Accurate measurements and reliable theoretical calculations of the properties of low-energy nuclear structure play a vital role in the modeling and comprehension of fundamental nuclear phenomena. These include β-decays, which are closely linked to stellar nucleosynthesis. The β-decay characteristics are calculated from the perspective of theory, and so serve as a standard for theoretical models as they are sensitive to the wave functions in the parent and daughter states. The unique features of low-lying excitation energies and decay patterns found in neutron-rich nuclei around A = 100 and possessing N ≈ 60 are especially intriguing, holding importance for both studies of nuclear structure and astrophysical investigations [1, 2]. In these nuclei, shape dynamics may occur. Furthermore, these properties contribute to the rapid neutron capture (r-) process, which accounts for the production of heavy elements via nucleosynthesis in explosive scenarios. The r-process is responsible for producing around 50% of the nuclei that are more massive than iron [3, 4]. Radioactive ion beams are widely used in major experimental facilities across the world to accurately measure the β-decay half-lives of neutron-rich nuclei. Researchers have studied the β-decay properties of several neutron-rich nuclei including Ge and As [5–7]. Due to limited data from experiments, theoretical models have been used to study the β-decay rates of neutron-rich nuclei. Theoretical attempts to define low-lying nuclear states and their decay properties have encountered substantial difficulties. The decay process has been studied theoretically utilizing several nuclear models. A few models that are often utilized to accomplish these computations include the interacting boson model (IBM) [8], the quasi-particle random phase approximation (QRPA) [9, 10], the large-scale shell model (LSSM) [11] and the potential model [12, 13]. The structure of odd-A energy levels and the electromagnetic properties of ^69,71,73As and ^69,71,73Ge were studied in the context of the proton–neutron interacting boson–fermion model (pn-IBFM) [14]. Furthermore, the β-decay properties of neutron-rich nuclei have been studied by employing the deformed pn-QRPA [15]. The impact of finite-range tensor forces for even–even semi-magic/magic nuclei on the low-lying Gamow–Teller (GT) state and β-decay half-lives [16–19]. The β-decay and electron capture (EC) rates of Mn isotopes in the mass range A = 53–63 have been determined using the pn-QRPA [20]. The nuclear structural characteristics and β-decay half-lives of exotic proton-rich waiting point nuclei in the mass range A ≈ 70 have been examined using both the pn-QRPA and the interacting boson model-1 (IBM-1) [21].

Using a microscopic nuclear theory, Klapdor-Kleingrothaus et al [22] were the first to determine β-decay rates for several nuclei that are far from the line of stability. Later, this framework was employed to compute the β-decay half-lives of over 6000 nuclei rich in neutrons, spanning a wide range from the line of stability to the neutron drip line [23]. A microscopic computation of the β⁺/electron capture rates of neutron-deficient nuclei with Z = 10–108, up to the proton drip line for over 2000 nuclei, was performed using the same pn-QRPA model, which has a separable and schematic interaction [24]. The modest interaction rates have greatly aided our comprehension of the r-process. Fuller et al [25] calculated rates at stellar temperatures and densities, focusing on decays from excited states of parent nuclei. We utilize the deformed basis pn-QRPA models to provide a reliable description of the nuclear structure and β-decay properties of As isotopes. The proposed model is capable of finding the β-decay properties [26] and microscopic calculation of weak rates for any arbitrary heavy nucleus [27, 28].

We have investigated the β-decay properties of selected As isotopes (including the terrestrial β-decay half-lives, GT strength distributions, log ft values and stellar rates) by focusing on their dependence on deformation by employing the theoretical framework of the deformed pn-QRPA model. The β-decay properties for each As isotope were analyzed as a function of the quadrupole deformation parameter β₂. In isotopes with mass numbers between 67 and 80 there exist prolate, spherical and oblate shapes, corresponding to β > 0, β = 0 and β < 0, respectively. Particular characteristics may be seen in the β-decay properties computed at each equilibrium deformation. The goal of this study is to identify possible indicators of nuclear shape in the β-decay characteristics of As nuclei within 67 ≤ A ≤ 80. The results of the present model-based analysis were compared with previously observed and predicted data.

The paper is arranged as follows. In section 2, we give a brief discussion of the pn-QRPA model used to compute nuclear structure and β-decay features. Section 3 offers our findings and pertinent discussion. Section 4 summarizes the results of the present examination.

The pn-QRPA model was used to study the β-decay characteristics of As nuclei. The Hamiltonian of the pn-QRPA model can be described using

The Hamiltonian for a single particle is denoted as ${\mathcal{H}}^{\mathrm{s}\mathrm{p}}$, while ${\mathcal{V}}^{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}$ represents the interaction between nucleons. The term ${\mathcal{V}}_{\mathrm{G}\mathrm{T}}^{\mathrm{p}\mathrm{p}}$ corresponds to the particle–particle (pp) interactions and ${\mathcal{V}}_{\mathrm{G}\mathrm{T}}^{\mathrm{p}\mathrm{h}}$ corresponds to the particle–hole (ph) GT interactions. The wave functions and energy of a single particle were determined utilizing the Nilsson model [29]. The oscillator constant for the nucleon was calculated as ℏω = (45A^−1/3 − 25A^−2/3). Other variables associated with the model include β₂, pairing gaps, Nilsson potential parameters (NPP), GT force variables and Q-values. The NPP was modified from [30] and Q-values were obtained using mass excess values [31]. To get proton and neutron quasi-particle energies and occupancy probability, the BCS computations were solved with pairing gaps obtained utilizing the relation. where S_p is the proton separation energy and S_n is the neutron separation energy. The current pn-QRPA model has the limitation of ignoring the p–n pairing effect, and incorporation of p–n pairing may be the topic of future work. The present paper included the nn and pp pairing correlations, which have only an isovector contribution. For the isoscalar (IS) part, one must include the p–n pairing correlations, which was not considered in the present manuscript. The conclusions of the study in [32] state that IS interaction behaves in a way like the tensor force (TF) interaction. The study was performed only for N = Z + 2 nuclei. The IS interaction may impact the GT computation in a way similar to that of TF interaction, provided that the conclusions of the study can be generalized to any arbitrary nucleus (including N= Z). The calculations of [32] showed that TF shifts the GT peak to low excitation energies. Incorporation of TF may result in lower centroid values of calculated GT strength distributions and could lead to higher values of calculated β-decay rates. The same effect of shifting calculated β-strength to lower excitation energies in the current pn-QRPA model was achieved by incorporation of particle–particle forces [24]. To summarize, our model has the shortcoming of not incorporating the p–n pairing effect. The pp forces in the GT residual interaction take care of this missing effect.

The β₂ values were obtained from [33]. The GT force parameters were taken from [34]. For details of the pairing force and residual interaction as well as solution of equation (1) one can consult [35].

The partial half-lives t_1/2 and log ft were determined as where C = 6143 s [36]. The ratio g_A/g_V = −1.2694 [37], E = (Q–ω) and Q was taken from [31]. ω is the excitation energy of the QRPA phonon determined by solving the random phase approximation matrix equation. f_V(A, Z, E) is the phase space factor for the vector transitions and f_A(A, Z, E) is the phase space integrals for the axial vector transitions. The phase space integrals contains the lepton kinematics. These were analyzed utilizing the formula mentioned in [38]. B_GT is the reduced GT transition probability and B_F is the reduced GT Fermi transition probability. The β-decay terrestrial half-lives were computed by adding up the inverses of the partial half-lives and taking the inverseFrom parent level n to daughter state m, the stellar weak rate was calculated utilizingOne may view more information on the solution to equation (1) in [39, 40]. The sum of the stellar rates was calculated usingwhere P_m, which was calculated using the Boltzmann distribution, is the occupancy probability of the parent excited state. We continued to sum the initial and final states until our rate computation reached the necessary degree of convergence.

The pn-QRPA model was applied to examine the β-decay half-lives of ⁶⁷⁻⁸⁰As nuclei. Table 1 displays the terrestrial decay modes, Q-values [31], deformation parameters β₂ and pairing gap values for ⁶⁷⁻⁸⁰As. Q-values drop for odd-A and odd–odd isotopes for β⁺ decay as the mass number increases. The Q-values begin to rise in the β⁻ direction for both the odd-A and odd–odd isotopes of As once they pass the stable ⁷⁵As nucleus. Self-consistent calculations of the β-decay characteristics were performed utilizing the pn-QRPA model with the nuclear deformations obtained from [33] as an input parameter.

Figure 1 compares the pn-QRPA-predicted GT+ strength with outcomes from experiments [41, 42] for ^69,70As isotopes. The measured data were accessible up to excitation energies of 0.39 MeV and 5.37 MeV for ⁶⁹As and ⁷⁰As, respectively. We have also shown the calculated GT distributions inside the cut-off energies (3.99 MeV for ⁶⁹As and 6.22 MeV for ⁷⁰As). The low-lying transitions are consistent with the observed data for ⁶⁹As. The observed data for the decay of ⁷⁰As range from 2.0 MeV to 5.5 MeV. The predicted GT distributions are not considerably split for the case of ⁷⁰As. The present pn-QRPA model only analyzes 1p–1h correlations. Figure 1 demonstrates that the pn-QRPA-predicted GT strength distributions are more consistent with the experimental data at lower excitation energies.

Figure 2 compares the pn-QRPA calculated half-lives with the measured data of [31] for the selected As isotopes. It is obvious that high GT intensities cause low values of half-life and vice versa. It is clear from figure 1 that for odd–odd cases the GT intensities are maximum, so for the odd–odd cases the β-decay half-lives are minimum. Figure 3 displays the ratios of pn-QRPA computed to measured half-lives. The calculated half-lives are reproduced within a factor of 10 of the measured half-lives. This indicates good accuracy of the underlying nuclear model. The pn-QRPA model accurately predicts β-decay half-lives for neutron-rich nuclei.

Table 2 compares the pn-QRPA-based computed log ft for β⁺/EC of odd-A As nuclei with those of observed data [43] and earlier predictions [pn-IBFM-2 [18] and self-consistent mean-field (IBFM-2-SCMF) calculations [44]]. The pn-QRPA-based calculated log ft values correlate well with the observed data. For most transitions, the IBFM-2 predictions are lower than the measured values. The log ft values estimated by the IBFM-2-SCMF model, which are sensitive to the wave functions for the initial and final states, were consistently greater than the IBFM-2 values. The largest deviation occurs, for example, in the case of ⁶⁷As($5/2_1^{+}$) → ⁶⁷Ge($3/2_2^{+}$) decay. In particular, the computed log ft is a factor of 1.52 larger than the experimental data because pn-QRPA predicted a GT strength as small as 0.000 01. The agreement with the experimental data was evaluated through the calculation of root mean square (rms) errors. The rms error for the present pn-QRPA-based prediction is 1.06 while for IBFM-2 and IBFM-2-SCMF the rms errors are 1.34 and 1.31, respectively. This indicates that pn-QRPA predictions are within the range of observed values. The IBFM-2 framework relies significantly on empirical variables for the even–even IBM-2 core Hamiltonian. As a result, phenomenological single-particle energies were employed in the IBFM-2 computation. The IBFM-2-SCMF model utilized energy density functional computations to estimate the majority of the parameters.

Similarly, table 3 compares predicted and observed log ft values for β⁺/EC transitions of odd–odd As nuclei. The IBFM-2 model was not capable of computing the log ft values for odd–odd nuclei. For the decay ⁶⁸As → ⁶⁸Ge, our calculated log ft values are smaller than the measured ones. This is due to the fact that GT intensities are comparatively higher, which inversely affects the half-lives and the log ft values. The pn-QRPA-calculated log ft are within the limit of measured data for ⁷⁰As. For example, the greatest discrepancy exists in the case of the ⁶⁸Ge($3_1^{+}$) → ⁶⁸As($2_4^{+}$) decay. In particular, the computed log ft is a factor of 1.71 smaller than the measured value. This is because the pn-QRPA-predicted the GT strength to be larger than 0.226 30. More fragmentations occur in this decay. The rms errors for pn-QRPA and IBFM-2-SCMF are 1.76 and 1.34, respectively.

The time derivative of the lepton-to-baryon ratio Y_e is an important quantity to keep track of throughout the pre-supernova evolution of a massive star. It is provided by where τ represents the abundance of the nucleus and A represents its mass number. PC is positron capture. It is seen that (β⁺ + EC) contributes negatively to ${\dot{Y}}_e$ whereas the (β⁻ + PC) rates contribute positively. For the analysis of stellar weak rates, we have utilized the pn-QRPA model with nuclear deformations taken from [33] to determine the β^±, EC and PC rates of ⁶⁷⁻⁸⁰As. We have calculated stellar weak rates that are completely microscopic in nature. The so-called Brink–Axel hypothesis [45] was not applied in our computation for the GT strength distributions of excited states. The stellar weak rates were compiled utilizing the independent-particle model (IPM) for 226 nuclei within A = 21–60 [25]. Later, the data were compiled via the same model within A = 65–80 [46]. The authors improved their treatment for high-temperature partition functions and adjusted the location of GT centroids from [25] in the updated computation. The computations, referred to as IPM-03 from now on, were based on the Brink–Axel hypothesis. The pn-QRPA stellar rates are calculated and displayed in tables 4 and 5. We display (β⁺ + EC) and (β⁻ + PC) along with the IPM-03 calculations at all densities and temperatures. It is obvious that the stellar rates increase with core temperature. This happens because a parent state’s occupancy probability increases with temperature, making a sufficient contribution to the weak rates. As the density of the star core evolves by orders of magnitude, the (β⁺+ EC) rates rise owing to an elevation in electron chemical potential. On the contrary, the (β⁻+ PC) rates decrease with increasing core densities due to a significant reduction in the accessible phase space (positrons have a negative degeneracy parameter). The (β⁺+ EC) rates are only significant at high temperatures and densities. The pn-QRPA rates are up to 22 times greater than the IPM-03 rates because electron and positron pair generation probabilities are maximum at higher temperatures, for example in ⁸⁰As. Similarly, table 5 displays that the (β⁻+ PC) rates make a finite contribution only at high temperatures and low densities. This is due to the Pauli blocking effect at large densities, as well as the substantial reduction in phase space as noted above. It is noted that the IPM-03 rates are lower than the pn-QRPA rates. In the IPM-03 calculation, the sum of the partition function had several states without accompanying weak interaction strengths. The likely cause of the lower rates in the IPM-03 analysis might be the handling of partition functions and the quenching of GT strength.

The β-decay properties were analyzed within the context of the pn-QPRA approach. It was found that the computed half-lives were within a factor of 10 of the observed data, and the computed GT distributions agreed with the measured data rather well. The predicted log ft values were compared with observed and previously computed data. The present prediction shows a decent agreement with the measured data. The stellar weak interaction rates for As isotopes were computed utilizing a complete microscopic framework, excluding the Brink–Axel hypothesis from the computation of excited state GT distributions. The model-based weak rates exceed the previous predictions (shell model) by a factor of 22. The defined stellar rates might be beneficial for r-process nucleosynthesis predictions and simulations of late-stage star evolution.

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TUDSPP-2024-33). J-U Nabi would like to acknowledge financial support of the Higher Education Commission Pakistan under Grant No. 20-15394/NRPU/R&D/HEC/2021.

This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-33).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.