Anisotropy of anomalous dispersion in X-ray diffraction

Polarization-dependent resonant Bragg diffraction in crystals is investigated both theoretically and experimentally. In order to describe the effects of anisotropic anomalous dispersion on intensity and polarization of kinematically diffracted X-radiation, a general scattering model is developed on the basis of site-symmetry-compatible second-rank scattering-factor tensors for the absorbing atoms. For conventional four-circle single-crystal diffractometry it is shown that intensity and polarization of the diffracted beam can be predicted as functions of both crystal orientation and polarization of the incident radiation. In principle, anisotropy of anomalous dispersion may affect any reflection. In particular, it can give rise to the observation of intensities for reflections being systematically extinct by space-group symmetry. Both effects are discussed. Experimental proof of the model's validity was obtained by synchrotron-radiation X-ray diffraction measurements of mainly `forbidden' reflections in cubic cuprite, Cu<sub>2</sub>O and tetragonal rutile type TiO<sub>2</sub> and MnF<sub>2</sub>. The experiments were carried out at the respective K-absorption edges of the cations using different instruments at HASYLAB/DESY during dedicated mode of DORIS II (3.78 GeV). Significant anisotropy of anomalous dispersion due to excitation of K electrons into p states was observed in each case, allowing studies of the dependence of `forbidden' reflection intensities on both radiation energy and rotation (Ψ) around the scattering vector h. Comparison of the observations with the analytical intensity functions derived from the scattering model shows full agreement on a relative scale. For cuprite, estimates of the anisotropies of the real and imaginary components of the anomalous dispersion of Cu were obtained from the allowed reflection 330. The values derived from two different experiments (energies) are f′ = −0.56, −0.35 and f′′ = −0.23, 0.0 electrons, respectively.