Quasiparticle random-phase approximation with quasiparticle-vibration coupling: Application to the Gamow-Teller response of the superfluid nucleus ${}^{120}\mathrm{Sn}$

We propose a self-consistent quasiparticle random-phase approximation (QRPA) plus quasiparticle-vibration coupling (QPVC) model with Skyrme interactions to describe the width and the line shape of giant resonances in open-shell nuclei, in which the effect of superfluidity should be taken into account in both the ground state and the excited states. We apply the new model to the Gamow-Teller resonance in the superfluid nucleus ${}^{120}\mathrm{Sn}$, including both the isoscalar spin-triplet and the isovector spin-singlet pairing interactions. The strength distribution in ${}^{120}\mathrm{Sn}$ is well reproduced and the underlying microscopic mechanisms, related to QPVC and also to isoscalar pairing, are analyzed in detail.

Figure 1

Diagrammatic representation of the four terms whose sum gives the matrix element $W_{ab,a^{\prime }{b}^{\prime }}^{\downarrow }$.

Figure 2

GT strength function calculated with the interaction SGII within the QRPA approach, using different configuration spaces associated with different threshold values for the product $\left|uv\right|$ of the occupation amplitudes in canonical basis.

Figure 3

GT strength function calculated with the interaction SGII within QRPA (a) or QRPA+QPVC (b) with different configuration spaces associated with different values of the energy cutoff $E_{\mathrm{cut}}$.

Figure 4

GT strength function calculated with the interaction SGII within QRPA+QPVC with different configuration spaces associated with different values of the strength cutoff $b_{\mathrm{cut}}$.

Figure 5

Ikeda sum rule fulfillment as a function of the number of QRPA basis states used for the QPVC calculation, in the case of the Gamow-Teller response of ${}^{120}\mathrm{Sn}$ calculated with the interaction SGII.

Figure 6

Gamow-Teller strength distribution in ${}^{120}\mathrm{Sn}$ calculated by means of the Skyrme QRPA+QPVC model with and without the approximation of Eq. (25) in the spreading matrix elements.

Figure 7

The Gamow-Teller strength distributions (a) and their cumulative sums (b) for ${}^{120}\mathrm{Sn}$ calculated by means of QRPA and QRPA+QPVC models, without and with subtraction method, using the Skyrme interaction SkM*.

Figure 8

The Gamow-Teller strength distributions for ${}^{120}\mathrm{Sn}$ calculated by QRPA and QRPA+QPVC models, with isoscalar pairing, using the interactions SAMi (a), SGII (b), and SkM*(c). The experimental GT peak energies [53] are denoted by dashed lines, and the length of each line is proportional to the cross section.

Figure 9

The Gamow-Teller strength distributions for ${}^{120}\mathrm{Sn}$ calculated by QRPA (a) and QRPA+QPVC (b) models, with and without isoscalar pairing, using the Skyrme interaction SkM*.

Figure 10

The cumulative sum of Gamow-Teller strength for ${}^{120}\mathrm{Sn}$, calculated by QRPA and QRPA+QPVC models, with and without isoscalar pairing, using the Skyrme interaction SkM*. The experimental results from $({}^3\mathrm{He},t)$ [53] and $(p,n)$ reactions [54] are shown for comparison. In panel (a), the $B\left(\text{GT}\right)$ strength from the $({}^3\mathrm{He},t)$ experiment is obtained by multiplying the cross section by a factor of 1.6 so that the main GTR strength exhausts $65\%$ of the Ikeda sum rule, while the $B\left(\text{GT}\right)$ strength for the $(p,n)$ experiment is obtained by dividing the cross section by the unit cross section. In panel (b), the two experimental cumulative GT sums are normalized to the same value as the theoretical one of QRPA+QPVC with $f=0$ at $E=25\mathrm{MeV}$.

Figure 11

The Gamow-Teller strength distributions for ${}^{120}\mathrm{Sn}$ calculated by QRPA and QRPA+QPVC models, with [(a) and without (b) isoscalar pairing, using the Skyrme interaction SkM*. The smearing parameter $\mathrm{\Delta }=0.5\mathrm{MeV}$ is used instead of $\mathrm{\Delta }=0.2\mathrm{MeV}$ used for Fig. 9. The experimental results from $({}^3\mathrm{He},t)$ and $(p,n)$ reactions are shown for comparison. The cross section from the $({}^3\mathrm{He},t)$ experiment is scaled by a factor of 1.6 so that the main GTR strength exhausts $65\%$ of Ikeda sum rule [53]. The cross section from the $(p,n)$ reaction is normalized by the unit cross section [54] (cf. the main text).