Magnetism in solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$

The theory of magnetism in solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is reviewed with emphasis on the multiple-spin-exchange theory of the magnetic interaction. Critical experiments are discussed briefly, and the most reliable present values for various quantities are chosen. Various theories of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ magnetism are considered, with the conclusion that multiple particle exchange is the most likely mechanism. The microscopic theory of exchange is presented mainly from the standpoint of tunneling ducts. The spin Hamiltonian which results from two-, three-, and four-particle ring exchange is derived and studied in the high-temperature limit, in mean-field approximation, and with spin waves. Two reasonable parameter sets are selected which fit the experimental data, with the conclusion that the exchanges are mainly three- and four-particle rings. A discussion is given of remaining problems together with some conjectures.