Shell model calculation on Sn isotopes and evolution of shell structure

We perform shell model calculations of low-lying excited states of $^{100-132}$Sn even-even isotopes, and discuss the properties of the $E2$ transition probabilities of low-lying excited states. Recent improvements of rare isotope beam technique provide us with the experimental information of proton-rich tin isotopes towards $^{100}$Sn, especially $B(E2;0_1^+ \rightarrow 2_1^+)$ values of $^{106-114}$Sn. These $E2$ probabilities show unexpectedly large and inconsistent with the prediction of the large scale shell model calculation based on $G$-matrix prescription. In this work, we adopt rather schematic interaction such as pairing-plus-quadrupole interaction and monopole interaction, which is considered to play an important role in shell evolution of proton-rich nuclei around $^{100}$Sn and $Z=N=50$ shell gaps, with utilizing angular-momentum and number projection techniques and the Monte-Carlo shell model instead of traditional $n$-particle $n$-hole truncation. We demonstrates how the shell gap and $\nu 0h11/2$ intruder orbit evolve with decreasing neutron numbers of Sn isotopes, and discuss the origin of the anomalous behaviour of $E2$ transition probabilities.