Beta Decay Properties of Waiting-Point N = 50 and 82 Isotopes

Abstract

We performed the microscopic calculation of $\beta$-decay properties for waiting-point nuclei with neutron-closed magic shells. Allowed Gamow–Teller (GT) and first-forbidden (FF) transitions were simulated using a schematic model (SM) for waiting-point N = $50,82$ isotopes in the framework of a proton–neutron quasiparticle random phase approximation ($pn$-QRPA). The Woods–Saxon (WS) potential basis was used in our calculations. The $pn$-QRPA equations of allowed (GT) and (FF) transitions were utilized in both the particle–hole ($ph$) and particle–particle ($pp$) channels in the SM. We solved the secular equations of the GT and FF transitions for eigenvalues and eigenfunctions of the corresponding Hamiltonians. A spherical shape was assigned to each waiting-point nucleus in all simulations. Significantly, this study marks the first time that $\beta$-decay analysis has been applied to certain nuclei, including ⁸²Ge₅₀, ⁸³As₅₀, ⁸⁴Se₅₀, ⁸⁵Br₅₀ and ⁸⁷Rb₅₀ with $N=50$ isotones, and ¹³²Sn₈₂, ¹³³Sb₈₂, ¹³⁴Te₈₂, ¹³⁵I₈₂ and ¹³⁷Cs₈₂ with $N=82$ isotones. Since there is no prior theoretical research on these nuclei, this work is a unique addition to the field. We compared our results with the previous calculations and measured data, and our calculations agree with the experimental data and the other theoretical results.

1. Introduction

Weak reaction $r$- and s-processes are considered among the important phenomena in the nucleosynthesis of heavy elements, especially after the efforts that have been made and are being made to understand the formation of stars [1,2,3]. Despite the limited experimental data on these properties [4,5], our comprehension of the process has evolved due to advancements in theoretical and computational investigations, as evidenced by more recent publications [6,7,8,9]. This matter was required by understanding the nuclear properties of many of the nuclei rich in neutrons, which are located at a large distance from the stability line. This is a basic requirement in the $r$-process, which also requires a neutron density of more than ${10}^{20}$ neutrons/${\mathrm{c}\mathrm{m}}^3$, in addition to data about entropy, density, and high temperature, which is approximately ${10}^9\mathrm{K}$, and other information about stellar matter [2,3,8,10,11]. In addition, data about the beta decay of the reactor core, resulting from the fission process and the acting radioactive ion-beam facilities (RIB), such as ISOLDE-CERN, ALTO, RIKEN, TRIUMF, and NSCL, are useful in the sector of nuclear safety in reactors [1].

The decay half-life and the probability of delayed neutron emission are crucial indicators of nuclear structural behavior, especially regarding the response of nuclear spin states throughout the decay process. Many questions raise interest in the development of the nuclear shell, in particular, when $N/\mathrm{Z}$ ratios are high. New subshells may be formed as a result of the inversion of the quasi-particle level and the emergence of new magic numbers, like $\mathrm{N}=34$ in calcium-54 [1,12]. Experimental data for waiting-point nuclei are limited, particularly for those with closed neutron shells at N = 50 and 82, where available literature provides insufficient information [13,14,15].

The flow of $r$-process material slows down by waiting for multiple decays before restarting rapid neutron capture. This process is faster than beta-decay processes, through which a neutron-rich nucleus is produced with a neutron separation energy greater than or equal to 3 MeV. Due to the high binding energy, discontinuities appear in the separation energy spectrum, especially at the closed nucleus $\mathrm{N}=50$, 82, and 128 isotones [7,16]. The half-life periods of the waiting-point nucleus are measured in terrestrial laboratories as a result of rapid development. Conversely, conditions in stellar environments are significantly distinct, and the current study aims to enhance comprehension of these phenomena. This is mainly due to the r-process necessitating exceedingly high temperatures and the parent waiting-point nuclei potentially existing in thermally excited states, elements that can significantly modify their decay rates. This demonstrates the importance of the GT transition in both the ground state and the excited state due to the possibility of capturing the electron in the nucleus of the waiting point, which contributes to increasing the rates of the weak stellar interaction and is expected to be an exchange of charges [17,18]. Experimentally, a few waiting cores depend on the ground state [19,20]. Borzov adopted the energy density functional model as well as the continuous quasi-particle random phase approximation (DF + CQRPA) model, which is based on the self-consistent theory of finite Fermi systems as an alternative to the Skyrme–Hartree–Fock (SHF) method [21]. Nabi et al. studied the GT and the unique first-forbidden transitions U1F for waiting-point $\mathrm{N}=50$ and 82 nuclei in the stellar environment using the $pn$-QRPA (N) model. The data collected included several aspects of beta decay, such as its properties, half-life, rates of delayed neutron emission probability, and associated energies, as indicated in reference [22]. Caroline et al. calculated the allowed and first-forbidden transitions in the series $N=82 \mathrm{a}\mathrm{n}\mathrm{d} 126$, which adopted the approach based on the quasi-particle random phase approximation (QRPA) with quasi-particle vibration coupling. It was shown that the available data agreed with the obtained data, especially when taking into account quasi-particle vibrational coupling [23]. Wang et al. [24] developed the projected shell model (PSM) to describe the allowed and first-forbidden transitions in nuclear beta-decay for a wide range of light and superheavy nuclei, including even–even, odd–odd, and odd-mass nuclei. They concentrated on calculating and systematically evaluating 35 significant first-forbidden transitions that are expected to play a major role in solving the problems with the reactor anti-neutrino spectrum. These computations were performed to test how well the PSM works [25]. The $\beta$-decay $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values for the region surrounding the doubly magic nucleus (e.g., ²⁰⁸Pb) were estimated using the proton–neutron quasi-particle random phase approximation ($pn$-QRPA) model. This region has consistently garnered interest for additional research due to its enhanced stability. The schematic model approach was utilized to resolve the $pn$-QRPA equations. The Woods–Saxon (WS) potential was utilized as a mean-field basis, and spherical nuclei were assumed. An investigation was conducted on allowed Gamow–Teller (GT) and first-forbidden (FF) transitions in the particle–hole ($ph$) channel. Furthermore, a deformed Nilsson basis was employed to process the random phase approximation (RPA) equations in the particle–particle (pp) and particle–hole ($ph$) channels. It was anticipated that the allowed $\beta$-decay and unique first-forbidden (U1F) rates would exist in stellar matter [26]. The rates of $\beta$-decay in isotonic nuclear chains were investigated in a study at four distinct neutron shell closures: $\mathrm{N}=$ $50,82,126$, and 184. The nucleosynthesis of the $r$-process was dependent on the closure of these casings. This endeavor aims to examine the correlations among nucleons and assess first-forbidden transitions that transpire outside the bounds of the relativistic QRPA. The many-body approach clarifies the relationship between individual and collective degrees of freedom by employing the relativistic QRPA. The study provides evidence that Gamow–Teller transitions take place within the energy range of decay, thereby confirming the decay rates that have been observed. As the system reaches a state of stability, the likelihood of decay via the FF transition diminishes due to the reduction in the coupling between vibrations and quasi-particles. Nevertheless, Gamow–Teller transitions occur at the decay $\mathrm{Q}-$value [27].

The paper is organized as follows. Section 2 briefly describes the formalism of the $pn$-QRPA model used in this paper. In Section 3, we show the calculated results of allowed GT and FF strength distributions and $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values for the neutron-closed magic shells of 50 and 82 isotopes. In Section 4, we finally give a summary of the paper.

2. Mathematical Formalism

We used the $pn-\mathrm{Q}\mathrm{R}\mathrm{P}\mathrm{A}\left(\mathrm{W}\mathrm{S}\right)$ model for the calculation of allowed GT and FF transitions. The formalism is briefly discussed in the following subsections. The $\mathrm{l}\mathrm{o}\mathrm{g} ft$ and half-life values for GT and FF transitions were calculated employing a spherical schematic model (SSM). The Woods–Saxon (WS) potential was used to calculate the single-particle energies and wave functions. The calculations were performed in the particle–hole $\left(ph\right)$, and particle–particle ($pp$) channels in allowed GT transition to calculate the eigenvalues and eigenfunctions of the Hamiltonian. The calculation was performed only in the particle–hole channel in the FF transitions to compute the eigenvalues and eigenfunctions of the Hamiltonian. The Gamow–Teller $\left(J^{\pi }=1^{+}\right)$ and first-forbidden $\left(J^{\pi }=0^{-},1^{-},2^{-}\right)$ excitations in odd–odd nuclei were generated from the correlated ground state of the parent nuclei through the charge-exchange spin–spin and spin–dipole interactions.

2.1. Gamow–Teller and First-Forbidden Transitions

The schematic model (SM) Hamiltonian for Gamow-Teller (GT) excitations in the quasi-particle representation is usually accepted in the following form:

$$H_{SSM}^{GT}=H_{sqp}+h_{ph}^{GT}+h_{pp}^{GT}$$

(1)

where $H_{sqp}$ is the corresponding single quasi-particle Hamiltonian of the system and the $h_{ph},{ h}_{pp}$ are the effective interaction operators in the particle–hole and the particle–particle channels, respectively. The single quasi-particle Hamiltonian of the system was given by

$$H_{sqp}=\sum _{j_{\tau }{m}_{\tau }} {\epsilon }_{j_{\tau }{m}_{\tau }}{\widehat{\alpha }}_{j_p{m}_p}^{†}{\widehat{\alpha }}_{j_n{m}_n},\mathrm{}(\tau =n,p)$$

(2)

where ${\epsilon }_{j_{\tau }{m}_{\tau }}$ is the single quasi-particle energy of the nucleons with angular momentum $j_{\tau }{m}_{\tau }$. The ${\widehat{\alpha }}_{j_p{m}_p}^{†}$ and ${\widehat{\alpha }}_{j_n{m}_n}$ are one quasi-particle creation and annihilation operators, respectively. In quasi-boson approximation, the spin–isospin effective interaction Hamiltonian in the two channels is written in terms of the quasi-boson creation and annihilation operators as follows:

$$h_{ph}^{GT}={\displaystyle \frac{2{\chi }_{ph}}{g_A^2}}\sum _{j_{\tau }{j}_{{\tau }^{\prime }}\mu {\mu }^{\prime }} \left[b_{j_p{j}_n}\left(\lambda \right)A_{j_p{j}_n}^{†}\left(\lambda \mu \right)+(-1{)}^{(\lambda +\mu )}{\bar{b}}_{j_p{j}_n}(\lambda \left)A_{j_p{j}_n}\right(\lambda -\mu )\right]$$

(3)

$$h_{pp}^{GT}=-{\displaystyle \frac{2{\chi }_{pp}}{g_A^2}}\sum _{j_{\tau }{j}_{{\tau }^{\prime }}\mu {\mu }^{\prime }} \left[d_{j_p{j}_n}\left(\lambda \right)A_{j_p{j}_n}^{†}\left(\lambda \mu \right)-(-1{)}^{(\lambda +\mu )}{\bar{d}}_{j_p{j}_n}(\lambda \left)A_{j_p{j}_n}\right(\lambda -\mu )\right]$$

(4)

Within the $pn$-QRPA framework, the Gamow–Teller (GT) residual interactions in the particle–hole ($ph$) and particle–particle ($pp$) channels are established by multiple fundamental parameters. The Hamiltonians $h_{ph}^{GT}$ and $h_{pp}^{GT}$ characterize the effective interactions that affect nuclear excitations and weak decays. Furthermore, the interaction strength constants, ${\chi }_{ph}$ and ${\chi }_{pp}$, govern the intensity of the residual forces in their respective channels. The axial–vector coupling constant $g_A$ is a crucial parameter in weak interactions, impacting transition rates. Moreover, the quantum numbers $j_{\tau }$ and $j_{{\tau }^{\prime }}$ demonstrate the total angular momentum of nucleon states (protons and neutrons). The multipolarity parameter λ indicates the angular momentum transfer in nuclear transitions, with $\lambda =1$ signifying Gamow–Teller (GT) transitions and $\lambda =2$ representing first-forbidden (FF) transitions. The magnetic substates $\mu$ and ${\mu }^{\prime }$ represent the projection of λ onto the quantization axis, assuming values ranging from $-\lambda$ to $+\lambda$. The reduced matrix elements $b_{j_p{j}_n}\left(\lambda \right)$ and ${\bar{b}}_{j_p{j}_n}\left(\lambda \right)$, along with $d_{j_p{j}_n}\left(\lambda \right)$ and ${\bar{d}}_{j_p{j}_n}\left(\lambda \right)$, denote transition strengths in particle–hole and particle–particle interactions, respectively. Meanwhile, the quasi-boson creation and annihilation operators, $A_{j_p{j}_n}^{†}\left(\lambda \mu \right)$ and $A_{j_p{j}_n}\left(\lambda \mu \right)$, characterize nuclear excitations throughout both interaction channels. Finally, the factor $(-1{)}^{(\lambda +\mu )}$ guarantees the appropriate symmetry characteristics of the interaction, preserving the mathematical integrity of the $pn$-QRPA formalism.

The Hamiltonian of the first-forbidden transitions was chosen as

$$H_{SSM}^{FF}=H_{sqp}+h_{ph}^{FF}$$

(5)

The charge-exchange spin–spin effective residual interaction was determined using

$$\begin{array}{rr}& h_{ph}^{FF}={\displaystyle \frac{2{\chi }_{ph}}{g_A^2}}\sum _{j_{\tau }{j}_{{\tau }^{\prime }}\mu {\mu }^{\prime }} \left[b_{j_p{j}_n}\left(\lambda \right)A_{j_p{j}_n}^{†}\left(\lambda \mu \right)+(-1{)}^{(\lambda +\mu )}{\bar{b}}_{j_p{j}_n}(\lambda \left)A_{j_p{j}_n}\right(\lambda -\mu )\right]\end{array}$$

(6)

The $A_{j_p{j}_n}^{†}(\lambda -\mu )$ and $A_{j_p{j}_n}\left(\lambda \mu \right)$ in Equations (3), (4), and (6) are the quasi-boson creation and annihilation operators, respectively, defined by

$$A_{j_p{j}_n}^{†}(\lambda -\mu )=\sqrt{{\displaystyle \frac{2\lambda +1}{2j_n+1}}}\sum _{m_n{m}_p} (-1{)}^{j_p-m_p}{\alpha }_{j_p{m}_p}^{†}{\alpha }_{j_n-m_n}$$

(7)

and

$$A_{j_p{j}_n}\left(\lambda \mu \right)={\left[A_{j_p{j}_n}^{†}(\lambda -\mu )\right]}^{†}.$$

The $b_{j_p{j}_n}\left(\lambda \right),{ \bar{b}}_{j_p{j}_n}\left(\lambda \right)$ in Equations (3) and (6) stand for the reduced matrix elements of the related multipole operators. These operators for $\mathsf{\Delta }{J}^{\pi }=1^{+}$ transitions were given by

$${\bar{b}}_{j_p{j}_n}(\lambda )={\displaystyle \frac{1}{\sqrt{2\lambda +1}}}⟨j_p\left(l_p{s}_p\right)‖{\sigma }_k{\tau }_k^{\pm }‖j_n\left(l_n{s}_n\right)⟩v_{j_n}{u}_{j_p}.$$

For $\mathsf{\Delta }{J}^{\pi }=0^{-},1^{-},2^{-}$, transitions were described in a general form by

$${\bar{b}}_{j_p{j}_n}(\lambda )={\displaystyle \frac{1}{\sqrt{2\lambda +1}}}⟨j_p\left(l_p{s}_p\right)‖{\tau }_k^{\pm }{r}_k{\left[Y_1\left({\widehat{r}}_k\right){\sigma }_k\right]}_{\lambda }‖j_n\left(l_n{s}_n\right)⟩v_{j_n}{u}_{j_p}$$

and

$$b_{j_p{j}_n}(\lambda )={\displaystyle \frac{{\bar{b}}_{j_p{j}_n}\left(\lambda \right)}{v_{j_n}{u}_{j_p}}}{u}_{j_n}{v}_{j_p}$$

where $v$ and $u$ are single-particle and hole amplitudes, respectively. The solution of Hamiltonian Equations (1) and (5) is briefly described below. The allowed (GT) and the charge-exchange vibration modes in odd–odd nuclei are considered phonon excitations and are described by

$$\left|{\mathsf{\Psi }}_i>=Q_i^{†}\right|0>={\mathsf{\Sigma }}_{j_{\tau }\mu }\left[{\psi }_{j_p{j}_n}^i{A}^{†}(\lambda \mu {)}_{j_p{j}_n}-{\phi }_{j_p{j}_n}^i{A}_{j_p{j}_n}(\lambda \mu )\right]\mid 0>$$

(8)

where $Q_i^{†}$ is the $pn$-QRPA phonon creation operator, $\mid 0>$ is the phonon vacuum which corresponds to the ground state of an even–even nucleus and fulfils $Q_i\mid 0>=0$ for all $i$. The ${\psi }_{j_p{j}_n}^i$ and ${\phi }_{j_p{j}_n}^i$ are forward and backward quasi-boson amplitudes. Employing the conventional procedure of the $pn-\mathrm{Q}\mathrm{R}\mathrm{P}\mathrm{A}$, we solved the equation of motion

$$\left[H,Q_i^{†}\right]\left|0>={\omega }_i{Q}_i^{†}\right|0>$$

(9)

The ${\omega }_i$ is the $i$th $0^{-},1^{-}$, and $2^{-}$ excitation energy in odd–odd daughters calculated from the ground state of the parent even–even nucleus. Further details of the formalism can be seen in refs [28,29,30,31,32].

2.2. Extension in the Model for Odd-A Nuclei

We summarize the necessary formalism for treating odd-A nuclei in our model. The Hamiltonian of the odd-A system is chosen as

$$H_{SSM}=H_{sqp}+h_{ph}^{CC}+h_{ph}^{CD}.$$

(10)

The effective residual interactions are determined using

$$\begin{gathered} h_{ph}^{CC}={\displaystyle \frac{2{\chi }_{ph}}{g_A^2}}\sum _{j_{\tau }{j}_{{\tau }^{\prime }}\mu {\mu }^{\prime }} \left[{\bar{b}}_{j_p{j}_n}\left(\lambda \right)C_{j_p{j}_n}^{†}\left(\lambda \mu \right)+(-1{)}^{\left(\lambda +\mu \right)}{b}_{j_p{j}_n}(\lambda \left)C_{j_p{j}_n}\right(\lambda -\mu )\right] \mathrm{} \\ \hspace{1em}\hspace{1em}\times \left[{\bar{b}}_{j_{p^{\prime }}{j}_{n^{\prime }}}\left(\lambda \right)C_{j_{p^{\prime }}{j}_{n^{\prime }}}\left(\lambda {\mu }^{\prime }\right)+b_{j_{p^{\prime }}{j}_{n^{\prime }}}\left(\lambda \right)C_{j_{p^{\prime }}{j}_{n^{\prime }}}^{†}\left(\lambda -{\mu }^{\prime }\right)\right] \end{gathered}$$

(11-a)

$$\begin{array}{c}{h}_{ph}^{CD}={\displaystyle \frac{2{\chi }_{ph}}{g_A^2}}{\displaystyle \sum _{j_{\tau }{j}_{{\tau }^{\prime }}\mu {\mu }^{\prime }}} \left[{\bar{b}}_{j_p{j}_n}\left(\lambda \right)C_{j_p{j}_n}^{†}\left(\lambda \mu \right)+(-1{)}^{(\lambda +\mu )}{b}_{j_p{j}_n}(\lambda \left)C_{j_p{j}_n}\right(\lambda -\mu )\right]\end{array}$$

$$\hspace{1em}\hspace{1em}\times \left[d_{j_{p^{\prime }}{j}_{n^{\prime }}}\left(\lambda \right)D_{j_{p^{\prime }}{j}_{n^{\prime }}}\left(\lambda {\mu }^{\prime }\right)+{\bar{d}}_{j_{p^{\prime }}{j}_{n^{\prime }}}\left(\lambda \right)D_{j_{p^{\prime }}{j}_{n^{\prime }}}^{†}\left(\lambda -{\mu }^{\prime }\right)\right]$$

(11-b)

The $C_{j_p{j}_n}^{†}\left(\lambda -\mu \right),{ D}_{j_p{j}_n}^{†}(\lambda -\mu )$ and $C_{j_p{j}_n}\left(\lambda \mu \right),D_{j_p{j}_n}\left(\lambda \mu \right)$ are the quasi-boson creation and annihilation operators, respectively, and given as

$$C_{j_p{j}_n}^{†}(\lambda -\mu )=\sqrt{{\displaystyle \frac{2\lambda +1}{2j_n+1}}}\sum _{m_n{m}_p} (-1{)}^{j_p-m_p}{\alpha }_{j_n{m}_n}^{†}{\alpha }_{j_p-m_p}^{†}$$

and

$$\begin{array}{rr}& C_{j_p{j}_n}\left(\lambda \mu \right)={\left[C_{j_p{j}_n}^{†}(\lambda -\mu )\right]}^{†}\end{array}$$

and

$$D_{j_p{j}_n}\left(\lambda \mu \right)={\left[D_{j_p{j}_n}^{†}(\lambda -\mu )\right]}^{†}.$$

The $b_{j_p{j}_n}\left(\lambda \right),{ \bar{b}}_{j_p{j}_n}\left(\lambda \right),{ d}_{j_p{j}_n}\left(\lambda \right),{\overline{ d}}_{j_p{j}_n}\left(\lambda \right)$ in Equations (6) and (7) stand for the reduced matrix elements of the related multipole operators. The charge-exchange spin–spin and spin–dipole transitions are defined by

$$\begin{array}{cc} & \hfill {\bar{d}}_{j_p{j}_n}(\lambda )={\displaystyle \frac{1}{\sqrt{2\lambda +1}}}⟨j_p\left(l_p{s}_p\right)‖{\sigma }_k{\tau }_k^{\pm }‖j_n\left(l_n{s}_n\right)⟩v_{j_n}{v}_{j_p}\hfill \\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\bar{d}}_{j_p{j}_n}(\lambda )={\displaystyle \frac{1}{\sqrt{2\lambda +1}}}⟨j_p\left(l_p{s}_p\right)‖{\tau }_k^{\pm }{r}_k{\left[Y_1\left({\widehat{r}}_k\right){\sigma }_k\right]}_{\lambda }‖j_n\left(l_n{s}_n\right)⟩v_{j_n}{v}_{j_p},\end{array}$$

(12)

and

$$d_{j_p{j}_n}(\lambda )={\displaystyle \frac{{\bar{d}}_{j_p{j}_n}\left(\lambda \right)}{v_{j_n}{v}_{j_p}}}{u}_{j_n}{u}_{j_p}$$

The wave function of the odd-A (with odd neutrons) nuclei is given by

$$\left|{\mathsf{\Psi }}_{j_k{m}_k}^j>={\mathsf{\Omega }}_{j_k{m}_k}^{j†}\right|0>=\left(N_{j_k}^j{\alpha }_{j_k{m}_k}^{†}+\sum _{j_{\nu }{m}_{\nu }} R_{k\nu }^{ij}{A}_i^{†}{\alpha }_{j_{\nu }{m}_{\nu }}^{†}\right)\mid 0>$$

(13)

where ${\mathsf{\Omega }}_{j_k{m}_k}^{j†}$ and $\mid 0>$ represent the phonon operator and phonon vacuum, respectively. $N_{j_k}^j$ and $R_{k\nu }^{ij}$ are the quasi-boson amplitudes corresponding to the states and fulfil the normalization conditions. The wave function was formed by the superposition of one and three-quasi-particle (one quasi-particle + one phonon) states. The equation of motion is defined by

$$\left[H_{SSM},{\mathsf{\Omega }}_{j_k{m}_k}^{j†}\right]\left|0>={\omega }_{j_k{m}_k}^j{\mathsf{\Omega }}_{j_k{m}_k}^{j†}\right|0>$$

(14)

The excitation energies ${\omega }_{j_k{m}_k}^j$ and the wave functions of the (GT) and (FF) excitations were obtained from the $pn-\mathrm{Q}\mathrm{R}\mathrm{P}\mathrm{A}\left(\mathrm{W}\mathrm{S}\right)$ equation of motion. The details of the solution for the odd-A nuclei can be seen from ref. [26].

2.3. Nuclear Matrix Elements

One of the characteristic quantities for the (GT) $1^{+}$ states occurring in the neighboring odd–odd nuclei is the (GT) transition matrix elements. The allowed (GT) ${\beta }^{\pm }$ transition matrix elements are calculated using the following expressions:

$$M_{{\beta }^{-}}^i\left(0^{+}\to 1_i^{+}\right)=\left.⟨1_i^{+},\mu \right|G_{1\mu }^{-}\left|0^{+}\right.⟩=⟨0\left|\left[Q_i\left(\mu \right),G_{1\mu }^{-}\right]\right|0⟩$$

(15)

$$M_{{\beta }^{+}}^i\left(0^{+}\to 1_i^{+}\right)=\left.⟨1_i^{+},\mu \right|G_{1\mu }^{+}\left|0^{+}\right.⟩=⟨0\left|\left[Q_i\left(\mu \right),G_{1\mu }^{+}\right]\right|0⟩$$

(16)

The ${\beta }^{\pm }$ reduced matrix elements are given by

$$B_{\mathrm{G}\mathrm{T}}^{(\pm )}\left({\omega }_i\right)=\sum _{\mu } {\left|M_{{\beta }^{\pm }}^i\left(0^{+}\to 1_i^{+}\right)\right|}^2$$

(17)

The ${\beta }^{\pm }$ transition strengths ($S^{\pm }$) must fulfil the Ikeda sum rule (ISR).

$$S^{\pm }=\sum _i B_{\mathrm{G}\mathrm{T}}^{(\pm )}\left({\omega }_i\right)$$

(18)

The FF transition consists of six nuclear matrix elements (NMEs), and these NMEs include relativistic and non-relativistic terms for the $\mathsf{\Delta }{J}^{\pi }=0^{-},1^{-}$ transitions. The relativistic NMEs were calculated directly without any assumptions. The contribution from the spin–orbit potential was included in the calculation of the relativistic matrix elements. The unique first-forbidden (U1F) transitions, $\mathsf{\Delta }{J}^{\pi }=2^{-}$, do not contain any relativistic term. The transitions probabilities $\widehat{B}\left({\lambda }^{\pi }=0^{-},1^{-},2^{-}\right)$ were specified as [33]

$$\widehat{B}\left({\lambda }^{\pi }=0^{-}\right)={\left|<0_i^{-}‖M^{\mathrm{rank} 0}‖0^{+}>\right|}^2$$

(19)

where

$$M^{rank0}=M\left({\rho }_A,\lambda =0\right)-i{\displaystyle \frac{m_{e}c}{h}}\xi \widehat{M}\left(j_A,\kappa =1,\lambda =0\right),$$

(20)

with

$$\begin{array}{cc}& M\left({\rho }_A,\lambda =0\right)={\displaystyle \frac{g_A}{(4\pi {)}^{1/2}c}}{\mathsf{\Sigma }}_k{\tau }_k^{\pm }\left({\sigma }_k\cdot {\vartheta }_k\right)\\ & \hspace{1em}\hspace{1em}{\left.M\left(j_A,\kappa =1,\lambda =0\right)=g_A{\mathsf{\Sigma }}_k{\tau }_k^{\pm }{r}_k\left[Y_1\left({\widehat{r}}_k\right){\sigma }_k\right)\right]}_{0\mu }\end{array}$$

(21)

where

$$\begin{array}{rr}& M^{rank1}=M\left(j_V,\kappa =0,\lambda =1,\mu \right)\pm i{\displaystyle \frac{m_{e}c}{\sqrt{3}\mathrm{\hslash }}}\xi M\left({\rho }_V,\lambda =1,\mu \right)\end{array}$$

(22)

with

$$\begin{array}{ll}& M\left(j_V,\kappa =0,\lambda =1,\mu \right)={\displaystyle \frac{g_V}{c\sqrt{4\pi }}}{\mathsf{\Sigma }}_k{\tau }_k^{\pm }{\left({\vartheta }_k\right)}_{1\mu }\\ & M\left({\rho }_V,\lambda =1,\mu \right)=g_V{\mathsf{\Sigma }}_k{\tau }_k^{\pm }{r}_k{Y}_{1\mu }\left({\widehat{r}}_k\right)\\ & M\left(j_A,\kappa =1,\lambda =1,\mu \right)=g_A{\mathsf{\Sigma }}_k{\tau }_k^{\pm }{r}_k{\left[Y_1\left({\widehat{r}}_k\right){\sigma }_k\right]}_{1\mu }\end{array}$$

(23)

where

$$M^{\mathrm{rank} 2}=M\left(j_A,\kappa =1,\lambda =2,\mu \right)=g_A{\mathsf{\Sigma }}_k{\tau }_k^{\pm }{r}_k{\left[Y_1\left({\widehat{r}}_k\right){\sigma }_k\right]}_{2\mu }$$

(24)

where ${\sigma }_k,{\vartheta }_k,Y_1\left({\widehat{r}}_k\right)$ and ${\tau }_k^{\pm }$ denote the Pauli matrices, velocity, spherical harmonics, and isospin creation (annihilation) operators, respectively. The ${\rho }_V\left({\rho }_A\right)$ and $j_V\left(j_A\right)$ are the vector (axial–vector) charge and current densities associated with a single nucleon, respectively. These variables are linear and depend only on the space point $\mathbf{r}$. They are independent of the velocity. For further details, we refer to Appendix 3D-Beta Interaction in ref. [33]. In Equations (22) and (24), the upper and lower signs refer to ${\beta }^{-}$- and ${\beta }^{+}$-decays, respectively. The multipole operators considered to calculate the reduced NMEs of the first forbidden transitions were defined using

$$M\left(j_A,\kappa =1,\lambda ,\mu \right)=g_A{\mathsf{\Sigma }}_k{\tau }_k^{\pm }{r}_k{\left[Y_1\left({\widehat{r}}_k\right){\sigma }_k\right]}_{\lambda \mu }$$

(25)

where $\lambda =0,1,2$ and $\mu =(0,\pm 1,\pm 2,\dots ,\pm \lambda )$ are the corresponding nuclear spin $\left({\lambda }^{\pi }=0^{-},1^{-},2^{-}\right)$ for the transition and its projection, respectively. The $ft$ values of the allowed GT and non-unique first-forbidden transitions (rank 0, rank 1) were calculated using

$$ft={\displaystyle \frac{D}{{\left(g_A/g_V\right)}^{2}4\pi \widehat{B}\left({\lambda }^{\pi }=1^{+},0^{-},1^{-}\right)}}$$

(26)

and for U1F transitions using [23]

$$t={\displaystyle \frac{(2n+1)!!}{(n+1!{)}^{2}n!}}{\displaystyle \frac{D}{{\left(g_A/g_V\right)}^{2}4\pi \widehat{B}\left({\lambda }^{\pi }=2^{-}\right)}}$$

(27)

where $D$ is a constant taken as 6295 s. The effective ratio of the axial and vector coupling constant was taken as $(g\_A/g\_V)=-1.254$ [34]. The effective/quenched axial–vector weak couplings may be considered depending on different nuclear models. But, the effective ratio of axial–vector weak couplings $0.7\times {\left(g_A/g_V\right)}^2$ was not used in our calculations.

3. Results and Discussion

The matrix elements of a single quasi-particle were calculated using the WS radial wave functions. The parameters were taken from ref. [12]. The proton and neutron pairing gaps were determined using ${\mathsf{\Delta }}_p=C_p/\sqrt{A}$ and ${\mathsf{\Delta }}_n=C_n/\sqrt{A}$, respectively. The pairing strength parameters ($C_p$ and $C_n$) were fixed to reproduce the experimental pairing gaps [35]. The study of the allowed GT decay nuclear matrix elements was carried out using the schematic Hamiltonian with the $ph$ and $pp$ channels. The parameter values ${\chi }_{ph}$ and ${\chi }_{pp}$ for the allowed GT transitions were 1.0 and 0.25 in schematic model calculations. The ${\chi }_{pp}$ effective interaction constant for allowed GT is taken as ${\chi }_{pp}=0.58A^{0.7}\mathrm{M}\mathrm{e}\mathrm{V}$. The ${\chi }_{ph}$ effective interaction constants for allowed GT, rank 0, rank 1, and rank 2 transitions are chosen as ${\chi }_{ph}=5.2A^{0.7}\mathrm{M}\mathrm{e}\mathrm{V},{\chi }_{ph}=30A^{-5/3}\mathrm{M}\mathrm{e}\mathrm{V}{\mathrm{f}\mathrm{m}}^{-2}$, ${\chi }_{ph}=90A^{-5/3}{\mathrm{MeVfm}}^{-2}$, and ${\chi }_{ph}=350A^{-5/3}{\mathrm{MeVfm}}^{-2}$, respectively. These values give better results and are close to the experimental values. The relativistic $\beta$—moment matrix elements in the first-forbidden $\mathsf{\Delta }J=0(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k} 0 )$ and $\mathsf{\Delta }J=1$ (rank 1$)$ transitions were calculated directly, without any assumption. Also, the beta-transition probabilities of $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k} 0$ and rank 1 transitions were performed within the $\xi$-approximation.

The computed $\beta$-decay half-lives including allowed GT and first-forbidden contributions for waiting-point nuclei having $\mathrm{N}=50$, 82 are shown in

Table 1. Comparison of our computed $\beta$-decay prior theoretical half-lives (in units of seconds) for $N=50$ waiting-point nuclei with previous calculations and experimental half-lives. Half-lives mentioned with an asterisk in the last column were adopted from [20].

Nuclei−A	Refs [31,32] GT	Ref [33] GT + FF	Ref [34]		Ref [22]		Present Work		Exp. [30]
			GT	GT + FF	GT	GT + FF	GT	GT + FF
$\mathrm{F}\mathrm{e}-76$	0.008	0.008	0.045	0.027	0.060	0.012	0.057	0.022	0.003
$\mathrm{C}\mathrm{o}-77$	0.016	0.016	0.013	0.014	0.025	0.016	0.021	0.017	0.015
$\mathrm{N}\mathrm{i}-78$	0.127	0.150	0.477	0.224	1.210	0.102	0.284	0.173	0.122
$\mathrm{C}\mathrm{u}-79$	0.222	0.270	0.430	0.157	0.436	0.235	0.321	0.223	0.241
$\mathrm{Z}\mathrm{n}-80$	0.432	0.530	3.068	1.260	0.851	0.557	0.735	0.544	0.562
$\mathrm{G}\mathrm{a}-81$	0.577	1.030	1.568	1.227	3.387	1.083	1.816	1.174	1.217
$\mathrm{G}\mathrm{e}-82$	-	-	-	-	-	-	4.82	4.25	4.31
$\mathrm{A}\mathrm{s}-83$	-	-	-	-	-	-	14.52	12.83	13.4
$\mathrm{S}\mathrm{e}-84$	-	-	-	-	-	-	211.4	178.6	195.6
$\mathrm{B}\mathrm{r}-85$	-	-	-	-	-	-	197.3	161.2	174
$\mathrm{R}\mathrm{b}-87$	-	-	-	-	-	-	51.6 Gy	47.3 Gy	49.7 Gy

and

Table 2. Comparison of our computed $\beta$-decay half-lives (in units of seconds) for $N=82$ waiting-point nuclei with previous calculations and experimental half-lives. Half-lives mentioned with an asterisk in the last column were adopted from [20].

Nuclei−A	Refs [31,32] GT	Ref [33] GT + FF	Ref [34]		Ref [22]		Present Work		Exp. [30]
			GT	GT + FF	GT	GT + FF	GT	GT + FF
$\mathrm{T}\mathrm{c}-125$	0.009	0.010	0.009	0.009	0.017	0.008	0.025	0.013	-
$\mathrm{R}\mathrm{u}-126$	0.020	0.020	0.034	0.030	0.027	0.017	0.038	0.021	-
$\mathrm{R}\mathrm{h}-127$	0.028	0.028	0.022	0.020	0.032	0.029	0.030	0.027	0.028
$\mathrm{P}\mathrm{d}-128$	0.046	0.047	0.125	0.074	0.042	0.035	0.073	0.044	0.035
$\mathrm{A}\mathrm{g}-129$	0.070	0.070	0.047	0.032	0.052	0.049	0.051	0.046	0.0499
$\mathrm{C}\mathrm{d}-130$	0.162	0.164	1.123	0.502	0.135	0.122	0.154	0.117	0.1268
$\mathrm{I}\mathrm{n}-131$	0.260	0.248	0.147	0.139	0.286	0.281	0.276	0.247	0.261
$\mathrm{S}\mathrm{n}-132$	-	-	-	-	-	-	41.2	38.3	39.7
$\mathrm{S}\mathrm{b}-133$	-	-	-	-	-	-	154.2	131.7	140.4
$\mathrm{T}\mathrm{e}-134$	-	-	-	-	-	-	2731.53	2401.24	2508.00
$\mathrm{I}-135$	-	-	-	-	-	-	27,123.12	22,314.56	23,688.00
$\mathrm{C}\mathrm{s}-137$	-	-	-	-	-	-	31.23 y	29.11 y	30.08 y

. The measured half-lives were taken from the recent available atomic mass data evaluation of reference [36]. It may be seen from Table 1 and Table 2 that our calculated half-life values are in very good agreement with the experimental data and the previous theoretical results. The calculated excitation energy $\left(\mathrm{M}\mathrm{e}\mathrm{V}\right)$ and $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values of $\mathrm{N}=50,82$ waiting-point nuclei with the $pn$-QRPA (WS) for allowed GT, rank 0, rank 1, and rank 2 transitions are given in

Table 3. Calculated $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values for $\mathrm{N}=50$ waiting-point nuclei with the $pn$-QRPA (WS).

Nuclei−A	GT					Rank 0					Rank 1					Rank 2
	ωi (MeV)	Logft	I (%)	E (MeV)	Logft	ωi (MeV)	Logft	I (%)	E (MeV)	Logft	ωi (MeV)	Logft	I (%)	E (MeV)	Logft	ωi (MeV)	Logft	I (%)	E (MeV)	Logft
$\mathrm{F}\mathrm{e}-76$	1.037	4.63	-	-	-	1.022	6.28	-	-	-	0.816	8.58	-	-	-	2.045	9.53	-	-	-
$\mathrm{C}\mathrm{o}-77$	0.902	4.57	-	-	-	1.487	5.84	-	-	-	0.684	8.42	-	-	-	2.314	9.72	-	-	-
$\mathrm{N}\mathrm{i}-78$	0.827	4.43	-	-	-	1.224	5.87	-	-	-	0.683	7.85	-	-	-	1.674	9.75	-	-	-
$\mathrm{C}\mathrm{u}-79$	0.287	4.77	-	-	-	1.146	6.02	-	-	-	0.793	8.71	-	-	-	1.204	9.61	-	-	-
$\mathrm{Z}\mathrm{n}-80$	0.832	4.58	34	1.449	4.61	0.517	5.73	-	-	-	0.562	8.31	-	-	-	0.865	9.87	-	-	-
$\mathrm{G}\mathrm{a}-81$	1.074	4.76	-	-	-	1.083	6.22	-	-	-	1.217	8.47	-	-	-	2.113	9.84	-	-	-
$\mathrm{G}\mathrm{e}-82$	0.892	4.57	-	-	-	1.032	6.24	-	-	-	0.836	8.43	-	-	-	1.345	9.73	-	-	-
$\mathrm{A}\mathrm{s}-83$	0.583	4.52	-	-	-	0.846	5.63	-	-	-	1.142	8.52	-	-	-	0.937	9.34	-	-	-
$\mathrm{S}\mathrm{e}-84$	1.208	4.37	100	0.408	4.04	1.265	6.04	-	-	-	0.583	7.68	-	-	-	1.562	9.22	$0.1$	0.0	8.5
$\mathrm{B}\mathrm{r}-85$	0.755	4.53	-	-	-	0.423	6.52	-	-	-	1.053	7.23	-	-	-	0.875	9.74	-	-	-
$\mathrm{R}\mathrm{b}-87$	0.748	4.81	-	-	-	1.132	6.13	-	-	-	0.614	8.07	-	-	-	1.026	9.85	-	-	-

and

Table 4. Calculated $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values for $\mathrm{N}=82$ waiting-point nuclei with the $pn$-QRPA (WS).

Nuclei−A	GT					Rank 0					Rank 1					Rank 2
	ωi (MeV)	Logft	I (%)	E (MeV)	Logft	ωi (MeV)	Logft	I (%)	E (MeV)	Logft	ωi (MeV)	Logft	I (%)	E (MeV)	Logft	ωi (MeV)	Logft	I (%)	E (MeV)	Logft
Tc − 125	2.356	4.87	-	-	-	1.207	5.83	-	-	-	2.173	7.56	-	-	-	1.058	8.87	-	-	-
Ru − 126	1.608	4.65	-	-	-	2.023	6.02	-	-	-	2.713	7.82	-	-	-	2.682	9.73	-	-	-
Rh − 127	0.984	4.73	-	-	-	2.124	5.76	-	-	-	1.756	8.45	-	-	-	2.583	9.21	-	-	-
Pd − 128	0.793	4.82	-	-	-	1.157	5.94	-	-	-	2.302	7.85	-	-	-	1.706	9.13	-	-	-
Ag − 129	0.858	4.58	-	-	-	1.014	5.76	-	-	-	1.925	8.21	-	-	-	1.314	9.26	-	-	-
Cd − 130	2.577	4.78	-	-	-	2.321	5.93	-	-	-	1.804	7.63	-	-	-	1.424	9.26	-	-	-
In − 131	0.538	4.32	90	2.434	4.40	1.274	6.04	-	-	-	1.835	7.12	<20	0.0	>5.6	2.341	8.94	-	-	-
Sn − 132	0.821	4.12	99	1.325	4.02	2.137	5.81	-	-	-	1.563	8.52	-	-	-	1.253	9.23	-	-	-
Sb − 133	1.472	4.68	-	-	-	0.846	6.04	-	-	-	1.673	7.68	-	-	-	2.517	9.42	-	-	-
Te − 134	1.275	4.57	42	0.846	4.65	0.748	5.92	-	-	-	1.576	7.23	-	-	-	2.031	9.56	-	-	-
I − 135	1.637	4.87	23.6	1.260	7.04	2.136	5.86	0.142	2.475	6.01	2.614	8.07	-	-	-	0.634	9.84	1.9	0.526	9.97
Cs − 137	1.032	4.68	-	-	-	1.172	6.72	-	-	-	0.542	7.86	-	-	-	0.931	9.57	94	0.661	9.625

, respectively. The ${\omega }_i$ denotes the excitation energy in the daughter nucleus in Table 3 and Table 4. The available measured values for each transition are given in Table 3 and Table 4. Here, the last column $I\left(\mathrm{\%}\right)$, $E\left(\mathrm{M}\mathrm{e}\mathrm{V}\right)$, and $\mathrm{l}\mathrm{o}\mathrm{g} ft$ represents the experimental intensity, energy, and $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values, and they were taken from NuDAT. The energy dependence of beta-decay probabilities of the first-forbidden transitions is generally $\left(14–32\right) \mathrm{M}\mathrm{e}\mathrm{V}$. The dominant contributions for the rank 0, rank 1, and rank 2 transitions in the waiting-point nuclei having $\mathrm{N}=$ 50, 82 are located at energies of the order $\left(18–25\right)$ MeV, (14–32) MeV, and (17–25) MeV, respectively. The Ikeda sum rule values calculated by a single quasi-particle (sqp) and pn-QRPA for waiting-point isotopes are shown in

Table 5. The Ikeda Sum Rule calculated by single quasi-particle (sqp) and $pn$-QRPA for waiting-point isotopes.

Waiting-Point Nuclei			Ikeda Sum Rule
Nuclei	$\mathit{Z}$	A	Single Quasi-Particle	$\mathit{p}\mathit{n}$-QRPA	Theoretical
Fe	26	76	71.9	72.0	72
Co	27	77	69.1	69.5	69
Ni	28	78	65.2	65.7	66
Cu	29	79	62.4	62.8	63
Zn	30	80	59.3	59.6	60
Ga	31	81	56.1	56.7	57
Ge	32	82	53.3	53.8	54
As	33	83	50.2	50.7	51
Se	34	84	47.1	47.7	48
Br	35	85	44.3	44.6	45
Rb	37	87	38.2	38.8	39
Tc	43	125	116.2	116.5	117
Ru	44	126	113.4	113.8	114
Rh	45	127	110.3	110.9	111
Pd	46	128	107.4	107.6	108
Ag	47	129	105.1	105.3	106
Cd	48	130	101.2	101.6	102
In	49	131	98.3	98.7	99
Sn	50	132	95.3	95.8	96
Sb	51	133	92.4	92.7	93
Te	52	134	89.1	89.4	90
I	53	135	87.5	87.8	87
Cs	55	137	80.2	80.7	81

. It may be seen that the Ikeda sum rule is fulfilled in both even–even and odd-A cases. The compliance is greater than $97\mathrm{\%}$. The calculated Ikeda sum rule values are model-independent. The calculated $\beta$-decay half-lives for waiting-point nuclei with $N=50$ and $N=82$, shown in Table 1 and Table 2, demonstrate significant concordance with experimental data and earlier theoretical models. The experimental half-lives, derived from the latest atomic mass evaluation [36], provide a reliable standard for assessing the precision of the current calculations.

This research utilizes the proton–neutron quasi-particle random phase approximation (pn-QRPA) in conjunction with the Woods–Saxon potential, drawing on theoretical perspectives from significant references. The authors of [37] established the foundational formalism for nuclear structure, whereas [38] offered essential values for axial–vector coupling constants, crucial for the accurate derivation of log ft values and transition probabilities.

Moreover, the values for pairing strength and effective interaction constants were derived from [24], guaranteeing alignment with experimentally reported pairing gaps. The findings strongly correspond with previous theoretical investigations based on pn-QRPA, including those in [29], which emphasizes the significance of both allowed Gamow–Teller (GT) and first-forbidden (FF) transitions in influencing $\beta$-decay properties.

As shown in Table 1, due to the lack of previous theoretical estimations, this study provided important new data by calculating β-decay half-lives for specific nuclei at the N = 50 waiting point for the first time. For nuclei like ⁸²Ge, ⁸³As, ⁸⁴Se, ⁸⁵Br and ⁸⁷Rb no computational results can be found in the literature, as indicated in Table 1. The pn-QRPA (WS) model-calculated half-lives are in good agreement with experimental observations from ref. [31]. The β-decay half-life for ⁸²Ge, for example, was found to be 4.82 s, which is quite close to the experimental result of 4.31 s.

⁸³As’s calculated half-life (12.83 s) and the experimental figure of 13.4 s both agree quite well. Interestingly, the measured values of 195.6 and 174 s, respectively, exhibit a good association with the half-lives of ⁸⁴Se (178.6 s) and ⁸⁵Br (161.2 s). The experimental result of 49.7 Gy for ⁸⁷Rb is in agreement with the calculated half-life of 47.3 Gy.

These findings close a major gap in the nuclear data landscape by representing the first theoretical study of the β-decay parameters for these nuclei. The strong agreement with experimental values highlights the pn-QRPA methodology’s dependability, especially its ability to forecast β-decay properties for nuclei without the need for previous theoretical research. In addition to confirming the durability of the chosen model, this work lays the groundwork for future research on neutron-rich nuclei in the N = 50 area, which is vital for comprehending r-process nucleosynthesis.

⁸⁷Rb’s unusual nuclear structure and decay mechanism are the main causes of its extraordinarily long β-decay half-life, which is measured in giga-years (Gy). This isotope’s exceptionally slow β-decay rate is caused by a number of reasons. Because of its unique nuclear shell structure and quantum states, ⁸⁷Rb has a strongly suppressed weak transition matrix element that controls the β-decay probability. Furthermore, ⁸⁷Rb’s half-life is much increased and the accessible phase space for decay is constrained by its comparatively small Q-value, which represents the energy difference between the parent and daughter nuclei. Strict spin-parity selection restrictions that limit permitted transitions exacerbate this suppression and further slowdown the decay rate.

Furthermore, ⁸⁷Rb has a very stable nuclear structure and a relatively low decay energy due to its proximity to the line of nuclear stability, which also adds to its long half-life. ⁸⁷Rb’s half-life is estimated by experimental measurements to be around 49.7 Gy, and the study’s computed value of 47.3 Gy is in great agreement with experimental data. The accuracy with which the pn-QRPA (WS) model captures the physics of ⁸⁷Rb decay is demonstrated by this consistency.

One feature that sets ⁸⁷Rb apart from the other nuclei under study is its incredibly lengthy half-life. In geochronology and cosmology, where ⁸⁷Rb is a crucial chronometer in the rubidium–strontium dating technique, which is frequently used to determine the age of rocks and minerals, and this special characteristic has important ramifications. These results demonstrate the significance of ⁸⁷Rb for both basic and applied research.

A shown in Table 2, this study filled a major gap in our understanding of the decay properties of N = 82 waiting-point nuclei, including ¹³²Sn, ¹³³Sb, ¹³⁴Te, ¹³⁵I and ¹³⁷Sc, by calculating their β-decay half-lives for the first time. This work is an important contribution to nuclear physics and astrophysics because there was no previous theoretical or computational data available for these nuclei. Table 2 displays the experimental data from ref. [30] in addition to the outcomes of the current computations, which are based on the pn-QRPA (WS) model.

The theoretical framework’s dependability was demonstrated by the calculated half-life for ¹³²Sn, which was 41.2 s and nearly matched the experimental value of 39.7 s. Likewise, the experimental value of 140.4 s is in good agreement with the estimated half-life of 131.7 s for ¹³³Sb. With theoretical half-lives of 2401.24 s and 22,314.56 s, respectively, the heavier isotopes ¹³⁴Te and ¹³⁵I show greater half-lives than their experimental equivalents, which are 2508.00 and 23688.00 s, respectively. The half-life of ¹³⁷Sc, a well-known isotope with important uses in nuclear technology, was determined to be 29.11 years, which is quite close to the experimental value of 30.08 years.

These results highlight the pn-QRPA (WS) model’s prediction ability, since it effectively depicts the β-decay processes of these nuclei. The model’s resilience in handling nuclei without any previous theoretical research is demonstrated by the agreement with experimental results. Additionally, these N = 82 waiting-point nuclei’s computed half-lives offer crucial inputs for r-process nucleosynthesis research, especially when simulating the creation of heavy elements in stellar environments. This groundbreaking study lays the groundwork for further investigation and provides important standards for astrophysical modeling and experimental confirmation.

A combination of nuclear structure, decay energetics, and transition characteristics can account for ¹³⁷Sc’s comparatively lengthy half-life (measured in years) in comparison to other N = 82 nuclei but its much shorter half-life (measured in giga-years) in comparison to ⁸⁷Rb.

First, ¹³⁷Sc slows its β-decay because of the increased stability that the closed neutron shell at N = 82 provides. However, because of higher favorable energetics, it decays more quickly than ⁸⁷Rb since it is located closer to the nuclear stability line. In particular, the β-decay Q-value of Rb-87 (around 0.28 MeV) is much smaller than that of ¹³⁷Sc (about 1.17 MeV). The half-life is shortened by this greater Q-value because it expands the phase space that is accessible for decay.

Furthermore, compared to the severely prohibited transitions seen in ⁸⁷Rb, the allowed or semi-forbidden transitions involved in the β-decay of ¹³⁷Sc are less repressed. As a result, the decay probability increases and ¹³⁷Sc’s half-life is significantly shortened. Because ¹³⁷Sc is closer to the stability line and has lower decay Q-values, which slow the decay process, it has a longer half-life than other N = 82 nuclei in this study, such ¹³⁴Te and ¹³⁵I.

¹³⁷Sc’s intermediate half-life emphasizes how important it is for real-world uses like radiological research and environmental monitoring. It is both scientifically intriguing and practically valuable because of its decay rate, which strikes a compromise between the extremely long half-life of isotopes like ⁸⁷Rb and the fast decay of highly unstable isotopes.

The strength distributions of permitted Gamow–Teller (GT) and first-forbidden (FF) transitions are significantly revealed by the calculated log ft values and excitation energies (${\mathsf{\omega }}_i$) for the waiting-point nuclei with N = 50 and N = 82, which are shown in Table 3 and Table 4. Reliability in capturing the nuclear structural effects governing β-decay is demonstrated by the successful reproduction of the log ft values by the pn-QRPA (WS) model.

According to the calculated log ft values for N = 50 isotones (Table 3), first-forbidden contributions play a secondary but not insignificant part in the decay process, while GT transitions dominate. Notably, the strength of acceptable transitions is confirmed by the log ft values for ⁸⁴Se and ⁸⁵Br falling within the anticipated range. The presence of collective nuclear excitations is confirmed by the excitation energies (${\mathsf{\omega }}_i$), which show that the dominant decay transitions take place at lower energy states. The calculated values for ⁸⁷Rb provide additional evidence that the model can include long-lived isotopes with significantly reduced rates of decay.

The log ft values show a more intricate interaction between allowed and forbidden transitions for N = 82 isotones (Table 4). Given the comparatively greater $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values and wider excitation energy distributions, the results imply that first-forbidden transitions play a major role in the decay process in heavier nuclei like ¹³⁴Te and ¹³⁵I. This outcome supports earlier theoretical predictions and emphasizes how crucial FF transitions are in heavier, neutron-rich nuclei. The pn-QRPA framework’s accuracy in forecasting decay properties across various nuclear mass regions is further supported by the calculated log ft values for ¹³²Sn and ¹³³Sb, which closely match experimental data.

Overall, the excitation energies and $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values shown in Table 3 and Table 4 demonstrate how well the chosen model reproduces the β-decay properties of neutron-rich nuclei. The findings highlight the need of incorporating both first-forbidden and allowed GT contributions in theoretical computations, especially for waiting-point nuclei that are essential for r-process nucleosynthesis.

The Ikeda sum rule (ISR) values for the waiting-point nuclei with N = 50 and N = 82 are shown in Table 5 and were determined using the single quasi-particle (sqp) and proton–neutron quasi-particle random phase approximation ($pn$-QRPA) approaches. Equation (20) is the Ikeda sum rule, which is a basic criterion for confirming that nuclear models in β-decay investigations are consistent. With compliance reaching 97% in both even–even and odd-A scenarios, Table 5’s results show that the computed ISR values are in great accord with theoretical expectations.

The calculated ISR values for nuclei like ⁸²Ge, ⁸³As, ⁸⁴Se and ⁸⁵Br for the N = 50 isotones substantially resemble the predicted theoretical values, demonstrating that the model faithfully captures the GT transition intensity. The consistency of the single quasi-particle (sqp) and $pn$-QRPA results emphasizes even more how well the chosen theoretical framework captures the collective nuclear excitations that cause β-decay.

Likewise, the ISR values for the N = 82 isotones ¹³²Sn₈₂, ¹³³Sb₈₂, ¹³⁴Te₈₂, ¹³⁵I₈₂ and ¹³⁷Cs₈₂ remain in line with theoretical expectations. The pn-QRPA model provides a thorough description of β-decay properties in neutron-rich nuclei by effectively accounting for both authorized GT and first-forbidden transitions, as evidenced by the reported high compliance rate.

All things considered, Table 5’s findings support the $pn$-QRPA (WS) model’s resilience in forecasting β-decay properties while meeting basic nuclear physics requirements. The approach’s validity is demonstrated by the high degree of agreement between calculated and theoretical ISR values, which makes it a trustworthy tool for examining the function of waiting-point nuclei in r-process nucleosynthesis.

This work validates the dependability of the $pn$-QRPA approach and its parameters by integrating these contributions, providing predicted insights into $\beta$-decay processes of astrophysical significance, especially for nuclei essential to r-process nucleosynthesis.

4. Conclusions

We present for the first time the allowed GT and all the first forbidden excited state half-lives, excitation energies, and $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values of $\mathrm{N}=50$ and 82 waiting-point nuclei using the schematic spherical $pn$-QRPA model. We did not use any quenching factor in our calculations. Our calculation fulfilled the model-independent Ikeda sum rule. We considered the relativistic terms in the non-unique first forbidden rank 0 and rank 1 transitions. The $\mathrm{l}\mathrm{o}\mathrm{g} ft$ values and half-lives obtained by the $pn$-QRPA (WS) calculation showed a decent agreement with the available experimental data and the other theoretical results. Our study shows that FF transitions play a substantial role in the total $\beta$-decay half-lives. The calculations support the argument that the pn-QRPA model gives a reliable prediction for neutron-rich nuclei. Pyatov’s restoration method in the framework $pn$-QRPA

(WS) model may further improve the computed half-lives, which we hope to report as a future assignment.