Microscopic study of $M1$ resonances in Sn isotopes

The magnetic dipole $\left(M1\right)$ resonances of even-even ${}^{112–120,124}\mathrm{Sn}$ isotopes are investigated in the framework of the self-consistent Skyrme Hartree-Fock + Bardeen-Cooper-Schrieffer (HF+BCS) and quasiparticle random phase approximation (QRPA). The Skyrme energy density functionals SLy5 and T11 with and without tensor terms are adopted in our calculations. The mixed type pairing interaction is used to take care of the pairing effect for open-shell nuclei both in the ground and excited states calculations. The calculated magnetic dipole strengths are compared with available experimental data. The QRPA results calculated by SLy5 and T11 with tensor force show a better agreement with the experimental data than those without the tensor force. By analyzing the HF and QRPA strength distributions of ${}^{112}\mathrm{Sn}$ and ${}^{124}\mathrm{Sn}$, we discuss the effect of tensor force on the $M1$ resonances in detail. It is found that the $M1$ resonance is sensitive to the tensor interaction, and favors especially a negative triplet-odd tensor one. Depending on the nucleus, a quenching factor of the $M1$ operator of about 0.71–0.95 is needed to reproduce the total observed transition strength. In our calculations, we also find some low-lying, pygmy-type magnetic dipole states distributed below 6.0 MeV, and they are formed mainly from the neutron configuration $\nu 2d_{5/2}\to \nu 2d_{3/2}$.

Figure 1

The $M1$ strength distributions of ${}^{120}\mathrm{Sn}$ in filling approximation and QRPA, respectively. The calculated strengths are convoluted by a Lorentzian shape with a width of 2.0 MeV.

Figure 2

The QRPA strength distributions of ${}^{124}\mathrm{Sn}$ calculated by using several $\mathrm{T}ij$ and other Skyrme EDFs. In (a)–(d), the $\mathrm{T}ij$ EDFs are employed: (a) by changing $\alpha$ for a fixed value $\beta =-60\mathrm{MeV}{\mathrm{fm}}^5$. (b) by changing $\beta$ for a fixed value $\alpha =-60\mathrm{MeV}{\mathrm{fm}}^5$. (c) by fixing $\alpha =\beta =(-60,60,180)\mathrm{MeV}{\mathrm{fm}}^5$ that corresponds to $i=j=(1,3,5)$. (d) for the cases $\alpha \ne \beta$ that are not shown in (a)–(c), including $\mathrm{T}24$, $\mathrm{T}42$, $\mathrm{T}35$, $\mathrm{T}53$, $\mathrm{T}46$, and $\mathrm{T}64$, except $\mathrm{T}11$. (e) and (f) show commonly used Skyrme EDFs without the tensor terms, except SLy5 which includes the tensor terms. Experimental data are taken from Bassauer et al. [32].

Figure 3

The energies of unperturbed $M1$ peaks for ${}^{124}\mathrm{Sn}$ as a function of $W_0$. In the calculations, the Skyrme EDFs without tensor terms are employed. The computed data points are labeled, here and in what follows, by numbers: 1 = T11, 2 = T12, 3 = T13, 4 = T14, 5 = T15, 6 = T16, 7 = T21, 8 = T22, 9 = T23, 10 = T24, 11 = T25, 12 = T26, 13 = T31, 14 = T32, 15 = T33, 16 = T34, 17 = T35, 18 = T36, 19 = T41, 20 = T42, 21 = T43, 22 = T44, 23 = T45, 24 = T46, 25 = T51, 26 = T52, 27 = T53, 28 = T54, 29 = T55, 30 = T56, 31 = T61, 32 = T62, 33 = T63, 34 = T64, 35 = T65, 36 = T66, 37 = SLy4, 38 = SLy5, 39 = SGII, 40 = SKM*, 41 = SKP, 42 = SIII, 43 = KDE0v, 44 = KDEv1, 45 = SKb, 46 = SKT6, 47 = MSK1, 48 = LNS1, 49 = MSK9, 50 = SLy6. The grey lines correspond to the results of the linear fits.

Figure 4

The energies of low-lying and high-lying unperturbed $M1$ peaks for ${}^{124}\mathrm{Sn}$ as a function of $W_0$, calculated by using $\mathrm{T}ij$ sets with tensor force.

Figure 5

The energies of low-lying and high-lying unperturbed $M1$ peaks for ${}^{124}\mathrm{Sn}$ are shown as a function of $\alpha$ by fixing $\beta$.

Figure 6

The QRPA energies of low-lying and high-lying $M1$ peaks for ${}^{124}\mathrm{Sn}$ as a function of $\alpha$ for each fixed $\beta$ (figure (a)), and as a function of $\beta$ for each fixed $\alpha$ (b). The theoretical values are calculated by using the $\mathrm{T}ij$ parameter sets with tensor force.

Figure 7

The difference of the centroid energies of the QRPA and unperturbed response for ${}^{124}\mathrm{Sn}$ as a function of $G_0+G_0^{\prime }$. All sets of $\mathrm{T}ij$ family members and other Skyrme forces are adopted in the QRPA calculations. The grey line corresponds to the result of linear fit.

Figure 8

The QRPA strength distributions of ${}^{112–120,124}\mathrm{Sn}$, calculated by using the SLy5 Skyrme interaction. The results with and without tensor terms are both shown, and compared with the experimental data [31, 32]. Calculated strengths are convoluted by a Lorentzian shape with a width of 2.0 MeV.

Figure 9

The same as Fig. 8, but calculated with T11.

Figure 10

The $M1$ Hartree-Fock and QRPA strength distributions of ${}^{112}\mathrm{Sn}$ obtained using the SLy5 and T11 parameter sets in the cases with and without tensor force.

Figure 11

The $M1$ Hartree-Fock and QRPA strength distributions of ${}^{124}\mathrm{Sn}$ obtained using theSLy5 and T11 parameter sets in the cases with and without tensor force.

Figure 12

The $M1$ Hartree-Fock and QRPA strength distributions of ${}^{112}\mathrm{Sn}$ and ${}^{124}\mathrm{Sn}$ obtained using the T15 parameter set in the cases with and without tensor force.