Self-consistent relativistic quasiparticle random-phase approximation and its applications to charge-exchange excitations

The self-consistent quasiparticle random-phase approximation (QRPA) approach is formulated in the canonical single-nucleon basis of the relativistic Hatree–Fock–Bogoliubov (RHFB) theory. This approach is applied to study the isobaric analog states (IASs) and Gamow–Teller resonances (GTRs) by taking Sn isotopes as examples. It is found that self-consistent treatment of the particle-particle residual interaction is essential to concentrate the IAS in a single peak for open-shell nuclei and the Coulomb exchange term is very important to predict the IAS energies. For the GTR, the isovector pairing can increase the calculated GTR energy, while the isoscalar pairing has an important influence on the low-lying tail of the Gamow–Teller transition.

Figure 1

Deviations of the theoretical nuclear masses from the experimental data [82] for the even-even Ca, Ni, and Sn isotopes. The theoretical results are calculated by the RHFB theory with the effective interaction PKO1 (open circles) or the RHB theory with the effective interaction DD-ME2 (open diamonds).

Figure 2

$Q_{\beta }$ values of the even-even Ca, Ni, and Sn isotopes (isobaric mass differences for stable nuclei). The RHFB calculations with PKO1 are denoted by the open circles. For comparison, the experimental data [82] and the calculated results by the RHB theory with DD-ME2 are shown by the filled squares and open diamonds, respectively.

Figure 3

Two-neutron separation energies of the even-even Ca, Ni, and Sn isotopes. The RHFB calculations with the effective interaction PKO1 are denoted by the open circles. For comparison, the experimental data [82] and the results calculated by RHB theory with the effective interaction DD-ME2 are shown by the filled squares and open diamonds, respectively.

Figure 4

Neutron-skin thicknesses ($r_n-r_p$) of the even-even Sn isotopes. Open circles and open diamonds show the results calculated by the RHFB theory with PKO1 and the RHB theory with DD-ME2, respectively. The experimental results from the spin-dipole resonance (SDR) [7], antiprotonic x-ray data [83], and proton elastic scattering [84] are shown by the filled squares, diamonds, and triangles, respectively.

Figure 5

Transition probabilities for the IAS in ${}^{114}\mathrm{Sn}$. The calculations are performed by the $\text{RHFB}+\text{QRPA}$ approach with PKO1, while the Coulomb interaction is switched off. The horizontal dotted line denotes the $N-Z$ sum rule. For comparison, the unperturbed result (labeled by RHFB) and the calculation without the pp residual interaction ($V^{pp}=0$) are shown by the dashed and dash-dotted lines, respectively.

Figure 6

Running sum of the GT transition probabilities for ${}^{118}\mathrm{Sn}$ calculated by the $\text{RHFB}+\text{QRPA}$ approach with PKO1. The dashed line shows the QRPA calculation with only the ph configurations from the Fermi states. The solid line corresponds to the calculation further including the configurations from the occupied Fermi states and the unoccupied Dirac states. The horizontal dotted line corresponds to the value $3(N-Z)$ of the Ikeda sum rule.

Figure 7

Transition probabilities for the IAS in ${}^{114}\mathrm{Sn}$ calculated with PKO1. The $\text{RHF}+\text{RPA}$, RHFB+RPA, $\text{RHFB}+\text{QRPA}$*, and $\text{RHFB}+\text{QRPA}$ calculations are shown in panels (a)–(d), respectively. See the text for details.

Figure 8

IAS excitation energies of the even-even Sn isotopes. The experimental data [90] are denoted by the filled squares. The self-consistent $\text{RHF}+\text{RPA}$ and $\text{RHFB}+\text{QRPA}$ calculations with PKO1 are shown by the open and filled circles, respectively, while the self-consistent $\text{RHB}+\text{QRPA}$ calculations with DD-ME2 are shown by the filled diamonds. For comparison, the results obtained with $\text{RHFB}+\text{QRPA}$ approach with PKO1 but excluding the Coulomb exchange term are denoted by the open squares.

Figure 9

GT strength distribution in ${}^{118}\mathrm{Sn}$ calculated by the $\text{RHFB}+\text{QRPA}$ approach with PKO1. The unperturbed (labeled by RHFB) strength, the calculation with only ph residual interactions of $\sigma$ and $\omega$ fields, and that with only ph residual interactions of $\sigma , \omega$, and $\rho$ fields (excluding $\pi$ field) are shown by the dotted, dashed, and dash-dotted lines, respectively. The experimental data [90] are shown with an arrow, whose width illustrates the width of the resonance.

Figure 10

GT strength distribution in ${}^{118}\mathrm{Sn}$ calculated by the $\text{RHF}+\text{RPA}$ (dotted line) and $\text{RHFB}+\text{QRPA}$ (solid line) approaches with PKO1.

Figure 11

GT strength distribution in ${}^{118}\mathrm{Sn}$ calculated by the $\text{RHFB}+\text{QRPA}$ approach with PKO1 for different values of $V_0$. The experimental data [90] are shown with arrows, whose widths illustrate the widths of the corresponding resonances.

Figure 12

GT excitation energies of the even-even Sn isotopes. The $\text{RHFB}+\text{QRPA}$ calculations without and with the $T=0$ pairing in Eq. (22) are shown by the open and filled circles, respectively. The experimental values in Ref. [90] are denoted by the filled squares.