Gunnar Thorkildsen

Angle calculations for a six-circle κ diffractometer

By parametrizing the conic surfaces spanned by the reciprocal-lattice vector under the action of the rotation axes, a new procedure for angle calculations for a κ diffractometer is presented. It is applied to obtain angle settings in various modes for a newly developed six-circle diffractometer.

Three-Beam Diffraction in Finite Perfect Crystals. I. Exploring Laue–Laue and Bragg–Laue Scattering Using a Series-Expansion Approach for the Solution of the Takagi–Taupin Equations

Using the boundary-value Green-function technique, analytical expressions in the form of finite series expansions are obtained for the relative change in the integrated power of the primary reflection due to the gradual excitation of a secondary reciprocal-lattice point on the Ewald sphere. Solutions are found for both a Laue-Laue and a Bragg-Laue case in finite shaped crystals confined by the scattering vectors. When the crystal sizes do not exceed the Pendell6sung length of the involved reflections, the 7z profiles exhibit the same qualitative features in the two cases. The solutions do however indicate a strong dependence on the outer crystal dimensions - which add a geometrical aspect to the interpretation of the Aufhellung and Umweganregung concepts.

Three-Beam Diffraction in Finite Perfect Crystals. II. Influence of Absorption, Resonant Scattering and Polarization

By using the boundary-value Green-function technique, effects of absorption, resonant scattering and polarization are incorporated in the solution of the Takagi–Taupin equations for three-beam diffraction in finite perfect crystals. It is shown how such features may influence the dynamical interactions. The potential of three-beam interference acting as a phase-sensitive probe for resonant scattering is demonstrated. Depending on the applied model used for the calculations of the corrections to the atomic scattering factor, profile asymmetry reversal is theoretically predicted to occur near the photoelectric threshold in a finite germanium crystal. Significant distortions of the three-beam profiles are found for polarized incident radiation, especially for reflection triplets having an invariant phase sum near 90°.

Primary Extinction in Cylinders and Spheres

By transforming the Takagi equations into a representation using angular coordinates, it is in principle possible to obtain analytical expressions for the coefficients in a series expansion for the primary extinction factor in perfect crystals with a circular diffraction plane. In practice, it has been possible to obtain the first five terms in the expansion. This involves establishing recurrence relations for the families of Bragg and Laue boundary-value Green functions combined with integrations over the entrance and exit surfaces. The calculations, which cover the whole range of values for the scattering angle, theta oh, are performed using the mathematical software systems Mathematica and Maple.

Absorption and Weighted Path Lengths in Cylinders and Spheres

By using the boundary-value Green-function technique for the Takagi equations, it is possible to calculate the normal absorption factor and weighted path length in cylinders and spheres as double integrals in angular coordinates of well defined functions. The results are easy to implement for numerical computations. Exact analytical expressions for both crystal geometries are found in the limits of the Bragg angle theta oh-->0 degree and theta oh-->90 degrees.

Approximate solution of the Takagi-Taupin equations for a semi-infinite crystal in the three-beam Laue-Laue case.

An analytical approximate solution of the Takagi-Taupin equations for a symmetrical three-beam Laue-Laue case in a perfect non-absorbing semi-infinite crystal slab has been obtained. The expression, a second-order expansion, is valid for situations where the effective crystal thickness does not exceed half the actual extinction length and it is shown to be in perfect agreement with the full numerical solution of the fundamental equations.

Three-Beam Diffraction in Finite Imperfect Crystals. An Analysis of Coherent Wavefield Interactions Using Statistical Dynamical Theory

By applying the concepts of statistical dynamical theory and the boundary-value Green-function technique, an analytical expression in the form of a series expansion has been obtained for the coherent contribution to the integrated power in a finite crystal geometry. The solution suggests that three-beam interference is likely to cause observable perturbations also in crystals with a high degree of imperfection. Owing to a first-order dependence on the invariant triplet phase sum, profile reversal may occur in cases where this sum is close to 90°. The result may serve as an experimental check on statistical dynamical parameters.

X-ray diffraction in perfect t × l crystals. Rocking curves

A general formalism, based on the Takagi-Taupin equations, for calculating rocking curves in perfect t x l crystals is presented. It includes nonsymmetrical scattering, refraction, and ordinary and anomalous absorption. t and l may be varied independently. In the limit of a semi-infinite crystal, the standard results from the fundamental theory are retrieved. For crystal dimensions less than the extinction length, the theory converges to the kinematical limit. Simulations for germanium and silicon show significant influence of crystal finiteness. When dynamical effects are prominent, the curves exhibit various degrees of asymmetry and the full width at half-maximum is generally larger than the corresponding Darwin width. This is attributed to combined Laue and Bragg contributions which are shifted with respect to each other owing to refraction.

Three-beam X-ray diffraction: profile analysis

An original functional description of the intensity perturbation of the two-beam diffracted power caused by an interfering three-beam interaction has been developed. By using this approach in the analysis of measured three-beam profiles of α-oxalic acid dihydrate, parameters related to the Darwin mosaic model are refined. The final results indicate an anisotropy in both the mean domain size, measured along the secondary beam, and the block orientation, measured as the angular spread in the location of the three-beam point. The presented method relies on a procedure for merging the contributions to the perturbation originating from dynamical (coherent) and kinematical (incoherent) scattering processes.

Three-beam X-ray diffraction in a semi-infinite perfect crystal

The approach to multibeam X-ray diffraction by Shen [Acta Cryst. (1986). A42, 525-533] has been carefully re-examined for three-beam Laue-Laue diffraction in a perfect non-absorbing semi-infinite crystal. A modified version of Shens theory is shown to be equivalent to the Takagi-Taupin formalism. The perturbation of the two-beam integrated power when a third beam is being excited is thus expressed by non-diverging terms. Both Aufhellung and Umweganregung terms are recovered.