Multiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a grid

The generalized Darwin–Hamilton equations [Wuttke (2014). Acta Cryst. A70, 429–440] describe multiple Bragg reflection from a thick, ideally imperfect crystal. These equations are simplified by making full use of energy conservation, and it is demonstrated that the conventional two-ray Darwin–Hamilton equations are obtained as a first-order approximation. Then an efficient numeric solution method is presented, based on a transfer matrix for discretized directional distribution functions and on spectral collocation in the depth coordinate. Example solutions illustrate the orientational spread of multiply reflected rays and the distortion of rocking curves, especially if the detector only covers a finite solid angle.

What is an `ideally imperfect' crystal? Is kinematical theory appropriate?

Most materials are crystalline because atoms and molecules tend to form ordered arrangements, and since the interatomic distances are comparable with the wavelength of X-rays, their interaction creates diffraction patterns. The intensity in these patterns changes with crystal quality. Perfect crystals,e.g. semiconductors, fit well to dynamical theory, whereas crystals that reveal the stereochemistry of complex biological molecules, the structure of organic and inorganic molecules and powders are required to be fragmented (termed `ideally imperfect') to justify the use of the simpler kinematical theory. New experimental results of perfect and imperfect crystals are interpreted with a fundamental description of diffraction, which does not need fragmented crystals but just ubiquitous defects. The distribution of the intensity is modified and can influence the interpretation of the patterns.

Synthesis and properties of high temperature phthalonitrile polymers based on o, m, p-dihydroxybenzene isomers

o-BDB, m-BDB and p-BDB monomers with imperfect crystal structures were synthesized by a nucleophilic substitution reaction.

Multiple Bragg reflection by a thick mosaic crystal

Symmetric Bragg-case reflections from a thick, ideally imperfect, crystal slab are studied mostly by analytical means. The scattering transfer function of a thin mosaic layer is derived and brought into a form that allows for analytical approximations or easy quadrature. The Darwin–Hamilton equations are generalized, lifting the restriction of wavevectors to a two-dimensional scattering plane. A multireflection expansion shows that wavevector diffusion can be studied independently of the real-space coordinate. Combining analytical arguments and Monte Carlo simulations, multiple Bragg reflections are found to result in a minor correction of the reflected intensity, a moderate broadening of the reflected azimuth angle distribution, a considerable modification of the polar angle distribution, and a noticeable shift and distortion of rocking curves.

The Concept of X-Ray Standing Wave

The concept of “Standing Waves” (SW) that arise in the crystal dynamically “attacked” by the frequency fields X is analytically analyzed towards expressing, in almost all the cases, the total intensity of the fields on dispersion branches in the perfect crystal and for the embedded layer on the imperfect crystal using various extensions of the semi-classical dynamic theory, adapted or reparameterized, depending on the specific conditions of analysis performed.

Preparation and characterization of supported magnetic nanoparticles prepared by reverse micelles

Monatomic (Fe, Co) and bimetallic (FePt and CoPt) nanoparticles were prepared by exploiting the self-organization of precursor loaded reverse micelles. Achievements and limitations of the preparation approach are critically discussed. We show that self-assembled metallic nanoparticles can be prepared with diameters d = 2–12 nm and interparticle distances D = 20–140 nm on various substrates. Structural, electronic and magnetic properties of the particle arrays were characterized by several techniques to give a comprehensive view of the high quality of the method. For Co nanoparticles, it is demonstrated that magnetostatic interactions can be neglected for distances which are at least 6 times larger than the particle diameter. Focus is placed on FePt alloy nanoparticles which show a huge magnetic anisotropy in the L10 phase, however, this is still less by a factor of 3–4 when compared to the anisotropy of the bulk counterpart. A similar observation was also found for CoPt nanoparticles (NPs). These results are related to imperfect crystal structures as revealed by HRTEM as well as to compositional distributions of the prepared particles. Interestingly, the results demonstrate that the averaged effective magnetic anisotropy of FePt nanoparticles does not strongly depend on size. Consequently, magnetization stability should scale linearly with the volume of the NPs and give rise to a critical value for stability at ambient temperature. Indeed, for diameters above 6 nm such stability is observed for the current FePt and CoPt NPs. Finally, the long-term conservation of nanoparticles by Au photoseeding is presented.