Effect of isoscalar spin-triplet pairings on spin-isospin responses in $sd$-shell nuclei

The spin magnetic dipole transitions and the neutron-proton spin-spin correlations in $sd\text{−}\mathrm{shell}$ even-even nuclei with $N=Z$ are investigated using shell-model wave functions. The isoscalar spin-triplet pairing correlation provides a substantial quenching effect on the spin magnetic dipole transitions, especially the isovector (IV) ones. Consequently, an enhanced isoscalar spin-triplet pairing interaction influences the proton-neutron spin-spin correlation deduced from the difference between the isoscalar (IS) and the IV sum rule strengths. The effect of the $\mathrm{\Delta }$ (${\mathrm{\Delta }}_{33}$ resonance)–hole coupling is examined in the IV spin transition and the spin-spin correlations of the ground states.

Figure 1

(a) IS and (b) IV spin-$M1$ transition strengths in ${}^{28}\mathrm{Si}$. Shell-model calculations are performed in the full $sd$-shell model space with an USDB effective interaction. In the results of ${\mathrm{USDB}}^{*}$, multiplying the relevant matrix elements by a factor of 1.2 compared to the original USDB interaction enhances the IS spin-triplet interaction. For the results of the IV spin-$M1$ transitions, a quenching factor $q=0.9$ is used for ${\mathrm{USDB}}^{**}$. Calculated results are smoothed by taking a Lorentzian weighting factor with the width of 0.5 MeV, while the experimental data are shown in the units of $B(\sigma )$ for the IS excitations and $B(\sigma \tau )$ for the IV excitations. Experimental data are from Ref. [20].

Figure 2

Accumulative sum of the IS spin-$M1$ strength (a) and the IV spin-$M1$ strength (b) as a function of the excitation energy in ${}^{28}\mathrm{Si}$. For ${\mathrm{USDB}}^{**}$, quenching factor $q=0.9$ is used for the results of the IV spin-$M1$ transitions. Calculated results are smoothed in the same manner in Fig. 1. A dot with a vertical error bar denotes the experimental accumulated sum of the strengths.

Figure 3

Sums of the spin-$M1$ transition strengths of IS (a) and IV (b). Experimental and theoretical data are summed up to $Ex=16$ MeV. Shell-model calculations are performed with the USDB effective interaction. In the results of ${\mathrm{USDB}}^{*}$, the IS spin-triplet interaction is enhanced by multiplying the relevant matrix elements by a factor of 1.2 compared to the original USDB interactions. For ${\mathrm{USDB}}^{**}$, a quenching factor $q=0.9$ is used for the results of the IV spin-$M1$ transitions. Experimental data are from Ref. [20]. Long thin error bars indicate the total experimental uncertainty, while the short thick error bars denote the partial uncertainty from the spin assignment.

Figure 4

Experimental and calculated proton-neutron spin-spin correlation ${\mathrm{\Delta }}_{\mathrm{S}}$. Spin $M1$ transition strengths are summed up to $Ex=16$ MeV. Shell-model calculations are performed with an effective interaction USDB. In the results of ${\mathrm{USDB}}^{*}$, the IS spin-triplet interaction is enhanced by multiplying the relevant matrix elements by a factor of 1.2 compared to the original USDB. A quenching factor $q=0.9$ is used for the results of the IV spin-$M1$ for ${\mathrm{USDB}}^{**}$. Experimental data are taken from Ref. [20]. See the caption of Fig. 3 for a description of the experimental error bars.