X-ray methods for strain energy evaluation of dislocated crystals

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.