Low-energy structure and $\beta$-decay properties of neutron-rich nuclei in the region of a shape phase transition

The low-energy structure and $\beta$-decay properties of the neutron-rich even-mass nuclei near the neutron number $N=60$ that are experimentally of much interest are investigated within the framework of the nuclear density functional theory and the interacting boson-fermion-fermion model. By using the results of the constrained self-consistent mean-field calculations based on the relativistic energy density functional, the interacting-boson Hamiltonian describing the even-even core nuclei, the boson-fermion, and the fermion-fermion interactions are determined. The Gamow-Teller transition strengths, with the corresponding operator being constructed without introducing further phenomenological adjustment, are computed by using the wave functions of the initial and final nuclei of the $\beta$ decay. The triaxial quadrupole potential energy surfaces computed for the $N=60$ even-even isotones suggest a pronounced $\gamma$ softness. The calculated energy spectra for the even-even and odd-odd nuclei in the Rb to Cd isotopic chains exhibit an abrupt change in nuclear structure around $N=60$, as suggested experimentally. The predicted $\beta$-decay ${log}_{10}ft$ values underestimate the measured values for the nuclei with low $Z$ and with $N⩽60$, exhibit a rapid increase for $N>60$, reflecting the nuclear structure evolution, and agree rather well with the measured values for those nuclei with $Z$ being not far from the proton major shell closure $Z=50$. Sensitivity of the predicted $\beta$-decay properties to the model assumptions and parameters employed in the nuclear structure calculations is discussed, specifically, by comparing results obtained based on the different choices of the underlying energy density functional.

Figure 1

Columns 1 and 2: Contour plots of the triaxial quadrupole $(\beta ,\gamma )$ PESs for the even-even $N=60$ isotones calculated within the constrained relativistic Hartree-Bogoliubov method employing the energy functional DD-PC1 and separable pairing force of finite range. Columns 3 and 4: The corresponding mapped PESs in the IBM-2. The global minimum is indicated by an open triangle, and the difference between the neighboring contours is 0.2 MeV.

Figure 2

Calculated excitation spectra of the $2_1^{+}, 4_1^{+}, 0_2^{+}$, and $2_2^{+}$ states of the even-even Kr, Sr, Zr, Mo, Ru, Pd, and Cd isotopes (solid symbols connected by solid lines). The experimental data (open symbols connected by dotted lines) are adopted from the NNDC database [73].

Figure 3

Adopted strength parameters of the IBM-2 Hamiltonian for the even-even nuclei studied in the present work.

Figure 4

Calculated and experimental [73] (a) $B(E2;2_1^{+}\to 0_1^{+})$, (b) $B(E2;4_1^{+}\to 2_1^{+})$, (c) $B(E2;0_2^{+}\to 2_1^{+})$, and (d) $B(E2;2_2^{+}\to 2_1^{+})$ transition strengths for the even-even $N=60$ isotones as functions of the proton number $Z$.

Figure 5

Calculated and experimental excitation energies of the low-spin positive-parity states of the odd-odd nuclei.

Figure 6

Fractions in percent of the configurations in terms of the pair components ${[\nu s_{1/2}\otimes \pi g_{9/2}]}^{\left(J\right)}, {[\nu d_{3/2}\otimes \pi g_{9/2}]}^{\left(J\right)}, {[\nu d_{5/2}\otimes \pi g_{9/2}]}^{\left(J\right)}$, and ${[\nu g_{7/2}\otimes \pi g_{9/2}]}^{\left(J\right)}$ that are coupled to even-even boson cores in the IBFFM-2 wave function of the $1_1^{+}$ state for the studied odd-odd nuclei.

Figure 7

The ${log}_{10}ft$ values for [(a)–(f)] the ${\beta }^{-}$ decay of the $0_1^{+}$ ground state of even-even nuclei into the $1_1^{+}$ state of odd-odd nuclei, and [(g)–(l)] the ${\beta }^{-}$ decay of the $1_1^{+}$ state of odd-odd nuclei into the $0_1^{+}$ state of even-even nuclei. Calculated ${log}_{10}ft$ values are represented by solid circles connected by lines, and open circles denote the experimental values [73].

Figure 8

$B\left(\mathrm{GT}\right)$ strength distribution for the ${\beta }^{-}$ decay $0_1^{+}\to 1_n^{+}$ of the even-even $N=60$ isotones in terms of the excitation energies of the $1_n^{+}$ states of the odd-odd final nuclei (left), their running sum (middle), and $B\left(\mathrm{GT}\right)$ strength distribution for the ${\beta }^{-}$ decay $1_n^{+}\to 0_1^{+}$ of the odd-odd $N=59$ isotones (right). The experimental $B(GT;0_1^{+}\to 1^{+})$ values extracted from the ${log}_{10}ft$ data [73] are shown as red squares. The dashed vertical lines in the middle panels indicate experimental $Q_{{\beta }^{-}}$ values [73]. Note that the $B\left(\mathrm{GT}\right)$ values for the ${}^{96}\mathrm{Rb}\left(1_n^{+}\right)\to {}^{96}\mathrm{Sr}\left(0_1^{+}\right)$ are too small, and they are scaled with a factor of 100 so that they are more visible.

Figure 9

$B\left(\mathrm{GT}\right)$ strength distribution for the ${\beta }^{-}$ decay $0_1^{+}\to 1_n^{+}$ of the Zr isotopes in terms of the excitation energies $E\left(1_n^{+}\right)$ of the final Nb nuclei (left), and their running sums (right).

Figure 10

Low-energy spectra for the ${}^{106}\mathrm{Pd}$ and ${}^{106}\mathrm{Rh}$ nuclei computed with the microscopic inputs provided from the DD-PC1 and Gogny-D1M EDFs. The experimental data are taken from Ref. [73], and the results for the Gogny-D1M EDF are exploited from Ref. [55].