A sphere-packing model for the prediction of lattice parameters and order in σ phases

Abstract

A model is presented for the determination of the unit-cell size for ordered and disordered binary o" phases from a geometry assuming spherical atoms. The basis of the calculations is the assumption of an average radius of the atoms occupying each of the five crystallographic sites in the unit cell. The model is then used to predict the changes in lattice parameters produced by fast-neutron irradiation and by the addition of a third element in some ternary a phases. The calculated parameters are compared with experimental results.

The ordering of the σ phases Cr2Ru and Cr2Os

“…It is worth pointing out that even those authors who base their predictions of ordering in a phases on size-factor considerations assume that some overlap occurs between spheres occupying the J sites [Wilson & Spooner, 1973, Fig. 2(b)]; the need for admitting that such deformed spheres occur can only be interpreted in terms of electronic interaction between neighbouring J sites.…”

“…(1) Kasper & Waterstrat (1956); (2) Spooner & Wilson (1964); (3) Pearson (1967); (4) Reuth & Waterstrat (1968); (5) Wilson & Spooner (1973); (6) Waterstrat & Dickens (1974); (7) Present work.…”

“…According to these authors, it would be desirable to re-examine Cr2Ru and Cr2Os for which the ratio RA/R B takes intermediate values, 0.96 and 0.95 respectively; this might help in establishing the size ratio for which the changeover from sizecontrolled to electronic-structure-controlled ordering occurs in o phases. Wilson & Spooner (1973) have proposed a model for the prediction of the unit-cell parameters and of the ordering in a phases, based on sphere-packing considerations. A comparison between the calculated and observed values of a and c has been made by these authors for a large number of o phases for which experimental data are available; the overall agreement was found to be better than 1%.…”

The effect of fast-neutron irradiation on Zr4Al3

“…It is shown in another publication (Wilson & Spooner, 1973) that calculations very similar to those above can be used to estimate the sizes of the ordered and disordered unit cells of binary a phases. These calculations lead to an increase in the c parameter and a very much smaller change, which is usually negative, in the a parameter of the tetragonal unit cell.…”

Characterization of the invariant reaction involving the L, η, γ and σ phases in the Ti–Al–Nb system

“…The volume fractions of the phases were measured from the SEM micrographs. The crystal structures, and the lattice parameters of each phase r [22][23][24], c [23,25,26] and g [27], were combined with the measured compositions to estimate the density and subsequently calculate the weight per cent (wt.%) of each phase from its volume per cent (vol.%). Comparison of the calculated wt.% (via compositional analysis) and the measured wt.% (via microstructural analysis) shown in Table 3 are generally in good agreement.…”