β-decay of N = 126 isotones for the r-process nucleosynthesis

Quest for a better understanding of the r-process continues to date. The interest is primarily due to the fact that nearly half of the heavy elements beyond iron is thought to be synthesized during the r-process [1, 2]. At closed-shell (N = 50, 82, 126), the r-process flow of matter decelerates. The corresponding nuclei have to wait for several β-decays to occur before capturing of neutrons resumes (also referred to as waiting points). The matter is therefore accumulated at these waiting points resulting in the well-known peaks in the observed r-process abundance distribution. The challenge posed to theorists is to come up with a microscopic r-process nucleosynthesis calculation reproducing this observed distribution pattern. The β-decay rate of waiting points is one of the key nuclear properties that can affect the r-process matter flow. The β-decay rates are important not only in the accurate determination of the structure of the stellar core but also play a vital role in the elemental abundance calculations and nucleosynthesis. A reliable calculation of β-decay half-lives for the waiting points is one of the pre-requisites for a better understanding of the r-process and reproduction of the observed abundance curve of the r-process nuclei. The experimental data is rather scarce but is expected to improve with results from new heavy-ion accelerator facility (e.g. RIKEN, GSI-FRS, GANIL-LISE and CERN-ISOLDE). Despite advances in measurements of β-decay half-lives, a better understanding of the r-process can be realized with reliable theoretical estimates of β-decay properties of waiting points specially under r-process physical conditions (T ∼10⁹ K, neutron densities >10²⁰cm⁻³). The observed r-process spectra of waiting-point nuclei may be affected by the presence of low-lying energy levels possessing different parities. This necessitates the incorporation of the first-forbidden (FF) chapter to the β-decay half-lives. For the N = 50 and 82 isotones, the past theoretical calculations of β-decays are in decent comparison with each other as against those for N = 126 isotones [3]. One important reason for disagreement between different theoretical calculations is the computation of FF transitions. Single-particle states appearing with unlike parity can significantly affect the calculated half-lives. The FF transitions, therefore, become important for the N = 126 isotones in addition to the allowed Gamow-Teller (GT) transitions. Few noticeable calculations of β-decay half-lives of N = 126 isotones include (i) quasiparticle random phase approximation (QRPA) calculation of GT [4] and gross theory calculation of GT and FF rates [5] (referred to as QRPA-FRDM), (ii) the continuum QRPA approach based on the self-consistent ground state description in the framework of the density functional theory [6, 7] (referred to as CQRPA-DF3), (iii) shell model rates [3] (referred to as SM), (iv) large-scale shell model rates [8] (referred to as LSSM) (v) an empirical formula based calculation [9] and calculations of QRPA within energy-density functional theory [10–12]. In this work, two different sets of the pn-QRPA model were employed to study N = 126 isotones. The first set of calculation solves the pn-QRPA equations using the spherical schematic model approach. The Woods-Saxon (WS) potential was employed as mean-field basis and deformation of waiting points were taken as zero. Allowed GT and FF (rank 0, 1 and 2) transitions were calculated separately for computing β-decay half-lives. The second set uses a pn-QRPA model in deformed Nilsson basis to calculate β-decay rates (allowed GT and unique first-forbidden (U1F)) under terrestrial and stellar conditions. The first and second sets of the pn-QRPA model are represented as pn-QRPA (WS) and pn-QRPA (N), respectively, throughout this manuscript. The paper is divided into four broad sections. Section 2 outlines the brief formalism involved in our calculation. Results and comparison with previous theoretical and measured data (wherever available) are presented in section 3. Conclusions are finally stated in section 4.

A brief formalism of both versions of the pn-QRPA model is presented here.

2.1. The pn-QRPA (WS) model

Allowed GT and the FF β-decay half-lives are calculated employing spherical schematic model (SSM) within the pn-QRPA framework. The Woods-Saxon potential was used as a mean-field basis. The eigenvalues and eigenfunctions of the Hamiltonian with separable residual GT and FF effective interaction were only calculated in particle-hole (ph) channel. We shall consider the GT 1⁺ excitations in odd–odd nuclei generated from the correlated ground state of the parent nucleus by the charge-exchange spin-spin forces and use the eigenstates of the single quasiparticle Hamiltonian H_sqp as a basis. The schematic method Hamiltonian for GT excitations in the neighboring odd–odd nuclei is given in the following form:

$$\begin{eqnarray}&&{H}_{{SSM}}={H}_{{sqp}}+{\hat{h}}_{{ph}},\end{eqnarray} \tag{ 1 }$$

where H_sqp is the single quasiparticle (sqp) Hamiltonian and ${\hat{h}}_{{ph}}$ is the GT effective interaction in the ph channel. Details of solution of allowed GT formalism are available in [13, 14].

A separable FF force with the ph channel was employed aiming to reduce the eigenvalue equation to an easily solvable algebraic equation of fourth order and minimize the computational effort. The spherical schematic method Hamiltonian for FF excitations is given by

$$\begin{eqnarray}&&{H}_{{SSM}}={H}_{{sqp}}+{\hat{h}}_{{ph}}.\end{eqnarray} \tag{ 2 }$$

The spin-isospin effective interaction denoted by ${\hat{h}}_{{ph}}$ causes 0⁻, 1⁻, 2⁻ vibrational modes in ph channel, and is specified as

$$\begin{eqnarray}&&{\hat{h}}_{{ph}}=2{\chi }_{{ph}}\sum _{{j}_{p}{j}_{n}{j}_{{p}^{{\prime} }}{j}_{{n}^{{\prime} }}}\left[{b}_{{j}_{p}{j}_{n}}{A}_{{j}_{p}{j}_{n}}^{\dagger }+{\bar{b}}_{{j}_{p}{j}_{n}}{A}_{{j}_{p}{j}_{n}}\right]\left[{b}_{{j}_{{p}^{{\prime} }}{j}_{{n}^{{\prime} }}}{A}_{{j}_{{p}^{{\prime} }}{j}_{{n}^{{\prime} }}}+{\bar{b}}_{{j}_{{p}^{{\prime} }}{j}_{{n}^{{\prime} }}}{A}_{{j}_{{p}^{{\prime} }}{j}_{{n}^{{\prime} }}}^{\dagger }\right],\end{eqnarray} \tag{ 3 }$$

where χ_ph stands for the ph effective interaction constant, which was taken as χ_GT = 5.2A^0.7 MeV, χ_rank0 = 30A^−5/3 MeVfm⁻², χ_rank1 = 99A^−5/3 MeVfm⁻² and χ_rank2 = 350A^−5/3 MeVfm⁻² for allowed GT, rank 0, rank 1 and rank 2 transitions, respectively. The effective interaction constant in the ph channel was fixed from the experimental value of the resonance energy.

The operator ${A}_{{j}_{p}{j}_{n}}^{\dagger }$ (${A}_{{j}_{p}{j}_{n}}$) is called the quasi-boson creation (annihilation) operator and specified as follows

$$\begin{eqnarray}&&{A}_{{j}_{p}{j}_{n}}^{\dagger }=\displaystyle \frac{1}{\sqrt{2j+1}}\sum _{m}{\left(-1\right)}^{j-m}{\alpha }_{{j}_{p}{m}_{n}}^{\dagger }{\alpha }_{{j}_{p}-{m}_{n}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{A}_{{j}_{p}{j}_{n}}={\left({A}_{{j}_{p}{j}_{n}}^{\dagger }\right)}^{\dagger }\end{eqnarray} \tag{ 5 }$$

where ${\alpha }_{{j}_{p}{m}_{n}}^{\dagger }$ presents the quasiparticle creation (annihilation) operator. The ${\bar{b}}_{{j}_{p}{j}_{n}}$, ${b}_{{j}_{p}{j}_{n}}$ appearing above stands for reduced matrix elements of the multipole operators for rank 0, 1 and 2 [15]. Here we have shown only the relevant equations and detailed formalism can be seen from [16, 17]. Details of rank 0 excitation computation are available in [18]. The nuclear matrix elements (both the relativistic and the non-relativistic, respectively) for λ^π = 0⁻ are given by

$$\begin{eqnarray}&&{M}^{\pm }(\lambda =0,{\rho }_{A})=\displaystyle \frac{{g}_{A}}{{\left(4\pi \right)}^{1/2}c}{{\rm{\Sigma }}}_{k}{t}_{\pm }(k)({\vec{\sigma }}_{k}\cdot {\vec{\vartheta }}_{k}),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{M}^{\pm }(\lambda =0,\kappa =1,{j}_{A})={g}_{A}{{\rm{\Sigma }}}_{k}{t}_{\pm }(k){r}_{k}{\left({Y}_{1}({r}_{k}){\sigma }_{k}\right)}_{0}.\end{eqnarray} \tag{ 7 }$$

where σ_k stands for the Pauli-spin matrices and t_±represents the iso-spin raising/lowering operator. Similarly, the relativistic and the non-relativistic matrix elements, for λ^π = 1⁻ are presented by

$$\begin{eqnarray}&&{M}^{\pm }(\lambda =1,\kappa =0,\mu ,{j}_{v})=\displaystyle \frac{{g}_{v}}{{\left(4\pi \right)}^{1/2}c}{{\rm{\Sigma }}}_{k}{t}_{\pm }(k){r}_{k}{\left({\vec{\vartheta }}_{k}\right)}_{1\mu },\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{M}^{\pm }(\lambda =1,{\rho }_{v},\mu )={g}_{v}{{\rm{\Sigma }}}_{k}{t}_{\pm }(k){r}_{k}{Y}_{1\mu }({r}_{k}),\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{M}^{\pm }(\lambda =1,\kappa =1,{j}_{v},\mu )={g}_{A}{{\rm{\Sigma }}}_{k}{t}_{\pm }(k){r}_{k}{\left({Y}_{1}({r}_{k}){\sigma }_{k}\right)}_{1\mu }.\end{eqnarray} \tag{ 10 }$$

Finally, the non-relativistic matrix element for λ^π = 2⁻ is specified by

$$\begin{eqnarray}&&{M}^{\pm }(\lambda =2,\kappa =1,{j}_{A},\mu )={g}_{A}{{\rm{\Sigma }}}_{k}{t}_{\pm }(k){r}_{k}{\left({Y}_{1}({r}_{k}){\sigma }_{k}\right)}_{2\mu }.\end{eqnarray} \tag{ 11 }$$

The transitions probabilities B(λ^π = 0⁻, 1⁻, 2⁻; β^±) are specified by [15]

$$\begin{eqnarray}&&B({\lambda }^{\pi }={0}^{-}{\beta }^{\pm })={\left|\left\langle {0}_{i}^{-}\parallel {M}_{{\beta }^{\pm }}^{0}\parallel {0}^{+}\right\rangle \right|}^{2},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{M}_{{\beta }^{\pm }}^{0}=\pm {M}^{\pm }(\lambda =0,{\rho }_{A})-i\displaystyle \frac{2\pi {m}_{e}c}{h}\xi {M}^{\pm }(\lambda =0,\kappa =1,{j}_{A}).\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&B({\lambda }^{\pi }={1}^{-},{\beta }^{\pm })={\left|\,,\left\langle {1}_{i}^{-}\parallel {M}_{{\beta }^{\pm }}^{1}\parallel {0}^{+}\right\rangle ,\,\right|}^{2},\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{M}_{{\beta }^{\pm }}^{1} & = & {M}^{\pm }(\lambda =1,\kappa =0,{j}_{v},\mu )\pm i\displaystyle \frac{2\pi {m}_{e}c}{{3}^{1/2}h}{M}^{\pm }(\lambda =1,{\rho }_{v},\mu )\\ & & +i{\left(\displaystyle \frac{2}{3}\right)}^{1/2}\displaystyle \frac{2\pi {m}_{e}c}{h}\xi {M}^{\pm }(\lambda =1,\kappa =1,{j}_{A},\mu ).\end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&B({\lambda }^{\pi }={2}^{-},{\beta }^{\pm })={\left|\left\langle {2}_{i}^{-}\parallel {M}_{{\beta }^{\pm }}^{2}\parallel {0}^{+}\right\rangle \right|}^{2},\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray}&&{M}_{{\beta }^{\pm }}^{2}={M}^{\pm }(\lambda =2,\kappa =1,{j}_{A},\mu ).\end{eqnarray} \tag{ 17 }$$

In equations (12), (14), the upper sign stands for β⁺ while the lower signs show the β⁻ decay. The ft values are specified by

$$\begin{eqnarray}&&{\left({ft}\right)}_{{\beta }^{\pm }}=\displaystyle \frac{D}{{\left({g}_{A}/{g}_{V}\right)}^{2}4\pi B({I}_{i}\longrightarrow {I}_{f},{\beta }^{\pm })}.\end{eqnarray} \tag{ 18 }$$

Transitions with λ = n + 1 are known as U1F transitions [15] and the ft values are calculated using

$$\begin{eqnarray}&&{\left({ft}\right)}_{{\beta }^{\pm }}=\displaystyle \frac{D}{{\left({g}_{A}/{g}_{V}\right)}^{2}4\pi B({I}_{i}\longrightarrow {I}_{f},{\beta }^{\pm })}\displaystyle \frac{(2n+1)!!}{{\left[(n+1)!\right]}^{2}n!},\end{eqnarray} \tag{ 19 }$$

where D = 6295sec. and the effective ratio of axial and vector coupling constants is taken as (g_A /g_V ) = − 1,254 [19]. No explicit quenching factor was introduced in our calculation. The pair correlation function was chosen as ${C}_{n}={C}_{p}=12/\sqrt{A}$ for the open shell nuclei. The energies were calculated from ground state of the daughter nuclei in all calculations. To obtain the beta transition probabilities, the nuclear matrix elements are calculated as a whole with the relativistic and the non-relativistic terms for the rank 0 and rank 1 transitions. Thus, FF contributions to the nuclear matrix element were computed by considering the relativistic correction terms Hence, the contributions coming from the virtual 0⁻ and 1⁻ intermediate states were obtained within ξ−approximation. As can be seen in equations (13) and (15), the ξ−approximation is only considered in the calculation of the non-relativistic terms in the rank 0 and rank 1 excitations. According to the Bohr-Mottelson model, the ξ−approximation is taken into account in the investigation of the first forbidden transitions. Of course, this approximation is fairly accurate for the investigated transitions, but it is important in the quantitative evaluation of the multipole moments to include the corrections due to the finite nuclear size. In the present paper, this approximation was applied for the first time in the N = 126 nuclei. The unique first forbidden 2⁻ contributions do not contain any relativistic term. The FF transitions become significant for neutron-rich isotopes. Therefore, the contribution of the FF excitations is very important when calculating the total β-decay half-lives.

2.1.1. Extension for odd-A nuclei

In this section, we present a brief summary of the necessary formalism for the odd-A nuclei. The wave function of the odd mass (with odd neutron) nuclei is given by in the pn-QRPA(WS) method

$$\begin{eqnarray}&&\left|{{\rm{\Psi }}}_{{j}_{k}{m}_{k}}^{j}\right\rangle ={{\rm{\Omega }}}_{{j}_{k}{m}_{k}}^{j\dagger }\left|0\right\rangle ={N}_{{j}_{k}}^{j}{\alpha }_{{j}_{k}{m}_{k}}^{\dagger }+\sum _{{j}_{\nu }{m}_{\nu }}{R}_{k\nu }^{{ij}}{A}_{i}^{\dagger }{\alpha }_{{j}_{\nu }{m}_{\nu }}^{\dagger }\left|0\right\rangle \end{eqnarray} \tag{ 20 }$$

where ${{\rm{\Omega }}}_{{j}_{k}{m}_{k}}^{j\dagger }$ and $\left|0\right\rangle$ present the phonon operator, the phonon vacumm, respectively. Also, ${N}_{{j}_{k}}^{j}$ and ${R}_{k\nu }^{{ij}}$ are the quasiboson amplitutes corresponding to the states and are fulfilled by the normalization conditions. It is that the wave function is formed by superposition of one and three quasiparticle (one quasiparticle + one phonon) states. The equation of motion the pn-QRPA (WS) is given by

$$\begin{eqnarray}&&\left[{H}_{{SSM}},{{\rm{\Omega }}}_{{j}_{k}{m}_{k}}^{j\dagger }\right]\left|0\right\rangle ={\omega }_{{j}_{k}{m}_{k}}^{j}{{\rm{\Omega }}}_{{j}_{k}{m}_{k}}^{j\dagger }\left|0\right\rangle \end{eqnarray} \tag{ 21 }$$

The excitation energies ${\omega }_{{j}_{k}{m}_{k}}^{j}$ and the wave functions of the GT and FF excitations are obtained from the pn-QRPA(WS) equation of motion. One-particle and one-hole nuclei allow of the simplest possible theoretical description of their states. The structure of one-particle nuclei within the simple mean-field approximation is the following. One-proton states $\left|\nu \right\rangle$ and one-neutron states $\left|k\right\rangle$ are described as

$$\begin{eqnarray*}&&\left|\nu \right\rangle ={A}_{\nu }^{\dagger }\left|{core}\right\rangle ,\left|k\right\rangle ={A}_{k}^{\dagger }\left|{core}\right\rangle \end{eqnarray*}$$

where $\left|{core}\right\rangle$ is the core with its Fermi level at some magic number. The detail solutions of the GT and FF transitions for the odd−A nuclei can be seen from [20, 21].

2.2. The pn-QRPA (N) model

Our second set of calculation involves solution of pn-QRPA equations in the deformed Nilsson basis. The Hamiltonian of the model is given by

$$\begin{eqnarray}&&{H}^{{QRPA}}={H}^{{sp}}+{V}^{{pair}}+{V}_{{GT}}^{{ph}}+{V}_{{GT}}^{{pp}}.\end{eqnarray} \tag{ 22 }$$

where H^sp , V^pair , ${V}_{{GT}}^{{ph}}$ and ${V}_{{GT}}^{{pp}}$ represent single-particle Hamiltonian, pairing potential, particle-hole GT force and particle-particle GT force, respectively. Single particle wave functions and energies were computed in the deformed Nilsson basis. Pairing was treated within the BCS formalism. It is to be noted that both particle-hole (ph) and particle-particle (pp) channels were considered in the GT force in the current model. The ph interaction strength χ was taken as 4.2/A MeV and 56.16/A MeV fm⁻² for the allowed GT and U1F transitions respectively, following 1/A dependence [22]. The pp interaction strength was taken as 0.0001. The functional form of χ is the same as previously used in [16, 17, 23]. The deformation parameter (β₂) was calculated using β₂ = $\tfrac{125{Q}_{2}}{1.44{{ZA}}^{2/3}}$ where the quadrupole moment Q₂ was taken from [24]. For the cases where the Q₂ values were missing in [24], the β₂ values were taken from [25]. Details of solution of the Hamiltonian [equation (19)] may be seen from [26]. Computation of terrestrial β-decay half-lives may be seen from [27]. Below we present a brief formalism involved in the calculation of stellar weak rates using the current model.

The stellar β-decay rates from the ith parent state to the jth daughter state of the nucleus is given by

$$\begin{eqnarray}&&{\lambda }_{{ij}}^{\beta }=\displaystyle \frac{{m}_{e}^{5}{c}^{4}}{2{\pi }^{3}{{\hslash }}^{7}}\sum _{{\rm{\Delta }}{J}^{\pi }}{g}^{2}{B}_{{ij}}({\rm{\Delta }}{J}^{\pi }){f}_{{ij}}({\rm{\Delta }}{J}^{\pi }).\end{eqnarray} \tag{ 23 }$$

In above equation B_ij (Δ J^π ) stands for β-decay reduced transition probability while f_ij (Δ J^π ) is the integrated Fermi function. g is the weak coupling constant which takes the value g_V or g_A according to whether the Δ J^π transition is associated with the vector or axial-vector weak interaction. The dynamics part of the rate equation is given by

$$\begin{eqnarray}&&{B}_{{ij}}({\rm{\Delta }}{J}^{\pi })=\displaystyle \frac{1}{12}{\zeta }^{2}({w}_{m}^{2}-1)-\displaystyle \frac{1}{6}{\zeta }^{2}{w}_{m}w+\displaystyle \frac{1}{6}{\zeta }^{2}{w}^{2},\end{eqnarray} \tag{ 24 }$$

where ζ is

$$\begin{eqnarray}&&\zeta =2{g}_{A}\displaystyle \frac{\left\langle f\parallel {\sum }_{k}{r}_{k}{\left[{{\bf{C}}}_{1}^{k}\times \sigma \right]}^{2}{{\bf{t}}}_{-}^{k}\parallel i\right\rangle }{\sqrt{2{J}_{i}+1}},\end{eqnarray} \tag{ 25 }$$

and

$$\begin{eqnarray}&&{{\bf{C}}}_{{lm}}=\sqrt{4\pi {(2l+1)}^{-1}}{{\bf{Y}}}_{{lm}},\end{eqnarray} \tag{ 26 }$$

the Y_lm represents the spherical harmonics.

The kinematics portion of equation (23) can be estimated using

$$\begin{eqnarray}\begin{array}{rcl}{f}_{{ij}} & = & {\displaystyle \int }_{1}^{{w}_{m}}w{\left({w}_{m}-w\right)}^{2}{\left({w}^{2}-1\right)}^{1/2}\left[{\left({w}_{m}-w\right)}^{2}{F}_{1}(w,Z)\right.\\ & & \left.+({w}^{2}-1){F}_{2}(w,Z)\right](1-{D}_{-}){dw}.\end{array}\end{eqnarray} \tag{ 27 }$$

The term w in the equation indicates the total energy of the electron (kinetic + rest mass energy). The total energy for β-decay is given by w_m = E_i − E_j + m_p − m_d , where E_i (E_j ) and m_p (m_d ) are excitation energy and mass of the parent (daughter) nucleus, respectively. D₋ are the electron distribution functions and are given by

$$\begin{eqnarray}&&{D}_{-}={\left[{\exp }\left(\displaystyle \frac{E-{E}_{f}}{{kT}}\right)+1\right]}^{-1},\end{eqnarray} \tag{ 28 }$$

where E_f and E = (w − 1) are the Fermi energy and kinetic energy of the electrons. The Fermi functions F₁(±Z, w) and F₂( ± Z, w) appearing in equation (27) were calculated using the recipe of [28].

Due to the prevailing high temperatures in the stellar core, there is a finite probability of occupation of parent excited states in the interior of massive stars. Hence β-decays have a finite contribution from these parent excited states. Assuming thermal equilibrium, the probability of occupation of parent state i may be estimated using

$$\begin{eqnarray}&&{P}_{i}=\displaystyle \frac{{\exp }(-{E}_{i}/{kT})}{{\sum }_{i=1}\exp (-{E}_{i}/{kT})}.\end{eqnarray} \tag{ 29 }$$

The total stellar β-decay rates were finally calculated using

$$\begin{eqnarray}&&{\lambda }^{\beta }=\sum _{{ij}}{P}_{i}{\lambda }_{{ij}}^{\beta }.\end{eqnarray} \tag{ 30 }$$

The summation runs upon all parent and daughter states until satisfactory convergence in rate calculation was obtained. A similar calculation was performed to compute continuum positron capture rates in high temperature-density environment.

In the pn-QRPA (N) model it was further assumed that all daughter excited states having energy larger than the neutron separation energy (S_n ), decayed through process of neutron emission. The energy rate for neutron emission from daughter states was computed using

$$\begin{eqnarray}&&{\lambda }^{n}=\sum _{{ij}}{P}_{i}{\lambda }_{{ij}}({E}_{j}-{S}_{n}),\end{eqnarray} \tag{ 31 }$$

for all E_j > S_n . The probability of β-delayed neutron emission, P_n , was calculated using

$$\begin{eqnarray}&&{P}_{n}=\displaystyle \frac{{\sum }_{{ij}^{\prime} }{P}_{i}{\lambda }_{{ij}^{\prime} }}{{\sum }_{{ij}}{P}_{i}{\lambda }_{{ij}}},\end{eqnarray} \tag{ 32 }$$

where j' indicates the energy levels of the daughter nucleus with E_j' > S_n . The λ_ij(') in equation (31) and equation (32), represents the sum of positron capture and β-decay rates, for transition from i → j(j') state.

2.2.1. Extension for odd-A nuclei

An extension of the pn-QRPA model is straight forward to GT transitions from nuclear excited states. The excited states are constructed as phonon-correlated multi-quasiparticle states. The transition amplitudes between the multi-quasiparticle states are reduced to those of one-quasiparticle states. The reduction is not specific to the GT transition but holds generally for any mode of charge-changing transitions.

For a nucleus with an odd nucleon, i.e. a proton and/or a neutron, some lowlying states are obtained by lifting the q.p. in the orbit of the smallest energy to higher-lying orbits. States of an odd-proton even-neutron nucleus are expressed by three-proton states or one-proton two-neutron states, corresponding to excitation of a proton or a neutron,

$$\begin{eqnarray}\begin{array}{rcl}| {p}_{1}{p}_{2}{p}_{3{corr}}\rangle & = & {a}_{{p}_{1}}^{\dagger }{a}_{{p}_{2}}^{\dagger }{a}_{{p}_{3}}^{\dagger }| -\rangle +\displaystyle \frac{1}{2}\displaystyle \sum _{{p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{n}^{{\prime} }\omega }{a}_{{p}_{1}^{{\prime} }}^{\dagger }{a}_{{p}_{2}^{{\prime} }}^{\dagger }{a}_{{n}^{{\prime} }}^{\dagger }{A}_{\omega }^{\dagger }(\mu )| -\rangle \\ & & \times \langle -| {\left[{a}_{{p}_{1}^{{\prime} }}^{\dagger }{a}_{{p}_{2}^{{\prime} }}^{\dagger }{a}_{{n}^{{\prime} }}^{\dagger }{A}_{\omega }^{\dagger }(\mu )\right]}^{\dagger }{H}_{31}{a}_{{p}_{1}}^{\dagger }{a}_{{p}_{2}}^{\dagger }{a}_{{p}_{3}}^{\dagger }| -\rangle {E}_{{p}_{1}{p}_{2}{p}_{3}}({p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{n}^{{\prime} },\omega )\end{array}\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}\begin{array}{rcl}| {p}_{1}{n}_{1}{n}_{2{corr}}\rangle & = & {a}_{{p}_{1}}^{\dagger }{a}_{{n}_{1}}^{\dagger }{a}_{{n}_{2}}^{\dagger }| -\rangle +\displaystyle \frac{1}{2}\displaystyle \sum _{{p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{n}^{{\prime} }\omega }{a}_{{p}_{1}^{{\prime} }}^{\dagger }{a}_{{p}_{2}^{{\prime} }}^{\dagger }{a}_{{n}^{{\prime} }}^{\dagger }{A}_{\omega }^{\dagger }(-\mu )| -\rangle \\ & & \times \langle -| {\left[{a}_{{p}_{1}^{{\prime} }}^{\dagger }{a}_{{p}_{2}^{{\prime} }}^{\dagger }{a}_{{n}^{{\prime} }}^{\dagger }{A}_{\omega }^{\dagger }(-\mu )\right]}^{\dagger }{H}_{31}{a}_{{p}_{1}}^{\dagger }{a}_{{n}_{1}}^{\dagger }{a}_{{n}_{2}}^{\dagger }| -\rangle \\ & & \times {E}_{{p}_{1}{n}_{1}{n}_{2}}({p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{n}^{{\prime} },\omega )+\displaystyle \frac{1}{6}\displaystyle \sum _{{n}_{1}^{{\prime} }{n}_{2}^{{\prime} }{n}_{3}^{{\prime} }\omega }{a}_{{n}_{1}^{{\prime} }}^{\dagger }{a}_{{n}_{2}^{{\prime} }}^{\dagger }{a}_{{n}_{3}^{{\prime} }}^{\dagger }{A}_{\omega }^{\dagger }(\mu )| -\rangle \\ & & \times \langle -| {\left[{a}_{{n}_{1}^{{\prime} }}^{\dagger }{a}_{{n}_{2}^{{\prime} }}^{\dagger }{a}_{{n}_{3}^{{\prime} }}^{\dagger }{A}_{\omega }^{\dagger }(\mu )\right]}^{\dagger }{H}_{31}{a}_{{p}_{1}}^{\dagger }{a}_{{n}_{1}}^{\dagger }{a}_{{n}_{2}}^{\dagger }| -\rangle {E}_{{p}_{1}{n}_{1}{n}_{2}}({n}_{1}^{{\prime} }{n}_{2}^{{\prime} }{n}_{3}^{{\prime} },\omega )\end{array}\end{eqnarray} \tag{ 34 }$$

with the energy denominators of first order perturbation,

$$\begin{eqnarray}&&{E}_{{abc}}({\text{}}{def},\omega )=\displaystyle \frac{1}{\left({\epsilon }_{a}+{\epsilon }_{b}+{\epsilon }_{c}\right)-\left({\epsilon }_{d}+{\epsilon }_{e}+{\epsilon }_{f}+\omega \right)}\end{eqnarray} \tag{ 35 }$$

Three-quasiparticle states of an even-proton odd-neutron nucleus are obtained from equations (33) and (34) by the exchange of proton states and neutron states, p ↔ n, and ${A}_{\omega }^{\dagger }(\mu )$ ↔ ${A}_{\omega }^{\dagger }(-\mu )$. Amplitudes of the quasiparticle transitions between the three-quasiparticle states are reduced to those for correlated one-quasiparticle states. For parent nuclei with an odd-proton and even-neutron,

$$\begin{eqnarray}&&\begin{array}{l}\langle {p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1}{p}_{2}{p}_{3{corr}}\rangle =\delta ({p}_{1}^{{\prime} },{p}_{2})\delta ({p}_{2}^{{\prime} },{p}_{3})\langle {n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1{corr}}\rangle \\ \ \ \ -\ \delta ({p}_{1}^{{\prime} },{p}_{1})\delta ({p}_{2}^{{\prime} },{p}_{3})\langle {n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{2{corr}}\rangle +\delta ({p}_{1}^{{\prime} },{p}_{1})\delta ({p}_{2}^{{\prime} },{p}_{2})\langle {n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{3{corr}}\rangle ,\end{array}\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&\begin{array}{l}\langle {p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {p}_{1}{n}_{1}{n}_{2{corr}}\rangle =\delta ({n}_{1}^{{\prime} },{n}_{2})\left[\delta ({p}_{1}^{{\prime} },{p}_{1})\langle {p}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1{corr}}\rangle \right.\\ \ \ \ \left.-\ \delta ({p}_{2}^{{\prime} },{p}_{1})\langle {p}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1{corr}}\rangle \right]-\delta ({n}_{1}^{{\prime} },{n}_{1})\left[\delta ({p}_{1}^{{\prime} },{p}_{1})\langle {p}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{2{corr}}\rangle \right.\\ \ \ \ \left.-\ \delta ({p}_{2}^{{\prime} },{p}_{1})\langle {p}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{2{corr}}\rangle \right],\end{array}\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&\begin{array}{l}\langle {n}_{1}^{{\prime} }{n}_{2}^{{\prime} }{n}_{3{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1}{n}_{1}{n}_{2{corr}}\rangle =\delta ({n}_{2}^{{\prime} },{n}_{1})\delta ({n}_{3}^{{\prime} },{n}_{2})\langle {n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1{corr}}\rangle \\ \ \ \ -\ \delta ({n}_{1}^{{\prime} },{n}_{1})\delta ({n}_{3}^{{\prime} },{n}_{2})\langle {n}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1{corr}}\rangle +\delta ({n}_{1}^{{\prime} },{n}_{1})\delta ({n}_{2}^{{\prime} },{n}_{2})\langle {n}_{3{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1{corr}}\rangle ,\end{array}\end{eqnarray} \tag{ 38 }$$

Similarly, for an odd-neutron even-proton nucleus,

$$\begin{eqnarray}&&\begin{array}{l}\langle {p}_{1}^{{\prime} }{n}_{1}^{{\prime} }{n}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1}{n}_{2}{n}_{3{corr}}\rangle =\delta ({n}_{1}^{{\prime} },{n}_{2})\delta ({n}_{2}^{{\prime} },{n}_{3})\langle {p}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1{corr}}\rangle \\ \ \ \ -\ \delta ({n}_{1}^{{\prime} },{n}_{1})\delta ({n}_{2}^{{\prime} },{n}_{3})\langle {p}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{2{corr}}\rangle +\delta ({n}_{1}^{{\prime} },{n}_{1})\delta ({n}_{2}^{{\prime} },{n}_{2})\langle {p}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{3{corr}}\rangle ,\end{array}\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&\begin{array}{l}\langle {p}_{1}^{{\prime} }{n}_{1}^{{\prime} }{n}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1}{p}_{2}{n}_{1{corr}}\rangle =\delta ({p}_{1}^{{\prime} },{p}_{2})\left[\delta ({n}_{1}^{{\prime} },{n}_{1})\langle {n}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1{corr}}\rangle \right.\\ \ \ \ \left.-\ \delta ({n}_{2}^{{\prime} },{n}_{1})\langle {n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{1{corr}}\rangle \right]-\delta ({p}_{1}^{{\prime} },{p}_{1})\left[\delta ({n}_{1}^{{\prime} },{n}_{1})\langle {n}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{2{corr}}\rangle \right.\\ \ \ \ \left.-\delta ({n}_{2}^{{\prime} },{n}_{1})\langle {n}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{-\mu }| {p}_{2{corr}}\rangle \right],\end{array}\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&\begin{array}{l}\langle {p}_{1}^{{\prime} }{p}_{2}^{{\prime} }{p}_{3{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {p}_{1}{p}_{2}{n}_{1{corr}}\rangle =\delta ({p}_{2}^{{\prime} },{p}_{1})\delta ({p}_{3}^{{\prime} },{p}_{2})\langle {p}_{1{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1{corr}}\rangle \\ \ \ \ -\ \delta ({p}_{1}^{{\prime} },{p}_{1})\delta ({p}_{3}^{{\prime} },{p}_{2})\langle {p}_{2{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1{corr}}\rangle +\delta ({p}_{1}^{{\prime} },{p}_{1})\delta ({p}_{2}^{{\prime} },{p}_{2})\langle {p}_{3{corr}}^{{\prime} }| {t}_{\pm }{\sigma }_{\mu }| {n}_{1{corr}}\rangle ,\end{array}\end{eqnarray} \tag{ 41 }$$

For further details we refer to [26].

The terrestrial β-decay half-lives, stellar weak rates, β-delayed neutron emission probabilities, phase space integrals and charge-changing strength distributions computed using the pn-QRPA (N) model include both allowed GT and U1F transitions. The pn-QRPA (N) model is well-known for its very good predictive power of estimating half-lives of unknown nuclei, especially for nuclei having smaller half-life values (nuclei far off from the line of stability) [16, 29]. It is worth noting that no quenching factor was utilized in the present calculation. In the recent past, we have reported β-decay half-lives, GT strength distributions, phase space and stellar weak rates of 13 closed-shell waiting point nuclei (N = 50, 82) employing the deformed pn-QRPA (N) model [30].

In this present project, we extend our calculation to N = 126 waiting point nuclei and select 17 cases given in table 1. The calculated β-decay half-lives using the pn-QRPA (N) and pn-QRPA (WS) models are shown in table 1. For the sake of comparison, previous calculations including allowed GT calculation of [4], QRPA-FRDM [5], CQRPA-DF3 [6, 7], SM [3], LSSM [8], empirical formula based calculation [9] and NUBASE2016 values [31] are also shown in the table. The calculation done by [4] includes only allowed GT contribution. QRPA-FRDM calculation comprises of both allowed GT and first-forbidden (FF) contributions where allowed GT part was computed via QRPA approximation and gross theory was used for the computation of FF part. It is to be noted that the experimental evaluations in [5] were taken from [32, 33]. The SM computed the half-lives (including allowed GT and FF transitions) employing a quenching factor of 0.7. The LSSM results contain allowed GT and FF rates including rank 0, 1 and 2 operator values. LSSM reported that FF rates provide a substantial reduction in the total calculated half-lives for the cases having N = 126. It was reported in [6] that FF contribution dominates over allowed GT for Z ≥ 76. It is to be noted that LSSM incorporated varying quenching values ranging from 0.38 to 1.266, whereas the pn-QRPA (N) and pn-QRPA (WS) models did not incorporate any explicit quenching factor as mentioned earlier. The LSSM calculated half-life values are smaller than pn-QRPA values. A fully converged computation of the FF transition strength in LSSM approach was denied due to computational constraints. They applied the Lanczos method to derive the strength but was again limited up to 100 iterations which were insufficient for converging the states above 2.5MeV excitation energies. The pn-QRPA approach had no such limitations and was able to converge the states for excitation energies well over 10 MeV. It may be noted from table 1 that the pn-QRPA (N) and pn-QRPA (WS) calculated half-lives are in decent comparison with the measured data. The FF (U1F) component significantly reduced the computed half-lives, efficiently for higher Z numbers, and improved the comparison with experimental data (at times also with the previous calculations).

Tables 2 and 3 display the calculated β-decay (electron emission) rates (allowed GT and U1F) for the selected N = 126 isotones. The rates are shown at stellar temperatures (T₉ = 1, 2, 5, 15 & 25) GK and densities (ρY_e = 10²), (ρY_e = 10⁶) and (ρY_e = 10¹⁰) g cm⁻³. The rates are tabulated in log to base 10 values (in units of s⁻¹). These rates were calculated for a broad range of temperature (T₉ = 0.01–30) GK including the typical range of temperature for r-process (T₉ = 1–3) GK. The tables show that the β-decay rates on the selected nuclei increase with a rise in the stellar temperature. This may be attributed to the fact that the occupation probability of parent excited states increases with temperature rise and thus contributes effectively to the total rates. It is found that the electron Fermi energy increases with an increase in the density of stellar core which leads to a reduction in the available phase space and a corresponding decrease in the calculated β-decay rates.

Table 4 shows estimated values of the β-delayed neutron emission probability for selected N = 126 nuclei using different models. Previous computations including QRPA-FRDM [5], CQRPA-DF3 [6] and LSSM [8] are also shown in table 4. The calculations are not in good agreement with each other. The main reason for the varying predictions would be the calculation of daughter energy levels and computation of charge-changing matrix elements between the parent and daughter states in the different models. All the calculated probabilities reduce, generally, towards higher Z numbers.

To investigate weak rates in stellar environment, we computed the β-decay and (continuum) positron capture rates for a wide range of density (10–10¹¹) g cm⁻³ and temperature (T₉ = 0.01–30) GK for the selected waiting point nuclei. Due to space considerations, we chose to display results of only four cases (two even–even and two odd-A nuclei). Figures (1)–(4) display the computed weak rates for ¹⁹⁰Gd, ¹⁹⁵Tm,²⁰²Os and ²⁰⁵Au, respectively. Each figure comprises of three panels. The upper panel displays the sum of β-decay and positron capture rates (in units of s⁻¹) as a function of stellar temperature. The middle panel depicts the computed β-delayed neutron energy rates (in units of MeV s⁻¹) while the lower panel shows the calculated β-delayed neutron emission probability (P_n ) values. A fixed core density of 10⁷ g cm⁻³ was assumed for all these calculations. The allowed GT and U1F rates are shown separately in all these figures.

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The allowed GT rates for ¹⁹⁰Gd, shown in figure 1 (upper panel), are roughly factor 4 greater than the U1F rates at low stellar temperatures (T₉ = 0.01–2) GK. However, the U1F rates increase more rapidly beyond (T₉ = 2) GK and exceed the allowed GT rates, by up to a factor 66, at higher T₉ values. The emission rate of β-delayed neutrons was found higher due to U1F transitions when compared against allowed GT transitions at low-temperature range (T₉ = 0.01–3) GK. Consequently, the energy rates (middle panel) due to U1F transitions are factor 4 higher than those due to allowed GT transitions at low T₉ values and roughly an order of magnitude bigger for higher temperatures. The corresponding β-delayed neutron emission probabilities are roughly an order of magnitude greater due to U1F transitions alone at low stellar temperatures. The probability values due to allowed GT transitions exceed those due to U1F transitions at high T₉ values. This change may be ascribed to the behavior of the available phase spaces for allowed GT and U1F transitions, which we discuss later.

In case of ¹⁹⁵Tm, shown in figure 2, the allowed GT rates are comparable to U1F rates at low temperatures and are orders of magnitude bigger than U1F rates for higher T₉ values (T₉ = 10–30) GK. Similarly, the energy rates due to allowed GT transitions are up to an order bigger than those due to U1F at low temperatures (T₉ = 0.01–1) GK while they are orders of magnitude bigger beyond (T₉ = 1) GK. Correspondingly, the neutron emission probability values (bottom panel) due to allowed GT transitions were found up to an order (two orders) of magnitude bigger than U1F for low (high) T₉ values.

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The β-decay rates and the energy rates of β-delayed neutron for the case of ²⁰²Os, shown in figure 3, due to U1F transitions are greater than the respective rates due to allowed GT transitions by up to 2 orders of magnitude at all temperatures. The corresponding β-delayed neutron emission probabilities due to U1F transitions are smaller than those due to the only allowed GT at low-temperatures. The rates are comparable for higher T₉ values.

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Similarly, the β-decay and energy rates of β-delayed neutron for ²⁰⁵Au (figure 4), due to U1F transitions, are up to a factor 2 bigger than the respective rates due to allowed GT transitions for all T₉ values. The probability values of β-delayed neutron emissions, both due to allowed GT and U1F transitions are comparable for all temperature values.

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The positron capture rates may be ignored as compared to β-decay rates for low temperature. They appear through e⁻–e⁺ pair creation at temperature beyond 1 MeV and compete well with β-decay rates at (T₉ = 30) GK. In fact for some cases (¹⁹⁴Er, ¹⁹⁶Yb, ²⁰²Os, ²⁰³Ir, ²⁰⁴Pt, ²⁰⁵Au and ²⁰⁶Hg) they are bigger up to a factor 2 than the competing β-decay rates. The weak rates are the product of phase space and reduced transition probabilities. The behavior of weak rates shown in figures (1)–(4) may be traced back to the strength distributions and phase space calculations which we discuss next.

The computed phase space factors (allowed GT and U1F) at fixed density of 10⁷ g cm⁻³ as a function of stellar temperature, for the waiting point nuclei, are displayed in figures (5)–(7). From these figures, it is noted that the allowed GT phase space is bigger than U1F phase space by up to an order of magnitude for the cases ¹⁹⁰Gd, ¹⁹¹Tb, ¹⁹²Dy, ¹⁹⁴Er, ¹⁹⁵Tm, ¹⁹⁶Yb, ¹⁹⁷Lu, ¹⁹⁸Hf, ¹⁹⁹Ta and ²⁰⁰W. The U1F phase is orders fo magnitude bigger than allowed GT phase space for ¹⁹³Ho, ²⁰²Os, ²⁰³Ir, ²⁰⁴Pt and ²⁰⁵Au where as both the phase spaces are comparable for ²⁰¹Re and ²⁰⁶Hg. Thus, we conclude that for all the cases shown in figures 5 and 6, the GT phase space is bigger than U1F (except for ¹⁹³Ho) and the allowed GT contribution in reducing the half-lives is bigger than U1F contribution. The stellar rates due to GT transitions are also bigger for these cases (see table 2). For the cases shown in figure 7, the U1F phase space is bigger than the GT phase space and contribute significantly to reducing half-lives and associated stellar rates.

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Figures (8–10) show the pn-QRPA (N) computed GT strength distributions for the selected N = 126 r-process waiting point nuclei. The allowed GT and U1F transitions are shown separately for each nucleus. The allowed GT strengths are given in units such that GT = 3 for neutron decay. We found that U1F transitions contributed significantly in reducing the total β-decay half-lives, for the cases shown in figure 10 whereas the U1F contributions are relatively smaller for the cases displayed in figures 8 and 9. The U1F transitions are relatively bigger in magnitude than allowed GT transitions for ²⁰²Os, ²⁰³Ir, ²⁰⁴Pt and ²⁰⁵Au. This is in line with the findings of [6] that FF contribution dominates over allowed GT for Z ≥ 76. Phase space amplification is yet another factor which contributes to the enhancement of U1F transitions. The strength distribution depends much on the choice of GT interaction strengths. Usually larger values of ph interaction strength shift the GT giant resonance to higher excitation energies. For the cases ¹⁹⁶Yb and ¹⁹⁸Hf, the values of particle-hole interaction strength are smaller as compared to other nuclei. That could be one probable reason why larger U1F strengths are available at lower excitation energies and little beyond 25MeV. For ²⁰⁶Hg, the U1F and allowed GT transitions are comparable in magnitude and the half-life is reduced by ∼46% when U1F transition was included (see table 1). Our calculation fulfilled the model-independent Ikeda Sum Rule (ISR).

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In this study, we investigated the β-decay properties of N = 126 waiting point nuclei using two versions of the pn-QRPA model. Allowed GT and U1F transitions were considered in the pn-QRPA (N) calculation, whereas the pn-QRPA (WS) calculation included allowed GT, rank 0, 1 and 2 excitations. The computed Ikeda sum rule was satisfied for all the cases (up to 1% deviation was noted in few cases of odd-A nuclei). The pn-QRPA computed half-lives were in decent agreement with the experimental data. The incorporation of FF transitions with rank 0 and 1 operator value in the pn-QRPA (N) model may further improve the computed half-lives, which we hope to report in near future. The stellar weak rates were computed as a function of core density and temperature values. These rates were found to increase (decrease) with the increase in stellar temperature (density). This study may contribute to accelerating the r-process nucleosynthesis calculation. The shorter half-lives reported in this work as compared to previous calculations may lead to re-adjustment of the third peak of the element abundances toward higher A. In the past a similar study of β-decays of the isotones with N = 126 was performed using shell-model calculations and a modest shift of the third peak of the element abundances in the r-process toward a higher mass region was reported [3]. We are in a process of computing the nuclear abundances (essentially by invoking the nuclear statistical equilibrium conditions and using the Sahas equation). To date we have performed the necessary (and more complicated) weak rate calculation. We would report further progress after completion of nuclear abundance calculation in near future.

N. Çakmak would like to thank Cevad Selam for very fruitful discussion on calculation of GT and FF transitions.

The data that support the findings of this study are available upon reasonable request from the authors.