Direct solution of the diffraction pattern of a crystal with planar faulting

Abstract

A direct formalism for the solution of the diffraction pattern from the faulted layer crystal is derived. The proposed method is not specific for any crystal structure. The solution avoids the need for specific planar faulting models and has a direct physical meaning. The correlation distribution function between lateral displaced layers can be directly obtained from the diffracted intensities. The solution was compared successfully to a Monte Carlo trial and error method for the fcc structure. The developed formalism was used to determine the layer-layer correlation distribution function of the Gd 2 Co 17 faulted layer structure.

Pairwise correlations in layered close-packed structures

“…Disordered layered materials are often studied via the pairwise correlations between layers, as these correlations are experimentally accessible from the Fourier analysis of a diffraction pattern (Estevez-Rams, Martinez et al, 2001;Estevez-Rams, Penton-Madrigual et al, 2001;Varn et al, 2002Varn et al, , 2013a, or directly from simulation studies (Kabra & Pandey, 1988Shrestha & Pandey, 1996a,b, 1997Shrestha et al, 1996;Varn & Crutchfield, 2004). Such studies yield important insights into the structural organization of materials.…”

“…In the following, we demonstrate how pairwise correlation functions can be calculated either analytically or to a high degree of numerical certainty for an arbitrary HMM and, thus, for an arbitrary "-machine. Previous researchers often calculated pairwise correlation functions for particular realizations of stacking configurations (Berliner & Werner, 1986;Kabra & Pandey, 1988;Shrestha & Pandey, 1997;Estevez-Rams, Martinez et al, 2001;Varn et al, 2013a) or from analytic expressions constructed for particular models (Tiwary & Pandey, 2007;Estevez-Rams et al, 2008;Varn & Crutchfield, 2004). The techniques developed here, however, are the first generally applicable methods that do not rely on samples of a stacking sequence.…”

The use of analytical peak profile functions to fit diffraction data of planar faulted layer crystals

“…This approach differs from other treatments in two main points 1) It avoids the need of any assumption about the specific planar disorder and can be used independently of the planar faulting density or interaction; 2) Does not assume an underlying stacking sequence "perturbed" by planar disorder. The formalism calculates from the diffraction data the correlation function giving the probability of finding two layers, ∆ layers apart and displaced one from another a given displacement vector [13]. Once the correlation functions are determined the characteristic length describing the range of correlation between the layers can be directly obtained.…”

“…The approach described in [13] should not be confused with Monte Carlo simulations based on Ising models such as those mentioned above. While Ising models try to explain why a particular stacking arrangement occurs based on the competing interaction of different near neighbors, the so called direct solution merely intends to describe directly from the experimental data which particular stacking order is occurring.…”

“…A layer correlation length, ∆c has been defined to take account of the lost of correlation between the stacks with increasing ∆ value [13]. This correlation length is similar to the one introduced by Shrestha and Pandey [11 and 12] in Ising model Monte Carlo simulations of stacking disorder.…”