Shape transitions in odd-mass $\gamma$-soft nuclei within the interacting boson-fermion model based on the Gogny energy density functional

The interacting boson-fermion model (IBFM), with parameters determined from the microscopic Hartree-Fock-Bogoliubov (HFB) approximation, based on the parametrization D1M of the Gogny energy density functional, is employed to study the structural evolution in odd-mass $\gamma$-soft nuclei. The deformation energy surfaces of even-even nuclei, single-particle energies, and occupation probabilities of the corresponding odd-mass systems have been obtained within the constrained HFB approach. Those building blocks are then used as a microscopic input to build the IBFM Hamiltonian. The coupling constants of the boson-fermion interaction terms are taken as free parameters, fitted to reproduce experimental low-lying spectra. The diagonalization of the IBFM Hamiltonian provides the spectroscopic properties for the studied odd-mass nuclei. The procedure has been applied to compute low-energy excitation spectra and electromagnetic transition rates, in the case of the $\gamma$-soft odd-mass systems ${}^{129-137}\mathrm{Ba}$, ${}^{127-135}\mathrm{Xe}$, ${}^{129-137}\mathrm{La}$, and ${}^{127-135}\mathrm{Cs}$. The calculations provide a reasonable agreement with the available experimental data and agree well with previous results based on the relativistic mean-field approximation.

Figure 1

The single-particle energies corresponding to the $3s_{1/2}$, $2d_{3/2}$, $2d_{5/2}$, $1g_{7/2}$, and $1h_{11/2}$ orbitals employed in the present study (denoted as “D1M”) for the considered odd-mass nuclei are plotted with respect to the $2d_{3/2}$ single-particle level. Results from Ref. [29] are also included in the plot (denoted as “DD-PC1”).

Figure 2

The same as in Fig. 1, but for occupation probabilities of the $3s_{1/2}$, $2d_{3/2}$, $2d_{5/2}$, $1g_{7/2}$, and $1h_{11/2}$ orbitals.

Figure 3

The strength parameter ${\mathrm{\Gamma }}_0$ from Eq. (5) (denoted by “D1M”) is plotted for both positive- (${\mathrm{\Gamma }}_0^{+}$) and negative-parity (${\mathrm{\Gamma }}_0^{-}$) states. Results from Ref. [29] are also included in the plot (denoted as “DD-PC1”).

Figure 4

The same as in Fig. 3, but for the strength parameter ${\mathrm{\Lambda }}_0$ of the exchange term in Eq. (6).

Figure 5

The Gogny-D1M energy surfaces for the even-even nuclei ${}^{128-136}\mathrm{Ba}$ (top) and ${}^{126-134}\mathrm{Xe}$ (bottom) are plotted up to 3 MeV above the absolute minimum. The difference between neighboring contours is 100 keV.

Figure 6

The same as in Fig. 5, but for the mapped IBM energy surfaces.

Figure 7

The low-energy excitation spectra obtained for ${}^{128-136}\mathrm{Ba}$ and ${}^{126-134}\mathrm{Xe}$ are plotted as functions of the neutron number $N$. Experimental data were taken from Ref. [51].

Figure 8

The low-energy positive- ($\pi =+1$) and negative-parity ($\pi =-1$) excitation spectra in the odd-mass isotopes ${}^{129-137}\mathrm{Ba}$ are plotted as functions of the neutron number $N$. Experimental data were taken from Ref. [51].

Figure 9

The same as in Fig. 8, but for the odd-mass Xe isotopes.

Figure 10

The same as in Fig. 8, but for the odd-mass La isotopes.

Figure 11

The same as in Fig. 8, but for the odd-mass Cs isotopes.

Figure 12

The transition probabilities $B(E2;{3/2}_1^{+}\to {1/2}_1^{+})$, $B(E2;{5/2}_1^{+}\to {1/2}_1^{+})$, and $B(E2;{5/2}_1^{+}\to {3/2}_1^{+})$ obtained for the odd-mass nuclei (a) ${}^{129-137}\mathrm{Ba}$ and (b) ${}^{127-135}\mathrm{Xe}$, and the transition probabilities $B(E2;{5/2}_1^{+}\to {1/2}_1^{+})$, $B(E2;{7/2}_1^{+}\to {3/2}_1^{+})$, and $B(E2;{7/2}_1^{+}\to {5/2}_1^{+})$ obtained for the odd-mass nuclei (c) ${}^{129-137}\mathrm{La}$ and (d) ${}^{127-135}\mathrm{Cs}$ are plotted as functions of the neutron number. The experimental values of $B(E2;{3/2}_1^{+}\to {1/2}_1^{+})$ (solid circles), $B(E2;{5/2}_1^{+}\to {1/2}_1^{+})$ (squares), $B(E2;{5/2}_1^{+}\to {3/2}_1^{+})$ (triangles), $B(E2;{7/2}_1^{+}\to {3/2}_1^{+})$ (diamonds), and $B(E2;{7/2}_1^{+}\to {5/2}_1^{+})$ (asterisks) are taken from Ref. [51].

Figure 13

The spectroscopic quadrupole moments $Q_{{3/2}_1^{+}}$ and $Q_{{11/2}_1^{-}}$ obtained for (a) ${}^{129-137}\mathrm{Ba}$ and (b) ${}^{127-135}\mathrm{Xe}$ and the spectroscopic quadrupole moments $Q_{{5/2}_1^{+}}$ and $Q_{{7/2}_1^{+}}$ obtained for (c) ${}^{129-137}\mathrm{La}$ and (d) ${}^{127-135}\mathrm{Cs}$ are plotted as functions of the neutron number. The experimental values of $Q_{{3/2}_1^{+}}$ (solid circles), $Q_{{11/2}_1^{-}}$ (squares), $Q_{{5/2}_1^{+}}$ (diamonds), and $Q_{{7/2}_1^{+}}$ (asterisks) are taken from Ref. [52].

Figure 14

The magnetic moments ${\mu }_{{1/2}_1^{+}}$, ${\mu }_{{3/2}_1^{+}}$, and ${\mu }_{{11/2}_1^{-}}$ obtained for (a) ${}^{129-137}\mathrm{Ba}$ and (b) ${}^{127-135}\mathrm{Xe}$ and the magnetic moments ${\mu }_{{1/2}_1^{+}}$, ${\mu }_{{5/2}_1^{+}}$, and ${\mu }_{{7/2}_1^{+}}$ obtained for (c) ${}^{129-137}\mathrm{La}$ and (d) ${}^{127-135}\mathrm{Cs}$ are plotted as functions of the neutron number. The experimental values of ${\mu }_{{1/2}_1^{+}}$ (solid triangles), ${\mu }_{{3/2}_1^{+}}$ (circles), ${\mu }_{{11/2}_1^{-}}$ (squares), ${\mu }_{{5/2}_1^{+}}$ (diamonds), and ${\mu }_{{7/2}_1^{+}}$ (asterisks) are taken from Ref. [52].

Figure 15

Detailed level scheme for the low-energy positive- and negative-parity states in ${}^{135}\mathrm{Ba}$. Experimental data are from Ref. [51].

Figure 16

The same as in Fig. 15, but for ${}^{129}\mathrm{Xe}$.

Figure 17

The same as in Fig. 15, but for ${}^{131}\mathrm{Xe}$.

Figure 18

The same as in Fig. 15, but for ${}^{133}\mathrm{La}$.

Figure 19

The same as in Fig. 15, but for ${}^{131}\mathrm{Cs}$.

Figure 20

The same as in Fig. 15, but for ${}^{133}\mathrm{Cs}$.