Shell-model study of quadrupole collectivity in light tin isotopes

A realistic shell-model study is performed for neutron-deficient tin isotopes up to mass $A=108$. All shell-model ingredients, namely, two-body matrix elements, single-particle energies, and effective charges for electric quadrupole transition operators, have been calculated by way of the many-body perturbation theory, starting from a low-momentum interaction derived from the high-precision CD-Bonn free nucleon-nucleon potential. The focus has been on the enhanced quadrupole collectivity of these nuclei, which is testified by the observed large $B(E2;0_1^{+}\to 2_1^{+})\mathrm{s}$. Our results give evidence of the crucial role played by the $Z=50$ cross-shell excitations that need to be taken into account explicitly to obtain a satisfactory theoretical description of light tin isotopes. We find also that a relevant contribution comes from the calculated neutron effective charges, whose magnitudes exceed the standard empirical values. An original double-step procedure has been introduced to reduce effectively the model space in order to overcome the computational problem.

Figure 1

(a) Experimental [4, 5, 6, 7, 8, 11] (red symbols) and calculated (black squares) excitation energies of the yrast $J^{\pi }=2^{+}$ states and (b) $B(E2;2_1^{+}\to 0_1^{+})$ transition rates for tin isotopes up to $N=58$, when using neutron effective charges (I) reported in Table 1. Blue squares refer to calculations with $e_n^{\mathrm{emp}}=0.5e$. Shell model calculations have been performed using only neutron degrees of freedom (see text for details).

Figure 2

Calculated proton effective single-particle energies of $H_{\mathrm{eff}}^{75}$ as a function of the atomic number $Z$.

Figure 3

Calculated neutron effective single-particle energies of $H_{\mathrm{eff}}^{75}$ as a function of $Z$.

Figure 4

Effective single-particle energies of tin isotopes as a function of $N$ calculated with $H_{\mathrm{eff}}^{75}$.

Figure 5

Experimental [30] and theoretical spectra of ${}^{102}\mathrm{Sn}$ calculated with $H_{\mathrm{eff}}^{45}$ and $H_{\mathrm{eff}}^{42}$ (see text for details).

Figure 6

Same as in Fig. 1, but with shell-model results obtained with $H_{\mathrm{eff}}^{42}$. Blue squares refer to calculations with $e_n^{\mathrm{emp}}=0.5,e_p^{\mathrm{emp}}=1.5$.