The Influence of Dipole-Dipole Coupling on the Specific Heat and Susceptibility of a Paramagnetic Salt

The mutual magnetic potential energy of atomic magnetic dipoles is unimportant in salts of high dilution at ordinary temperatures, but becomes important in determining the temperature scale in the new very low region obtained by magnetic cooling. This ``dipole-dipole'' energy is not to be confused with exchange interaction which is important in concentrated magnetic materials and which is there responsible for ferromagnetism at ordinary temperatures. The partition function, and hence the entropy, specific heat, and susceptibility are calculated for a paramagnetic solid inclusive of dipole and simultaneously also feeble exchange coupling. In Sections 3–4 the computation is made for atoms otherwise free, but in Section 6 they are subjected in addition to a crystalline Stark field. Comparison with experiment is made in the following paper by Hebb and Purcell. Our method of partition functions is to be contrasted with the usual, essentially static Lorentz method of representing dipole-dipole coupling by a local field, e.g., H+4πM/3 for a long test body. The Lorentz procedure is shown to be only a first approximation, which is really warranted if the density is so low or the temperature so high that one may neglect all terms but the first in the development of the partition function in 1/T. Otherwise the usual results of the local field method are obtained only by an extrapolation which is comparable with the assumption in Heisenberg's theory of ferromagnetism of identical energy for all states with the same crystalline spin. Two other types of extrapolation, based on a second approximation, are obtained which correspond respectively to assuming a Gaussian distribution of energies for these states and to use of the local field proposed by Onsager. The latter seems to be much the more satisfactory of the two.