Dislocation densities, arrangements and character from X-ray diffraction experiments

Abstract

X-ray diffraction peak profile analysis has become a powerful tool during the last two decades for the characterisation of microstructure either in the bulk or in loose powder materials. The evaluation and modelling procedures have developed together with the experimental techniques. It will be shown that the different features of diffraction peak profiles such as (i) broadening, (ii) asymmetric peak shape, (iii) peak shifts and (iv) anisotropic broadening provide a variety of microstructural parameters by modelling crystallite size and strain. Modelling strain by assuming dislocations will be more extensive. Two different procedures will be considered: (1) evaluation by using characteristic parameters of individual peak profiles, especially the FWHM, the integral breadths and the Fourier coefficients and (2) multiple whole profile fitting (MWPF) procedure using ab initio size and strain functions scaled by the contrast factors of dislocations. The two procedures will be discussed and illustrated by different case studies.

Introduction

Dislocations are a general feature in crystalline materials. In metals and alloys they are always present. In semiconductors dislocations are often a nuisance where technology tries to avoid them by any means. In ionic crystals they are a curiosity and in geological crystals they can tell us about the history of stones. The wide variety of their appearance and the difference in their usefulness or harm invoked a number of experimental techniques in order to keep track of their properties and presence. X-ray diffraction techniques and instrumentation have made tremendous developments during the last two decades due to high resolution techniques and synchrotron sources enabling to measure diffraction profiles with high accuracy and to resolve effects on profile tails caused by coherent elastic or diffuse scattering. Different techniques of reciprocal space mapping make it possible to separate between contributions of microtilts and lattice spacing changes which is especially informative in the analysis of the dislocation structures in thin films or multilayers. In low quality thin films the dislocation structures resemble that of chaotically distributed dislocations in bulk materials. The large variety of dislocation structures: (i) low density misfit dislocations in nearly perfect thin films and multilayers, (ii) dislocation clusters in low quality layered structures, (iii) special dislocation structures in intermetallic alloys revealing planar glide, (iv) interface dislocations in heterogeneous microstructures or (v) chaotically distributed dislocations in bulk materials, especially metals and alloys, all need special X-ray techniques and/or special methods, theories and models for the evaluation of experimental data. Taking into account all X-ray diffraction methods for studying dislocations from topography through reciprocal lattice mapping to peak broadening dislocation densities from 100 to 10¹⁸ m⁻² are accessible, indicating the powerfulness of X-ray diffraction methods. In the present paper only the methods, theories and models relevant for chaotically distributed dislocations in bulk materials will be considered. It is important to note, however, that even in this type of dislocation structures high resolution X-ray diffraction techniques have revealed interface dislocations due to the heterogeneity of the microstructures. This aspect will also be discussed briefly. Direct methods cluster around transmission electron microscopy (TEM). The observation of dislocations and Burgers vector analysis by conventional TEM is more or less routine up to about 10¹⁴ m⁻² local densities. This limit can easily be extended up to $\text{5×10}{}^{14}\text{m}{}^{\mathrm{-2}}$ local densities by weak beam techniques. Lattice imaging, when possible, enables observation up to even higher densities. Apart from their presence, dislocations have a number of other properties related to interaction and collective behaviour. Screening of strain fields, long-range internal stresses, dipole character, fractal character, etc., all these properties are tedious to receive from TEM micrographs. A large number of other, nondirect methods, like X-ray diffraction, neutron diffraction, convergent beam electron diffraction, electron back scattering diffraction, magnetic Barkhausen noise, texture analysis, small angle scattering or electrical resistivity measurements, are complementary to TEM observations and provide useful information about the different physical properties, such as screening, or character of dislocation systems. In the present account X-ray or neutron diffraction peak profile analysis will be treated focussing on the information which can be derived from relevant experiments. The most important results will be illustrated by case studies.