Using the $K$ matrix as computed in the study of nuclear matter and liquid ${\mathrm{He}}^3$ as the effective interaction at the Fermi surface, the possible superfluidity of these systems has been investigated. A theory of the cooperative phenomenon valid for particle-particle interaction in states of arbitrary angular momentum has been developed following the methods of Bardeen, Cooper, and Schrieffer. It is found for states of relative angular momentum other than $l=0\mathrm{}$ that the particle pairs must be correlated with respect to an arbitrary direction in the medium. As a result the change of structure of the Fermi surface is angularly dependent. An energy gap does not occur other than for $l=0\mathrm{}$, the particle excitation energy vanishing for certain orientations of the momentum. It is also shown that the specific heat shows a discontinuity at the transition temperature, but somewhat different from the case of $l=0\mathrm{}$.

Application of these results to liquid ${\mathrm{He}}^3$ shows that the cooperative effects arise from the interaction in the state with $l=2\mathrm{}$, and that the transition temperature is at about 0.1°K. In nuclear matter the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ interaction is very weak and probably repulsive at the Fermi surface, and the attractive ${}_{}{}^1{}_{}{}^{}D_2^{}{}_{}{}^{}$ interaction gives a negligible energy shift. The ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ interaction is attractive and in nuclear matter gives a few-tenths of an Mev energy gap. These results suggest that in finite nuclei, with pairing of identical nucleons in the same shell, the cooperative effects are not strictly analogous to those in nuclear matter but instead are closely associated with the finite level spacing.

• Received 15 January 1960

DOI:https://doi.org/10.1103/PhysRev.118.1442