Global study of separable pairing interaction in covariant density functional theory

A systematic global investigation of pairing properties based on all available experimental data on pairing indicators has been performed for the first time in the framework of covariant density functional theory. It is based on the separable pairing interaction of Tian, Ma, and Ring [Phys. Lett. B 676, 44 (2009)]. The optimization of the scaling factors of this interaction to experimental data clearly reveals its isospin dependence in the neutron subsystem. However, the situation is less certain in the proton subsystem since similar accuracy of the description of pairing indicators can be achieved both with isospin-dependent and mass-dependent scaling factors. The differences in the functional dependencies of scaling factors lead to uncertainties in the prediction of proton and neutron pairing properties which are especially pronounced at high isospin and could have a significant impact on some physical observables. For a given part of the nuclear chart the scaling factors for spherical nuclei are smaller than those for deformed ones; this feature exists also in nonrelativistic density functional theories. Its origin is traced back to particle-vibration coupling in odd-$A$ nuclei which is missing in all existing global studies of pairing. Although the present investigation is based on the NL5(E) covariant energy density functional (CEDF), its general conclusions are expected to be valid also for other CEDFs built at the Hartree level.

Figure 1

Experimental neutron pairing indicators ${\mathrm{\Delta }}_{\nu }^{\left(3\right)}(Z,N), {\mathrm{\Delta }}_{\nu }^{\left(4\right)}(Z,N)$, and ${\mathrm{\Delta }}_{\nu }^{\left(5\right)}(Z,N)$ as a function of the neutron number $N$ for indicated isotopic chains of spherical nuclei.

Figure 2

The same as Fig. 1 but for proton indicators ${\mathrm{\Delta }}_{\pi }^{\left(3\right)}(Z,N), {\mathrm{\Delta }}_{\pi }^{\left(4\right)}(Z,N)$, and ${\mathrm{\Delta }}_{\pi }^{\left(5\right)}(Z,N)$ as a function of proton number $Z$ for indicated isotonic chains of spherical nuclei.

Figure 3

Experimental neutron ${\mathrm{\Delta }}_{\nu }^{\left(5\right)}(Z,N)$ indicators as a function of neutron number $N$. Pairing indicators based both on measured and estimated binding energies (see discussion in the beginning of Sec. 4) are included. The data are grouped into six panels according to increasing proton number. This is done to illustrate both the isospin dependence and the impact of different shell closures on the ${\mathrm{\Delta }}_{\nu }^{\left(5\right)}(Z,N)$ indicators. The ranges of the neutron number and ${\mathrm{\Delta }}_{\nu }^{\left(5\right)}(Z,N)$ changes are the same in panels (b)–(f); this allows direct comparison of the slopes of the ${\mathrm{\Delta }}_{\nu }^{\left(5\right)}(Z,N)$ values as a function of neutron number in these panels. Note that these ranges are different in panel (a). The experimental data on binding energies are taken from the AME2016 mass evaluation (see Ref. [72]).

Figure 4

The same as in Fig. 3 but for the proton ${\mathrm{\Delta }}_{\pi }^{\left(5\right)}(Z,N)$ indicators as a function of proton number $Z$.

Figure 5

The distribution of the scaling factors $f_{\nu }(Z,N)$ of neutron pairing in the nuclear chart. Panel (a) shows all nuclei for which experimental ${\mathrm{\Delta }}_{\nu }^{\left(5\right)}(Z,N)$ indicators can be defined. Note, however, that we exclude such indicators for $N=20,50,82$, and 126. Panels (b), (c), and (d) show scaling factors only for the nuclei whose ground states are either deformed, spherical, or transitional according to the calculations. The nuclei with equilibrium deformations ${\beta }_2$ satisfying the conditions $|{\beta }_2|\le 0.02, 0.02<|{\beta }_2|<0.15$, and $|{\beta }_2|\ge 0.15$ are considered to be spherical, transitional, and deformed, respectively.

Figure 6

The same as Fig. 5 but for proton scaling factors $f_{\pi }(Z,N)$ of proton pairing. Note, however, that we exclude proton pairing indicators for $Z=20,50$, and 82.

Figure 7

Proton $\left[f_{\pi }(Z,N)\right]$ and neutron $\left[f_{\nu }(Z,N)\right]$ scaling factors as a function of mass number $A$. Open circles, squares, and diamonds are used for spherical, transitional, and deformed nuclei, respectively. Top panels include all available scaling factors, while bottom panels show only those for the nuclei with $Z>20,N>20$. Solid green lines correspond to global functional dependencies given by Eqs. (13) with the parameters from Table 1 for top panels and from Table 3 for bottom panels.

Figure 8

The same as Fig. 7 but as a function of isospin factor $|N-Z|$. Solid green lines correspond to global functional dependencies given by Eqs. (15) with the parameters from Table 1 for top panels and from Table 3 for bottom panels.

Figure 9

The same as Fig. 7 but as a function of isospin factor $I=\frac{|N-Z|}{N+Z}$. Solid green lines correspond to global functional dependencies given by Eqs. (17) with the parameters from Table 1 for top panels and from Table 3 for bottom panels.

Figure 10

The variations of neutron and proton scaling factors as a function of neutron number $N$ in the Yb ($Z=70$) isotopes. Functional dependencies of Eqs. (13, 14, 15, 16, 17) are given here in inset with the parameters defined in Table 3. The pairs of functional dependencies with the best and worst RMSEs are shown by thick solid and thin dashed lines. The dot-dashed thin line is used for the functional dependence located (in terms of RMSE) in between of these pairs. Vertical orange dashed lines are used to indicate the span of neutron numbers for which experimental information on binding energies are available in the Yb isotopes.