Paramagnetism of Liquid Helium Three

The object of the present paper is the elaboration of a theoretical model describing the paramagnetic properties of liquid ${\mathrm{He}}^3$ throughout the region of existence of this phase. The model is based on a molecular field theoretical type of approach. The ratios of the actual paramagnetic susceptibility of liquid ${\mathrm{He}}^3$ to the one it would have if it were an ideal paramagnet are predicted to be representable through a unique function depending on the reduced temperature variable. The latter contains the characteristic temperature of the nuclear spin system, which, at the present time, is only available empirically through the susceptibility ratio data. In the susceptibility ratio-reduced temperature representation all susceptibility ratios of liquid ${\mathrm{He}}^3$ fall on a single curve. This theoretical ratio curve describes very closely the experimental ratios, available through the work of the Duke University investigators, up to values of the reduced temperature of 0.90-1.0. Beyond this range the experimental susceptibility ratios become systematically larger than the calculated ratios, the differences between them being small. Quantitative arguments will be advanced which appear to explain satisfactorily these discrepancies and to indicate that the theory should be valid throughout the whole range of the natural reduced temperature variable. The spin entropy-spin susceptibility relation, established and used previously, yields, with the theoretical paramagnetism model, rigorous lower limits of the entropy, heat capacity and expansion coefficient of the liquid throughout the region of existence of this phase. With the recent extension of the melting pressure data to quite low temperatures by the University of Illinois investigators, the rigorous spin entropy of the liquid along the melting line allows one to estimate the entropy of the solid along the melting line and at low temperatures. Here the solid entropy turns out to be less than $R\ln 2$, per mole, yielding the temperature of its heat capacity anomaly to be below one hundredth of a degree Kelvin. The discussion of the liquid ${\mathrm{He}}^3$-solid ${\mathrm{He}}^3$ equilibrium, on the basis of the above results, seems to render questionable any analysis of the thermal properties of the solid which ignores the existence of its nuclear spin system even at medium temperatures. Finally, a semiquantitative description of the entropy-pressure diagram of ${\mathrm{He}}^3$ discloses various singular characteristics of the liquid entropy along the melting line in its pressure dependence, as well as the peculiar features of the solid entropy at and around the melting pressure anomaly.