Shell evolution above $Z,N=50$ within Skyrme density functional theory: The impact of deformation and tensor interactions

Background: Recent years have seen considerable effort in associating the shell evolution (SE) for a chain of isotones or isotopes with the underlying nuclear interactions. In particular, it has been fairly well established that the tensor part of the Skyrme interaction is indispensable for understanding certain SE above $Z,N=50$ shell closures, as a function of nucleon numbers.

Purpose: The purpose of the present work is twofold: (1) to study the effect of deformation due to blocking on the SE above $Z,N=50$ shell closures and (2) to examine the optimal parametrizations in the tensor part which gives a proper description of the SE above $Z,N=50$ shell closures.

Methods: I use the Skyrme-Hartree-Fock-Bogoliubov (SHFB) method to compute the even-even vacua of the $Z=50$ isotopes and $N=50$ isotones. For Sb and odd-$A$ Sn isotopes, I perform calculations with a blocking procedure which accounts for the polarization effects, including deformations.

Results: The blocking SHFB calculations show that the light odd-$A$ Sb isotopes, with only one valence proton occupying down-sloping $\mathrm{\Omega }=11/2^{-}$ and $\mathrm{\Omega }=7/2^{+}$ Nilsson orbits, assume finite oblate deformations. This reduces the energy differences between $11/2^{-}$ and $7/2^{+}$ states by about 500 keV for ${}_{51}{\mathrm{Sb}}_{56-66}$, bringing the energy-difference curve closer to the experimental one. With une2t1 energy density functional (EDF), which differs from unedf2 parametrization by tensor terms, a better description of the slope of $\mathrm{\Delta }e(\pi 1h_{11/2}-\pi 1g_{7/2})$ as a function of neutron number has been obtained. However, the trend of $\mathrm{\Delta }e(\pi 1g_{7/2}-\pi 2d_{5/2})$ curve is worse using une2t1 EDF. $\mathrm{\Delta }e(\nu 3s_{1/2}-\nu 2d_{5/2})$ and $\mathrm{\Delta }e(\nu 1g_{7/2}-\nu 2d_{5/2})$ curve for $N=50$ isotones using une2t1 seems to be consistent with experimental data. The neutron SE of $\mathrm{\Delta }e(\nu 1h_{11/2}-\nu 1g_{7/2})$ and $\mathrm{\Delta }e(\nu 1g_{7/2}-\nu 2d_{5/2})$ for Sn isotopes are shown to be sensive to ${\alpha }_T$ tensor parameter.

Conclusions: Within the Skyrme self-consistent mean-field model, the deformation degree of freedom has to be taken into account for Sb isotopes, $N=51$ isotones, and odd-$A$ Sn isotopes when discussing variation of quantities like shell gap etc. The tensor terms are important for describing the strong variation of $\mathrm{\Delta }E({\mathrm{\Omega }}^{\pi }=11/2^{-}-7/2^{+})$ in Sb isotopes. The SE of $1/2^{+}$ and $7/2^{+}$ states in $N=51$ isotones may show signature for the existence of tensor interaction. The experimental excitation energies of $11/2^{-}$ and $7/2^{+}$ states in odd-$A$ Sn isotopes close to ${}^{132}\mathrm{Sn}$ give prospects for constraining the ${\alpha }_T$ parameter.

Figure 1

Mean-field single-proton levels for ${}^{112}\mathrm{Sn}$ (a) and single-neutron levels for ${}^{90}\mathrm{Zr}$ (b) as a function of ${\beta }_2$ deformation, calculated with unedf2 parametrization. For the definition of ${\beta }_2$, see Ref. [39].

Figure 2

Energy differences between spherical (a) $\pi 1h_{11/2}$ and $\pi 1g_{9/2}$ shells and between (b) $\pi 1g_{9/2}$ and $\pi 2d_{5/2}$ shells calculated with unedf1, unedf2, and unedf1${}^{\mathrm{SO}}$ EDFs. Experimental data are taken from Ref. [5].

Figure 3

Total energy differences between (a) $\pi 11/2^{-}\left[505\right]$ and $\pi 7/2^{+}\left[404\right]$ states and between (c) $\pi 7/2^{+}\left[404\right]$ and $\pi 5/2^{+}\left[402\right]$ states for ${}_{51}\mathrm{Sb}$ isotopes. Lower panels show (b) ${\beta }_2$ deformations for $\pi 11/2^{-}\left[505\right], \pi 7/2^{+}\left[404\right]$, and (d) $\pi 5/2^{+}\left[402\right]$ states with unedf2 for ${}_{51}\mathrm{Sb}$ isotopes.

Figure 4

$C_0^J$ and $C_1^J$ values for selected Skyrme parametrizations [8, 9, 11, 14, 42].

Figure 5

Energy contribution due to $J^2$ term (a) and the total energy as a function of $Q_{20}$ for ${}^{100}\mathrm{Zr}$ calculated with unedf2, une2t1, and une2t2 EDFs.

Figure 6

S.p. spectra for ${}^{100}\mathrm{Sn}$ and ${}^{132}\mathrm{Sn}$, calculated with unedf1 (1), unedf1${}^{\mathrm{SO}}$ (2), unedf2 (3), une2t1 (4), and une2t2 (5) EDFs.

Figure 7

Same as Fig. 2, except for une2t1 and une2t2 parameters described in the text.

Figure 8

Same as Fig. 3, except for une2t1 and une2t2 parameters described in the text.

Figure 9

(a) Mean-field energy differences between $\nu 1g_{7/2}$ and $\nu 2d_{5/2}$ levels and between $\nu 3s_{1/2}$ and $\nu 2d_{5/2}$ levels. (b) Same as in panel (a), but with une2t1 and une2t2 parametrizations. (c) Experimental data are taken from Ref. [57]. The g.s. are $5/2^{+}$ states.

Figure 10

Mean-field energy differences (a) between $\nu 1h_{11/2}$ and $\nu 1g_{7/2}$ levels and (b) between $\nu 1g_{7/2}$ and $\nu 2d_{5/2}$ levels calculated with various EDFs.

Figure 11

Excitation energies of neutron ${\frac{11}{2}}^{-}$ (a) and ${\frac{7}{2}}^{+}$ (b) states in Sn isotopes, calculated with various EDFs. The experimental data (crosses) are taken from Ref. [57]. See text for details about configuration assignments.