Two-neutron transfer reactions and shape phase transitions in the microscopically formulated interacting boson model

Two-neutron transfer reactions are studied within the interacting boson model based on the nuclear energy density functional theory. Constrained self-consistent mean-field calculations with the Skyrme energy density functional are performed to provide microscopic input to completely determine the Hamiltonian of the IBM. Spectroscopic properties are calculated only from the nucleonic degrees of freedom. This method is applied to study the $(t,p)$ and $(p,t)$ transfer reactions in the assorted set of rare-earth nuclei ${}^{146-158}\mathrm{Sm}$, ${}^{148-160}\mathrm{Gd}$, and ${}^{150-162}\mathrm{Dy}$, where spherical-to-axially deformed shape phase transition is suggested to occur at the neutron number $N\approx 90$. The results are compared with those from the purely phenomenological IBM calculations, as well as with the available experimental data. The calculated $(t,p)$ and $(p,t)$ transfer reaction intensities, from both the microscopic and phenomenological IBM frameworks, signal the rapid nuclear structural change at particular nucleon numbers.

Figure 1

Potential energy surfaces for the nuclei ${}^{148-154}\mathrm{Sm}$ plotted within the $(\beta ,\gamma )$ deformation space and with up to 5 MeV from the global minimum. The energy difference between neighboring contours is 250 keV. See the main text for details.

Figure 2

Same as described in the caption of Fig. 1, but for the nuclei ${}^{150-156}\mathrm{Gd}$.

Figure 3

Same as described in the caption of Fig. 1, but for the nuclei ${}^{152-158}\mathrm{Dy}$.

Figure 4

Excitation energies of the low-lying states $0_2^{+}$ (a), $0_3^{+}$ (b), $2_1^{+}$ (c), $2_2^{+}$ (d), $2_3^{+}$ (e), and $4_1^{+}$ (f) for the ${}^{150-158}\mathrm{Sm}$ isotopes are plotted against the neutron number $N$. The results of the three versions of the IBM calculations, i.e., $m$-IBM, $p$-IBM, and CQF, are compared with each other and with the experimental data [35].

Figure 5

Same as described in the caption of Fig. 4, but for the ${}^{148-160}\mathrm{Gd}$ isotopes.

Figure 6

Same as described in the caption of Fig. 4, but for the ${}^{150-162}\mathrm{Dy}$ isotopes.

Figure 7

The $(t,p)$ transfer reaction intensities for the ${}^{146-156}\mathrm{Sm}$ isotopes. The results of the $m$-IBM, $p$-IBM, and CQF calculations are compared with each other and with the experimental data [6]. The scale factors $t_0$ and $t_2$ in the $(t,p)$ transfer operators have been fitted to the experimental data for the $0_1^{+}\left({}^{152}\mathrm{Sm}\right)\to 0_1^{+}({}^{154}\mathrm{Sm}$) and $0_1^{+}{(}^{152}\mathrm{Sm})\to 2_1^{+}({}^{154}\mathrm{Sm}$) transfer reactions, respectively.

Figure 8

Similar to the description in the cation of Fig. 7, but for the $(p,t)$ transfer reaction intensities for the ${}^{146-156}\mathrm{Sm}$ isotopes. The experimental data have been taken from Ref. [37]. The scale factors $t_0$ and $t_2$ in the $(p,t)$ transfer operators have been fitted to the experimental data for the $0_1^{+}\left({}^{150}\mathrm{Sm}\right)\to 0_1^{+}({}^{148}\mathrm{Sm}$) and $0_1^{+}{(}^{152}\mathrm{Sm})\to 2_1^{+}({}^{150}\mathrm{Sm}$) transfer reactions, respectively.

Figure 9

Same as described in the caption of Fig. 7, but for the ${}^{158-158}\mathrm{Gd}$ isotopes. The experimental data have been taken from Refs. [10, 11, 12]. The scale factors $t_0$ and $t_2$ have been fitted to the experimental data for the $0_1^{+}\left({}^{154}\mathrm{Gd}\right)\to 0_1^{+}({}^{156}\mathrm{Gd}$) and $0_1^{+}{(}^{156}\mathrm{Gd})\to 2_1^{+}({}^{158}\mathrm{Gd}$) transfer reactions, respectively.

Figure 10

Same as described in the caption of Fig. 8, but for the ${}^{158-158}\mathrm{Gd}$ isotopes. The experimental data have been taken from Ref. [9]. The scale factors $t_0$ and $t_2$ have been fitted to the experimental data for the $0_1^{+}\left({}^{156}\mathrm{Gd}\right)\to 0_1^{+}({}^{154}\mathrm{Gd}$) and $0_1^{+}{(}^{154}\mathrm{Gd})\to 2_1^{+}({}^{152}\mathrm{Gd}$) transfer reactions, respectively.

Figure 11

Same as described in the caption of Fig. 7, but for the ${}^{150-160}\mathrm{Dy}$ isotopes. The experimental data have been taken from Ref. [38]. The scale factors $t_0$ and $t_2$ have been fitted to the experimental data for the $0_1^{+}\left({}^{156}\mathrm{Dy}\right)\to 0_1^{+}({}^{158}\mathrm{Dy}$) and $0_1^{+}{(}^{156}\mathrm{Dy})\to 2_1^{+}({}^{158}\mathrm{Dy}$) transfer reactions, respectively.

Figure 12

Same as described in the caption of Fig. 8, but for the ${}^{150-160}\mathrm{Dy}$ isotopes. The experimental data have been taken from Ref. [39] for ${}^{158}\mathrm{Dy}(p,t){}^{156}\mathrm{Dy}$ and Ref. [40] for ${}^{162}\mathrm{Dy}(p,t){}^{160}\mathrm{Dy}$ and ${}^{160}\mathrm{Dy}(p,t){}^{158}\mathrm{Dy}$ reactions. The scale factors $t_0$ and $t_2$ have been fitted to the experimental data for the $0_1^{+}\left({}^{158}\mathrm{Dy}\right)\to 0_1^{+}({}^{156}\mathrm{Dy}$) and $0_1^{+}{(}^{160}\mathrm{Dy})\to 2_1^{+}({}^{158}\mathrm{Dy}$) transfer reactions, respectively.