Evolution of magnetic dipole strength in ${}^{100-140}\mathrm{Sn}$ isotope chain and the quenching of nucleon $g$ factors

The evolution of electromagnetic transitions along isotope chains is of particular importance for the nuclear structure and dynamics, as well as for the r-process nucleosynthesis. Recent measurement of inelastic proton scattering on even-even ${}^{112-124}\mathrm{Sn}$ isotopes provides novel insight into the isotopic dependence of E1 and M1 strength distributions. We investigate M1 transitions in even-even ${}^{100-140}\mathrm{Sn}$ isotopes from a theoretical perspective, based on the relativistic nuclear energy density functional. The M1 transition strength distribution is characterized by an interplay between single- and double-peak structures that can be understood from the evolution of single-particle states, their occupations governed by the pairing correlations, and two-quasiparticle transitions involved. It is shown that the discrepancy between model calculations and experiments for the M1 transition strength is considerably more reduced than previously known, and the quenching of the $g$ factors for the free nucleons needed to reproduce the experimental data on M1 transition strength amounts $g_{\mathrm{eff}}/g_{\mathrm{free}}=0.80–0.93$. Because some of the M1 strength above the neutron threshold may be missing in the inelastic proton scattering measurement, further experimental studies are required to confirm if only small modifications of the bare $g$ factors are actually needed when applied in finite nuclei.

Figure 1

The occupation probabilities of $\pi \left(1g_{9/2}\right), \nu \left(1g_{9/2}\right), \nu \left(2d_{5/2}\right)$, and $\nu \left(1h_{11/2}\right)$ orbits in our RHB-GS solutions for Sn isotopes.

Figure 2

The M1 transition strength function $R_{\mathrm{M}1}\left(E\right)$ for Sn isotopes, including the full RQRPA and the unperturbed (RHB) responses. Calculations are based on the DD-PC1 parametrization of the RNEDF and Gogny-D1S force for the pairing correlations.

Figure 3

The partial M1 transition strengths $b_{ph}^{\pi \left(\nu \right)}\left(E\right)$ for protons ($\pi$) and neutrons ($\nu$) of even-even ${}^{100-140}\mathrm{Sn}$ isotopes. Partial strengths are evaluated at lower $E_{<}$ (upper panel) and higher energy peak $E_{>}$ (lower panel) of the double-peaked response function $R_{\mathrm{M}1}\left(E\right)$.

Figure 4

The M1 transition strengths $B_{\mathrm{M}1}\left(E\right)$ for the two main peaks in even-even ${}^{100-140}\mathrm{Sn}$ isotopes, calculated using the R(Q)RPA with the DD-PC1 functional. The $E_{<}$ and $E_{>}$ energies correspond to the low- and high-energy dominant peaks in the response function $R_{\mathrm{M}1}\left(E\right)$.

Figure 5

The M1-excitation energies of ${}^{100-140}\mathrm{Sn}$ isotopes and the corresponding SO-splitting energies $\mathrm{\Delta }{E}_{\mathrm{LS}}$. The calculations are based on the RHB plus RQRPA using DD-PC1 parametrization and D1S pairing force. The respective (nlj) quantum numbers are denoted for the SO-partner levels. $\mathrm{\Delta }{E}_{{\mathrm{LS}}_{<}}$ and $\mathrm{\Delta }{E}_{{\mathrm{LS}}_{>}}$ represent the SO splitting contributions at lower and higher RPA energies, respectively.

Figure 6

The total M1 transition strengths for even-even nuclei in the ${}^{100-140}\mathrm{Sn}$ isotope chain. The result is obtained with the R(Q)RPA using the DD-PC1 functional. The unperturbed RHB response is also shown.

Figure 7

(Top) Energy-weighted summation $m_1\left(\mathrm{M}1\right)$. For ${}^{124-132}\mathrm{Sn}$, the corresponding results with Kurath's sum rule are also displayed. (Bottom) M1 strength distribution of ${}^{124}\mathrm{Sn}$ with and without the pairing interaction.