Thermodynamic Behavior of Liquid Helium-Three in Its Possible Superfluid Phase. I

Abstract

The thermodynamic behavior of liquid ${\mathrm{He}}^3$ in its possible superfluid phase is investigated by extending the methods of Brueckner et al. They suggest that such a correlated phase can exist at very low temperatures due to the fact that there exist attractive $D$-state interactions near the Fermi surface. The free energy and the energy gap of the system for $D$-state interactions corresponding to different pure azimuthal modes are calculated at different temperatures. It is found that $l=2\mathrm{}$, $m=2\mathrm{}$ and $l=2\mathrm{}$, $m=1\mathrm{}$ modes correspond to the lowest free energy of the system near the critical temperature. In the intermediate range of temperatures the free-energy curves for the two modes, when the computations are made numerically, come out to be very nearly the same. But actually it can be shown by an analytical method that they are identical. The $l=2\mathrm{}$, $m=0\mathrm{}$ mode yields a higher free energy for all temperatures less than the critical temperature. The mixing of modes is investigated near the critical temperature. Any linear combination of all the modes $l=2\mathrm{}$, $m=0, 1, -1, 2, \mathrm{and} -2\mathrm{}$ does not seem to lead to a lower free energy than that of the $l=2\mathrm{}$, $m=\pm 2$, and $m=\pm 1$ modes. The correlation lengths at different temperatures are also analyzed. The specific heat and entropy curves for the $l=2\mathrm{}$, $m=2\mathrm{}$ mode are given.