Anisotropic Anomalous Scattering in Rutile TiO₂

Based on an experiment of A. Kirfel and A. Petcov on rutile TiO₂, space group (136) P4₂ mnm in 1989 [1], which verified the anisotropic anomalous scattering (AAS) in rutile through measuring ‘Forbidden Reflection near Edge Diffraction’ (FRED) intensity at the 100 and 001 forbidden reflections just above the Ti-K absorption edge, we aimed to extend the results by measuring AAS not only for forbidden but also for allowed reflections, to find effects of anisotropy also in this case and to fit the model of anisotropic scattering [2] to the experimental data.

Experiments were carried out at DESY/HASYLAB beamline C1 (CEMO) using a (111) double crystal monochromator at the Ti resonance energy E = 4985 eV. An automated optimization and Y-scan routine called ‘Peak-Optimized-Psi-Scan (POPS)’ which we programmed with LabView for a sample setup with rotating degree of freedom was tested for the first time for synchrotron conditions.

The rutile samples investigated were three 10 x 10 x 1 mm³ wafers synthesized by Crystec GmbH in (001), (110) and (111) orientation and Y-scans were measured for the reflections 001, 220, 110 and 111, respectively. Since the Ti in rutile occupies Wyckoff site 2a, its tensor symmetry must follow the local symmetry m.mm leaving 3 independent complex elements for the symmetric scattering tensor, f ₁₁, f ₁₂ and f ₃₃. Calculations and simulations with the developed ‘TensorScattering’ code (Fig. 1) showed following dependencies: 001 to f ₁₂, 220 to f ₁₁- f ₃₃, 110 to f ₁₁- f ₃₃ and 111 to f ₁₂. For the 001 reflection intensity the real and imaginary parts correlate, but the 111 intensity shows asymmetric influence so that the ambiguity should be separable.

The measured data show clear evidence of AAS and the 001 FRED and 111 reflection intensities could be fitted quite reasonably within the model. They were fitted simultaneously because both only depend on f ₁₂, since the 111 is also forbidden within the Ti partial structure and the isotropic flat contribution is caused only by the oxygen partial structure. Within the fit the isotropic part of the 111 reflection intensity locks the constant and the asymmetry detangles f ’ ₁₂ and f ” ₁₂. They were evaluated due to refinement as:

f ’ ₁₂ = f ’ _{Ti(E=4985eV) *} d ’ ₁₂ = -5.44 el _* 0.58(2)

f ” ₁₂ = f ” _{Ti(E=4985eV) *} d ” ₁₂ = 3.98 el _* 1.42(0)

References

[1] A. Kirfel, A. Petcov, Anisotropy of anomalous dispersion. II. FRED (Forbidden Reflection near Edge Diffraction) in Rutile, TiO₂, HASYLAB/DESY Jahresbericht (1989) 385-386.

[2] A. Kirfel, W. Morgenroth, Anisotropy of anomalous scattering in X-ray diffraction. III. `Forbidden' axial reflections in space groups up to orthorhombic symmetry, Acta Crystallogr. A 49 (1993) 35-45.