Theory of Exchange in CrystallineHe3

In this work a theory of exchange in crystalline ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is developed. A spin Hamiltonian is deduced and its matrix elements calculated by cluster-expansion methods. A Heisenberg form of the spin Hamiltonian is found adequate, but with the customary exchange integral replaced by an exchange operator, which has matrix elements in phonon space. The diagonal elements of the exchange operator are used to derive an expression for the exchange frequency which depends explicitly on the phonon spectrum of the crystal and on the short-range (hard-core) correlations among particles. Simple arguments are used to show that this expression gives the two outstanding features of exchange in crystalline ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, viz., that the exchange frequency increases exponentially as the lattice constant increases and that it is antiferromagnetic. The density and the temperature dependence of the exchange frequency are calculated using a phonon spectrum and short-range correlation function obtained in a self-consistent fashion from a variational treatment. Various approximations to the phonon spectrum and the short-range correlation function are used in the calculation of exchange frequency, and the results compared. For example, it is found that the correct form of the short-range correlation function for small interparticle distance and the anisotropy of the pair-distribution function must be considered for good agreement with experimentally deduced results.