Pairing vibrations in the interacting boson model based on density functional theory

We propose a method to incorporate the coupling between shape and pairing collective degrees of freedom in the framework of the interacting boson model (IBM), based on the nuclear density functional theory. To account for pairing vibrations, a boson-number nonconserving IBM Hamiltonian is introduced. The Hamiltonian is constructed by using solutions of self-consistent mean-field calculations based on a universal energy density functional and pairing force, with constraints on the axially symmetric quadrupole and pairing intrinsic deformations. By mapping the resulting quadrupole-pairing potential energy surface onto the expectation value of the bosonic Hamiltonian in the boson condensate state, the strength parameters of the boson Hamiltonian are determined. An illustrative calculation is performed for ${}^{122}\mathrm{Xe}$, and the method is further explored in a more systematic study of rare-earth $N=92$ isotones. The inclusion of the dynamical pairing degree of freedom significantly lowers the energies of bands based on excited $0^{+}$ states. The results are in quantitative agreement with spectroscopic data, and are consistent with those obtained using the collective Hamiltonian approach.

Figure 1

The potential energy surface (PES) of ${}^{122}\mathrm{Xe}$ in the $(\beta ,\alpha )$ plane, calculated by the constrained $\mathrm{RMF}+\mathrm{BCS}$ with the PC-PK1 energy density functional and separable pairing interaction. All energies (in MeV) in the PES are normalized with respect to the binding energy of the absolute minimum. The contours join points on the surface with the same energies, and energy difference between the neighboring contours is 200 keV.

Figure 2

Same as in the caption to Fig. 1 but for the IBM energy surface.

Figure 3

Excitation spectra for the unperturbed boson configurations $[n-1]$, $\left[n\right]$ and $[n+1]$, and the final one obtained after mixing the three configurations.

Figure 4

Excitation spectra of ${}^{122}\mathrm{Xe}$ resulting from the IBM calculation with a single boson-number configuration $[n=9]$ (left panel), and including configuration mixing between $[n=8]$, $[n=9]$, and $[n=10]$ boson spaces ($\mathit{p}v$-IBM). Experimental states are from Refs. [61, 62] (right panel). The $0^{+}$ band-head states of the $K^{\pi }=0^{+}$ bands are highlighted with thick lines.

Figure 5

Probabilities of the $[n-1]$, $\left[n\right]$, and $[n+1]$ components in the wave functions of the five lowest-energy $0^{+}$, $2^{+}$, and $4^{+}$ states in ${}^{122}\mathrm{Xe}$.

Figure 6

Matrix elements of the monopole pair transfer operator ${\widehat{H}}_{\mathrm{mix}}$ [last term in Eq. (6)] between the unperturbed $[n-1]$ and $\left[n\right]$ configurations, and between the unperturbed $\left[n\right]$ and $[n+1]$ configurations, for the lowest three even-spin states up to $I=8^{+}$.

Figure 7

Same as in the caption to Fig. 1 but for the $N=92$ isotones ${}^{152}\mathrm{Nd}$, ${}^{154}\mathrm{Sm}$, ${}^{156}\mathrm{Gd}$, and ${}^{158}\mathrm{Dy}$.

Figure 8

Same as in the caption to Fig. 7, but for the IBM energy surfaces.

Figure 9

Low-energy $K^{\pi }=0^{+}$ bands of ${}^{152}\mathrm{Nd}$, calculated using the IBM without and with the dynamical pairing degree of freedom, in comparison to available data [61]. The corresponding spectrum obtained with the quadrupole-pairing collective Hamiltonian (QPCH) model is included for comparison.

Figure 10

Same as described in the caption to Fig. 9, but for ${}^{154}\mathrm{Sm}$.

Figure 11

Same as described in the caption to Fig. 9, but for ${}^{156}\mathrm{Gd}$.

Figure 12

Same as described in the caption to Fig. 9, but for ${}^{158}\mathrm{Dy}$.

Figure 13

Probabilities of the $[n-1]$, $\left[n\right]$, and $[n+1]$ components in the $\mathit{p}v$-IBM wave functions of the four lowest $0^{+}$ states in the $N=92$ isotones.