Simultaneous description of $\beta$ decay and low-lying structure of neutron-rich even- and odd-mass Rh and Pd nuclei

In this study, we use the neutron-proton IBM (IBM-2) [57, 58]. In this model both neutron and proton monopole ($s_{\nu}$ and $s_{\pi}$), and quadrupole ($d_{\nu}$ and $d_{\pi}$) bosons are considered as fundamental degrees of freedom. From a microscopic point of view [58, 57], the $s_{\nu}$ ($s_{\pi}$) and $d_{\nu}$ ($d_{\pi}$) bosons are associated with the collective $S_{\nu}$ ($S_{\pi}$) and $D_{\nu}$ ($D_{\pi}$) pairs of valence neutrons (protons) with angular momenta and parity $0^{+}$ and $2^{+}$, respectively. In comparison with the simpler IBM-1, in which the neutrons and protons are not distinguished, the IBM-2 appears to be more suitable to treat $\beta$ decay, since in this process both proton and neutron degrees of freedom should be explicitly taken into account. For the model space the neutron $N=$ 50-82 and proton $Z=$ 28-50 major shells are used. Hence for ^104-124Pd, the number of neutron bosons, $N_{\nu}$, varies within the range $2\leqslant N_{\nu}\leqslant 8$, while the number of the proton bosons is fixed, $N_{\pi}=2$.

To deal with even-even, odd-mass, and odd-odd nuclei on an equal footing, both collective and single-particle degrees of freedom are treated within the framework of the neutron-proton IBFFM (IBFFM-2). The IBFFM-2 Hamiltonian reads

$\displaystyle\hat{H}=\hat{H}_{\mathrm{B}}+\hat{H}_{\mathrm{F}}^{\nu}+\hat{H}_{\mathrm{F}}^{\pi}+\hat{V}_{\mathrm{BF}}^{\nu}+\hat{V}_{\mathrm{BF}}^{\pi}+\hat{V}_{\nu\pi},$

(1)

where $\hat{H}_{\mathrm{B}}$ is the IBM-2 Hamiltonian representing the bosonic even-even core, $\hat{H}_{\mathrm{F}}^{\nu}$ ($\hat{H}_{\mathrm{F}}^{\pi}$) is the one-body, single-neutron (-proton) Hamiltonian, and $\hat{V}_{\mathrm{BF}}^{\nu}$ ($\hat{V}_{\mathrm{BF}}^{\pi}$) stands for the interaction between the odd neutron (proton) and the even-even IBM-2 core. The last term $\hat{V}_{\nu\pi}$ represents the residual interaction between the odd neutron and the odd proton.

The IBM-2 Hamiltonian takes the form

$\displaystyle\hat{H}_{\mathrm{B}}=\epsilon_{d}(\hat{n}_{d_{\nu}}+\hat{n}_{d_{\pi}})+\kappa\hat{Q}_{\nu}\cdot\hat{Q}_{\pi},$

(2)

where in the first term, $\hat{n}_{d_{\rho}}=d^{\dagger}_{\rho}\cdot\tilde{d}_{\rho}$ ($\rho=\nu$ or $\pi$) is the $d$-boson number operator, with $\epsilon_{d}$ the single $d$-boson energy relative to the $s$-boson one, and $\tilde{d}_{\rho\mu}=(-1)^{\mu}d_{\rho-\mu}$. The second term stands for the quadrupole-quadrupole interaction between neutron and proton boson systems with strength $\kappa$, and $\hat{Q}_{\rho}=d_{\rho}^{\dagger}s_{\rho}+s_{\rho}^{\dagger}\tilde{d}_{\rho}+\chi_{\rho}(d^{\dagger}_{\rho}\times\tilde{d}_{\rho})^{(2)}$ represents the bosonic quadrupole operator, with the dimensionless parameter $\chi_{\rho}$.

The single-nucleon Hamiltonian $\hat{H}_{\mathrm{F}}^{\rho}$ takes the form

$\displaystyle\hat{H}_{\mathrm{F}}^{\rho}=-\sum_{j_{\rho}}\epsilon_{j_{\rho}}\sqrt{2j_{\rho}+1}(a_{j_{\rho}}^{\dagger}\times\tilde{a}_{j_{\rho}})^{(0)},$

(3)

where $\epsilon_{j_{\rho}}$ stands for the single-particle energy of the odd neutron $(\rho=\nu)$ or proton ($\rho=\pi$) orbital $j_{\rho}$. $a_{j_{\rho}}$ and $a_{j_{\rho}}^{\dagger}$ are annihilation and creation operators of the single particle, respectively. The operator $\tilde{a}_{j_{\rho}}$ is defined as $\tilde{a}_{j_{\rho}m_{\rho}}=(-1)^{j_{\rho}-m_{\rho}}a_{j_{\rho}-m_{\rho}}$.

In this study, we employ the following boson-fermion interaction $\hat{V}_{\mathrm{BF}}^{\rho}$ [31]

$$\hat{V}_{\mathrm{BF}}^{\rho}=\Gamma_{\rho}\hat{V}_{\mathrm{dyn}}^{\rho}+\Lambda_{\rho}\hat{V}_{\mathrm{exc}}^{\rho}+A_{\rho}\hat{V}_{\mathrm{mon}}^{\rho}.$$

(4)

The first, second, and third terms are dynamical quadrupole, exchange, and monopole interactions, respectively. Within the generalized seniority scheme [59, 31], the dynamical and exchange terms are assumed to be dominated by the interaction between unlike particles. On the other hand, the monopole term is assumed to be dominated by the interaction between like particles. The explicit form of the different terms in Eq. (4) then read

		$\displaystyle\hat{V}_{\mathrm{dyn}}^{\rho}=\sum_{j_{\rho}j_{\rho}^{\prime}}\gamma_{j_{\rho}j_{\rho}^{\prime}}(a^{\dagger}_{j_{\rho}}\times\tilde{a}_{j_{\rho}^{\prime}})^{(2)}\cdot\hat{Q}_{\rho^{\prime}},$		(5)
		$\displaystyle\hat{V}^{\rho}_{\mathrm{exc}}=-\left(s_{\rho^{\prime}}^{\dagger}\times\tilde{d}_{\rho^{\prime}}\right)^{(2)}\cdot\sum_{j_{\rho}j_{\rho}^{\prime}j_{\rho}^{\prime\prime}}\sqrt{\frac{10}{N_{\rho}(2j_{\rho}+1)}}\beta_{j_{\rho}j_{\rho}^{\prime}}\beta_{j_{\rho}^{\prime\prime}j_{\rho}}$
		$\displaystyle{\quad}:\left((d_{\rho}^{\dagger}\times\tilde{a}_{j_{\rho}^{\prime\prime}})^{(j_{\rho})}\times(a_{j_{\rho}^{\prime}}^{\dagger}\times\tilde{s}_{\rho})^{(j_{\rho}^{\prime})}\right)^{(2)}:+(\text{H.c.}),$		(6)
		$\displaystyle\hat{V}_{\mathrm{mon}}^{\rho}=-\hat{n}_{d_{\rho}}\sum_{j_{\rho}}\sqrt{2j_{\rho}+1}(a_{j_{\rho}}^{\dagger}\times\tilde{a}_{j_{\rho}})^{(0)},$		(7)

where the coefficients $\gamma_{j_{\rho}j_{\rho}^{\prime}}=(u_{j_{\rho}}u_{j_{\rho}^{\prime}}-v_{j_{\rho}}v_{j_{\rho}^{\prime}})Q_{j_{\rho}j_{\rho}^{\prime}}$, and $\beta_{j_{\rho}j_{\rho}^{\prime}}=(u_{j_{\rho}}v_{j_{\rho}^{\prime}}+v_{j_{\rho}}u_{j_{\rho}^{\prime}})Q_{j_{\rho}j_{\rho}^{\prime}}$ are proportional to the matrix elements of the fermion quadrupole operator in the single-particle basis $Q_{j_{\rho}j_{\rho}^{\prime}}=\braket{\ell_{\rho}\frac{1}{2}j_{\rho}}{Y^{(2)}}{\ell^{\prime}_{\rho}\frac{1}{2}j_{\rho}^{\prime}}$. The operator $\hat{Q}_{\rho^{\prime}}$ in Eq. (5) is the same boson quadrupole operator as in the boson Hamiltonian (2). In Eq. (II.1) the notation $:(\cdots):$ stands for normal ordering. Within this formalism, the single-particle energy $\epsilon_{j_{\rho}}$ in Eq. (3) is replaced with the quasiparticle energy $\tilde{\epsilon}_{j_{\rho}}$.

For the residual neutron-proton interaction $\hat{V}_{\nu\pi}$ in Eq. (1), we adopt the form [60]

	$\displaystyle\hat{V}_{\nu\pi}=4\pi{v_{\mathrm{d}}}$	$\displaystyle\delta(\bm{r})\delta(\bm{r}_{\nu}-r_{0})\delta(\bm{r}_{\pi}-r_{0})$
		$\displaystyle+v_{\mathrm{t}}\left[\frac{3({\bm{\sigma}}_{\nu}\cdot{\bf r})({\bm{\sigma}}_{\pi}\cdot{\bf r})}{r^{2}}-{\bm{\sigma}}_{\nu}\cdot{\bm{\sigma}}_{\pi}\right],$		(8)

where the first and second terms are surface-delta and tensor interactions with strength parameters $v_{\mathrm{d}}$, and $v_{\mathrm{t}}$, respectively. Note that $\bm{r}=\bm{r}_{\nu}-\bm{r}_{\pi}$ and $r_{0}=1.2A^{1/3}$ fm.

Table 1 summarizes the even-even Pd core nuclei, neighboring odd-$A$ Pd and Rh, and odd-odd Rh nuclei considered in this study.

In the initial step a set of constrained HFB calculations for even-even Pd isotopes based on the parametrization D1M of the Gogny-EDF is carried out to obtain the microscopic input to build the IBFFM-2 Hamiltonian. For each even-even Pd isotope, those calculations provide the corresponding energy surfaces, i.e., the total mean-field energies as functions of the triaxial quadrupole deformations $\beta$ and $\gamma$ [61]. For each nucleus, the Gogny-D1M HFB energy surface is mapped onto the expectation value of the IBM-2 Hamiltonian $\hat{H}_{\mathrm{B}}$ (2) in the boson condensate state [62]. This procedure specifies the parameters of the boson Hamiltonian, i.e., $\epsilon_{d}$, $\kappa$, $\chi_{\nu}$, and $\chi_{\pi}$. For more details about the mapping procedure, the reader is referred to Refs. [63, 64].

Next, the Hamiltonian $\hat{H}_{\mathrm{F}}$ of Eq. (3) and the boson-fermion interactions $\hat{V}_{\mathrm{BF}}$ of Eq. (4) are determined using the procedure of Refs. [65, 66]. The single-particle energies $\epsilon_{j_{\rho}}$ of the odd nucleon are obtained from HFB calculations constrained to zero quadrupole deformation. Once the single-particle energies are available, the quasiparticle energies $\tilde{\epsilon}_{j_{\rho}}$ and occupation probabilities $v^{2}_{j_{\rho}}$ are computed within the BCS approximation, separately for neutron and proton single-particle spaces. The empirical pairing gap $12A^{-1/2}$ is used. We include in the BCS calculations the $2s_{1/2}$, $1d_{3/2}$, $1d_{5/2}$, $0g_{7/2}$, $0g_{9/2}$, and $0h_{11/2}$ orbitals for the odd neutron, and the $1d_{5/2}$, $0g_{7/2}$, $0g_{9/2}$, $2p_{1/2}$, $2p_{3/2}$, and $1f_{5/2}$ orbitals for the odd proton. The corresponding quasiparticle energies $\tilde{\epsilon}_{j_{\nu}}$ ($\tilde{\epsilon}_{j_{\pi}}$), and occupation probabilities $v^{2}_{j_{\nu}}$ ($v^{2}_{j_{\pi}}$) for the odd neutron (proton) $2s_{1/2}$, $1d_{3/2}$, $1d_{5/2}$, and $0g_{7/2}$ ($1d_{5/2}$, $0g_{7/2}$, and $0g_{9/2}$) orbitals are taken as the inputs to $\hat{H}_{\mathrm{F}}^{\nu}$ ($\hat{H}_{\mathrm{F}}^{\pi}$) and $\hat{V}_{\mathrm{BF}}^{\nu}$ ($\hat{V}_{\mathrm{BF}}^{\pi}$), respectively. The strength parameters $\Gamma_{\rho}$, $\Lambda_{\rho}$, and $A_{\rho}$ for $\hat{V}_{\mathrm{BF}}^{\rho}$ are then fixed so that the observed low-energy positive-parity levels for the odd-$A$ Pd ($\rho=\nu$) or odd-$Z$ Rh ($\rho=\pi$) nuclei are reproduced reasonably well.

Finally, the parameters $v_{\mathrm{d}}$ and $v_{\mathrm{t}}$ for the residual neutron-proton interaction in Eq. (II.1) are determined [67] so that the observed low-lying positive-parity states for each odd-odd Rh nucleus are reasonably well reproduced. Note that the same strength parameters as those obtained in the previous step for the neighboring odd-$A$ nuclei are employed in the IBFFM-2 calculations for odd-odd nuclei. On the other hand, the quasiparticle energies and occupation probabilities of the odd particles are independently computed.

Figure 1 shows the neutron and proton spherical single-particle energies ($\epsilon_{j_{\nu}}$ and $\epsilon_{j_{\pi}}$), resulting from the Gogny-HFB calculations, and the quasiparticle energies ($\tilde{\epsilon}_{j_{\nu}}$ and $\tilde{\epsilon}_{j_{\pi}}$) and occupation probabilities ($v^{2}_{j_{\nu}}$ and $v^{2}_{j_{\pi}}$) used in the IBFM-2 and IBFFM-2 calculations.

Theories with effective degrees of freedom, like the IBFFM, require the definition of transition operators to be used in the evaluation of electromagnetic transition probabilities. For the electric $E2$ transition the operator $\hat{T}^{(E2)}$ to be used in the IBFFM-2 takes the form [31]

$\displaystyle\hat{T}^{(E2)}=\hat{T}^{(E2)}_{\text{B}}+\hat{T}^{(E2)}_{\text{F}}\;,$

(9)

where the first and second terms are the boson and fermion parts, respectively. They are given by

$\displaystyle\hat{T}^{(E2)}_{\mathrm{B}}=\sum_{\rho=\nu,\pi}e_{\rho}^{\mathrm{B}}\hat{Q}_{\rho}\;,$

(10)

and

	$\displaystyle\hat{T}^{(E2)}_{\mathrm{F}}=-\frac{1}{\sqrt{5}}$	$\displaystyle\sum_{\rho=\nu,\pi}\sum_{j_{\rho}j_{\rho}^{\prime}}(u_{j_{\rho}}u_{j_{\rho}^{\prime}}-v_{j_{\rho}}v_{j_{\rho}^{\prime}})$
		$\displaystyle\times\left\langle\ell_{\rho}\frac{1}{2}j_{\rho}\bigg{\|}e^{\mathrm{F}}_{\rho}r^{2}Y^{(2)}\bigg{\|}\ell_{\rho}^{\prime}\frac{1}{2}j_{\rho}^{\prime}\right\rangle(a_{j_{\rho}}^{\dagger}\times\tilde{a}_{j_{\rho}^{\prime}})^{(2)}\;.$		(11)

The fixed values $e^{\mathrm{B}}_{\nu}=e^{\mathrm{B}}_{\pi}=0.1$ $e$b for the boson effective charges are taken so that the experimental $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ transition probabilities are reproduced for even-even Pd isotopes. The standard neutron and proton effective charges $e^{\mathrm{F}}_{\nu}=0.5$ $e$b $e^{\mathrm{F}}_{\pi}=1.5$ $e$b are employed for all the studied odd-nucleon systems. The $M1$ transition operator $\hat{T}^{(M1)}$ is defined as

	$\displaystyle\hat{T}^{(M1)}=\sqrt{\frac{3}{4\pi}}$	$\displaystyle\sum_{\rho=\nu,\pi}\Biggl{[}g_{\rho}^{\mathrm{B}}\hat{L}_{\rho}-\frac{1}{\sqrt{3}}\sum_{j_{\rho}j_{\rho}^{\prime}}(u_{j_{\rho}}u_{j_{\rho}^{\prime}}+v_{j_{\rho}}v_{j_{\rho}^{\prime}})$
		$\displaystyle\times\left\langle j_{\rho}\|g_{l}^{\rho}{\bf l}+g_{s}^{\rho}{\bf s}\|j_{\rho}^{\prime}\right\rangle(a_{j_{\rho}}^{\dagger}\times\tilde{a}_{j_{\rho}^{\prime}})^{(1)}\Biggr{]}.$		(12)

The empirical $g$ factors $g_{\nu}^{\mathrm{B}}=0\,\mu_{N}$ (nuclear magneton) and $g_{\pi}^{\mathrm{B}}=1.0\,\mu_{N}$, are adopted for the neutron and proton bosons. For the neutron (proton) $g$ factors, the standard Schmidt values $g_{l}^{\nu}=0\,\mu_{N}$ and $g_{s}^{\nu}=-3.82\,\mu_{N}$ ($g_{l}^{\pi}=1.0\,\mu_{N}$ and $g_{s}^{\pi}=5.58\,\mu_{N}$) are used, with $g_{s}^{\rho}$ quenched by 30% with respect to the free value.

As in the electromagnetic case, the transition operators for allowed $\beta$ decay have to be redefined in terms of the relevant degrees of freedom of the model. The Gamow-Teller $\hat{T}^{\mathrm{GT}}$ and Fermi $\hat{T}^{\mathrm{F}}$ transition operators take the form

		$\displaystyle\hat{T}^{\rm GT}=\sum_{j_{\nu}j_{\pi}}\eta_{j_{\nu}j_{\pi}}^{\mathrm{GT}}\left(\hat{P}_{j_{\nu}}\times\hat{P}_{j_{\pi}}\right)^{(1)},$		(13)
		$\displaystyle\hat{T}^{\rm F}=\sum_{j_{\nu}j_{\pi}}\eta_{j_{\nu}j_{\pi}}^{\mathrm{F}}\left(\hat{P}_{j_{\nu}}\times\hat{P}_{j_{\pi}}\right)^{(0)},$		(14)

with the coefficients

	$\displaystyle\eta_{j_{\nu}j_{\pi}}^{\mathrm{GT}}$	$\displaystyle=-\frac{1}{\sqrt{3}}\left\langle\ell_{\nu}\frac{1}{2}j_{\nu}\bigg{\|}{\bm{\sigma}}\bigg{\|}\ell_{\pi}\frac{1}{2}j_{\pi}\right\rangle\delta_{\ell_{\nu}\ell_{\pi}},$		(15)
	$\displaystyle\eta_{j_{\nu}j_{\pi}}^{\mathrm{F}}$	$\displaystyle=-\sqrt{2j_{\nu}+1}\delta_{j_{\nu}j_{\pi}}.$		(16)

In Eqs. (13) and (14), $\hat{P}_{j_{\rho}}$ represents one of the one-particle creation operators

		$\displaystyle A^{\dagger}_{j_{\rho}m_{\rho}}=\zeta_{j_{\rho}}a_{{j_{\rho}}m_{\rho}}^{\dagger}+\sum_{j_{\rho}^{\prime}}\zeta_{j_{\rho}j_{\rho}^{\prime}}s^{\dagger}_{\rho}(\tilde{d}_{\rho}\times a_{j_{\rho}^{\prime}}^{\dagger})^{(j_{\rho})}_{m_{\rho}}$		(17a)
		$\displaystyle B^{\dagger}_{j_{\rho}m_{\rho}}=\theta_{j_{\rho}}s^{\dagger}_{\rho}\tilde{a}_{j_{\rho}m_{\rho}}+\sum_{j_{\rho}^{\prime}}\theta_{j_{\rho}j_{\rho}^{\prime}}(d^{\dagger}_{\rho}\times\tilde{a}_{j_{\rho}^{\prime}})^{(j_{\rho})}_{m_{\rho}},$		(17b)
and the annihilation operators
		$\displaystyle\tilde{A}_{j_{\rho}m_{\rho}}=(-1)^{j_{\rho}-m_{\rho}}A_{j_{\rho}-m_{\rho}}$		(17c)
		$\displaystyle\tilde{B}_{j_{\rho}m_{\rho}}=(-1)^{j_{\rho}-m_{\rho}}B_{j_{\rho}-m_{\rho}}.$		(17d)

The operators in Eqs. (17a) and (17c) conserve the boson number, whereas those in Eqs. (17b) and (17d) do not. The operators $\hat{T}^{\rm GT}$ and $\hat{T}^{\rm F}$ are expressed as a combination of two of the operators in Eqs. (17a)-(17d), depending on the type of the $\beta$ decay studied (i.e., $\beta^{+}$ or $\beta^{-}$) and on the particle or hole nature of the valence nucleons. In the present case,

$\displaystyle\hat{P}_{j_{\nu}}=\left\{\begin{array}[]{cc}\tilde{B}_{j_{\nu},m_{\nu}}&(N\leqslant 66)\\ \tilde{A}^{\dagger}_{j_{\nu},m_{\nu}}&(N\geqslant 68)\\ \end{array}\right.$

(20)

for the $\beta^{-}$ decay of the odd-$A$ Rh, while

$\displaystyle\hat{P}_{j_{\nu}}=\left\{\begin{array}[]{cr}\tilde{A}_{j_{\nu},m_{\nu}}&(N\leqslant 65)\\ \tilde{B}^{\dagger}_{j_{\nu},m_{\nu}}&(N\geqslant 67)\\ \end{array}\right.$

(23)

for the $\beta^{-}$ decay of the even-$A$ Rh. On the other hand, $\hat{P}_{j_{\pi}}=\tilde{A}_{j_{\pi},m_{\pi}}$ for all the considered $\beta^{-}$ decays. Note, that Eqs. (17a)–(17d) are simplified forms of the most general one-particle transfer operators in the IBFM-2 [31].

By using the generalized seniority scheme, the coefficients $\zeta_{j}$, $\zeta_{jj^{\prime}}$, $\theta_{j}$, and $\theta_{jj^{\prime}}$ in Eqs. (17a) and (17b) can be written as [68]

	$\displaystyle\zeta_{j_{\rho}}$	$\displaystyle=u_{j_{\rho}}\frac{1}{K_{j_{\rho}}^{\prime}},$		(24a)
	$\displaystyle\zeta_{j_{\rho}j_{\rho}^{\prime}}$	$\displaystyle=-v_{j_{\rho}}\beta_{j_{\rho}^{\prime}j_{\rho}}\sqrt{\frac{10}{N_{\rho}(2j_{\rho}+1)}}\frac{1}{KK_{j_{\rho}}^{\prime}},$		(24b)
	$\displaystyle\theta_{j_{\rho}}$	$\displaystyle=\frac{v_{j_{\rho}}}{\sqrt{N_{\rho}}}\frac{1}{K_{j_{\rho}}^{\prime\prime}},$		(24c)
	$\displaystyle\theta_{j_{\rho}j_{\rho}^{\prime}}$	$\displaystyle=u_{j_{\rho}}\beta_{j_{\rho}^{\prime}j_{\rho}}\sqrt{\frac{10}{2j_{\rho}+1}}\frac{1}{KK_{j_{\rho}}^{\prime\prime}}.$		(24d)

The factors $K$, $K_{j_{\rho}}^{\prime}$, and $K_{j_{\rho}}^{\prime\prime}$ are defined as

		$\displaystyle K=\left(\sum_{j_{\rho}j_{\rho}^{\prime}}\beta_{j_{\rho}j_{\rho}^{\prime}}^{2}\right)^{1/2},$		(25a)
		$\displaystyle K_{j_{\rho}}^{\prime}=\left[1+2\left(\frac{v_{j_{\rho}}}{u_{j_{\rho}}}\right)^{2}\frac{\braket{(\hat{n}_{s_{\rho}}+1)\hat{n}_{d_{\rho}}}_{0^{+}_{1}}}{N_{\rho}(2j_{\rho}+1)}\frac{\sum_{j_{\rho}^{\prime}}\beta_{j_{\rho}^{\prime}j_{\rho}}^{2}}{K^{2}}\right]^{1/2},$		(25b)
		$\displaystyle K_{j_{\rho}}^{\prime\prime}=\left[\frac{\braket{\hat{n}_{s_{\rho}}}_{0^{+}_{1}}}{N_{\rho}}+2\left(\frac{u_{j_{\rho}}}{v_{j_{\rho}}}\right)^{2}\frac{\braket{\hat{n}_{d_{\rho}}}_{0^{+}_{1}}}{2j_{\rho}+1}\frac{\sum_{j_{\rho}^{\prime}}\beta_{j_{\rho}^{\prime}j_{\rho}}^{2}}{K^{2}}\right]^{1/2},$		(25c)

where $\hat{n}_{s_{\rho}}$ is the number operator for the $s_{\rho}$ boson and $\braket{\cdots}_{0^{+}_{1}}$ stands for the expectation value of a given operator in the $0^{+}_{1}$ ground state of the even-even nucleus. The amplitudes $v_{j_{\rho}}$ and $u_{j_{\rho}}$ appearing in Eqs. (24a)-(24d) and (25a)-(25c) are the same as those used in the IBFM-2 (or IBFFM-2) calculations for the odd-mass (or odd-odd) nuclei. No additional parameter is introduced for the GT and Fermi operators. For a more detailed account on $\beta$-decay operators within the IBFM-2 or IBFFM-2 framework, the reader is also referred to Refs. [68, 6, 31].

The $\beta$-decay $ft$ values are given by

$\displaystyle ft=\frac{K}{|M({\mathrm{F}})|^{2}+\left(\frac{g_{\mathrm{A}}}{g_{\mathrm{V}}}\right)^{2}|M({\mathrm{GT}})|^{2}},$

(26)

where the numeric constant $K$ takes the value $K=6163$ s. The quantities $M({\mathrm{F}})$ and $M({\mathrm{GT}})$ are the reduced matrix elements of the operators $\hat{T}^{\mathrm{F}}$ of Eq. (14) and $\hat{T}^{\mathrm{GT}}$ of Eq. (13), respectively. Here $g_{\mathrm{V}}$ and $g_{\mathrm{A}}$ are the vector and axial-vector coupling constants, respectively. In this study, we use the free nucleon values, $g_{\mathrm{V}}=1$ and $g_{\mathrm{A}}=1.27$, for the $\beta$ decays of both even- and odd-$A$ Rh.

The Gogny-D1M HFB and mapped IBM-2 potential energy surfaces are shown in Fig. 2 as functions of the $(\beta,\gamma)$ deformation parameters for the even-even ^104-124Pd nuclei. The variation of the HFB potential energy surfaces as functions of the neutron number suggests a transition from prolate (for $N\lesssim 62$) to $\gamma$-soft ($64\lesssim N\lesssim 70$), and to nearly spherical ($N\gtrsim 72$) shapes. In particular, both ^112,114Pd exhibit rather flat potential energy surfaces along the $\gamma$ direction. This is what is expected in the $\gamma$-unstable O(6) limit of the IBM [29]. In the case of ¹¹⁶Pd, a flat-bottomed potential with a weak $\gamma$ dependence, characteristic of the E(5) critical-point symmetry [69], is obtained.

For each of the considered nuclei, the Gogny-HFB and IBM-2 energy surfaces display a similar topology in the neighborhood of the global minimum (the location of the minimum, and the softness in the $\beta$ and $\gamma$ directions are similar). However, the mapped IBM-2 surfaces generally become flat at large $\beta$ deformation ($\beta\gtrsim 0.4$). This difference is a consequence of the fact that in the HFB approach all nucleonic degrees of freedom are taken into account while the IBM-2 is built on the more limited model (valence) space of nucleon pairs. However, since the mean-field configurations most relevant to the low-energy collective excitations are those in the vicinity of the global minimum, the mapping is considered specifically in that region [63, 64].

The mapped IBM-2 excitation energies of the $2^{+}_{1}$, $4^{+}_{1}$, $0^{+}_{2}$, and $2^{+}_{2}$ states in the even-even ^104-124Pd nuclei are shown in Fig. 3 as functions of the neutron number $N$. Results obtained using the conventional IBM-2 approach (hereinafter referred to as phenomenological IBM-2), with parameters adopted from the earlier phenomenological study [37], are also included in the plot. As can be seen from the figure, the excitation energies decrease toward the middle of the major shell, i.e., $N=66$. For $N\leqslant 64$, the mapped IBM-2 $2^{+}_{1}$ and $4^{+}_{1}$ excitation energies underestimate the experimental ones while the energies of the non-yrast $0^{+}_{2}$ and $2^{+}_{2}$ states are overestimated. In the mapped (phenomenological) IBM-2 approach the ratios $R_{4/2}$ of the $4^{+}_{1}$ to $2^{+}_{1}$ excitation energies are 2.96 (2.43), 2.86 (2.39), and 2.69 (2.34) for ¹⁰⁴Pd, ¹⁰⁶Pd, and ¹⁰⁸Pd, respectively. These values should be compared with the experimental ratios of 2.38, 2.40, and 2.41. Thus, the mapped IBM-2 provides excitation spectra which are more rotational in character than the phenomenological IBM-2 and experimental ones. Around the neutron midshell $N=66$, both the predicted and experimental $2^{+}_{2}$ levels have the lowest energies, being even below the $4^{+}_{1}$ state. The $2^{+}_{2}$ state is the bandhead of the quasi-$\gamma$ band, and the lowering of this state reflects an emergence of pronounced $\gamma$ softness.

The IBM-2 parameters obtained for the even-even Pd isotopes from the mapping procedure, and those determined phenomenologically are shown in Fig. 4. The phenomenological IBM-2 parameters are extracted from earlier fitting calculations for Pd and Ru isotopes [37]. In Ref. [37], in addition to the terms that appear in Eq. (2), the like-boson interactions, and the so-called Majorana terms were included in the model Hamiltonian. These terms were, however, shown to play a minor role [37], and are omitted in the present study. From Fig. 4, one sees that the single-$d$ boson energy $\epsilon_{d}$ and the strength $\kappa$ have similar nucleon-number dependence for both the mapped and phenomenological IBM-2 models. A notable quantitative difference is that the derived $\kappa$ values for the former are $\approx$ 1.4 larger in magnitude than for the latter. The behavior of the parameter $\chi_{\nu}$ is different in the two approaches for $N\geqslant 70$. The sign and absolute value of the sum $\chi_{\nu}+\chi_{\pi}$ reflect the extent of $\gamma$ softness and whether the nucleus is prolate or oblate deformed. In both calculations, the sum is negative, $\chi_{\nu}+\chi_{\pi}<0$, for $N\lesssim 64$, indicating prolate deformation, and takes nearly vanishing values, $\chi_{\nu}+\chi_{\pi}\approx 0$, around the neutron midshell $N=66$, reflecting $\gamma$ softness. However, for $N\geqslant 70$, the sum is negative (positive) in the mapped (phenomenological) calculations, implying prolate (oblate) deformation. Note that a fixed value $\chi_{\pi}=0.2$ is employed in the phenomenological IBM-2 calculations, whereas in the mapped approach this parameter exhibits a strong nucleon number dependence.

The $B(E2)$ transition probabilities, computed within the mapped and phenomenological IBM-2 models, are plotted in Fig. 5 as functions of the neutron number $N$. The same $E2$ effective boson charge is used for the quadrupole operators in the two sets of the IBM-2 calculations. The $B(E2;2^{+}_{1}\to 0^{+}_{1})$ and $B(E2;4^{+}_{1}\to 2^{+}_{1})$ values obtained in the mapped IBM-2 calculations agree reasonably well with the experiment, exception made of ¹¹²Pd. Both the mapped and phenomenological IBM-2 calculations predict $B(E2;0^{+}_{2}\to 2^{+}_{1})$ and $B(E2;2^{+}_{2}\to 2^{+}_{1})$ rates with similar trends as functions of $N$. However, the mapped IBM-2 scheme provides smaller $B(E2;0^{+}_{2}\to 2^{+}_{1})$ values for Pd isotopes with $58\leqslant N\leqslant 62$. The enhancement of the predicted $B(E2;2^{+}_{2}\to 2^{+}_{1})$ transition rates around the midshell $N=66$ [see Fig. 5(d)] can be considered as another signature of $\gamma$ soft deformation.