Symmetry breaking of Gamow-Teller and magnetic-dipole transitions and its restoration in calcium isotopes

Nuclear magnetic-dipole ($M1$) and Gamow-Teller (GT) transitions provide insight into the spin-isospin properties of atomic nuclei. By considering them as unified spin-isospin transitions, the $M1$/GT transition strengths and excitation energies are subject to isospin symmetry. The excitation properties associated to the $M1$/GT symmetry need to be clarified within a consistent theoretical approach. In this work, the relationship between the $M1$ and GT transitions in Ca isotopes is investigated in a unified framework based on the relativistic energy-density functional (REDF) with point-coupling interactions, using the relativistic quasiparticle random-phase approximation (RQRPA). It is shown that the isovector-pseudovector (IV-PV) residual interaction affects both transitions, and the symmetry of $M1$ and giant-GT transitions is disrupted by this interaction in closed-shell nuclei. In open-shell Ca isotopes, the proton-neutron pairing in the residual RQRPA interaction also plays a role in GT transitions. Due to the interplay between these interactions, the $M1$/GT symmetry can be restored especially in the ${}^{42}\mathrm{Ca}$ nucleus, i.e., the giant-GT strength can become comparable to that of the $M1$ mode in terms of the unified spin-isospin transitions by adjusting the proton-neutron pairing strength to reproduce the experimental low-lying GT-excitation energies. The mirror symmetry of both $M1$ and GT transitions is also demonstrated for open-shell mirror partners, ${}^{42}\mathrm{Ca}$ and ${}^{42}\mathrm{Ti}$. Further improvements are required to achieve simultaneous reproduction of $M1$ and GT transition energies in the REDF framework.

Figure 1

The nonperturbed IV-spin-$M1$ and $\mathrm{GT}(\pm$) strength distributions obtained with the DD-PC1 interaction. The $\mathrm{GT}(\pm$)-excitation energies $E_{\mathrm{x}}$ are presented with respect to the daughter nuclei [73].

Figure 2

Top: The IV-spin-$M1$ and $\mathrm{GT}(-$) strength distributions for ${}^{48}\mathrm{Ca}$ with the DD-PC1 and DD-PCX interactions supplemented with the IV-PV residual interaction in the RRPA. The arrows indicate the experimental $M1$ and $\mathrm{GT}(-$) energies: $E_{\mathrm{M}1}=10.23$ MeV [79]; $E_{\mathrm{GT}}=2.517$ MeV [75] and $\cong 10.8$ MeV [80]. Several values for the IV-PV coupling constant are used, i.e., ${\alpha }_{\mathrm{IV}\text{−}\mathrm{PV}}=0$, 1.0, 1.7, and $2.1{\mathrm{fm}}^2$. For the $\mathrm{GT}(-$) mode, the excitation energy is presented with respect to the ground state of the daughter ${}^{48}\mathrm{Sc}$ nucleus. Bottom: The same plot but for ${}^{208}\mathrm{Pb}$. The experimental $M1$ and $\mathrm{GT}(-$) energies are measured as $E_{\mathrm{M}1}\cong 7.3$ MeV [81] and $E_{\mathrm{GT}}=15.6$ MeV [82, 83].

Figure 3

The $M1$ and $\mathrm{GT}(-$) excitation energies for ${}^{48}\mathrm{Ca}$ obtained with the DD-PC1 and DD-PCX interactions supplemented with the IV-PV residual interaction for the range of values of its coupling constant ${\alpha }_{\mathrm{IV}\text{−}\mathrm{PV}}$. The $\mathrm{\Delta }{E}_{\mathrm{GT}(-)}$ denotes the gap between two major $\mathrm{GT}(-$) energies. The experimental $M1$ energy, 10.23 MeV [79], is indicated by a dashed line. The experimental low-lying and giant $\mathrm{GT}(-$) energies, 2.517 MeV [75] and 10.8 MeV [80], respectively, are indicated by solid lines.

Figure 4

Cumulative $\mathrm{GT}(-$) strength for ${}^{48}\mathrm{Ca}$ obtained from our RRPA method. The Ikeda-Fujii-Fujita sum rule as in Eq. (14) is plotted by the dotted line. The arrow indicates the experimental value of $\sum B_{\mathrm{GT}(-)}=15.3/4$ up to 30 MeV [80].

Figure 5

The IV-spin-$M1$ and $\mathrm{GT}(-$) strength distributions for ${}^{42-46}\mathrm{Ca}$ nuclei with the DD-PC1 and the IV-PV interaction with ${\alpha }_{\mathrm{IV}\text{−}\mathrm{PV}}=1.7{\mathrm{fm}}^2$. The $\mathrm{GT}(-$) response is shown for three values of PN pairing strength, $f=0$, 0.81, and 1.1. Vertical arrows indicate the low-lying $\mathrm{GT}(-$) energies in experiment [55, 75]: for ${}^{42}\mathrm{Ca}⟶{}^{42}\mathrm{Sc}$, 0.611 MeV; for ${}^{44}\mathrm{Ca}⟶{}^{44}\mathrm{Sc}$, 0.667 MeV.

Figure 6

Cumulative $\mathrm{GT}(-$) strength for ${}^{42}\mathrm{Ca}$ obtained with the DD-PC1 plus IV-PV interaction with ${\alpha }_{\mathrm{IV}\text{−}\mathrm{PV}}=1.7{\mathrm{fm}}^2$. The Ikeda-Fujii-Fujita sum rule as in Eq. (14) is plotted by the dotted line. The arrow indicates the experimental value of $\sum B_{\mathrm{GT}(-)}=2.6/4$ [55].

Figure 7

The IV-spin-$M1$ and GT strength distributions of ${}^{42}\mathrm{Ca}$ and ${}^{42}\mathrm{Ti}$ obtained with the DD-PC1 plus IV-PV interaction with ${\alpha }_{\mathrm{IV}\text{−}\mathrm{PV}}=1.7{\mathrm{fm}}^2$. For $\mathrm{GT}(\pm$) modes, their excitation energies are presented with respect to the common daughter nucleus ${}^{42}\mathrm{Sc}$. The arrow indicates the experimental low-lying $\mathrm{GT}(-$) energy 0.611 MeV in ${}^{42}\mathrm{Sc}$ [75].

Figure 8

The IV-$M1$ response functions for ${}^{42,48}\mathrm{Ca}$ nuclei.