Quadrupole Collectivity in ${}^{32,34,36,38}\mathrm{Si}$ and the $\mathit{N}=20\mathrm{}$ Shell Closure

The $\beta$-unstable nuclei ${}^{32,34,36,38}\mathrm{Si}$ have been produced by projectile fragmentation and studied by in-beam Coulomb excitation. Excited states at $1399\pm 25$ keV and $1084\pm 20$ keV have been identified for the first time in ${}^{36}\mathrm{Si}$ and ${}^{38}\mathrm{Si}$, respectively, and tentatively assigned $J^{\pi }{=2\mathrm{}}^{+}$. The $B({E2;0\mathrm{}}_1^{+}\to 2_1^{+})$ values leading to these states and the previously identified $2_1^{+}$ states in ${}^{32,34}\mathrm{Si}$ have been measured, and are compared to shell model calculations. Our results indicate that the $2_1^{+}$ state in ${}^{34}\mathrm{Si}$ has a large $\mathrm{fp}$-shell intruder component, and that the $2_1^{+}$ states in the $N>\mathrm{}20\mathrm{}$ silicon isotopes can be reproduced assuming an $N=20\mathrm{}$ shell closure.