Pair Distribution Function for Interacting Bosons and the Ground-state Energy of Solid Helium-4

Abstract

The pair distribution function g(r) for a system of bosons interacting through a two-body potential composed of a hard core followed by a square well has been obtained using the reaction matrix formalism for the energy excitation spectrum Ek and the fundamental definition relating the structure factor S(k) and g(r). This has been used in obtaining the ground state energy of solid ⁴He using the kinetic energy and potential energy expressions of Hansen and Levesque. The most stable ground state of solid ⁴He corresponds to a potential width of b = 3.8 Å, but the corresponding ⟨E⟩/N for the density, ρ, of solid ⁴He is very large compared with the experimental value. However, for b = 4.81 Å and ρ = 2.8 × 10²² particles/cm³, ⟨E⟩/N = -11.24 K = -93.4 Joule/mol, and for b = 4.82 Å and ρ = 2.5 × 10²² particles/cm³, ⟨E⟩/N = -6.84 K ≈ -56.8 Joule/mol. The experimental value for the ground state energy of solid ⁴He is -59.5 J/mol.