Search

Two X-ray methods are applied to estimate the strain energy of crystals containing dislocations, a simpler method based on the full width at half-maximum (FWHM) of the diffraction peaks, and asymptotic line profile analysis (LPA), which exploits the functional form of the Fourier transform corresponding to small Fourier parameters. It is shown analytically that, in the single-defect approximation, the modified Williamson–Hall (mWH) plot of the FWHMs is linear and the slope of the line is directly related to the strain energy of the dislocation system. Evaluation of the numerically generated peaks for randomly arranged edge dislocation dipoles shows that the method based on the mWH plot gives accurate strain energy, while asymptotic LPA overestimates it by about 50%. The accurate result given by the mWH plot is explained by the long correlation distance associated with the FWHM, which better captures the dislocation arrangement over large distances. By contrast, asymptotic LPA is related to atomic correlations over distances smaller than the mean dislocation–dislocation spacing, where the displacement gradient is mainly determined by the field of single dislocations. Therefore, asymptotic LPA leads to a very accurate dislocation density (with error less than 1%) and the result is independent of the dislocation arrangement. However, these short-range correlations overestimate the outer cut-off radius by one order of magnitude.

Three-dimensional X-ray diffraction was applied to characterize the strain/stress evolution in individual grains of an Al–0.3 wt% Mn polycrystal deformedin situat a synchrotron source. Methodological aspects concerning the calibration of the geometrical setup and the evaluation of the strain/stress tensors are discussed. A two-step calibration method separately treating the detector and the rotation axis allows one to determine the centre-of-mass position and crystallographic orientation of grains with standard errors of about 1.5 µm and 0.02°, respectively. Numerical simulations indicate that the error of normal strain components (about 1 × 10−4) is mainly caused by calibration errors, while the error of shear components (about 0.5 × 10−4) is largely influenced by counting statistics and random spot-centre errors due to detector distortion. The importance of monitoring the beam energy is emphasized.

The calibration of high-energy X-ray diffraction setups using an area detector and a rotation axis is discussed. The characterization of the tilt and spatial distortions of an area detector was discussed in part one of this series [Borbély, Renversade, Kenesei & Wright (2014).J. Appl. Cryst.47, 1042–1053]. Part II links the detector frame to the laboratory frame comprising an additional rotation axis and introduces a general diffractometer equation accounting for all sources of misalignment. Additionally, an independent high-accuracy method for the evaluation of the crystallographic orientation and cell parameters of the undeformed reference crystal is presented. Setup misalignments are mainly described in terms of a residual strain tensor, considered as a quality label of the diffractometer. The method is exemplified using data sets acquired at beamlines ID11 (European Synchrotron Radiation Facility) and 1-ID (Advanced Photon Source) on Al and W single crystals, respectively. The results show that the residual strain tensor is mainly determined by the detector spatial distortion, and values as small as 1–2 × 10−4can be practically achieved.

Two different methods of diffraction profile analysis are presented. In the first, the breadths and the first few Fourier coefficients of diffraction profiles are analysed by modified Williamson–Hall and Warren–Averbach procedures. A simple and pragmatic method is suggested to determine the crystallite size distribution in the presence of strain. In the second, the Fourier coefficients of the measured physical profiles are fitted by Fourier coefficients of well establishedab initiofunctions of size and strain profiles. In both procedures, strain anisotropy is rationalized by the dislocation model of the mean square strain. The procedures are applied and tested on a nanocrystalline powder of silicon nitride and a severely plastically deformed bulk copper specimen. The X-ray crystallite size distributions are compared with size distributions obtained from transmission electron microscopy (TEM) micrographs. There is good agreement between X-ray and TEM data for nanocrystalline loose powders. In bulk materials, a deeper insight into the microstructure is needed to correlate the X-ray and TEM results.

It has been shown recently that in many cases strain anisotropy in powder diffraction can be well accounted for by the dislocation model of the mean square strain. The practical application assumes knowledge of the individual contrast factorsCof dislocations related to particular Burgers, line and diffraction vectors or to the average contrast factorsC¯. A simple procedure for the experimental determination ofC¯has been worked out, enabling the determination of the character of the dislocations in terms of a simple parameterq. The values of the individualCfactors were determined numerically for a wide range of elastic constants for cubic crystals. TheC¯factors andqparameters were parametrized by simple analytical functions, which can be used in a straightforward manner in numerical analyses, ase.g.in Rietveld structure refinement procedures.

Electrodeposited nanocrystalline Ni foils were studied by high-resolution X-ray diffractometry. The full width at half-maximum and Fourier coefficients were found to vary rather anisotropically as a function of diffraction order. The modified Williamson–Hall plot and the modified Warren–Averbach analysis, developed recently by taking into account the dislocation contrast in peak broadening, have been applied to interpret this anisotropic behaviour in terms of grain size, dislocation densities and twin boundaries. The average grain size has been found to range between 50 and 12 nm, in good agreement with transmission electron microscopy observations. The average dislocation density has been found to be 4.9 (5) × 1015 m−2 and the dislocations are of the screw character.

The computer programANIZChas been developed using the Pascal programming language for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals. The contrast factor is obtained numerically by integrating the angular part of the distortion tensor in the slip plane. The distortion tensor is calculated by solving the sextic equation provided by the mechanical equilibrium of a single dislocation in an infinite anisotropic medium. The contrast factors can be used for the interpretation of strain anisotropy as obtained from peak profile measurements made on either single crystals, textured polycrystals or powders.

The latest virtual special issue of Journal of Applied Crystallography includes some highlights of the 12th Conference on High-Resolution X-ray Diffraction and Imaging (XTOP). This biennial conference was held in Villard-de-Lans and Grenoble in September 2014, and was organized by the ESRF, Universite Grenoble-Alpes, the CEA and CNRS. A refresher school for PhD students and postdoctoral scientists took place on the first day, with lectures on new diffraction and imaging techniques. The first XTOP conference (‘X-ray Topography and High Resolution Diffraction’) took place in Marseille in 1992. The original topics covered were diffraction topography, double- and triple-crystal diffractometry of epitaxial layers, reflectometry, and the standing wave technique. Many experimental techniques have been added since, which cover today a wide range to reflect the scientific evolution of the XTOP community: coherent imaging, X-ray phase contrast imaging (radiography and microtomography), microdiffraction (monochromatic and Laue), grazing incidence and resonant diffraction, fast (time-resolved) tomography etc. The topics covered now include both synchrotron and laboratory instrumentation, including industrial developments. The materials studied include single crystals and multilayers, nanostructures (metals, oxides, semiconductors), and organic, biological and other natural (sea shells, fossils etc.) samples. A few presentations in 2014 covered advances in the theory of high-resolution X-ray diffraction, and with the occasion of the International Year of Crystallography, Andre Authier gave a presentation on the Early Days of X-ray Diffraction and the Birth of the Dynamical Theory. Among the topics presented during the XTOP conference, the most noteworthy included the development of high temporal and spatial resolution for diffraction and imaging, the use of X-ray coherent beams (including ptychography and X-ray free electron lasers), and applications to nanostructured materials. Following the traditional publication policy of the XTOP conferences, the articles presented in this issue are not selected proceedings but rather original research articles, which were first presented at the XTOP conference. These include laboratory and synchrotron studies of semiconductors, software for fast data analysis or instrumental function simulation, micro-Laue analysis of oxide fuel cells and bent nanowires, microtomography of bone structure, X-ray phase contrast imaging of bacterial endospores, and Bragg imaging of ice under applied stress. Over 200 participants attended the five-day conference, which included six tutorial lectures, 13 invited and 44 contributed presentations, and 144 posters.