Skyrme energy functional and low lying $2^{+}$ states in Sn, Cd, and Te isotopes

We study the predictive power of Skyrme forces with respect to low lying quadrupole spectra along the chains of Sn, Cd, and Te isotopes. Excitation energies and $B\left(E2\right)$ values for the lowest quadrupole states are computed from a collective Schrödinger equation which is deduced through a collective path generated by constraint Skyrme-Hartree-Fock (SHF) plus self-consistent cranking for the dynamical response. We compare the results from four different Skyrme forces, all treated with two different pairing forces (volume versus density-dependent pairing). The region around the neutron shell closure $N=82$ is very sensitive to changes in the Skyrme while the midshell isotopes in the region $N<82$ depend mainly on the adjustment of pairing. The neutron rich isotopes are most sensitive and depend on both aspects.

Figure 1

The raw collective potential $\mathcal{V}$ and the effective potential $V$ including the zero-point energy corrections (A6), both drawn as function of the intrinsic axial quadrupole momentum $a_{20}$. Test case is ${}^{132}\mathrm{Sn}$ computed with DI pairing and SkI3. The position of the $0_0^{+}$ ground state and the first $2^{+}$ state are indicated by horizontal bars. The difference between the minimum of $\mathcal{V}$ and the $0_0^{+}$ energy is the correlation energy $\Delta E_{\mathrm{corr}}$.

Figure 2

Energies $E\left(2^{+}\right)$ and $B\left(E2\right)\uparrow$ values $(={\mathbf{\mid }⟨0^{+}\mid Q_{2M}\mid 2^{+}⟩\mathbf{\mid }}^2)$ along the chain of Sn isotopes calculated using the four different Skyrme interactions as indicated. The experimental results are taken from [38].

Figure 3

Energies $E\left(2^{+}\right)$, $B\left(E2\right)\uparrow$ values $(={\mathbf{\mid }⟨0^{+}\mid Q_{2M}\mid 2^{+}⟩\mathbf{\mid }}^2)$, and average neutron-pairing gaps at spherical shape along the chain of Sn isotopes calculated with SkI3 and three different pairing recipes: standard DI (circles), DDDI (boxes), and DI with 25% enhanced strength. The experimental results are taken from [38].

Figure 4

As Fig. 2, but now with DDDI pairing.

Figure 5

Energies $E\left(2^{+}\right)$ and $B\left(E2\right)\uparrow$ values $(={\mathbf{\mid }⟨0^{+}\mid Q_{2M}\mid 2^{+}⟩\mathbf{\mid }}^2)$ along the chain of Sn isotopes computed with SkI3 and DI pairing, once with (solid line, full circles) and once without (dashed line, open circle) particle number restoration as outlined in Appendix (A4). The experimental results (dotted lines, full squares) are taken from [38].

Figure 6

Systematics of the energies of the first excited $2^{+}$ states calculated with the microscopic Bohr-Hamiltonian (A9) using the interaction SkI3 for the nearest even-even neighbors of ${}^{116}\mathrm{Sn}$. The experimental results are taken from [38].

Figure 7

Energies $E\left(2^{+}\right)$ and $B\left(E2\right)\uparrow$ values $(={\mathbf{\mid }⟨0^{+}\mid Q_{2M}\mid 2^{+}⟩\mathbf{\mid }}^2)$ along the chain of Cd isotopes (upper panels) and Te isotopes (lower panels) calculated with different Skyrme forces as indicated. The experimental results are taken from [38].

Figure 8

As Fig. 7, but now for DDDI pairing.

Figure 9

Isotope shifts of the charge r.m.s radii, $\delta {⟨r_{\mathrm{rms}}^2⟩}^{66,N}$, relative to ${}^{116}\mathrm{Sn}$. Upper panel: comparison of results from the four forces SkI3, SLy6, SkM*, and SkO including collective ground-state correlations. The expectation value $⟨r^2⟩$ is calculated here according to Appendix (A6). Lower panel: Comparison of pure mean field result with those including correlations for the force SkI3. The experimental data are taken from [44].