Structure of the quartetting ground state of $N=Z$ nuclei

Abstract

The formal equivalence between the quartetting picture and the symmetry restored BCS picture is established for the ground state correlations induced by the general isovector-isoscalar pairing interaction. Multiple ground state structures compatible with the particle number and isospin symmetries are evaluated. The competition of isovector and isoscalar correlations is discussed for the $N=Z$ nuclei above $^{100}$Sn.

Appendix A

We present below the results obtained using a weaker pairing interaction with respect to the main text, $V_{0}^{T=1}=V_{0}^{T=0}=300$ MeV fm$^3$. The Tables 4, 5 and 6 below are constructed in an analogous manner with the Tables 1, 2 and 3 in Sect. 3.

We note that the values of the correlation energies in Table 4 are about half of those in Table 1, in better agreement with the combined contributions to the binding energy of like-particle pairing and proton-neutron correlations (see also the discussion below Eq. 14).

For most nuclei the $|PQCM\rangle$, ${\mathcal {P}}_{NT}|BCS\rangle$, $|\text {iv}\oplus \text {is}\rangle$ and $|\text {iv}\rangle$ ansatzes become numerically equivalent in the present weaker pairing regime, with differences in the correlation energies of 0.01 MeV or less, and overlaps of 99.9% or more with the $|PQCM\rangle$ state, as seen in Tables 4 and 5.

For the nuclei above $^{100}$Sn, the errors for the pure isoscalar solution $|\text {is}\rangle$ in Table 6 are smaller than those of Table 3, but they still do not show any improvement with increasing mass number.

Table 4 Correlation energies (in MeV) calculated with the states $|PQCM\rangle$ of Eq. (2), ${\mathcal {P}}_{NT}|BCS\rangle$ of Eq. (6), $|\text {iv} \oplus \text {is}\rangle$ of Eq. (10) and with $|\text {iv}\rangle$ and $|\text {is}\rangle$ of Eq. (9), for $V_{0}^{T=1}=V_{0}^{T=0}=300$ MeV fm$^3$

Table 5 Overlaps (in percentages) between the $|PQCM\rangle$ state of Eq. (2) and the states ${\mathcal {P}}_{NT}|BCS\rangle$ of Eq. (6), $|\text {iv} \oplus \text {is}\rangle$ of Eq. (10), $|\text {iv}\rangle$ and $|\text {is}\rangle$ of Eq. (9), for $V_{0}^{T=1}=V_{0}^{T=0}=300$ MeV fm$^3$

Table 6 Correlation energies (in MeV) for the states $|\text {iv}\oplus \text {is}\rangle$ of Eq. (10), $|\text {iv}\rangle$ and $|\text {is}\rangle$ of Eq. (9) for 1–7 quartets above $^{100}$Sn, together with the relative error of the $|\text {is}\rangle$ correlation energies with respect to the $|\text {iv}\oplus \text {is}\rangle$ values, for $V_{0}^{T=1}=V_{0}^{T=0}=300$ MeV fm$^3$