Novel Shape Evolution in Sn Isotopes from Magic Numbers 50 to 82

A novel shape evolution in the Sn isotopes by the state-of-the-art application of the Monte Carlo shell model calculations is presented in a unified way for the ${}^{100–138}\mathrm{Sn}$ isotopes. A large model space consisting of eight single-particle orbits for protons and neutrons is taken with the fixed Hamiltonian and effective charges, where protons in the $1g_{9/2}$ orbital are fully activated. While the significant increase of the $B(E2;0_1^{+}\to 2_1^{+})$ value, seen around ${}^{110}\mathrm{Sn}$ as a function of neutron number ($N$), has remained a major puzzle over decades, it is explained as a consequence of the shape evolution driven by proton excitations from the $1g_{9/2}$ orbital. A second-order quantum phase transition is found around $N=66$, connecting the phase of such deformed shapes to the spherical pairing phase. The shape and shell evolutions are thus described, covering topics from the Gamow-Teller decay of ${}^{100}\mathrm{Sn}$ to the enhanced double magicity of ${}^{132}\mathrm{Sn}$.

Figure 1

(a) Two-neutron separation energy, $S_{2n}(N)$, and (b) $2_1^{+}$ levels are shown, as a function of $N$, for experimental [9] and present values by blue dashed and red solid lines, respectively. The inset of the panel (b) displays the $R_{4/2}$ ratio: the $4_1^{+}$ level over the $2_1^{+}$ level energy. In (a), $\mathrm{\Delta }{S}_{2n}\times 10$ is shown with averages by the dashed-dotted lines. In (b), alternative values are shown by green dashed-dotted lines.

Figure 2

(a) Calculated and measured $B(E2)$ values, and (b) E2 matrix elements and proton ratio (%). Experimental data are indicated by symbols shown in the inset with the correspondence: RIKEN [23], ORNL [11, 12, 19, 24], GSI [13, 16, 18, 22], GSI_ANU [20], ISOLDE [14, 17], LBNL [25], MSU [15, 21], IUAC [26], TUD [28], Koeln [29].

Figure 3

Occupation numbers of (a) proton and (b) neutron holes in the $1g_{9/2}$ orbit. For each value of $N$, histograms for the $0_1^{+}(\text{blue})$, $2_1^{+}(\text{red})$, and $4_1^{+}(\text{green})$ states are shown from left to right, except for the missing $4_1^{+}$ for $N=50$.

Figure 4

(a)–(h) $T$-plots for $0_1^{+}$ and $2_1^{+}$ states of selected Sn isotopes and (i) $T$-plot for $1_1^{+}$ state of ${}^{100}\mathrm{In}$, with the present interaction. (j)–(l) the same as (a)–(h), with the alternative interaction.