Cooper pairs in atomic nuclei

We consider the development of Cooper pairs in a self-consistent Hartree-Fock mean field for the even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS treatment and an exact treatment using the Richardson ansatz. For a fixed value of the pairing strength, projected BCS captures much of the pairing correlation energy that is absent from BCS, but still misses a sizable component, typically of order 1 MeV. Furthermore, because it does not average over the properties of the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper wave function and of the fraction of pairs that are collective.

Figure 1

The modified Yang prescription for the BCS treatment of pairing (smooth curve) and the alternative prescription discussed in the text (sawtooth curve) for the exact Richardson treatment. $G_0=0.106$ MeV denotes the physical value of the pairing strength and ${\epsilon }_1=\mu$ denotes the strength at which the whole system binds.

Figure 2

Pair energies (in MeV) for the exact Cooper pairs that emerge from four calculations of the ${}^{154}\mathrm{Sm}$ isotope. $G=0.106$ MeV is the physical value of the pairing strength. In that panel, we denote the most collective pairs as ${\text{C}}_i$, for subsequent notational purposes.

Figure 3

Square of the wave function of the most collective Cooper pairs in ${}^{154}\mathrm{Sm}$ (denoted ${\text{C}}_1$, ${\text{C}}_2$, ${\text{C}}_3$, ${\text{C}}_4$, and ${\text{C}}_5$) and the pair correlator (BCS) versus the single-particle levels.