Equilibrium concentration of point defects in crystalline⁴He at O K

Abstract

We calculate the concentrations of vacancies and intersitials in the ground state of a Bose solid which models⁴He. Because ground-state boson wave functions are nodeless, their probability densities correspond to classical Boltzmann factors, and properties of Bose solids, such as the concentration of vacancies and interstitials, can be calculated using classical statistical mechanics. We model the ground-state wave function of⁴He with the product (Jastrow) form that corresponds to a classical 1/r ^b pair potential, and use a quasiharmonic approximation to calculate the concentrations of vacancies and interstitials in an fcc lattice with this potential. We find that the fractional concentration of vacancies at the melting point is 1.60×10⁻⁵ for 1/r ⁹ and 6.36×10⁻⁶ for 1/r ⁶, while the interstitial fractional concentrations are 1.32×10⁻³ and 1.08×10⁻⁵, respectively; the defect concentrations decrease by 7–16 orders of magnitude when the crystal density increases by 50%. At the same density, and with the same 1/r ⁹ potential, the concentration of vacancies in an hcp lattice is essentially the same as in an fcc lattice, but the interstitial concentration is much lower, apparently because the fcc lattice contains a more favorable split-interstitial site than does hcp. Therefore, our fcc vacancy results should be directly relevant for (hcp)⁴He, providing what we think is a lower bound on the vacancy concentration, while the interstitial concentration in⁴He is probably much lower than our results.