Pair distribution function for interacting bosons and the ground-state energy of solid helium-4

Abstract

The pair distribution function <i>g</i>(<i>r</i>) 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 <i>E</i>_kand the fundamental definition relating the structure factor <i>S</i>(<i>k</i>) and <i>g</i>(<i>r</i>) . 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 <i>b</i>=3.8Ǻ ,but the corresponding <<i>E</i>>/<i>N</i> for the density, ρ, of solid ⁴He is very large compared with the experimental value. However, for <i>b</i> =4.81Ǻ and ρ=2.8×10²² particles/ cm³, <<i>E</i>>/<i>N</i> =-11.24 K= -93.4Joule/mol. and for <i>b</i>=4.82Ǻ and ρ=2.5×10²² particles/cm³, <<i>E</i>>/<i>N</i> = -6.84 K=56.8 Joule/mol. The experimental value for the ground state energy of solid ⁴He =-59.5J/mol.