Exchange in solid ³He and the Heisenberg Hamiltonian

The nuclear spins in solid 3He are coupled to one another as a result of the Pauli exclusion principle. The exchange energy is so small that the Pauli principle can be treated rigourously as a boundary perturbation, and the coupling of the spins is given correctly by the Heisenberg Hamiltonian. The coupling is antiferromagnetic in sign if, as is to be expected, two-particle exchange predominates, but may be ferromagnetic if three-particle exchange is more important. Only if four or more neighbouring particles can change places simultaneously must a more complicated form of the Heisenberg Hamiltonian be used. Herring's formula for the exchange energy is generalized to include exchange induced transitions between nearly degenerate levels of the system (spin-lattice coupling). It is demonstrated that coupling between the spin system and the lattice can be calculated if the exchange frequency is known as a function of strain, and a rough argument is given to show how different components of the strain tensor may affect this frequency. Temperature dependence of the exchange frequency is discussed. It is shown that at moderate temperatures thermally excited vacancies produce a new mechanism for spin exchange, which may work towards parallel alignment of the spins in the body-centred phase, but towards anti-parallel alignment in the hexagonal phase.