The RMF model is a theoretical tool used for the description of nuclear structure properties of nuclei (see [29] and related references). The preliminary model [30] struggled to describe nuclear surface features and the incompressibility of nuclear matter. To address this, a nonlinear model was developed [29]. The later versions of the model were termed covariant density functional theory and included a density-dependent meson-exchange model [31]. In the present investigation, the ground state parameters for nuclei have been determined by employing the density-dependent meson-exchange (DD-ME2) [32] version of the RMF framework. According to the RMF model, nucleons interact by exchanging various mesons and photons [30]. The first version of the RMF model ran into several issues while attempting to describe the incompressibility of nuclear matter and the surface characteristics of nuclei. This led to the introduction of the model’s nonlinear variant [29]. Subsequent versions of the RMF framework, known as covariant density functional theory, developed with elements including point coupling (PC) and meson-exchange (ME) [31, 33, 34]. We utilized the density-dependent -(ME) framework in our analysis. The density-dependent meson-exchange variant of the RMF model considers the isoscalar scalar σ meson, the isoscalar vector ω meson, and the isovector vector ρ meson fields for the analysis of nuclear matter and single-particle nuclear properties. In the DD-ME model, the coupling constants are self-consistently governed by nuclear functions. The definition of the Lagrangian density is given below. L DDME = ψ ̄ γ μ i ∂ μ − Γ ω ω μ − Γ ρ τ ⋅ ρ μ − M − Γ σ σ − e γ μ A μ 1 − τ 3 2 ψ − 1 2 ∂ μ σ ∂ μ σ − m σ 2 σ 2 − 1 4 Ω μ ν Ω μ ν + 1 2 m ω 2 ω μ ω μ − 1 4 R ⃗ μ ν ⋅ R ⃗ μ ν + 1 2 m ρ 2 ρ ⃗ μ ⋅ ρ ⃗ μ − 1 4 F μ ν ⋅ F μ ν . (1) In Equation 1, the terms Γ σ , Γ ω , and Γ ρ depend on the (nucleon) density, which is further defined by Equation 2: Γ i ρ = Γ i ρ sat f i x , (2) where i is generalized for the ρ , ω , and σ , respectively. Similarly, x is the ratio of ρ b (density of baryon), and ρ sat (density of baryon at saturation). f ( x ) is defined for the σ or ω meson by Equation 3: f j x = a j 1 + b j x + d j 2 1 + c j x + d j 2 ,   j = ω ,   σ (3) whereas for the ρ meson, f ( x ) is defined by Equation 4: f ρ x = e − a ρ x − 1 . (4) The DD-ME2 interactions ([35–38]) are often employed as covariant density functionals in the DD-ME model.

Studying even-even systems within the mean field approach is a good approximation. In this case, the configurations, neglected above the mean field ground state, are 4- or higher-quasiparticle (qp) configurations. The 2-qp configurations do not couple to the Hamiltonian ( H 20 = 0 ) . Mixing configurations are relatively few and separated by the pairing gaps. The exact solution has only a small admixture of higher qp configurations (4-qp and higher). On the other hand, investigating odd-A nuclei using the RMF model is rather challenging. In this case, there are many 3-qp states in the region close to the ground state, which may mix. The pairing gap even increases the level density of neighboring 1-qp states. Only a few of the H 31 matrix elements vanish. The mean field approximation in odd-A cases is not as good as in even-even nuclei. Of course, sometimes symmetries help ( K -value in deformed nuclei), but this is not always the case. We performed the HFB calculations in odd-A systems by using the blocking technique. The blocking was carried out by replacing one U -vector with the corresponding V -vector (see Section 6.3.2 of [39]). We carried out the blocking calculations with small modifications of the HFB code [40]. Pairing correlations play an important role for open-shell nuclei, and the Bardeen–Cooper–Schrieffer (BCS) approximation was used to tackle these correlations. Furthermore, constant G approximation [41] was used for the PEC calculations.

The pn-QRPA model is employed to analyze GT strength distributions and stellar weak rates. The Hamiltonian configuration in the pn-QRPA model may be characterized using Equation 5: H QRPA = H s p + V pair + V G T p p + V G T p h . (5) The Hamiltonian for a single particle is denoted as H s p , while V pair  represents the interaction between nucleons. The terms V G T p p and V G T p h correspond to interactions involving particle-particle ( p p ) and particle-hole ( p h ) GT interactions, respectively. Wave functions and energies of individual particles are calculated via the Nilsson model [42]. The oscillator constant is determined using ℏ ω = 45   A − 1 / 3 − 25   A − 2 / 3 . Other crucial factors affecting weak-interaction rates consist of β 2 , the nucleon pairing gap ( Δ nucleon ) , parameters of the Nilsson potential (PNP), and the Q -values. The PNP parameters were sourced from [43], and Q -values were derived through the calculation of mass excess values as presented in the compilation [44]. In order to obtain proton and neutron quasiparticle energies and occupation probabilities, the BCS equations were solved with pairing gaps computed using Equations 6, 7: Δ n n = 1 8 − 1 A − Z + 1 2 S n A + 1 , Z − 4 S n A , Z + 2 S n A − 1 , Z , (6) Δ p p = 1 8 − 1 1 + Z 2 S p A + 1 , Z + 1 − 4 S p A , Z + 2 S p A − 1 , Z − 1 , (7) where S p ( S n ) is the separation energy of protons and neutrons, respectively. As mentioned earlier, the β 2 values were determined using the RMF model. β 2 is determined by using Equation 8: β 2 = 125 Q 2 1.44 A 2 / 3 Z , (8) where Q 2 denotes the electric quadrupole moment chosen from [45] or the RMF framework. In the pn-QRPA model, the charge-changing reaction transitions are defined by phonon creation operators. The pn-QRPA phonons are given as Equation 9: A ω + μ = ∑ p n X ω p n μ a p + a n ̄ + − Y ω p n μ a n a p ̄ . (9) Here, the summation is taken on all the p-n pairs having μ = m p  − M n = 0, ± 1, and m n ( m p ) represents the angular momentum third component of the neutron (proton). The operators a n ( p ) + signify the creation of a quasiparticle (q.p) state for either a neutron or a proton, and p ̄ represents the time-reversed state of p. In the context of QRPA phonons, the theory defines the ground state as the vacuum, symbolized by A ω ( μ ) | Q R P A 〉 = 0. The phonon operator’s excitation energy ( ω ) and amplitudes ( X ω , Y ω ) are acquired by solving Equation 10, which is the well-known RPA equation. C D − D − C X Y = ω X Y , (10) Here, X(Y) represent forward (backward) amplitudes. The ω shows the energy eigenvalues of the eigenfunctions, and the two sub-matrices are specified by Equations 11, 12: C p n , p ′ n ′ = V p n , p ′ n ′ p p u p u n u p ′ u n ′ + v p v n v p ′ v n ′ + V p n ′ , p ′ n ′ p h u p v n u p ′ u n ′ + v p u n v p ′ v n ′ + δ p n , p ′ n ′ ε n + ε p , (11) D p n , p ′ n ′ = + V p n , p ′ n ′ p p u p u n v p ′ v n ′ + v p v n u p ′ u n ′ − V p n , p ′ n ′ p h u p v n v p ′ u n ′ + v p u n u p ′ v n ′ . (12) Here, the ε n ( p ) shows the q.p energies of the neutron (proton), whereas v p ( n ) and u p ( n ) represent the occupation and unoccupation amplitudes and are treated in the BCS theory. The detailed solution of the RPA matrix equation can be seen in [46]. The pairing force is calculated using the BCS approximation. These calculations were performed separately for both proton and neutron. We took a constant pairing force of strength G ( G p and G n for protons and neutrons, respectively), V pair = − G ∑ j k j ′ k ′ − 1 l + j − k s j k † s j − k † × − 1 l ′ + j ′ − k ′ s j ′ − k ′ s j ′ k ′ , (13) where l is orbital angular momentum, and summation over k and k ′ is restricted to positive values. The proton-neutron residual interactions take place through the pp and ph GT forces, which were characterized by interaction constants κ and χ , respectively, in the pn-QRPA framework. The p p GT force is explained using Equations 14, 15:              V G T p p = − 2 κ ∑ μ = − 1 1 − 1 μ O μ † O − μ , (14) with O μ † = ∑ j p k p j n k n < j n k n ∣ t ± σ μ † ∣ j p k p > × − 1 l n + j n − k n s j p k p † s j n − k n † . (15) In order to calculate the p h GT force, we used Equations 16, 17:              V G T p h = + 2 χ ∑ μ = − 1 1 − 1 μ U μ U − μ † , (16) with U μ = ∑ j p k p j n k n < j p k p ∣ t ±   σ μ ∣ j n k n > s j p k p † s j n k n . (17) The κ and χ interaction strengths were determined using the relation 0.58 / A 0.7 and 5.2 / A 0.7 , respectively, taken from [47]. Our results fulfilled the model-independent Ikeda sum rule [48]. The reduced GT transition probabilities were calculated using Equation 18: B G T ω = | ⟨ ω , μ ‖ τ ± σ μ ‖ Q R P A ⟩ | 2 , (18) where σ μ is the spin operator, and τ ± = τ x ± ι τ y are the isospin raising and lowering operators, respectively. The model-independent Ikeda sum rule may be evaluated using Equation 19 for the operators: S − + S + = ∑ f | 〈 f | O ^ − | i 〉 | 2 + ∑ f | 〈 f | O ^ + | i 〉 | 2 = 3 N − Z , (19) where | i 〉 represents the parent state, and | f 〉 represents the daughter state connected via the GT operator. For further details, see [49–51].

The stellar β -decay (we subsequently refer to this as electron emission E E ) and positron capture ( P C ) rates between parent level n and daughter state m have been determined utilizing Equation 20:               λ m n E E / P C = ln ⁡ 2 f m n E E / P C ρ , T , E f f t m n , (20) where ( f t ) m n corresponds to the GT and Fermi transitions shown in Equations 21–23:               B m n = g A / g V 2 B G T m n + B F m n , (21)               B G T m n = 1 2 J m + 1 ⟨ n ‖ ∑ k τ ± k σ ⃗ k ‖ m ⟩ | 2 , (22)               B F m n = 1 2 J m + 1 ⟨ n ‖ ∑ k τ ± k ‖ m ⟩ | 2 . (23) The construction of low-lying excited levels and computation of nuclear matrix elements in our present analysis may be found in [46]. f m n is the phase space and depends on the core temperature ( T ) , core density ( ρ ) , and Fermi energy ( E f ) . It was calculated using Equation 24: f m n E E = ∫ 1 E β E k E k 2 − 1 E β − E k 2 F + Z , E k 1 − R − d E k , (24) for E E decay rates. The f n m for P C were computed using Equation 25: f m n P C = ∫ E l ∞ E k E k 2 − 1 E β + E k 2 F − Z , E k R + d E k , (25) where E k is the kinetic energy of the electron, and E l is the total capture threshold energy. The Fermi functions, F ( ± Z , E k ) , were calculated using the method described in [52]. The total β decay energy was determined using Equation 26:               E β = m p − m d + E m − E n , (26) where E n is the excitation energy of the daughter nucleus having mass m d , while E m is the corresponding quantities of parent nucleus with mass m p . The distribution functions have been determined with Equations 27, 28:               R − = exp E k − E f k β T + 1 − 1 , (27)               R + = exp E k + 2 − E f k β T + 1 − 1 , (28) where k β is the Boltzmann constant. The electron number density, which is related to nuclei and protons, was determined using Equation 29: ρ Y e N A = 1 π 2 m e c ℏ 3 ∫ 0 ∞ R − − R + p 2 d p . (29) Here, N A represents the Avogadro number, Y e is the ratio of the electron number to the baryon number, and p represents the momentum of positron/electron. The total stellar weak rates were computed using Equation 30:               λ E E / P C = ∑ m n P m λ m n E E / P C , (30) where P m , which was calculated using the Boltzmann distribution, is the occupancy probability of the parent excited states. We continued to sum the initial and final states until our rate computation reached the necessary degree of convergence.