X-Ray Diffraction Line Broadening: Modeling and Applications to High-T_c Superconductors

A method to analyze powder-diffraction line broadening is proposed and applied to some novel high-T_c superconductors. Assuming that both size-broadened and strain-broadened profiles of the pure-specimen profile are described with a Voigt function, it is shown that the analysis of Fourier coefficients leads to the Warren-Averbach method of separation of size and strain contributions. The analysis of size coefficients shows that the "hook" effect occurs when the Cauchy content of the size-broadened profile is underestimated. The ratio of volume-weighted and surface-weighted domain sizes can change from ~1.31 for the minimum allowed Cauchy content to 2 when the size-broadened profile is given solely by a Cauchy function. If the distortion co-efficient is approximated by a harmonic term, mean-square strains decrease linearly with the increase of the averaging distance. The local strain is finite only in the case of pure-Gauss strain broadening because strains are then independent of averaging distance. Errors of root-mean-square strains as well as domain sizes were evaluated. The method was applied to two cubic structures with average volume-weighted domain sizes up to 3600 Å, as well as to tetragonal and orthorhombic (La-Sr)₂CuO₄, which exhibit weak line broadenings and highly overlapping reflections. Comparison with the integral-breadth methods is given. Reliability of the method is discussed in the case of a cluster of the overlapping peaks. The analysis of La₂CuO₄ and La_1.85M_0.15CuO₄(M = Ca, Ba, Sr) high-T_c superconductors showed that microstrains and incoherently diffracting domain sizes are highly anisotropic. In the superconductors, stacking-fault probability increases with increasing T_c; microstrain decreases. In La₂CuO₄, different broadening of (h00) and (0k0) reflections is not caused by stacking faults; it might arise from lower crystallographic symmetiy. The analysis of Bi-Cu-O superconductors showed much higher strains in the [001] direction than in the basal a-b plane. This may be caused by stacking disorder along the c-axis, because of the two-dimensional weakly bonded BiO double layers. Results for the specimen containing two related high-T_c phases indicate a possible mechanism for the phase transformation by the growth of faulted regions of the major phase.