Nuclear Pairing Energies with Simple Interactions

A brief review is given of the estimation of nuclear pairing energies using interactions with constant matrix elements, in analogy with the theory of superconductivity. More realistic potentials are discussed in the solution of the nuclear-matter problem, and an analytic expression for the pairing parameter $\Delta$ is obtained using a simple separable $S$-state interaction which approximately fits the two-nucleon scattering data up to the point where the phase shift turns negative. The pairing energies obtained are of the same order as those obtained by other authors, and smaller than those found experimentally for finite nuclei. It is argued that since the interaction is a decreasing function of the kinetic energy, the average coupling will be increased by the low-density surface region, and the effect of the latter is then taken into account by assuming the validity of the Thomas-Fermi approximation. An average coupling constant is calculated using the same separable interaction as was used in the nuclear-matter case, and an expression for $\Delta$ in the case of uniformly spaced single-particle levels is obtained by making a simple approximation for the off-diagonal matrix elements. It is found that for a trapezoidal one-body potential $\Delta \approx 4.0A^{-\frac{1}{3}}$ in agreement with the empirical values, and this result is discussed in relation to the approximations made during its derivation, and to the various parameters entering into the theory.