X-ray diffraction line broadening: modeling and applications to high-Tc superconductors

Abstract

A method to analyze powder-diffraction line broadening is proposed and applied to some novel high-Tc 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 La1.85M0.15CuO4(M = Ca, Ba, Sr) high-Tc superconductors showed that microstrains and incoherently diffracting domain sizes are highly anisotropic. In the superconductors, stacking-fault probability increases with increasing Tc; 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-Tc phases indicate a possible mechanism for the phase transformation by the growth of faulted regions of the major phase.

Incipient low-temperature formation of MAX phase in Cr–Al–C films

“…The integral breadth method summarized in Ref. [33], calculates the root-mean-square strain (RMSS) and both surface-and volume-weighted domain sizes according to the "double-Voigt" method [34,35], which is equivalent to the Warren-Averbach approach [36].…”

Crystal structure of isotibolone: a major degradation product of tibolone

“…17, 18 The peak profiles were modeled by the Double-Voigt approach 15,19 with anisotropic peak broadening adjusted using spherical harmonics.…”

Microstrain Measurement in Plastically Deformed Austenitic Steel

“…To obtain the original sample broadened profile, which contains the information about the microstructural parameters of the sample, different approaches can be used. The most rigorous one is the Stokes deconvolution method combined with the Warren-Averbach analysis, since no assumption about the analytical form of diffraction-peak shape is required in this case (see Warren, 1959 (Balzar, 1993 (e2) 1/2 by the following formulas:…”

“…More recently, the neutron diffraction technique has also been applied for the determination of plastic strains which result from microstructural defects and lead to peak broadening (Macek et al, 1996). Different approaches are described in literature allowing to relate the diffraction profile parameters obtained by refinement with the microstructural parameters (for a review see Balzar, 1993). In the present study Keijser's approach was used to separate the line broadening due to the domain size and the mean-square microstrains.…”