Calculation of the phase diagram of−43He solid and liquid mixtures

The phase diagram of ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures has been calculated for temperatures from ∼0.02 to ∼1 K using well-known phenomenological models to describe the liquid and crystalline phases. For the bcc and hcp phases the regular solution model is shown to agree with measurements of the isotopic phase separation and the hcp-bcc transformation. The theory can be applied in thermodynamic equilibrium or when the crystalline transformation is too slow for complete equilibrium to be reached. For the liquid phases the theory is confined to fairly dilute solutions of either ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ or ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, where the models of Landau and Pomeranchuk and Zharkov and Silin can be applied. The input parameters for the calculation are taken mostly from measurements made on the pure isotopes or single-phase mixtures. The analysis includes the determination of the free-energy difference at T=0 between the bcc and hcp structures in both pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The calculated phase diagram agrees well with experiment, mostly within the experimental uncertainties which are sometimes quite large. The comparison with experiment includes measurements of freezing and melting curves, univariants (three-phase equilibrium lines) and the two quadruple points. Parts of the phase diagram which have not yet been measured are described in detail.