Constraining cross sections for unstable ^153,159Gd(n,γ) and their astrophysical implications

Abstract

Neutron capture (n,\gamma ) cross sections of gadolinium (Gd) isotopes are critical to astrophysics research, nuclear reactor designs, and medical applications. However, the available (n,\gamma ) data on unstable Gd isotopes are scarce and direct measurement is challenging. In this work, we propose an approach to infer the (n,\gamma ) cross sections for unstable ^153,159Gd isotopes by constraining both the \gamma-ray strength functions (\gammaSFs) and nuclear level densities (NLDs). Specifically, the key \gammaSF parameters are adjusted to match the available experimental data, and the NLD parameters are determined by renormalizing microscopic level densities through a Bayesian optimization method. Our approach is verified by comparing our predictions with the experimental (n,\gamma ) data for the stable ^155,157Gd isotopes. We then infer the unstable {}^{153,159}\text{Gd}(n,\gamma ) cross sections within the neutron energy range of 0.01–5.0 MeV. The resulting uncertainty intervals are evaluated to be 23.5%–40.4% for ¹⁵³Gd and 30.8%–51.1% for ¹⁵⁹Gd, which are about two times smaller than those obtained without experimental constraints. We further calculate the astrophysical reaction rates for the {}^{153,159}\text{Gd} isotopes. It is found that the {}^{159}\text{Gd}(n,\gamma ) rate is larger by a factor of \sim2.9 than the JINA REACLIB recommendation. This suggests a larger branching ratio for ¹⁵⁹Gd and may result in a visible increase in the ¹⁶⁰Gd abundance through neutron capture reaction. Our approach is promising for constraining (n,\gamma ) data for a wider range of unstable isotopic chains and for improving astrophysical reaction-network calculations and related nuclear-science applications.

Graphical abstract

Keywords

gadolinium isotopes / neutron capture cross section / gamma-ray strength function / nuclear level density / astrophysical reaction rate / s-process nucleosynthesis

1 Introduction

Most of the elements heavier than iron are produced by a sequence of neutron capture reactions and \beta-decays during different phases of stellar evolution. The two dominant processes involved are the rapid (r) and the slow (s) neutron capture processes [1–4]. In particular, unlike most of the isotopes, ¹⁵²Gd and ¹⁵⁴Gd receive contributions only from the s-process. These so-called “s-only” isotopes are crucial for testing s-process models. As illustrated in Fig. 1, they are shielded against the \beta-decay chains from the r-process region by stable samarium isobars [4–6]. The determination of their final abundances, however, crucially depends on the branchings at unstable nuclei, where neutron-capture rates compete with \beta-decay rates [5, 7]. In the standard s-process, the branching at unstable isotope ¹⁵³Gd is relatively minor [5]. In contrast, under the higher neutron density conditions of the intermediate neutron capture process (i-process), the contribution from ¹⁵³Gd(n,\gamma ) may become non-negligible. Similarly, a comparable situation exists for the unstable isotope ¹⁵⁹Gd, which thus requires further analysis. Furthermore, ¹⁵⁵Gd and ¹⁵⁷Gd account for the majority of the natural Gd cross section, with ¹⁵⁷Gd notably possessing the highest thermal cross section of 254,000 barns, making them pivotal for reactor designs and medical applications such as Gadolinium Neutron Capture Therapy [8–10]. Consequently, obtaining reliable neutron capture cross sections and reaction rates for Gd isotopes is critical for stellar nucleosynthesis models, reactor designs, and medical applications.

While experimental cross section data for stable Gd isotopes are generally available, experimental data for the crucial unstable isotopes ¹⁵³Gd and ¹⁵⁹Gd are scarce and direct measurements remain extremely challenging. Specifically, measurements involving radioactive targets are hindered by the high radioactivity of the samples. Furthermore, experiments in inverse kinematics are currently not feasible for (n,\gamma ) reactions because free-neutron targets are not available [11, 12]. In the absence of experimental data, most neutron capture cross sections and reaction rates required for stellar modeling rely heavily on theoretical model predictions. However, these predictions often suffer from large uncertainties due to difficulties in accurately describing the deexcitation process of the nucleus formed after neutron capture. Indeed, this process is ruled by fundamental nuclear properties (\gamma-ray strength functions (\gammaSF), nuclear level densities (NLD), etc.), for which existing nuclear models yield widely conflicting results in the absence of experimental constraints. As a result, the predicted cross sections can exhibit significant divergence, reaching variations of up to several orders of magnitude [13–16]. Consequently, these theoretical uncertainties propagate directly into astrophysical reaction rates, limiting the reliability of nucleosynthesis simulations for unstable isotopes. To reduce these uncertainties, indirect approaches have become an important alternative when direct (n,\gamma ) measurements are not feasible [11, 12, 17–20].

In this work, we present an effective approach to infer the (n,\gamma ) cross sections for unstable ^153,159Gd isotopes by constraining both the \gammaSFs and NLDs. The constrained \gammaSFs and NLDs are implemented in the nuclear reaction code TALYS-2.0 [21]. We validate the approach using the stable isotopes ^155,157Gd, for which experimental data are available, and then infer the (n,\gamma ) cross sections for the unstable isotopes ^153,159Gd. Furthermore, we calculate the resulting astrophysical reaction rates for ^153,159Gd and explore the implications for the s-process branching at ¹⁵⁹Gd and the production of ¹⁶⁰Gd. The remainder of this paper is structured as follows: In Section 2, we detail our approach for constraining the \gammaSF and NLD parameters. In Section 3, we validate our approach using stable ^155,157Gd and present the calculated (n,\gamma ) cross sections for the unstable ^153,159Gd isotopes. In Section 4, we discuss the astrophysical implications, including the reaction rates for ^153,159Gd, the s-process branching at ¹⁵⁹Gd, and the impact on the production of ¹⁶⁰Gd. In Section 5, we conclude and provide an outlook.

2 Extraction of \gamma-ray strength function and nuclear level density

2.1 E1 \gamma-ray strength function

The \gammaSF is important for the description of any transition involving gamma rays in nuclear reactions, providing essential information for predicting capture cross sections, \gamma-ray production spectra, and competition between \gamma-ray and particle emission [22]. In this section, we extract the E1 \gammaSF parameters for the ^{154,156,158,160}Gd isotopes, which serve as the compound nuclei for the neutron capture reactions calculated in this work. These parameters are constrained using experimental \gammaSF data from the IAEA Photon Strength Function Database [23], derived from photo-absorption and photo-neutron cross section measurements [24–28]. Specifically, we use the data in the giant dipole resonance (GDR) region above the neutron separation energy (S_n) to extract the E1 \gammaSF parameters.

The Standard Lorentzian (SLO) model [29, 30] is probably the most widely adopted approach for describing photo-absorption data of medium-weight and heavy nuclei [31–33]. The SLO form for the \gamma-ray strength is given by

(1)f_{X\ell }(E_{\gamma })=K_{X\ell }{\displaystyle \sum _{i=1}^{N_{\text{peak}}}{\displaystyle \frac{{\sigma }_i{E}_{\gamma }{\mathrm{\Gamma }}_i^2}{(E_{\gamma }^2-E_i^2{)}^2+E_{\gamma }^2{\mathrm{\Gamma }}_i^2}},}

where K_{X\ell }=[(2\ell +1){\pi }^2(\hslash c{)}^2{]}^{-1}, X denotes either electric (E) or magnetic (M) radiation, and \ell is the multipolarity of the transition. E_i, {\mathrm{\Gamma }}_i, and {\sigma }_i denote the centroid energy, width, and peak cross section of the i-th resonance, respectively. According to the Reference Input Parameter Library (RIPL-3, Ref. [22]), nuclei with a quadrupole deformation parameter {\beta }_2>0.01 are considered to be deformed. The Gd isotopes studied here have relatively large {\beta }_2 values ranging from 0.30 to 0.35 [34]. For such deformed nuclei, their GDR shapes are generally described by two resonance components [27, 35]. As a result, the E1 \gammaSF in Eq. (1) is parameterized with two Lorentzian components (N_{\text{peak}}=2).

The parameter set is \theta =({\sigma }_1,E_1,{\mathrm{\Gamma }}_1,{\sigma }_2,E_2,{\mathrm{\Gamma }}_2), and it was determined by minimizing the chi-square defined as

(2){\chi }^2(\theta )={\displaystyle \sum _k{\displaystyle \frac{{\left[f_{\text{exp}}(E_k)-f_{E1}(E_k;\theta )\right]}^2}{{\sigma }_k^2}},}

where f_{\text{exp}}(E_k) is the experimental \gammaSF value at the k-th energy point E_k, and {\sigma }_k is the corresponding experimental uncertainty. The minimization was performed using nonlinear least-squares method.

Figure 2 shows our E1\gammaSF fits for ^{154,156,158,160}Gd compared with two microscopic predictions: the Skyrme-HFB+QRPA strength [36] and the D1M-Gogny HFB+QRPA strength [37]. The fitted curves reproduce the experimental data well over the GDR energy range. In contrast, the microscopic QRPA predictions reproduce the gross trend. However, they systematically underestimate the peak magnitude near the GDR centroids and overestimate the strength on the high-energy tail. The peak parameters extracted from our fits are listed in Table 1. However, due to the lack of direct measurements below the S_n, the low-energy E1\gammaSF remains an additional source of uncertainty that requires further experimental constraints.

The fitted parameters show systematic trends across the Gd isotopic chain. The central energies E_1 and E_2 are relatively stable at approximately 12 MeV and 15 MeV, respectively. This two-peak structure is a characteristic feature of the GDR splitting in prolate deformed nuclei, where E_1 is attributed to oscillations along the nuclear symmetry axis and the higher-energy peak E_2 to oscillations perpendicular to it. The {\sigma }_2 is generally greater than {\sigma }_1 due to the difference in the degrees of freedom for these two oscillation modes. The widths, {\mathrm{\Gamma }}_1 and {\mathrm{\Gamma }}_2, ranging from 2.7 to 5.4 MeV, reflect the intrinsic damping of the collective dipole motion [32, 38].

2.2 M1 \gamma-ray strength function

Although E1 transitions dominate the \gammaSF, the M1 component may also contribute non-negligibly in the 2–10 MeV region and affects neutron capture cross sections. We consider three phenomenological M1 models, and their impact on the calculated cross sections is discussed in Section 3.

(i) SLO model. This prescription corresponds to the \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{M}11 option in TALYS. It adopts the RIPL-3 systematic formulae [22] for the M1 resonance parameters:

(3)\begin{array}{rl}{f}_{M1}(E_{\gamma }=7\text{MeV})& =1.58\times {10}^{-9}{A}^{0.47},\\ E_{M1}& =41A^{-1/3}\text{MeV},{\mathrm{\Gamma }}_{M1}=4\text{MeV},\end{array}

where A is the mass number, the {\sigma }_{M1} is determined by inverting Eq. (1) at E_{\gamma }=7 MeV. This prescription serves as the default M1 input for TALYS calculations.

(ii) Renormalization based on E1 strength. This prescription corresponds to the \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{M}12 option in TALYS. It utilizes the empirical coefficient 0.0588A^{0.878} given in the RIPL-2 handbook [39] to normalize the M1 strength:

(4)f_{M1}(E_{\gamma })={\displaystyle \frac{f_{E1}(E_{\gamma })}{0.0588A^{0.878}}}.

(iii) Spin-flip and scissors-mode. This prescription corresponds to the \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{M}13 option in TALYS. It models the total M1 strength as a superposition of two components. The Spin-flip component represents a broad resonance parameterized based on the nuclear mass. In contrast, the Scissors mode accounts for the extra strength observed in the E_{\gamma }\sim 2–4 MeV region. The scissors mode was first predicted by Iudice and Palumbo [40] within a semiclassical two-rotor model. This collective excitation mode was subsequently observed experimentally in ¹⁵⁶Gd by Bohle et al. [41] Since its strength is directly proportional to {\beta }_2, for the deformed Gd nuclei studied here ({\beta }_2\approx 0.3–0.35) this low-energy contribution cannot be neglected [42].

As shown in Fig. 2, these three M1 prescriptions are compared explicitly with the available experimental data. For the renormalization based on E1 strength [method (ii)], the calculation uses the best-fit E1 parameters from Table 1. All three models exhibit a resonance peak centered near E_{\gamma }\approx 7.6 MeV, with the method (i) predicting the lowest strength and the second model predicting the highest. The third model is distinguished by an additional pronounced enhancement in the low-energy region, E_{\gamma }\sim 2–4 MeV. Such a low-energy upturn can impact the capture calculations. These differences, particularly at low energies, illustrate the model uncertainty in the M1 strength function. This uncertainty propagates to the (n,\gamma ) cross section calculations.

2.3 Nuclear level density normalization

At the neutron-separation energy S_n, the average s-wave neutron-resonance spacings D_0 [22] are usually used to calculate the total level density \rho (S_n). For the case of Gd, there are no experimental D_0 data for ¹⁶⁰Gd, because ¹⁵⁹Gd is unstable. To estimate a reasonable D_0 for ¹⁶⁰Gd, we considered systematics of s-wave resonance spacings for this mass region by using the most recent evaluation of the RIPL-3. In addition, to determine \rho (S_n), systematic errors arising from the spin distribution at S_n must be reasonably accounted for [43].

Following Refs. [44, 45], we first describe the excitation-energy dependence of the spin-cutoff parameter {\sigma }^2 with two phenomenological Fermi-gas (FG) prescriptions. We first introduce the rigid-body approximation of Ref. [44], which reads (FG05):

(5){\sigma }_{\mathrm{F}\mathrm{G}05}^2(E_x)=0.0146A^{5/3}{\displaystyle \frac{1+\sqrt{1+4a(E_x-E_1)}}{2a}},

where E_x is the excitation energy, a is the NLD parameter, and E_1 is the excitation-energy back-shift determined from global systematics of Ref. [44]. We then consider the FG spin-cutoff parameter of Ref. [45] (FG09):

(6){\sigma }_{\mathrm{F}\mathrm{G}09}^2(E_x)=0.391A^{0.675}{\left(E_x-{\displaystyle \frac{1}{2}}P{a}^{\ast }\right)}^{0.312},

with P{a}^{\ast } denoting the pairing energy for deuteron as defined in Ref. [45]. For the phenomenological {\sigma }^2, the spin distribution is given by the standard expression [46]:

(7)g(E_x,J)={\displaystyle \frac{2J+1}{2{\sigma }^2(E_x)}}\exp \left[-{\displaystyle \frac{(J+{\displaystyle \frac{1}{2}}{)}^2}{2{\sigma }^2(E_x)}}\right],

where J is the spin of the levels at E_x. By using the phenomenological spin-cutoff parameters and assuming equal parity populations at S_n, the total level density is obtained from D_0 through

(8)\rho (S_n)={\displaystyle \frac{2{\sigma }^2(S_n)/D_0}{(J_{\text{t}}+1)\exp \left[-{\displaystyle \frac{(J_{\text{t}}+1{)}^2}{2{\sigma }^2(S_n)}}\right]+J_{\text{t}}\exp \left[-{\displaystyle \frac{J_{\text{t}}^2}{2{\sigma }^2(S_n)}}\right]}},

where J_{\text{t}} is the ground-state spin of the target nucleus.

For the ^{154,156,158,160}Gd nuclei, we investigated the spin distribution using {\sigma }_{\mathrm{F}\mathrm{G}05}^2, {\sigma }_{\mathrm{F}\mathrm{G}09}^2. We observe that the spin distribution calculated using {\sigma }_{\mathrm{F}\mathrm{G}05}^2 is significantly broader and centered at higher spins compared to that using {\sigma }_{\mathrm{F}\mathrm{G}09}^2. This finding is consistent with the results reported by Larsen et al. [43]. Therefore, we consider {\rho }_{\text{FG05}} (calculated using {\sigma }_{\text{FG05}}^2) as the upper limit and {\rho }_{\text{FG09}} (calculated using {\sigma }_{\text{FG09}}^2) as the lower limit. Consequently, we establish an uncertainty range for \rho (S_n) using {\rho }_{\text{FG05}} and {\rho }_{\text{FG09}}. Based on this, we adopt the HFB+c level densities, which are renormalized to reproduce the known discrete levels and the \rho (S_n) derived from Eq. (8).

As mentioned before, the experimental D_0 data for ¹⁶⁰Gd is unavailable. Given the limited experimental D_0 data for the Gd isotopic chain, we estimate it by a linear fit of \ln (D_0) versus A using these neighboring nuclei (Eu, Gd, Tb) from RIPL-3. As shown in Fig. 3, a strong odd-even effect is observed in the D_0 systematics. Therefore, separate fits were performed for odd-A and even-A nuclei. For ¹⁶⁰Gd, the even-A fit yields D_0=7.8\pm 2.4 eV. As a validation, we derived a theoretical D_0 value for ¹⁶⁰Gd based on the \rho (S_n) predicted by the microscopic HFB+c model, which is consistent with the linearly extrapolated result within the 1\sigma uncertainty. Subsequently, with this estimated value and the experimental D_0 for the other isotopes, we calculated \rho (S_n) for ^{154,156,158,160}Gd using Eq. (8), with spin cutoff parameters from Eqs. (5) and (6).

We adopt the HFB+c level densities provided by Goriely et al. [47] and renormalize them via a Bayesian optimization (BO) method as follows:

(9){\rho }_{\text{norm}}(E_x,J,\pi )=\exp (c\sqrt{E_x-\delta }){\rho }_{\text{HFB}}(E_x-\delta ,J,\pi ),

where \delta is an energy shift, and constant c acts as a slope correction, playing a role similar to that of the NLD parameter a of phenomenological models.

The optimal values of parameters c and \delta are determined through BO, with the objective of simultaneously satisfying two constraints. Specifically, at low excitation energies (1.05\le E_x\le 2.05 MeV), {\rho }_{\text{norm}} is fitted to level densities calculated from known levels using the binning method. The low-energy level density is defined as \rho (E_{\text{center}})=\mathrm{\Delta }N/\mathrm{\Delta }E, where \mathrm{\Delta }N is the number of levels in a bin of width \mathrm{\Delta }E=0.1 MeV centered at E_{\text{center}}. Simultaneously, at S_n, {\rho }_{\text{norm}} is fitted to the \rho (S_n) calculated from Eq. (8). To achieve this, we employ a Gaussian process surrogate model, minimizing the total {\chi }^2 of both constraints. The search ranges for parameters c and \delta are both set to [-1.5,1.5]. The optimization is performed independently for the {\rho }_{\text{FG05}} and {\rho }_{\text{FG09}} of \rho (S_n), yielding two (c,\delta ) parameter sets per isotope. These define the NLD uncertainty band. Final optimized parameters are listed in Table 2. As expected, {\rho }_{\text{FG05}} yields more positive \delta and larger c than {\rho }_{\text{FG09}}, since a larger target value requires a steeper level density curve. Figure 4 shows a fit for ¹⁵⁶Gd as an example, where the renormalized curves reproduce the low-energy discrete levels while passing through the respective \rho (S_n) anchors. Similar fit quality is obtained for the other isotopes. The uncertainty band bounded by these two curves is propagated to the cross section calculations.

3 Cross section calculations

The (n,\gamma) cross sections in TALYS are calculated within the Hauser−Feshbach (HF) statistical model [48] and depend on the neutron and \gamma-ray transmission coefficients [21, 22]. In the energy region just above the resonance range and before the opening of the inelastic channel, the neutron transmission coefficient is much larger than the \gamma-ray transmission coefficient. In this case, the HF expression can be written as

(10){\sigma }_{n,\gamma }(E_n)={\displaystyle \frac{T_n{T}_{\gamma }}{T_n+T_{\gamma }}}\approx T_{\gamma },

where T_n and T_{\gamma } denote the neutron and total \gamma-ray transmission coefficients, respectively. T_n is determined by the neutron optical model potential (OMP), and in the present work it is calculated using the Koning−Delaroche (KD) [49] OMP. T_{\gamma } can be written as

(11)\begin{array}{rl}{T}_{\gamma }=& \sum _{J,\pi }\sum _{X\ell }\sum _{I^{\prime }=|J-\ell |}^{J+\ell }\sum _{{\pi }^{\prime }}{\int }_0^{S_n}\mathrm{d}{E}_{\gamma }2\text{π}{E}_{\gamma }^{2\ell +1}{f}_{X\ell }(E_{\gamma })\\ & \times \rho (S_n-E_{\gamma },I^{\prime },{\pi }^{\prime })F(X,{\pi }^{\prime },\ell ),\end{array}

where I^{\prime } and {\pi }^{\prime } denote the spin and parity of the final states, respectively, and F(X,{\mathrm{\Pi }}^{\prime },\ell ) is the selection-rule factor. In the present work, the E1 \gammaSF is given by Eq. (1), and \rho is taken from the renormalized HFB+c level densities obtained using Eq. (9). Therefore, the calculated (n,\gamma ) cross sections depend explicitly on the \gammaSF and the NLD, and the corresponding uncertainties are propagated through f_{X\ell }(E_{\gamma }) and \rho. The calculation procedure is summarized as follows:

(i) Cross sections were calculated using the combinations of NLD and \gammaSF (both E1 and M1 components) models provided in TALYS. This procedure yields an estimate of the model uncertainty range, defined by the lower and upper limits from the theoretical models, for all isotopes studied here.

(ii) Cross sections were calculated using the default models and parameters in TALYS.

(iii) Cross sections and reaction rates were calculated by implementing the constrained \gammaSF and NLD parameters. The results for the stable isotopes ^155,157Gd serve as a validation, supporting the reliability of the predictions for the unstable isotopes ^153,159Gd.

3.1 Cross sections predictions for model uncertainties

We performed calculations for various combinations of the available NLD models and \gammaSF prescriptions (E1 and M1) in TALYS. In these calculations, the KD OMP [49] was adopted. To assess the sensitivity of the inferred data to the choice of OMP, we compared the results obtained with the KD and Jeukenne−Lejeune−Mahaux (JLM) [50] OMPs. The calculated results show that the data obtained with the KD potential is slightly higher (by a factor of 1.2) than those obtained with the JLM potential. Since this sensitivity is relatively modest, in the following we employ the KD OMP by default to calculate the (n, \gamma) cross sections of the above-mentioned four Gd isotopes. The resulting uncertainty range is shown as the gray shaded bands in Fig. 5. The combinations yielding the minimum and maximum values are:

(i) Minimum cross section: calculated using the temperature-dependent RMF E1 strength function model from Daoutidis and Goriely [51], the SLO M1 strength function based on the RIPL-3 systematic formulae [22], and the constant temperature + Fermi gas model for the NLD as introduced by Gilbert and Cameron [52] (TALYS keywords \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}7, \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{M}11, \mathrm{l}\mathrm{d}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}1).

(ii) Maximum cross section: calculated using the Brink-Axel Lorentzian E1 strength function [29, 30], the M1 strength function including spin-flip and scissors-mode contributions, and the NLD is based on temperature-dependent Hartree−Fock−Bogoliubov calculations using the Gogny force [53] (TALYS keywords \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}2, \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\mathrm{M}13, \mathrm{l}\mathrm{d}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}6).

To characterize the uncertainty of the inferred cross sections, we define the uncertainty interval as ({\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x},i}-{\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n},i})/2⟨{\sigma }_i⟩, where {\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x},i}, {\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n},i}, and ⟨{\sigma }_i⟩ denote the maximum, minimum, and mean (n,\gamma ) cross sections for a given isotope at the i-th neutron-energy point. Using various combinations of NLD, E1, and M1 \gammaSF models in TALYS, the uncertainty intervals over E_n = 0.01–5 MeV are calculated to be 68.26%–89.73% for ¹⁵³Gd, 45.64%–88.59% for ¹⁵⁵Gd, 63.03%–93.01% for ¹⁵⁷Gd, and 76.11%–90.28% for ¹⁵⁹Gd. These results underscore the importance of experimental constraints on both the \gammaSF and NLD to reduce the inherent model uncertainties in cross section predictions. This is particularly relevant for the energy region that dominates astrophysical s-process conditions and thermal reactor applications.

To further assess the sensitivity of the calculated cross sections to individual model components, we varied the \gammaSF and NLD models separately while keeping the other model components fixed, with the uncertainty factor is defined as the ratio of the maximum to minimum cross sections. In the E_n = 0.01–5 MeV energy range, NLD variations yield an average uncertainty factor of \sim2.8 for the four Gd isotopes. The largest factors are obtained for ^157,159Gd, reaching 3.4 and 3.2, respectively. The E1 \gammaSF produces an average uncertainty factor of \sim3.3, ranging from 2.7 for ¹⁵³Gd to 4.4 for ¹⁵⁹Gd. In contrast, the M1 strength function has a minimal impact, with a factor of only 1.1. This is attributed to the significantly smaller magnitude of the M1 strength compared to the dominant E1 component, thereby limiting its influence on the (n,\gamma ) cross sections.

3.2 TALYS default predictions

For comparison, we also performed calculations using the default models and parameters of TALYS. These are the Simplified Modified Lorentzian model [66] for E1 \gammaSF, the Spin-flip and scissors mode for M1 \gammaSF [40], and the Constant Temperature + Fermi gas model [67] for the NLD. The results are displayed as black lines in Fig. 5. For the stable isotopes ^155,157Gd, significant discrepancies are observed between the TALYS default predictions and the experimental data. For ¹⁵³Gd, the TALYS default prediction is higher than the major evaluated libraries when MeV. In the high-energy region, the TALYS default falls in between the evaluated libraries. For ¹⁵⁹Gd, the TALYS default prediction is in good agreement with the ENDF/B-VIII.1 and TENDL-2023 libraries.

3.3 Results from constrained nuclear model parameters

Finally, we employ our constrained nuclear model parameters to calculate the {}^{153,155,157,159}\text{Gd}(n,\gamma ) cross sections. For the neutron transmission coefficients, we adopted the KD optical potential [49]. The total uncertainty in our results accounts for three contributions: (i) E1 \gammaSF: Uncertainties are propagated via Monte Carlo sampling of the double-Lorentzian parameters (Table 1) based on their covariance matrix. (ii) M1 \gammaSF: The uncertainty is defined by the envelope of predictions from the three phenomenological models discussed in Section 2.2. (iii) NLD: The uncertainty is propagated from the renormalized NLD bands derived in Section 2.3 (Table 2).

We first calculate the cross sections for stable isotopes ^155,157Gd(n,\gamma ), for which abundant experimental data exist, to validate our approach. As shown in Figs. 5(a) and (b), for the stable isotope ¹⁵⁵Gd, our results show good agreement with both experimental data and most of the evaluated libraries across the entire energy range. The only exception is TENDL-2023, which begins to deviate from our results above 0.1 MeV. Similarly, for ¹⁵⁷Gd, our predictions agree well with the experimental data although they are slightly lower than part of the dataset. Based on this, we further calculate the ^153,159Gd(n,\gamma ) cross sections, where no experimental data are available. As shown in Figs. 5(c) and (d), for ¹⁵³Gd, our calculated cross sections are consistent with the ENDF/B-VIII.1 evaluation below 0.3 MeV. In the energy range of 0.3–1.2 MeV, our results lie between the JEFF-4 and JENDL-5 predictions. Above 1.2 MeV, our calculations agree well with TENDL-2023 and JEFF-4. Finally, for ¹⁵⁹Gd, our calculations are slightly higher than most evaluations across the considered energy range. However, the lower bound of our calculated band shows good agreement with ENDF/B-VIII.1 and TENDL-2023. Overall, the uncertainty intervals of the extracted ^{153,155,157,159}Gd data over E_n = 0.01–5 MeV are estimated to be 23.54%–40.37%, 7.63%–39.38%, 14.32%–36.64%, and 30.82%–51.05%, respectively. These uncertainties are narrower than those obtained with the TALYS models without experimental constraints.

4 Astrophysical implication

Based on the constrained cross sections, we calculate the astrophysical reaction rates for {}^{153,159}\text{Gd}(n,\gamma ), as shown in Fig. 6(a). These are compared with the recommended values from the JINA REACLIB database [68]. Note that the ks03 evaluation [69] is used for ¹⁵³Gd and the thra evaluation [70] is used for ¹⁵⁹Gd in the database. For ¹⁵³Gd, the JINA recommended values agree with the upper limit of our uncertainty band. In contrast, our calculated rates for ¹⁵⁹Gd are notably higher than the JINA REACLIB recommended values, approximately by a factor of 2.9 at the mean value. The enhanced (n,\gamma ) rate found in this work implies a stronger reaction flux through this channel than JINA REACLIB estimated, potentially facilitating the synthesis of heavier isotopes such as ¹⁶⁰Gd.

4.1 The branching at ¹⁵³Gd and ¹⁵⁹Gd

As mentioned in Section 1, the final abundances of the s-only Gd isotopes are sensitive to branchings at unstable nuclei, where neutron capture competes with \beta decay. We investigate the branchings at ¹⁵³Gd and ¹⁵⁹Gd, which influence the production of the s-only isotope ¹⁵⁴Gd and the s-process contribution to ¹⁶⁰Gd, respectively. This competition can be quantified by the neutron capture branching ratio f_{n,\gamma }, defined as

(12)f_{n,\gamma }={\displaystyle \frac{{\lambda }_{n,\gamma }}{{\lambda }_{n,\gamma }+{\lambda }_{\beta }}}={\displaystyle \frac{n_n⟨\sigma v⟩}{n_n⟨\sigma v⟩+\ln 2/t_{1/2}}}.

Here, {\lambda }_{n,\gamma } and {\lambda }_{\beta } denote the rates for neutron capture and \beta-decay, respectively. The n_n is the neutron density, and t_{1/2} is the \beta-decay half-life. The ⟨\sigma v⟩ denotes the reaction rate per particle pair (in {\text{cm}}^3\cdot {\text{s}}^{-1}). Note that the astrophysical reaction rates presented in Fig. 6(a) are given as N_A⟨\sigma v⟩ (in {\text{cm}}^3\cdot {\text{mol}}^{-1}\cdot {\text{s}}^{-1}), and thus must be divided by Avogadro’s constant N_A for the evaluation of Eq. (12).

We calculated f_{n,\gamma } over the temperature range of 0.1–2 GK, with a typical s-process neutron density of n_n={10}^8{\text{cm}}^{-3}. The ratios f_{n,\gamma }/f_{n,\gamma }^{\mathrm{J}\mathrm{I}\mathrm{N}\mathrm{A}} for ¹⁵³Gd and ¹⁵⁹Gd are shown in Fig. 6(b). For ¹⁵³Gd (t_{1/2}=240.4 d), we obtain f_{n,\gamma }\approx 70\mathrm{\%} at T=0.2 GK, meaning that neutron capture is the dominant channel, and the ratio f_{n,\gamma }/f_{n,\gamma }^{\mathrm{J}\mathrm{I}\mathrm{N}\mathrm{A}} remains close to unity. In contrast, for ¹⁵⁹Gd (t_{1/2}=18.5 h), the short half-life leads to a much smaller branching ratio, f_{n,\gamma }\approx 0.3\mathrm{\%} at T=0.2 GK, i.e., \beta decay dominates. Nevertheless, our constrained rates yield a larger branching ratio by a factor of about 2–3 compared to JINA REACLIB. This notable enhancement suggests a stronger neutron-capture flow toward ¹⁶⁰Gd compared to the JINA REACLIB prediction.

4.2 Impact on ¹⁶⁰Gd production

The enhanced branching at ¹⁵⁹Gd is expected to increase the s-process flow toward ¹⁶⁰Gd. To illustrate this sensitivity, we perform nucleosynthesis calculations as described below. We performed s-process nucleosynthesis simulations using the single-zone reaction network code \mathrm{N}\mathrm{u}\mathrm{c}\mathrm{N}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{s} [71]. The calculations adopted fixed temperature, density, and neutron abundance. Specifically, the temperature was maintained at T=0.2 GK, and the matter density was set to \rho ={10}^3 g/cm³. The neutron abundance was fixed at Y_n=1.66\times {10}^{-19}, corresponding to approximately n_n\approx {10}^8 cm⁻³. The solar system elemental abundances from Ref. [72] were adopted as the initial composition. The reaction network was evolved for a duration of 3.15\times {10}^{12} s (\approx {10}^5 yr), sufficient to reach equilibrium for the relevant isotopes.

As shown in Fig. 7, within the adopted single-zone calculation the ¹⁶⁰Gd abundance predicted in this work is consistently higher than the default prediction based on the JINA REACLIB rate after 2\times {10}^{-2} Myr. In the intermediate and late stages of the evolution, our results exhibit a stable enhancement, stabilizing at a factor of approximately 2 compared to the default calculation. However, it should be noted that the above ¹⁶⁰Gd abundance enhancement can only be regarded as a trend observed in the present sensitivity study, not yet a definitive prediction for realistic stellar models.

5 Summary and outlook

In summary, we have successfully inferred the unstable ^153,159Gd(n,\gamma) cross sections by constraining the key \gammaSF and NLD inputs. Over E_n = 0.01–5 MeV, the uncertainty intervals for these inferred data are evaluated to be 23.5%–40.4% for ¹⁵³Gd and 30.8%–51.1% for ¹⁵⁹Gd, which are approximately half of those obtained without experimental constraints. Note that constraining these uncertainties is particularly is particularly important for investigating s-process branching points and the i-process, where experimental data are scarce, but accurate cross sections are essential. Furthermore, we derived the astrophysical reaction rates for ^153,159Gd isotopes. The calculated ¹⁵³Gd(n,\gamma ) reaction rate is comparable to the JINA REACLIB recommendation, while that for ¹⁵⁹Gd(n,\gamma ) is higher by a factor of \sim2.9. This is expected to be associated with a larger branching ratio for ¹⁵⁹Gd, which in turn results in an approximately twofold increase in the ¹⁶⁰Gd abundance through neutron capture reaction in our simplified one-zone setup.

Our study enriches the (n,\gamma) data for the Gd isotopic chain, thereby providing useful input for astrophysical reaction-network calculations and related applications involving Gd isotopes. In the future, we will apply our approach to derive (n,\gamma ) cross sections for other unstable isotopic chains. Sm and Er isotopes are of our priority, since their (n,\gamma ) data play an important role in determining cosmic-ray exposure conditions for planetary materials [73]. Moreover, their accurate astrophysical reaction rates are indispensable for resolving the competition between (n,\gamma ) reactions and \beta-decays during the r-process freeze-out, which shapes the final rare-earth abundance pattern [74]. We also should note that in order to infer the (n,\gamma ) data for these unstable isotopes of interest, we still need reliable experimental data of neighboring stable isotopes, which are useful for constraints of their \gammaSFs and NLDs. As a result, we call for more experimental measurements on (n,\gamma ) cross sections at operational and under-construction facilities including CSNS [75, 76], n_TOF [77], LANSCE [78], J-PARC [79], GELINA [80], and SARAF-II [81].

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