The experimental solubility limit of Zr in Al is well known. ${\mathrm{Al}}_3\mathrm{Zr}$ has a stable structure ${\mathrm{DO}}_{23}$ and a metastable one $\mathrm{L}{1}_2.$ Consequently there is a metastable solubility limit for which only few experimental data are available. The purpose of this study is to obtain, by ab initio calculations, the solubility limit of Zr in Al for stable as well as metastable phase diagrams. The formation energies of several ordered compounds ${\mathrm{Al}}_x{\mathrm{Zr}}_{\left(1\mathrm{}-x\right)},$ all based on a fcc underlying lattice, were calculated using the FP-LMTO (full-potential linear-muffin-tin-orbital) method. Taking into account all the relaxations allowed by the symmetry, we found the ${\mathrm{DO}}_{23}$ structure to be the stable one for ${\mathrm{Al}}_3\mathrm{Zr}.$ This set of results was then used with the cluster expansion in order to fit a generalized Ising model through the inverse method of Connolly and Williams. Different ways to consider volume relaxations were examined. This allowed us to calculate, in the Bragg-Williams approximation, the configurational free energy at finite temperature. According to the previous FP-LMTO calculations the free energy due to electronic excitations can be neglected. For the vibrational free energy of ordered structures, we compared results obtained from a calculation of the elastic constants used with the Debye model and results obtained from a calculation of the phonon spectrum. All these different steps lead to a calculation of the solubility limit of Zr in Al which is found to be lower than the experimental one. The solubility limit in the metastable phase diagram is calculated in the same way, and thus can be compared to the stable one.

• Received 17 September 2001

DOI:https://doi.org/10.1103/PhysRevB.65.094105