Phase-sensitive x-ray diffraction in the expanded distorted-wave approximation

Based on an expanded distorted-wave approximation, analytical intensity expressions are derived for phase-sensitive reference-beam x-ray diffraction in both the transmission and the reflection cases. Results from this approach are compared with the rigorous n-beam dynamical theory calculations and are shown to be very accurate if the crystal is in the thin kinematic limit. The method represents a unified theoretical approach for both the incoherent standing-wave effect and the phase-sensitive three-beam diffraction of the internal wave fields. The simple formulas for thin crystals are particularly useful for phase determination in biological crystals as well as in semiconductor thin-film structures