the first-order solution are also discussed.