A statistical mechanical study on the melting lines of ³He and ⁴He

Abstract

A theoretical melting line is sensitive to a small difference in solid and fluid free energies. Such a difference may arise from different approximations made in fluid and solid theories and also from inaccuracies in handling anharmonic lattice vibrations. Statistical mechanical calculations have been made for liquid and solid ⁴He and ³He and their melting lines, using a new perturbation theory (PT) which does not suffer from the above limitations. These calculations, which use the Aziz potential for helium, are augmented by additional calculations using quasi-harmonic lattice dynamics and Mansoori—Canfield—Ross theory (with an exponential-6 potential). Comparisons of these results and available Monte Carlo and real experimental data show that the PT can accurately predict solid, fluid, and the melting data of a model of helium (based on the Aziz potential) and that the melting line of real ⁴He is insensitive to many-body effects up to 200 K, but their influence grows gradually with pressure (≈6% in melting pressure at 300 K). An effective pair potential is suggested which can handle many-body contributions over an extended density range. The calculations on the isotopic pressure shift, δP _m = P _m(³He)-P _m(⁴He), along the melting lines of helium isotopes show that δP _m > 0 at T < 100K in agreement with experiment at T ≈ 30 K and δP _m < 0 at T > 100 K in agreement with path integral Monte Carlo data. The PT with the first-order quantum correction was found to explain the experimental melting data surprisingly well; i.e., beyond an estimated range of applicability of the Wigner—Kirkwood expansion. It can imply a rapid convergence of the Wigner—Kirkwood expansion along the melting line of He.