Stacking and twin faults in close-packed crystal structures: exact description of random faulting statistics for the full range of faulting probabilities

Abstract

The classical model of independent random single deformation faults and twin faulting in face-centered-cubic and hexagonal close packing is revisited. The model is extended to account for the whole range of faulting probabilities. The faulting process resulting in the final stacking sequences is described by several equivalent computational models. The probability sequence tree is established. Random faulting is described as a finite-state automaton machine. An expression giving the percent of hexagonality from the faulting probabilities is derived. The average sizes of the cubic and hexagonal domains are given as a function of single deformation and twinning fault probabilities. An expression for the probability of finding a given sequence within the complete stacking arrangement is also derived. The probability P(0)(Delta) of finding two layers of the same type Delta layers apart is derived. It is shown that previous generalizations did not account for all terms in the final probability expressions. The different behaviors of the P(0)(Delta) functions are discussed.

Pairwise correlations in layered close-packed structures

“…and 2H CPSs: the RGDF process Estevez- Rams et al (2008) recently showed that simultaneous random growth and deformation stacking faults (SFs) in 2H and 3C CPSs can be modeled for all values of the fault parameters by a simple HMM, and this is shown in Fig. 8.…”

“…Warren (1969) uses P 0 m ; P þ m and P À m ; Kabra & Pandey (1988) call these PðmÞ; QðmÞ and RðmÞ; and Estevez- Rams et al (2008) use P 0 ðÁÞ; P f ðÁÞ and P b ðÁÞ. Since we prefer to reserve the symbol 'P' for other probabilities previously established in the literature, here and elsewhere we follow the notation of Yi & Canright (1996), with a slight modification of replacing 'Q r ðnÞ' with 'Q a ðnÞ'.…”

“…We refer to this process as the Random Growth and Deformation Faults (RGDF) process. 13 As has become convention (Warren, 1969;Estevez-Rams et al, 2008), refers to deformation faulting and refers to growth faults.…”

“…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.…”

“…Tiwary & Pandey (2007) calculated the size of domains in a model of randomly faulted close-packed structures (CPSs) by calculating the (exponential) decay rate of pairwise correlation functions between layers. Recently Estevez-Rams et al (2008) derived analytical expressions for the correlation functions for CPSs that contained both random growth and deformation faults, and Beyerlein et al (2011) demonstrated that correlation functions in finite-sized face-centered cubic crystals depend not only on the kind and amount of faulting, but additionally on their placement.…”