Comparison of the Site Occupancies Determined by Combined Rietveld Refinement and Density Functional Theory Calculations: Example of the Ternary Mo–Ni–Re σ Phase

Abstract

The site occupancies of the Mo-Ni-Re σ phase have been studied as a function of the composition in the ternary homogeneity domain by both experimental measurements and calculations. Because of the possible simultaneous occupancy of three elements on the five sites of the crystal structure, the experimental determination of the site occupancies was achieved by using combined Rietveld refinement of X-ray and neutron diffraction data, whereas calculation of the site occupancies was carried out by using the density functional theory results of every ordered (i.e., 3(5) = 243) configuration appearing in the ternary system. A comparison of the experimental and calculation results showed good agreement, which suggests that the topologically close-packed phases, such as the σ phase, could be described by the Bragg-Williams approximation (i.e., ignoring the short-range-order contributions). On the other hand, the atomic distribution on different crystallographic sites of the Mo-Ni-Re σ phase was found to be governed by the atomic sizes. Ni, having the smallest atomic size, showed a preference for low-coordination-number (CN) sites, whereas Mo, being the largest in atomic size, preferred occupying high-CN sites. However, the preference of Re, having intermediate atomic size, varied depending on the composition, and a clear reversal in the preference of Re as a function of the composition was evidenced in both the calculated and experimental site-occupancy results.

Non-Stoichiometry and Calphad Modeling of Frank-Kasper Phases

“…Increased computer performance has allowed researchers to investigate more complex problems. For the first time, the total energies of every ordered configuration of a ternary σ phase, i.e., 3 5 = 243 configurations in the Cr-Mo-Re were calculated in [46], followed by studies of σ Cr-Ni-Re [47] and σ Mo-Ni-Re [20]. For the latter, the use of BWA showed a very good agreement with the experimental ternary site occupations.…”

“…Therefore, combined analysis of different diffraction data sets in which the contrast is different has to be used. Two techniques are in this case available: resonant diffraction [18,19] and combined X-ray and neutron analysis [20].…”

Irregular Homogeneity Domains in Ternary Intermetallic Systems

“…This technique was used in the case of Zr-W-Re [3], Mo-Re-Ni [4], and Re-Cr-Co systems (this work) in order to obtain the ternary site occupancies for the χ and the σ phases. In each case the neutron diffraction measurements were performed at Laboratoire Léon Brillouin (common laboratory CEA-CNRS) in Saclay, France using the instrument 3T2.…”

“…For each ordered configuration, both the internal atomic coordinates and the lattice parameters were fully relaxed using successive steps, keeping the original cell structure. The conditions for the calculations (method, accuracy) have been detailed previously [3,4]. All the calculations have been handled using the ZenGen code [6] (generation of input files, job monitoring, extraction of results).…”

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

“…In the 1950s Frank and Kasper (3,4) recognized complex tetrahedral atomic-and molecular-packing geometries that bridge the familiar close-packed crystals [e.g., face-centered cubic (FCC), hexagonally close-packed (HCP), and body-centered cubic (BCC) structures] characterized by periodic order, and quasiperiodic crystals (QCs) that extend crystallography beyond the 230 space groups relevant to periodic crystals (5,6). The scientific literature is replete with examples of Frank-Kasper phases in hard materials, particularly in the area of intermetallics (7)(8)(9), but also in a few complex elemental crystals, including manganese (10,11) and uranium (12). Recently, this class of crystalline order has cropped up in a host of soft materials, including dendrimers (13), surfactant solutions (14), and block polymers (15,16), often in close proximity to QC phases (17)(18)(19).…”