Astrophysical Reaction Rates of ¹⁰⁶Cd(α,γ)¹¹⁰Sn

Astrophysical Reaction Rates of ¹⁰⁶Cd(α,γ)¹¹⁰Sn | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Astrophysical Reaction Rates of ¹⁰⁶Cd(α,γ)¹¹⁰Sn Caner YALÇIN, Recep Taygun GÜRAY, Ezgi TANTOĞLU, Tuğba GÜLTEKİN, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7180952/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract The alpha induced nuclear reaction cross sections were evaluated over an energy range of 5–20 MeV for the target nucleus 106 Cd. In order to evaluate the experimental results, a total of 432 combinations of eight optical potentials, six level density models and nine gamma strength functions were analyzed. Subsequently, the astrophysical reaction rates were determined by ascertaining the model parameters that optimally represented the experimental outcomes, employing the Threshold Logic Unit (TLU) method. The calculated reaction rates were compared with the existing reaction rate libraries REACLIB and STARLIB, and new reaction rates were proposed for the ¹⁰⁶Cd(α,γ)¹¹⁰Sn reaction. Reaction rate Nuclear reaction cross section p-process Cd-106 optic model potentials Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 INTRODUCTION The formation of heavy atomic nuclei is a multifaceted astrophysical process shaped by the life cycles of stars and explosive events such as supernovae. These events have significantly shaped the elemental structure of the universe and the chemical diversity of the Earth [ 1 ]. Recent advances in nuclear astrophysics are shedding light on the complex mechanisms behind element formation, deepening our understanding of nucleosynthesis and the production of rare isotopes [ 2 ]. Fusion reactions in stars of different masses produce iron and lighter elements while heavier nuclei emerge primarily through neutron capture processes and then through β-decay [ 3 ]. Neutron capture nucleosynthesis is divided into two regimes based on the relative timescales of neutron capture and β-decay. The slow neutron capture process (s-process), operating when neutron capture rates are slower than β-decay, produces approximately half of the isotopes between iron and bismuth. In contrast, the rapid neutron capture process (r-process), active under conditions of extremely high neutron flux where capture outpaces β-decay, generates the remaining isotopes, including those beyond bismuth [ 4 , 5 ]. The s-process occurs predominantly in low-mass asymptotic giant branch (AGB) stars (primary s-process) and massive red giants (weak s-process), with neutrons supplied via (α, n) reactions on 13 C, 17 O, and 22 Ne during helium and carbon burning. The r-process, however, is associated with explosive environments such as Type II supernovae and neutron star mergers, where extreme neutron densities enable rapid captures [ 6 , 7 ]. In the proton-rich domain of the nuclear landscape, approximately 30 stable p-nuclei between selenium (Se) and mercury (Hg) exist, inaccessible to the s- and r-processes. These isotopes are synthesized primarily via photodisintegration reactions (γ-process), where pre-existing seed nuclei undergo photon-induced neutron, proton, or α-particle ejection. As neutron separation energies rise with successive (γ, n) reactions, competing (γ, p) and (γ, α) channels divert the reaction flow toward lower mass regions [ 8 , 9 ]. The γ-process requires temperatures of 2–3 GK, attainable during explosive oxygen/neon burning in massive stars or in sub-Chandrasekhar-mass carbon-oxygen white dwarfs undergoing Type Ia supernovae [ 10 – 12 ]. Although the γ-process dominates p-nuclei production, supplementary mechanisms such as the rp-process (rapid proton capture), νp-process (neutrino-driven proton capture), pn-process (proton-neutron sequential capture), and ν-process (neutrino interactions) contribute to specific isotopes. Notably, 138 La and 180 Ta receive significant yields from the ν-process. Charged-particle-induced reactions, particularly (γ,α) and their inverses, are pivotal for modeling medium-to-heavy p-nuclei. However, experimental constraints on these reactions remain sparse, especially for nuclei beyond iron. Astrophysical reaction networks demand precise cross-section data for thousands of neutron-, proton-, and α-induced reactions [ 13 – 15 ], yet few measurements exist for proton captures (e.g., [ 16 – 20 ] in Ref. [ 21 ]) or α-captures (e.g., [ 22 – 27 ] in Ref. [ 28 ]). Consequently, γ-process simulations rely heavily on theoretical cross sections derived from Hauser-Feshbach statistical models. The accuracy of such models is paramount, as deviations in predicted rates can propagate significant uncertainties into abundance calculations [ 29 ]. Validating these models against experimental data is thus essential for refining nucleosynthetic predictions. A critical test case is the p-nucleus 106 Cd, which is synthesized via the 110 Sn(γ,α) 106 Cd reaction. The closed proton shell (Z = 50) in 110 Sn reduces nuclear level densities, potentially challenging the assumptions of statistical models. To evaluate the robustness of Hauser-Feshbach predictions, this study computes α-capture cross sections for 106 Cd using the TALYS code, incorporating 432 combinations of eight optical potentials, six level density models, and nine gamma strength functions. These theoretical results are benchmarked against experimental data spanning the astrophysically relevant Gamow window [ 30 , 31 ]. Optimal parameter combinations were identified via the threshold logic unit (TLU) method [ 32 , 33 ], and resultant reaction rates were compared with those from established nuclear databases [ 34 , 35 ]. This approach aims to constrain uncertainties in γ-process nucleosynthesis and enhance the reliability of astrophysical models. METHOD In this research, the TALYS nuclear reaction code (version 1.95/2.0) was employed to theoretically determine the cross-sections of nuclear reactions occurring in astrophysical environments [ 36 ]. TALYS is a highly versatile computational tool capable of simulating nuclear reactions across a broad energy spectrum and for various target-projectile interactions. The code follows a comprehensive approach by integrating the optical model, the pre-equilibrium model, and the Hauser-Feshbach statistical model. Each of these models contributes to a detailed description of the nuclear reaction process and its mechanisms. Following the formation of a compound nucleus, the nucleus undergoes decay by emitting a series of particles or gamma rays. TALYS simulates this statistical decay process using the Hauser-Feshbach formalism [ 37 ] which determines the likelihood of each decay channel based on transmission coefficients and level densities. The transmission coefficients are computed using the optical model potential, while the level densities are determined through the Gilbert and Cameron (1965) [ 38 ] formulation. Additionally, gamma-ray decay is incorporated using the giant dipole resonance (GDR) model developed by Brink (1955) [ 39 ]. Key elements such as optical model potentials (OMP), nuclear level density models (LDM), and strength function models (SFM) are essential for precise theoretical calculations of reaction cross-sections. Further details regarding these models are available in the relevant literature. OPTICAL MODEL POTENTIALS In TALYS, the interaction between the incoming projectile and the target core is characterized by an optical model potential (OMP), which depends on variables such as energy, target mass, and isospin value. The choice of optical potential parameters plays a crucial role in accurately modeling both reaction processes and direct interaction channels. The OMPs utilized in these calculations are labeled OMP-1 through OMP-8 (Table 1 ). This set includes the standard alpha potential by Watanabe (1958) [ 42 ], the McFadden and Satchler (1966) potential [ 43 ], contributions from Demetriou et al. (2002) [ 44 ], Avrigeanu et al. (2014) [ 45 ] —which serves as the default selection in TALYS— alongside potentials derived from the work of Nolte et al. (1987) [ 46 ] and an additional study by Avrigeanu et al. (1994) [ 47 ]. Table 1 Optical model potentials (OMP), which are available in the TALYS code. The default options for OMP is the Avrigeanu et al. (2014) (OMP-6). Model no . Optical model potential OMP-1 Normal alpha potential [ 40 ] OMP-2 McFadden and Satchler [ 41 ] OMP-3 Demetriou et al. [ 42 ] (Table 1 ) OMP-4 Demetriou et al. [ 42 ] (Table 2 ) OMP-5 Demetriou et al. [ 42 ] (dispersive model) OMP-6 Avrigeanu et al. [ 43 ] OMP-7 Nolte et al. [ 44 ] OMP-8 Avrigeanu et al. [ 45 ] NUCLEAR LEVEL DENSITY MODELS Nuclear level densities are critical for the analysis of the statistical distribution of excited states, a fundamental component of the modeling of compound nuclear reactions. Phenomenological approaches, such as the Fermi Gas Model, employ empirical parameters calibrated to align with experimental data to describe level densities. The back-shifted Fermi Gas Model enhances predictions through the incorporation of shell corrections, whereas the Constant Temperature Model operates on the assumption of an exponential energy dependence. The collective enhancements from rotational and vibrational modes are incorporated via empirical multiplicative factors calibrated to empirical observations, including neutron resonance ranges. Microscopic models, derive level densities from single-particle spectra, explicitly accounting for shell effects, pairing, and deformation. These approaches are validated against experimental data, enabling reliable extrapolations for astrophysical applications. Advances in computational frameworks now integrate self-consistent nuclear interactions, enhancing predictions for exotic nuclei. Three macroscopic and three phenomenological density models were considered for the calculations (Table 2 ). The phenomenological LDMs include the constant temperature + Fermi gas model [ 38 ], the back-shifted Fermi gas model [ 46 , 47 ], and the generalized superfluid model [ 48 , 49 ], labeled as LDM-1 through LDM-3. Additionally, two macroscopic LDMs were derived using the Skyrme force from Goriely's tables (LDM-4) [ 50 ] and Hilaire's tables (LDM-5) [ 51 ], while a third macroscopic LDM was obtained using the Gogny force from Hilaire's combinatorial tables (LDM-6) [ 52 ]. The default model in TALYS is the constant temperature Fermi gas model (LDM-1) [ 38 ]. Table 2 Level density models (LDM), which are available in the TALYS code. The default options for LDM is constant temperature + Fermi gas model (LDM-1) Model no . Level density model LDM-1 Constant temperature + Fermi gas model [ 38 ] LDM-2 Back-shifted Fermi gas model [ 46 , 47 ] LDM-3 Generalized superfluid model [ 48 , 49 ] LDM-4 Microscopic level densities (Skyrme force) [ 50 ] from Goriely’s tables LDM-5 Microscopic level densities (Skyrme force) [ 51 ] from Hilaire’s combinatorial tables LDM-6 Microscopic LD (temp. dependent HFB, Gogny force) from Hilaire’s combinatorial tables [ 52 ] STRENGTH FUNCTION MODELS Gamma strength functions are of critical importance in the description of electromagnetic transition probabilities in the context of nuclear reactions. A multitude of scientific models have been developed to facilitate the analysis of energy-dependent γ-ray strength functions. Among these, the Standard Lorentzian, Generalized Lorentzian, and hybrid models are particularly prevalent. The Standard Lorentzian model [ 53 ] is based on the assumption of a Lorentzian shape for the fixed-width giant dipole resonance. The Generalized Lorentzian model [ 54 ] introduces a temperature-dependent width in order to account for thermal broadening. The hybrid model integrates the Standard Lorentzian and Generalized Lorentzian approaches, offering a holistic depiction of both low-energy and high-energy photon emission. The parameters for these models, which include resonance energies, widths, and peak cross sections, are generally derived from experimental photoabsorption data and systematics [ 55 ]. Furthermore, the Simplified Modified Lorentzian model [ 56 ] incorporates microscopic corrections to enhance the capture of the low-energy behavior of gamma strength functions, particularly in the context of neutron-rich nuclei. It is evident that microscopic models are founded upon a more fundamental representation of gamma strength functions through the integration of nuclear structure calculations. Consequently, this methodological approach affords a more comprehensive and quantitative understanding of the phenomena in question. Microscopic models provide a more fundamental description of gamma strength functions by integrating nuclear structure calculations. The Skyrme-Hartree-Fock (SHF) approach combined with Quasiparticle Random Phase Approximation (QRPA) [ 57 ] and the Relativistic Mean Field (RMF) + QRPA framework [ 58 ] are commonly employed to compute dipole excitations. The employment of these models has been shown to enhance the accuracy of predictions for exotic cores by taking into account both ground state deformations and shell effects. The Gogny-Hartree-Fock-Bogoliubov (HFB) + QRPA model [ 59 , 60 ] incorporates the necessary matching correlations to identify open-shell nuclei. Beyond the GDR, additional dipole contributions emerge from low-energy enhancements ascribed to the oscillations of neutron shells in neutron-rich isotopes and pygmy resonances. It is imperative to acknowledge the significance of these refined gamma strength functions, as they play a pivotal role in the estimation of radiative capture cross sections and astrophysics reaction rates. Nine distinct strength function models (SFM) are considered for the calculations, labeled SFM-1 through SFM-9, as presented in Table 3 . The default SFM model in TALYS is the Brink–Axel Lorentzian model (SFM-2) [ 55 , 56 ]. The alpha induced nuclear reaction cross sections were evaluated over an energy range of 5–20 MeV for the target nucleus 106 Cd. The input parameters were set following the models described above. To analyze the sensitivity of these parameters on the reaction cross-sections, calculations were conducted for multiple combinations involving eight OMPs, six LDMs, and nine SFMs, as detailed in Tables 1 , 2 , and 3 . Table 3 Gamma-ray strength function models (SFM) which are available in the TALYS code. The default options for SFM is the Brink–Axel Lorentzian model (SFM-2) Model no . Strength function model SFM-1 Kopecky–Uhl generalized Lorentzian [ 53 , 54 ] SFM-2 Brink–Axel Lorentzian [ 55 , 56 ] SFM-3 Hartree–Fock BCS tables [ 61 ] SFM-4 Hartree–Fock–Bogoliubov tables [ 57 ] SFM-5 Goriely’s hybrid model [ 62 ] SFM-6 Goriely T-dependent HFB [ 57 ] SFM-7 T-dependent RMF [ 58 ] SFM-8 Gogny D1M HFB + QRPA [ 59 , 60 ] SFM-9 Simplified Modified Lorentzian (SMLO) [ 63 ] Threshold Logic Unit Method In order to ascertain the most compatible input parameter sets for cross section calculations, the threshold logic unit (TLU) method was employed to evaluate 432 combinations of eight OMPs, six LDMs and nine SFMs for the alpha-activated reaction of 106 Cd isotope. The TLU method is predicated on the concept of a binary threshold function. Initially, the cross sections calculated as outlined in Eq. 1 are binarised by comparing them with the experimental results [ 32 ]. The determination of (X i ) is made by ascertaining whether the calculated TALYS cross section values (σ Ti ) fall within a range that is twice the uncertainties of the experimental cross section results (∆σ Ei ). If the calculated cross section falls within this range, the result of Eq. 1 is (X i ) = 1; otherwise, it is (X i ) = 0. $$\:{X}_{i}=\left\{\begin{array}{c}1\:if\:{\sigma\:}_{Ei}-2\varDelta\:{\sigma\:}_{Ei}\:\le\:\:{\sigma\:}_{Ti}\le\:\:{\sigma\:}_{Ei}+2\varDelta\:{\sigma\:}_{Ei}\\\:0\:otherwise\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\end{array}\right.$$ 1 Each (X i ) is then multiplied by a weight, and the sum of these weighted inputs is compared with a threshold value as expressed in Eq. ( 2 ) and referenced in [ 33 ]. Consequently, the optimal model combinations (BMC) can be ascertained as follows: $$\:BMC=\left\{\begin{array}{c}1\:\:\:if\:\:\:{\sum\:}_{i=0}^{n}{\theta\:}_{i}\:{X}_{i}\:\ge\:\tau\:\\\:0\:\:\:otherwise\:\:\:\:\:\:\:\:\:\:\:\:\end{array}\right.$$ 2 where the weighting factors (θ i ) are considered as one, under the assumption that the weights of the cross sections are equivalent at all energies, n is the number of energies at which experiments are conducted, and the threshold τ is the number of experimental energies anticipated within twice the uncertainty of the TALYS results with experimental values. The cross sections of the 106 Cd(α,γ) 110 Sn reaction were measured at 20 different energies, which implies that n is equal to 20. The TLU method was then applied separately for three thresholds τ = 20, 19 and 18. RESULTS AND DISCUSSION In the context of astrophysics, nuclear reactions occurring within a specified energy range are of primary importance. To address this, cross-section calculations were performed in 0.1 MeV increments for 432 combinations in the range 5.6 to 20.0 MeV. This range encompasses the astrophysical energy region known as the Gamow window and represents the overlap between the Maxwell-Boltzmann distribution and the Coulomb barrier effects described in Eq. (4). The Gamow window for the 106Cd(a,g)110Sn reaction at a temperature of 3.0 GK extends from 6.05 MeV to 9.44 MeV [ 31 ]. The TLU method was employed to ascertain the concordance between the calculated and the experimental cross-sections. The calculation yielded no model combinations across all 20 experimental energies. However, one model (MP-542) combination for 19 energies and two model (MP-558 and MP-562) combinations for 18 energies were found to be compatible with the experimental cross sections. In the MP-xyz notation, x denotes OMP, y denotes LDM and z denotes SFM, as delineated in Tables 1 , 2 and 3 , respectively. Figure 1 demonstrates these three theoretical cross-section calculations (MP-542, MP-558 and MP-562) and experimental cross-section graphs. The effective laboratory energies are calculated by taking into account the energy lost by the alpha beam in the target, and the experimental results are compared. The effective beam energy is defined as the thickness at which half of the reactions occurring in the target occur [ 64 ]. In addition, a comparison was made between the models in the best model combination (MP-542) and all other models in order to understand how the OMP, LDM, and SFM affect the calculations. The effect of different models on the cross-section calculation is demonstrated in Figs. 2 , 3 and 4 . In this analysis, the parameter of interest is varied while maintaining constant values for all other parameters. A thorough analysis of the figures reveals that the most significant parameter influencing the cross-section calculation is OMP (Fig. 2 ). This parameter has been found to effect a change in the cross section of almost 1000 times (for OMP-7). The OMP-5 potential shows excellent agreement with the experimental data. Among the other optical model potentials, OMP-4 also shows strong agreement with the OMP-5 results, particularly within the Gamow window. Figure 2 shows that the MP-442/MP-542 ratio is close to unity. OMP-1, OMP-2 and OMP-3 only demonstrate compatibility at specific energies, after 10 MeV. OMP-7, conversely, exhibits a failure to reproduce experimental results at nearly all energies. It is notable that the disparity between the OMPs diminishes at 16 MeV. However, the cross sections are found to be highly dependent on the OMPs within the astrophysically significant Gamow window range. Within the 6–20 MeV energy range, Level Density emerges as the second most effective parameter in cross section calculations, while SFM is identified as the least effective parameter. As demonstrated in Fig. 3 , the cross-section ratios have been calculated utilising a variety of Level Density Models. At energies lower than 10 MeV, the ratio approaches 1 for all LDMs. However, at higher energies, a sharp increase is observed in the calculations with most LDMs. A divergence of a significant magnitude is evident after 12 MeV, with a discrepancy of almost a factor of 7 observed in calculations employing LDM-5 at around 16 MeV. The ratio of the cross sections calculated with different Strength Function Models to MP-542 begins close to 1 for most SFMs at low energies (less than 8 MeV). However, as the energy increases, a sharp decrease is observed, especially between 8 MeV and 11 MeV. Beyond 11 MeV, a divergence in the curves is evident, accompanied by a gradual increase in the cross-section ratios calculated for certain SFMs. Within the Gamow window, the choice of LDM appears to have a minimal effect on the calculated cross sections, as evidenced by ratios close to one for all LDMs (see Fig. 3 ). Although SFMs are not as dominant as OMPs, some models can deviate 10–20% from MP-542 results at the upper energy limit of the Gamow window (Fig. 4 ). Therefore, while the OMP remains the dominant factor, the choice of SFM can introduce non-negligible uncertainty in this particular region. As Table 4 shows, the cross section values calculated with the three best model combinations are compared with the experimental data. It is evident that MP-542 failed to correctly predict only the experimental point with an energy of 11.321 MeV. A similar failure was observed with MP-558 and MP-562, which also failed at the point with an energy of 10.763 MeV. Table 4 Comparison of cross section values calculated with the best 3 model combinations with experimental data. In addition, the discrepancy between the experimental and theoretical results is expressed as a percentage. \(\:{\varvec{E}}_{\varvec{L}\varvec{a}\varvec{b}.}^{\varvec{E}\varvec{f}\varvec{f}.}\) (MeV) E_err (MeV) σ exp (µb) Err(σ exp ) (µb) MP-542 (µb) Deviation % MP-558 (µb) Deviation % MP-562 (µb) Deviation % 7.852 0.01 0.078 0.01 0.09 13.21 0.09 13.20 0.09 13.21 8.344 0.01 0.48 0.05 0.52 9.16 0.52 9.13 0.52 9.17 8.430 0.029 0.85 0.37 0.70 -17.14 0.70 -17.17 0.70 -17.13 8.835 0.011 2.59 0.26 2.68 3.49 2.68 3.36 2.68 3.53 8.958 0.026 4.87 0.55 3.95 -18.85 3.94 -19.00 3.95 -18.82 9.332 0.012 11.80 1.20 12.23 3.68 12.18 3.22 12.24 3.73 9.452 0.049 22.80 2.90 17.30 -24.12 17.20 -24.56 17.31 -24.10 9.824 0.012 46.40 4.60 48.05 3.55 47.47 2.31 48.00 3.44 9.828 0.012 48.30 4.80 48.56 0.53 47.97 -0.68 48.50 0.42 9.962 0.03 76.30 4.51 68.73 -9.93 67.69 -11.29 68.59 -10.11 10.283 0.036 147 15 150.90 2.65 147.32 0.22 150.08 2.09 10.763* 0.033 234 24 187.29 -19.96 164.84 -29.56 177.93 -23.96 10.823 0.014 244 124 200.56 -17.80 175.73 -27.98 188.84 -22.61 11.182 0.033 298 31 303.96 2.00 269.08 -9.71 281.66 -5.48 11.321** 0.014 430 40 329.73 -23.32 294.13 -31.60 302.33 -29.69 11.589 0.034 488 36 465.41 -4.63 423.85 -13.15 424.22 -13.07 11.815 0.015 596 61 612.86 2.83 567.92 -4.71 557.37 -6.48 11.98 0.035 601 62 705.01 17.31 688.12 14.50 668.44 11.22 12.505 0.036 1270 150 1062.27 -16.36 1348.43 6.18 1300.85 2.43 12.513 0.036 1280 133 1068.93 -16.49 1360.73 6.31 1313.40 2.61 * MP-558 and MP-562 model combinations failed at this energy. ** All three model combinations failed at this energy. The astrophysical reaction rates calculated with these three models and their averages are presented in the following section. Table 5 provides a comprehensive list of the best models (MP-542, MP-558 and MP-562) and their average reaction rates, together with the reaction rates of the available reaction rate libraries (STARLIB and REACLIB) in the temperature range 0.8 to 10 GK. At astrophysical p-process temperatures of between 1 and 3 GK, the results obtained from the three most accomplished models (MP-542, MP-558 and MP-562) are found to be almost indistinguishable from each other with regard to reaction rate. In Fig. 5 , a comparison is made between these models and existing reaction rate libraries. Figure 6 also provides a visual representation of the 1–3 GK temperature range, thereby facilitating a more nuanced analysis of the p-process temperature dynamics. Table 5 Comparison of the three best models calculated with TALYS and their average with REACLIB [ 34 ] and STARLIB [ 35 ]. Reaction rates are given in cm 3 s − 1 mol − 1 units. T (GK) REACLIB STARLIB TALYS MP-558 TALYS MP-558 TALYS MP-562 TALYS Average 0.8 1.33E-30 1.93E-32 2.47E-36 2.47E-36 2.47E-36 2.47E-36 0.9 3.47E-28 5.93E-30 1.93E-32 1.93E-32 1.93E-32 1.93E-32 1 4.13E-26 8.29E-28 2.09E-29 2.09E-29 2.09E-29 2.09E-29 1.5 7.47E-19 3.59E-20 3.71E-20 3.71E-20 3.71E-20 3.71E-20 2 2.25E-14 2.26E-15 3.00E-15 3.00E-15 3.00E-15 3.00E-15 2.5 2.85E-11 4.54E-12 5.31E-12 5.31E-12 5.31E-12 5.31E-12 3 5.61E-09 1.14E-09 1.38E-09 1.37E-09 1.38E-09 1.38E-09 3.5 3.28E-07 6.92E-08 1.05E-07 1.04E-07 1.05E-07 1.05E-07 4 8.18E-06 1.41E-06 3.30E-06 3.25E-06 3.28E-06 3.28E-06 5 8.94E-04 6.82E-05 5.28E-04 5.18E-04 5.25E-04 5.24E-04 6 2.06E-02 8.55E-04 1.81E-02 1.81E-02 1.82E-02 1.81E-02 7 1.68E-01 6.02E-03 2.43E-01 2.55E-01 2.52E-01 2.50E-01 8 6.45E-01 2.94E-02 1.80E + 00 1.99E + 00 1.93E + 00 1.91E + 00 9 1.39E + 00 1.06E-01 8.79E + 00 1.04E + 01 9.80E + 00 9.67E + 00 10 1.91E + 00 2.97E-01 3.20E + 01 4.05E + 01 3.71E + 01 3.65E + 01 At temperatures lower than 1.5 GK, both REACLIB [ 34 ] and STARLIB [ 35 ] predict reaction rates that are considerably high. In the p-process temperature range from approximately 1.5 to 3.5 GK, reaction rates from STARLIB have been observed to be largely consistent with the TALYS average, typically 10–30% below the TALYS average (Fig. 7 ). At temperatures in excess of 3.5 GK, STARLIB demonstrates a marked tendency to underestimate the TALYS average. REACLIB rates approach the TALYS average most closely around 3 GK, but generally remain significantly higher than the TALYS average over the entire temperature range examined. CONCLUTION In this study, the reaction cross sections of ¹⁰⁶Cd(α,γ)¹¹⁰Sn, which are of significance for the p-process core synthesis, were determined. This was achieved by comparing them with experimental cross sections and including the entire Gamow window. Furthermore, reaction rates within the temperature range of 0.8 to 10 GK were determined and subsequently compared with the currently available reaction rate libraries, REACLIB and STARLIB. Cross section calculations at astrophysical energies are found to be highly dependent on the optical model potentials. The best agreement with the experimental cross sections was found with Demetriou's dispersive model (OMP-5) [ 42 ]. At astrophysical energies, within the Gamow window, the contribution of LDM selection to cross section calculations is limited. The influence of SFMs is also less dominant than that of OMPs; however, certain SFM choices can cause moderate deviations towards the upper energy limit of the Gamow window. This suggests that their contribution should not be completely ignored in precision calculations. The reaction rate calculations performed with the Best Combinations were found to differ from the results of existing reaction libraries. In particular, the results of the REACLIB reaction rate library are considerably higher than the reaction rates calculated in this study. Consequently, the utilization of the reaction rates provided in Table 5 is strongly advocated within the context of reaction networks. This study, in conjunction with previous similar studies [ 33 , 65 ], underscores the imperative for comprehensive investigation of all potential reactions through this method. The Experimental Nuclear Physics group at Kocaeli University has been engaged in two primary research projects. Firstly, the group has been working on the systematic evaluation of reactions that are important for the p-process. Secondly, the group has been working on the creation of a new reaction rate library, which will be available soon. Declarations The authors have no competing interests to declare that are relevant to the content of this article. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Author Contribution C.Y. and R.T.G. conceived the idea and outlined the main framework of the article. C.Y. and E.T. calculated the theoretical cross sections. C.Y., E.T., T.G., and E.S.D. applied the TLU method and analyzed the results. E.T., T.G., and E.S.D. calculated the reaction rates and prepared all graphs and tables. R.T.G. supported the research and reviewed the study's findings. All authors discussed the results and contributed to the final article. Data Availability Statement Data sets generated during the current study are available from the corresponding author on reasonable request. References Jones, S., Côté, B., Röpke, F. K., & Wanajo, S. 2019, The Astrophysical Journal, 882, 170, https://doi.org/10.3847/1538-4357/ab384e Arcones, A., & Thielemann, F.-K. 2022, The Astronomy and Astrophysics Review, 31, https://doi.org/10.1007/s00159-022-00146-x Cowan, J. J., Sneden, C., Lawler, J. E., et al. 2021, Reviews of Modern Physics, 93, https://doi.org/10.1103/revmodphys.93.015002 Busso, M., Gallino, R., & Wasserburg, G. 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T. 2023, Nuclear Science and Techniques, 34, https://doi.org/10.1007/s41365-023-01301-4 Cyburt, R. H., Amthor, A. M., Ferguson, R., et al. 2010, The Astrophysical Journal Supplement Series, 189, 240, https://doi.org/10.1088/0067-0049/189/1/240 Sallaska, A. L., Iliadis, C., Champange, A. E., et al. 2013, The Astrophysical Journal Supplement Series, 207, 18, https://doi.org/10.1088/0067-0049/207/1/18 Koning, A., Hilaire, S., & Goriely, S. 2023, The European Physical Journal A, 59, https://doi.org/10.1140/epja/s10050-023-01034-3 Hauser, W., & Feshbach, H. 1952, Physical Review, 87, 366, https://doi.org/10.1103/physrev.87.366 Gilbert, A., & Cameron, A. G. W. 1965, Canadian Journal of Physics, 43, 1446, https://doi.org/10.1139/p65-139 D. M. Brink, Ph.D thesis, University of Oxford, 1955 Watanabe, S. 1958, Nuclear Physics, 8, 484, https://doi.org/10.1016/0029-5582(58)90180-9 McFadden, L., & Satchler, G. R. 1966, Nuclear Physics, 84, 177, https://doi.org/10.1016/0029-5582(66)90441-x Demetriou, P., Grama, C., & Goriely, S. 2002, Nuclear Physics A, 707, 253, https://doi.org/10.1016/s0375-9474(02)00756-x Avrigeanu, V., Avrigeanu, M., & Mănăilescu, C. 2014, Physical Review C, 90, https://doi.org/10.1103/physrevc.90.044612 Nolte, M., Machner, H., & Bojowald, J. 1987, Physical Review C, 36, 1312, https://doi.org/10.1103/physrevc.36.1312 Avrigeanu, V., Hodgson, P. E., & Avrigeanu, M. 1994, Physical Review C, 49, 2136, https://doi.org/10.1103/physrevc.49.2136 Dilg, W., Schantl, W., Vonach, H., & Uhl, M. 1973, Nuclear Physics A, 217, 269, https://doi.org/10.1016/0375-9474(73)90196-6 Demetriou, P., & Goriely, S. 2001, Nuclear Physics A, 695, 95, https://doi.org/10.1016/s0375-9474(01)01095-8 Ignatyuk, A. V., Istekov, K. K., & Smirenkin, G. N. 1979, Sov J Nucl Phys (Engl Transl); (United States), http://www.osti.gov/scitech/biblio/5770504-role-collective-effects-systematics-nuclear-level-densities Ignatyuk, A. V., Weil, J. L., Raman, S., & Kahane, S. 1993, Physical Review C, 47, 1504, https://doi.org/10.1103/physrevc.47.1504 Goriely, S., Hilaire, S., & Koning, A. J. 2008, Physical Review C, 78, https://doi.org/10.1103/physrevc.78.064307 Hilaire, S., & Goriely, S. 2006, Nuclear Physics A, 779, 63, https://doi.org/10.1016/j.nuclphysa.2006.08.014 Hilaire, S., Girod, M., Goriely, S., & Koning, A. J. 2012, Physical Review C, 86, https://doi.org/10.1103/physrevc.86.064317 Kopecky, J., & Uhl, M. 1990, Physical Review C, 41, 1941, https://doi.org/10.1103/physrevc.41.1941 Kopecky, J., Uhl, M., & Chrien, R. E. 1993, Physical Review C, 47, 312, https://doi.org/10.1103/physrevc.47.312 Brink, D. M. 1957, Nuclear Physics, 4, 215, https://doi.org/10.1016/0029-5582(87)90021-6 Axel, P. 1962, Physical Review, 126, 671, https://doi.org/10.1103/physrev.126.671 Goriely, S., Khan, E., & Samyn, M. 2004, Nuclear Physics A, 739, 331, https://doi.org/10.1016/j.nuclphysa.2004.04.105 Daoutidis, I., & Goriely, S. 2012, Physical Review C, 86, https://doi.org/10.1103/physrevc.86.034328 Goriely, S., Hilaire, S., Péru, S., & Sieja, K. 2018, Physical Review C, 98, https://doi.org/10.1103/physrevc.98.014327 Martini, M., Hilaire, S., Goriely, S., Koning, A. J., & Péru, S. 2014, Nuclear Data Sheets, 118, 273, https://doi.org/10.1016/j.nds.2014.04.056 Goriely, S., & Khan, E. 2002, Nuclear Physics A, 706, 217, https://doi.org/10.1016/s0375-9474(02)00860-6 Goriely, S. 1998, Physics Letters B, 436, 10, https://doi.org/10.1016/s0370-2693(98)00907-1 Plujko, V., Gorbachenko, O., & Solodovnyk, K. 2019, The European Physical Journal A, 55, https://doi.org/10.1140/epja/i2019-12899-6 Kutlu, S., GüRay, R. T., ÖZkan, N., & YalçIn, C. 2007, AIP Conference Proceedings, https://doi.org/10.1063/1.2733288 Yalçın, C. 2017, Nuclear Science and Techniques, 28, https://doi.org/10.1007/s41365-017-0267-y Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 17 Aug, 2025 Reviews received at journal 16 Aug, 2025 Reviewers agreed at journal 05 Aug, 2025 Reviewers invited by journal 05 Aug, 2025 Editor assigned by journal 31 Jul, 2025 Submission checks completed at journal 30 Jul, 2025 First submitted to journal 21 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7180952","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":496266027,"identity":"dfe4bb6a-71fb-4fcb-8635-f2c2379b0ae1","order_by":0,"name":"Caner YALÇIN","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVRIie3RMUvEMBQH8BcK7XLYNUW0X6GhcAqt+FVaDtqlOHcQySGki9L1vobfoEegt8Q9m8qB0w1100V8lk5S21Uw/yUhyQ/eewEwMfnLOQLCoWu+txbHxZonNhKy6QkZiD1L8NliIDBF3Opxuy9B+rYjxT5Wse9W63UDZZRy//55jFB1tWIKJBOLtAoLnbGN2mJhKk+57QRjJIBi6XGQRFAijotOJqBT3hAhkYxXFriHsw8klz05R+I/vSD5nCC0WGK7Mu0JaJkEGidG+O+E6kPo8SBfYS/Cu1MZe1BYWNLmobCz8YnVBXvjZXRRO7tX+t7G/ulOyq67jk5qqx0lQ3m3Pw4SmP3Jm+lrExMTk/+dL6v7YZHHrPBWAAAAAElFTkSuQmCC","orcid":"","institution":"Kocaeli University","correspondingAuthor":true,"prefix":"","firstName":"Caner","middleName":"","lastName":"YALÇIN","suffix":""},{"id":496266028,"identity":"807839bb-b9f7-40cf-a067-11e463cc8211","order_by":1,"name":"Recep Taygun GÜRAY","email":"","orcid":"","institution":"Kocaeli University","correspondingAuthor":false,"prefix":"","firstName":"Recep","middleName":"Taygun","lastName":"GÜRAY","suffix":""},{"id":496266030,"identity":"48e43779-bb7a-425a-a60e-c723bb8bb67d","order_by":2,"name":"Ezgi TANTOĞLU","email":"","orcid":"","institution":"Kocaeli University","correspondingAuthor":false,"prefix":"","firstName":"Ezgi","middleName":"","lastName":"TANTOĞLU","suffix":""},{"id":496266032,"identity":"ed3d121e-4595-4848-ac83-26a02807e35c","order_by":3,"name":"Tuğba GÜLTEKİN","email":"","orcid":"","institution":"Kocaeli University","correspondingAuthor":false,"prefix":"","firstName":"Tuğba","middleName":"","lastName":"GÜLTEKİN","suffix":""},{"id":496266033,"identity":"9baa14b1-b329-4a77-af19-a1743bbcdfd6","order_by":4,"name":"Esma Sude DURSUN","email":"","orcid":"","institution":"Vehbi Koç Vakfı","correspondingAuthor":false,"prefix":"","firstName":"Esma","middleName":"Sude","lastName":"DURSUN","suffix":""}],"badges":[],"createdAt":"2025-07-21 21:53:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7180952/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7180952/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":88627820,"identity":"ecc015f3-9a43-4973-bf21-f9c9eb489698","added_by":"auto","created_at":"2025-08-08 13:15:59","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":145734,"visible":true,"origin":"","legend":"\u003cp\u003eBest three theoretical models and experimental cross-section. The boundaries of the Gamow window for 3GK temperature are indicated by red dashed lines.\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/71277e7b65e4ccb1c87c42ff.png"},{"id":88627020,"identity":"6e3079f1-d95a-47a2-bc12-22c4937f96eb","added_by":"auto","created_at":"2025-08-08 13:07:59","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":180285,"visible":true,"origin":"","legend":"\u003cp\u003eRatios of different OMPs to the best model combination (MP-542). The Level Density Model was set to LDM-4 and the Strength Function Model was set to SFM-2. The boundaries of the Gamow window for 3GK temperature are indicated by red dashed lines.\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/cb8a9e46aabf49921b6640a0.png"},{"id":88627821,"identity":"56d38fcf-6f08-4245-b5d0-2a643d246b6a","added_by":"auto","created_at":"2025-08-08 13:15:59","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":171678,"visible":true,"origin":"","legend":"\u003cp\u003eRatios of different LDMs to the best model combination (MP-542). The Optic Model Parameters was set to OMP-5 and the Strength Function Model was set to SFM-2. The boundaries of the Gamow window for 3GK temperature are indicated by red dashed lines.\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/b8f3530efed71abea7eb9795.png"},{"id":88627023,"identity":"b4ad369d-6c55-4184-8902-5c63f217c654","added_by":"auto","created_at":"2025-08-08 13:07:59","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":193689,"visible":true,"origin":"","legend":"\u003cp\u003eRatios of different SFMs to the best model combination (MP-542). The Optic Model Parameters was set to OMP-5 and the Level Density Model was set to LDM-4. The boundaries of the Gamow window for 3GK temperature are indicated by red dashed lines.\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/61cc52a78699519989c00a5b.png"},{"id":88627038,"identity":"c61f7482-87b1-4126-94f6-5fc029b48997","added_by":"auto","created_at":"2025-08-08 13:08:00","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":99146,"visible":true,"origin":"","legend":"\u003cp\u003eThe average reaction rates calculated with the three sets of input parameters that best match the measurements.\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/9ff5373003889fb66b1bc58c.png"},{"id":88627029,"identity":"710fa5f5-4606-420c-8eab-2ea3b5ca1933","added_by":"auto","created_at":"2025-08-08 13:08:00","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":162437,"visible":true,"origin":"","legend":"\u003cp\u003eAs in Figure 5, but with greater focus on the p-process temperatures.\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/2501524a66544de57ceac907.png"},{"id":88627037,"identity":"ea81fa75-4a59-46d4-88c9-e37fdfee9aad","added_by":"auto","created_at":"2025-08-08 13:08:00","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":139540,"visible":true,"origin":"","legend":"\u003cp\u003eRatio of average reaction rates calculated with the three sets of input parameters that best match the experimental measurement results.\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/22589e77104b475a382331f9.png"},{"id":88629276,"identity":"c1cd5f15-815b-4ca0-97bc-a240cc89c336","added_by":"auto","created_at":"2025-08-08 13:32:15","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2882948,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7180952/v1/310c6287-c036-4799-ab78-0577feefe8b0.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Astrophysical Reaction Rates of ¹⁰⁶Cd(α,γ)¹¹⁰Sn","fulltext":[{"header":"INTRODUCTION","content":"\u003cp\u003eThe formation of heavy atomic nuclei is a multifaceted astrophysical process shaped by the life cycles of stars and explosive events such as supernovae. These events have significantly shaped the elemental structure of the universe and the chemical diversity of the Earth [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Recent advances in nuclear astrophysics are shedding light on the complex mechanisms behind element formation, deepening our understanding of nucleosynthesis and the production of rare isotopes [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Fusion reactions in stars of different masses produce iron and lighter elements while heavier nuclei emerge primarily through neutron capture processes and then through β-decay [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eNeutron capture nucleosynthesis is divided into two regimes based on the relative timescales of neutron capture and β-decay. The slow neutron capture process (s-process), operating when neutron capture rates are slower than β-decay, produces approximately half of the isotopes between iron and bismuth. In contrast, the rapid neutron capture process (r-process), active under conditions of extremely high neutron flux where capture outpaces β-decay, generates the remaining isotopes, including those beyond bismuth [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. The s-process occurs predominantly in low-mass asymptotic giant branch (AGB) stars (primary s-process) and massive red giants (weak s-process), with neutrons supplied via (α, n) reactions on \u003csup\u003e13\u003c/sup\u003eC, \u003csup\u003e17\u003c/sup\u003eO, and \u003csup\u003e22\u003c/sup\u003eNe during helium and carbon burning. The r-process, however, is associated with explosive environments such as Type II supernovae and neutron star mergers, where extreme neutron densities enable rapid captures [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eIn the proton-rich domain of the nuclear landscape, approximately 30 stable p-nuclei between selenium (Se) and mercury (Hg) exist, inaccessible to the s- and r-processes. These isotopes are synthesized primarily via photodisintegration reactions (γ-process), where pre-existing seed nuclei undergo photon-induced neutron, proton, or α-particle ejection. As neutron separation energies rise with successive (γ, n) reactions, competing (γ, p) and (γ, α) channels divert the reaction flow toward lower mass regions [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. The γ-process requires temperatures of 2–3 GK, attainable during explosive oxygen/neon burning in massive stars or in sub-Chandrasekhar-mass carbon-oxygen white dwarfs undergoing Type Ia supernovae [\u003cspan additionalcitationids=\"CR11\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e–\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Although the γ-process dominates p-nuclei production, supplementary mechanisms such as the rp-process (rapid proton capture), νp-process (neutrino-driven proton capture), pn-process (proton-neutron sequential capture), and ν-process (neutrino interactions) contribute to specific isotopes. Notably, \u003csup\u003e138\u003c/sup\u003eLa and \u003csup\u003e180\u003c/sup\u003eTa receive significant yields from the ν-process.\u003c/p\u003e\u003cp\u003eCharged-particle-induced reactions, particularly (γ,α) and their inverses, are pivotal for modeling medium-to-heavy p-nuclei. However, experimental constraints on these reactions remain sparse, especially for nuclei beyond iron. Astrophysical reaction networks demand precise cross-section data for thousands of neutron-, proton-, and α-induced reactions [\u003cspan additionalcitationids=\"CR14\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e–\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], yet few measurements exist for proton captures (e.g., [\u003cspan additionalcitationids=\"CR17 CR18 CR19\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e–\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] in Ref. [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]) or α-captures (e.g., [\u003cspan additionalcitationids=\"CR23 CR24 CR25 CR26\" citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e–\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e] in Ref. [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]). Consequently, γ-process simulations rely heavily on theoretical cross sections derived from Hauser-Feshbach statistical models. The accuracy of such models is paramount, as deviations in predicted rates can propagate significant uncertainties into abundance calculations [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]. Validating these models against experimental data is thus essential for refining nucleosynthetic predictions.\u003c/p\u003e\u003cp\u003eA critical test case is the p-nucleus \u003csup\u003e106\u003c/sup\u003eCd, which is synthesized via the \u003csup\u003e110\u003c/sup\u003eSn(γ,α)\u003csup\u003e106\u003c/sup\u003eCd reaction. The closed proton shell (Z = 50) in \u003csup\u003e110\u003c/sup\u003eSn reduces nuclear level densities, potentially challenging the assumptions of statistical models. To evaluate the robustness of Hauser-Feshbach predictions, this study computes α-capture cross sections for \u003csup\u003e106\u003c/sup\u003eCd using the TALYS code, incorporating 432 combinations of eight optical potentials, six level density models, and nine gamma strength functions. These theoretical results are benchmarked against experimental data spanning the astrophysically relevant Gamow window [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. Optimal parameter combinations were identified via the threshold logic unit (TLU) method [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e], and resultant reaction rates were compared with those from established nuclear databases [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]. This approach aims to constrain uncertainties in γ-process nucleosynthesis and enhance the reliability of astrophysical models.\u003c/p\u003e"},{"header":"METHOD","content":"\u003cp\u003eIn this research, the TALYS nuclear reaction code (version 1.95/2.0) was employed to theoretically determine the cross-sections of nuclear reactions occurring in astrophysical environments [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e]. TALYS is a highly versatile computational tool capable of simulating nuclear reactions across a broad energy spectrum and for various target-projectile interactions. The code follows a comprehensive approach by integrating the optical model, the pre-equilibrium model, and the Hauser-Feshbach statistical model. Each of these models contributes to a detailed description of the nuclear reaction process and its mechanisms.\u003c/p\u003e\u003cp\u003eFollowing the formation of a compound nucleus, the nucleus undergoes decay by emitting a series of particles or gamma rays. TALYS simulates this statistical decay process using the Hauser-Feshbach formalism [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] which determines the likelihood of each decay channel based on transmission coefficients and level densities. The transmission coefficients are computed using the optical model potential, while the level densities are determined through the Gilbert and Cameron (1965) [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e] formulation. Additionally, gamma-ray decay is incorporated using the giant dipole resonance (GDR) model developed by Brink (1955) [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eKey elements such as optical model potentials (OMP), nuclear level density models (LDM), and strength function models (SFM) are essential for precise theoretical calculations of reaction cross-sections. Further details regarding these models are available in the relevant literature.\u003c/p\u003e\u003cp\u003e\u003cem\u003eOPTICAL MODEL POTENTIALS\u003c/em\u003e\u003c/p\u003e\u003cp\u003eIn TALYS, the interaction between the incoming projectile and the target core is characterized by an optical model potential (OMP), which depends on variables such as energy, target mass, and isospin value. The choice of optical potential parameters plays a crucial role in accurately modeling both reaction processes and direct interaction channels.\u003c/p\u003e\u003cp\u003eThe OMPs utilized in these calculations are labeled OMP-1 through OMP-8 (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). This set includes the standard alpha potential by Watanabe (1958) [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e], the McFadden and Satchler (1966) potential [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e], contributions from Demetriou et al. (2002) [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e], Avrigeanu et al. (2014) [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e] —which serves as the default selection in TALYS— alongside potentials derived from the work of Nolte et al. (1987) [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e] and an additional study by Avrigeanu et al. (1994) [\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eOptical model potentials (OMP), which are available in the TALYS code. The default options for OMP is the Avrigeanu et al. (2014) (OMP-6).\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel no .\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eOptical model potential\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eNormal alpha potential [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMcFadden and Satchler [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eDemetriou et al. [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e] (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eDemetriou et al. [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e] (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eDemetriou et al. [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e] (dispersive model)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eAvrigeanu et al. [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eNolte et al. [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eOMP-8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eAvrigeanu et al. [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eNUCLEAR LEVEL DENSITY MODELS\u003c/p\u003e\u003cp\u003eNuclear level densities are critical for the analysis of the statistical distribution of excited states, a fundamental component of the modeling of compound nuclear reactions. Phenomenological approaches, such as the Fermi Gas Model, employ empirical parameters calibrated to align with experimental data to describe level densities. The back-shifted Fermi Gas Model enhances predictions through the incorporation of shell corrections, whereas the Constant Temperature Model operates on the assumption of an exponential energy dependence. The collective enhancements from rotational and vibrational modes are incorporated via empirical multiplicative factors calibrated to empirical observations, including neutron resonance ranges.\u003c/p\u003e\u003cp\u003eMicroscopic models, derive level densities from single-particle spectra, explicitly accounting for shell effects, pairing, and deformation. These approaches are validated against experimental data, enabling reliable extrapolations for astrophysical applications. Advances in computational frameworks now integrate self-consistent nuclear interactions, enhancing predictions for exotic nuclei.\u003c/p\u003e\u003cp\u003eThree macroscopic and three phenomenological density models were considered for the calculations (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The phenomenological LDMs include the constant temperature + Fermi gas model [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], the back-shifted Fermi gas model [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e, \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e], and the generalized superfluid model [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e, \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e], labeled as LDM-1 through LDM-3. Additionally, two macroscopic LDMs were derived using the Skyrme force from Goriely's tables (LDM-4) [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e] and Hilaire's tables (LDM-5) [\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e], while a third macroscopic LDM was obtained using the Gogny force from Hilaire's combinatorial tables (LDM-6) [\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e]. The default model in TALYS is the constant temperature Fermi gas model (LDM-1) [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eLevel density models (LDM), which are available in the TALYS code. The default options for LDM is constant temperature + Fermi gas model (LDM-1)\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel no .\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eLevel density model\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLDM-1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eConstant temperature + Fermi gas model [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLDM-2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eBack-shifted Fermi gas model [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e, \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLDM-3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eGeneralized superfluid model [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e, \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLDM-4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMicroscopic level densities (Skyrme force) [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e] from Goriely’s tables\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLDM-5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMicroscopic level densities (Skyrme force) [\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e] from Hilaire’s combinatorial tables\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLDM-6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMicroscopic LD (temp. dependent HFB, Gogny force) from Hilaire’s combinatorial tables [\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eSTRENGTH FUNCTION MODELS\u003c/p\u003e\u003cp\u003eGamma strength functions are of critical importance in the description of electromagnetic transition probabilities in the context of nuclear reactions. A multitude of scientific models have been developed to facilitate the analysis of energy-dependent γ-ray strength functions. Among these, the Standard Lorentzian, Generalized Lorentzian, and hybrid models are particularly prevalent. The Standard Lorentzian model [\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e] is based on the assumption of a Lorentzian shape for the fixed-width giant dipole resonance. The Generalized Lorentzian model [\u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e] introduces a temperature-dependent width in order to account for thermal broadening. The hybrid model integrates the Standard Lorentzian and Generalized Lorentzian approaches, offering a holistic depiction of both low-energy and high-energy photon emission. The parameters for these models, which include resonance energies, widths, and peak cross sections, are generally derived from experimental photoabsorption data and systematics [\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e]. Furthermore, the Simplified Modified Lorentzian model [\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e] incorporates microscopic corrections to enhance the capture of the low-energy behavior of gamma strength functions, particularly in the context of neutron-rich nuclei.\u003c/p\u003e\u003cp\u003eIt is evident that microscopic models are founded upon a more fundamental representation of gamma strength functions through the integration of nuclear structure calculations. Consequently, this methodological approach affords a more comprehensive and quantitative understanding of the phenomena in question. Microscopic models provide a more fundamental description of gamma strength functions by integrating nuclear structure calculations. The Skyrme-Hartree-Fock (SHF) approach combined with Quasiparticle Random Phase Approximation (QRPA) [\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e] and the Relativistic Mean Field (RMF) + QRPA framework [\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e] are commonly employed to compute dipole excitations. The employment of these models has been shown to enhance the accuracy of predictions for exotic cores by taking into account both ground state deformations and shell effects. The Gogny-Hartree-Fock-Bogoliubov (HFB) + QRPA model [\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e, \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e] incorporates the necessary matching correlations to identify open-shell nuclei. Beyond the GDR, additional dipole contributions emerge from low-energy enhancements ascribed to the oscillations of neutron shells in neutron-rich isotopes and pygmy resonances. It is imperative to acknowledge the significance of these refined gamma strength functions, as they play a pivotal role in the estimation of radiative capture cross sections and astrophysics reaction rates.\u003c/p\u003e\u003cp\u003eNine distinct strength function models (SFM) are considered for the calculations, labeled SFM-1 through SFM-9, as presented in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The default SFM model in TALYS is the Brink–Axel Lorentzian model (SFM-2) [\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e, \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe alpha induced nuclear reaction cross sections were evaluated over an energy range of 5–20 MeV for the target nucleus \u003csup\u003e106\u003c/sup\u003eCd. The input parameters were set following the models described above. To analyze the sensitivity of these parameters on the reaction cross-sections, calculations were conducted for multiple combinations involving eight OMPs, six LDMs, and nine SFMs, as detailed in Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, and \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eGamma-ray strength function models (SFM) which are available in the TALYS code. The default options for SFM is the Brink–Axel Lorentzian model (SFM-2)\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eModel no .\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eStrength function model\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eKopecky–Uhl generalized Lorentzian [\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e, \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eBrink–Axel Lorentzian [\u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e55\u003c/span\u003e, \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e56\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eHartree–Fock BCS tables [\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eHartree–Fock–Bogoliubov tables [\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eGoriely’s hybrid model [\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-6\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eGoriely T-dependent HFB [\u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e57\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-7\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eT-dependent RMF [\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e58\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eGogny D1M HFB + QRPA [\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e59\u003c/span\u003e, \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSFM-9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSimplified Modified Lorentzian (SMLO) [\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e]\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eThreshold Logic Unit Method\u003c/b\u003e\u003c/p\u003e\u003cp\u003eIn order to ascertain the most compatible input parameter sets for cross section calculations, the threshold logic unit (TLU) method was employed to evaluate 432 combinations of eight OMPs, six LDMs and nine SFMs for the alpha-activated reaction of \u003csup\u003e106\u003c/sup\u003eCd isotope. The TLU method is predicated on the concept of a binary threshold function. Initially, the cross sections calculated as outlined in Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e are binarised by comparing them with the experimental results [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]. The determination of (X\u003csub\u003ei\u003c/sub\u003e) is made by ascertaining whether the calculated TALYS cross section values (σ\u003csub\u003eTi\u003c/sub\u003e) fall within a range that is twice the uncertainties of the experimental cross section results (∆σ\u003csub\u003eEi\u003c/sub\u003e). If the calculated cross section falls within this range, the result of Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e is (X\u003csub\u003ei\u003c/sub\u003e) = 1; otherwise, it is (X\u003csub\u003ei\u003c/sub\u003e) = 0.\u003c/p\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{X}_{i}=\\left\\{\\begin{array}{c}1\\:if\\:{\\sigma\\:}_{Ei}-2\\varDelta\\:{\\sigma\\:}_{Ei}\\:\\le\\:\\:{\\sigma\\:}_{Ti}\\le\\:\\:{\\sigma\\:}_{Ei}+2\\varDelta\\:{\\sigma\\:}_{Ei}\\\\\\:0\\:otherwise\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eEach (X\u003csub\u003ei\u003c/sub\u003e) is then multiplied by a weight, and the sum of these weighted inputs is compared with a threshold value as expressed in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) and referenced in [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e]. Consequently, the optimal model combinations (BMC) can be ascertained as follows:\u003c/p\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:BMC=\\left\\{\\begin{array}{c}1\\:\\:\\:if\\:\\:\\:{\\sum\\:}_{i=0}^{n}{\\theta\\:}_{i}\\:{X}_{i}\\:\\ge\\:\\tau\\:\\\\\\:0\\:\\:\\:otherwise\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003ewhere the weighting factors (θ\u003csub\u003ei\u003c/sub\u003e) are considered as one, under the assumption that the weights of the cross sections are equivalent at all energies, n is the number of energies at which experiments are conducted, and the threshold τ is the number of experimental energies anticipated within twice the uncertainty of the TALYS results with experimental values.\u003c/p\u003e\u003cp\u003eThe cross sections of the \u003csup\u003e106\u003c/sup\u003eCd(α,γ)\u003csup\u003e110\u003c/sup\u003eSn reaction were measured at 20 different energies, which implies that n is equal to 20. The TLU method was then applied separately for three thresholds τ = 20, 19 and 18.\u003c/p\u003e"},{"header":"RESULTS AND DISCUSSION","content":"\u003cp\u003eIn the context of astrophysics, nuclear reactions occurring within a specified energy range are of primary importance. To address this, cross-section calculations were performed in 0.1 MeV increments for 432 combinations in the range 5.6 to 20.0 MeV. This range encompasses the astrophysical energy region known as the Gamow window and represents the overlap between the Maxwell-Boltzmann distribution and the Coulomb barrier effects described in Eq.\u0026nbsp;(4). The Gamow window for the 106Cd(a,g)110Sn reaction at a temperature of 3.0 GK extends from 6.05 MeV to 9.44 MeV [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e].\u003c/p\u003e\u003cp\u003eThe TLU method was employed to ascertain the concordance between the calculated and the experimental cross-sections. The calculation yielded no model combinations across all 20 experimental energies. However, one model (MP-542) combination for 19 energies and two model (MP-558 and MP-562) combinations for 18 energies were found to be compatible with the experimental cross sections. In the MP-xyz notation, x denotes OMP, y denotes LDM and z denotes SFM, as delineated in Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and \u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, respectively.\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e demonstrates these three theoretical cross-section calculations (MP-542, MP-558 and MP-562) and experimental cross-section graphs. The effective laboratory energies are calculated by taking into account the energy lost by the alpha beam in the target, and the experimental results are compared. The effective beam energy is defined as the thickness at which half of the reactions occurring in the target occur [\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn addition, a comparison was made between the models in the best model combination (MP-542) and all other models in order to understand how the OMP, LDM, and SFM affect the calculations. The effect of different models on the cross-section calculation is demonstrated in Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. In this analysis, the parameter of interest is varied while maintaining constant values for all other parameters. A thorough analysis of the figures reveals that the most significant parameter influencing the cross-section calculation is OMP (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). This parameter has been found to effect a change in the cross section of almost 1000 times (for OMP-7). The OMP-5 potential shows excellent agreement with the experimental data. Among the other optical model potentials, OMP-4 also shows strong agreement with the OMP-5 results, particularly within the Gamow window. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows that the MP-442/MP-542 ratio is close to unity. OMP-1, OMP-2 and OMP-3 only demonstrate compatibility at specific energies, after 10 MeV. OMP-7, conversely, exhibits a failure to reproduce experimental results at nearly all energies. It is notable that the disparity between the OMPs diminishes at 16 MeV. However, the cross sections are found to be highly dependent on the OMPs within the astrophysically significant Gamow window range. Within the 6\u0026ndash;20 MeV energy range, Level Density emerges as the second most effective parameter in cross section calculations, while SFM is identified as the least effective parameter.\u003c/p\u003e\u003cp\u003eAs demonstrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the cross-section ratios have been calculated utilising a variety of Level Density Models. At energies lower than 10 MeV, the ratio approaches 1 for all LDMs. However, at higher energies, a sharp increase is observed in the calculations with most LDMs. A divergence of a significant magnitude is evident after 12 MeV, with a discrepancy of almost a factor of 7 observed in calculations employing LDM-5 at around 16 MeV.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eThe ratio of the cross sections calculated with different Strength Function Models to MP-542 begins close to 1 for most SFMs at low energies (less than 8 MeV). However, as the energy increases, a sharp decrease is observed, especially between 8 MeV and 11 MeV. Beyond 11 MeV, a divergence in the curves is evident, accompanied by a gradual increase in the cross-section ratios calculated for certain SFMs. Within the Gamow window, the choice of LDM appears to have a minimal effect on the calculated cross sections, as evidenced by ratios close to one for all LDMs (see Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). Although SFMs are not as dominant as OMPs, some models can deviate 10\u0026ndash;20% from MP-542 results at the upper energy limit of the Gamow window (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). Therefore, while the OMP remains the dominant factor, the choice of SFM can introduce non-negligible uncertainty in this particular region.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAs Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows, the cross section values calculated with the three best model combinations are compared with the experimental data. It is evident that MP-542 failed to correctly predict only the experimental point with an energy of 11.321 MeV. A similar failure was observed with MP-558 and MP-562, which also failed at the point with an energy of 10.763 MeV.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eComparison of cross section values calculated with the best 3 model combinations with experimental data. In addition, the discrepancy between the experimental and theoretical results is expressed as a percentage.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"10\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{E}}_{\\varvec{L}\\varvec{a}\\varvec{b}.}^{\\varvec{E}\\varvec{f}\\varvec{f}.}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003cp\u003e(MeV)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eE_err\u003c/p\u003e\u003cp\u003e(MeV)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eσ\u003csub\u003eexp\u003c/sub\u003e\u003c/p\u003e\u003cp\u003e(\u0026micro;b)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eErr(σ\u003csub\u003eexp\u003c/sub\u003e) (\u0026micro;b)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eMP-542\u003c/p\u003e\u003cp\u003e(\u0026micro;b)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003eDeviation\u003c/p\u003e\u003cp\u003e%\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003eMP-558\u003c/p\u003e\u003cp\u003e(\u0026micro;b)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c8\"\u003e\u003cp\u003eDeviation\u003c/p\u003e\u003cp\u003e%\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c9\"\u003e\u003cp\u003eMP-562\u003c/p\u003e\u003cp\u003e(\u0026micro;b)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c10\"\u003e\u003cp\u003eDeviation\u003c/p\u003e\u003cp\u003e%\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e7.852\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.078\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.09\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e13.21\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.09\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e13.20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e0.09\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e13.21\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e8.344\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.48\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.05\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.52\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e9.16\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.52\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e9.13\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e0.52\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e9.17\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e8.430\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.029\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e0.85\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.37\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e0.70\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-17.14\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e0.70\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-17.17\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e0.70\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-17.13\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e8.835\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.011\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.59\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.26\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e2.68\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e3.49\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e2.68\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e3.36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e2.68\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e3.53\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e8.958\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.026\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e4.87\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0.55\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e3.95\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-18.85\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e3.94\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-19.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e3.95\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-18.82\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e9.332\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.012\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e11.80\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e1.20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e12.23\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e3.68\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e12.18\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e3.22\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e12.24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e3.73\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e9.452\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.049\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e22.80\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e2.90\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e17.30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-24.12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e17.20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-24.56\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e17.31\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-24.10\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e9.824\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.012\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e46.40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e4.60\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e48.05\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e3.55\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e47.47\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e2.31\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e48.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e3.44\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e9.828\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.012\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e48.30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e4.80\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e48.56\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e0.53\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e47.97\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-0.68\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e48.50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e0.42\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e9.962\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.03\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e76.30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e4.51\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e68.73\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-9.93\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e67.69\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-11.29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e68.59\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-10.11\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e10.283\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.036\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e147\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e150.90\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e2.65\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e147.32\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e0.22\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e150.08\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e2.09\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e10.763*\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.033\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e234\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e187.29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-19.96\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e\u003cb\u003e164.84\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e\u003cb\u003e-29.56\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e\u003cb\u003e177.93\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e\u003cb\u003e-23.96\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e10.823\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.014\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e244\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e124\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e200.56\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-17.80\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e175.73\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-27.98\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e188.84\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-22.61\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e11.182\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.033\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e298\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e31\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e303.96\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e2.00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e269.08\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-9.71\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e281.66\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-5.48\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e11.321**\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.014\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e430\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e\u003cb\u003e329.73\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e\u003cb\u003e-23.32\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e\u003cb\u003e294.13\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e\u003cb\u003e-31.60\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e\u003cb\u003e302.33\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e\u003cb\u003e-29.69\u003c/b\u003e\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e11.589\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.034\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e488\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e465.41\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-4.63\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e423.85\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-13.15\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e424.22\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-13.07\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e11.815\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.015\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e596\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e61\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e612.86\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e2.83\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e567.92\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e-4.71\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e557.37\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e-6.48\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e11.98\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.035\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e601\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e62\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e705.01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e17.31\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e688.12\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e14.50\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e668.44\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e11.22\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e12.505\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.036\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1270\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e150\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e1062.27\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-16.36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e1348.43\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e6.18\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e1300.85\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e2.43\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e12.513\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0.036\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1280\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e133\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e1068.93\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e\u003cp\u003e-16.49\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e\u003cp\u003e1360.73\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e\u003cp\u003e6.31\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e\u003cp\u003e1313.40\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e\u003cp\u003e2.61\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e* MP-558 and MP-562 model combinations failed at this energy.\u003c/p\u003e\u003cp\u003e** All three model combinations failed at this energy.\u003c/p\u003e\u003cp\u003eThe astrophysical reaction rates calculated with these three models and their averages are presented in the following section. Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e provides a comprehensive list of the best models (MP-542, MP-558 and MP-562) and their average reaction rates, together with the reaction rates of the available reaction rate libraries (STARLIB and REACLIB) in the temperature range 0.8 to 10 GK. At astrophysical p-process temperatures of between 1 and 3 GK, the results obtained from the three most accomplished models (MP-542, MP-558 and MP-562) are found to be almost indistinguishable from each other with regard to reaction rate. In Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, a comparison is made between these models and existing reaction rate libraries. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e also provides a visual representation of the 1\u0026ndash;3 GK temperature range, thereby facilitating a more nuanced analysis of the p-process temperature dynamics.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eComparison of the three best models calculated with TALYS and their average with REACLIB [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e] and STARLIB [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e]. Reaction rates are given in cm\u003csup\u003e3\u003c/sup\u003e s\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e mol\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e units.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eT\u003c/p\u003e\u003cp\u003e(GK)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eREACLIB\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eSTARLIB\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003eTALYS\u003c/p\u003e\u003cp\u003eMP-558\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eTALYS\u003c/p\u003e\u003cp\u003eMP-558\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e\u003cp\u003eTALYS\u003c/p\u003e\u003cp\u003eMP-562\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e\u003cp\u003eTALYS\u003c/p\u003e\u003cp\u003eAverage\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e0.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1.33E-30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1.93E-32\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e2.47E-36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e2.47E-36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e2.47E-36\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e2.47E-36\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e0.9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e3.47E-28\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e5.93E-30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e1.93E-32\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e1.93E-32\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e1.93E-32\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e1.93E-32\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e4.13E-26\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e8.29E-28\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e2.09E-29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e2.09E-29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e2.09E-29\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e2.09E-29\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e1.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e7.47E-19\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e3.59E-20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e3.71E-20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e3.71E-20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e3.71E-20\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e3.71E-20\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e2\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e2.25E-14\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.26E-15\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" 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colname=\"c7\"\u003e\u003cp\u003e1.91E\u0026thinsp;+\u0026thinsp;00\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e9\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1.39E\u0026thinsp;+\u0026thinsp;00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e1.06E-01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e8.79E\u0026thinsp;+\u0026thinsp;00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e1.04E\u0026thinsp;+\u0026thinsp;01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e9.80E\u0026thinsp;+\u0026thinsp;00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e9.67E\u0026thinsp;+\u0026thinsp;00\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e10\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e1.91E\u0026thinsp;+\u0026thinsp;00\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e\u003cp\u003e2.97E-01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e3.20E\u0026thinsp;+\u0026thinsp;01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u003cp\u003e4.05E\u0026thinsp;+\u0026thinsp;01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e\u003cp\u003e3.71E\u0026thinsp;+\u0026thinsp;01\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e\u003cp\u003e3.65E\u0026thinsp;+\u0026thinsp;01\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eAt temperatures lower than 1.5 GK, both REACLIB [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e] and STARLIB [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] predict reaction rates that are considerably high. In the p-process temperature range from approximately 1.5 to 3.5 GK, reaction rates from STARLIB have been observed to be largely consistent with the TALYS average, typically 10\u0026ndash;30% below the TALYS average (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e). At temperatures in excess of 3.5 GK, STARLIB demonstrates a marked tendency to underestimate the TALYS average. REACLIB rates approach the TALYS average most closely around 3 GK, but generally remain significantly higher than the TALYS average over the entire temperature range examined.\u003c/p\u003e"},{"header":"CONCLUTION","content":"\u003cp\u003eIn this study, the reaction cross sections of \u0026sup1;⁰⁶Cd(α,γ)\u0026sup1;\u0026sup1;⁰Sn, which are of significance for the p-process core synthesis, were determined. This was achieved by comparing them with experimental cross sections and including the entire Gamow window. Furthermore, reaction rates within the temperature range of 0.8 to 10 GK were determined and subsequently compared with the currently available reaction rate libraries, REACLIB and STARLIB.\u003c/p\u003e\u003cp\u003eCross section calculations at astrophysical energies are found to be highly dependent on the optical model potentials. The best agreement with the experimental cross sections was found with Demetriou's dispersive model (OMP-5) [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. At astrophysical energies, within the Gamow window, the contribution of LDM selection to cross section calculations is limited. The influence of SFMs is also less dominant than that of OMPs; however, certain SFM choices can cause moderate deviations towards the upper energy limit of the Gamow window. This suggests that their contribution should not be completely ignored in precision calculations. The reaction rate calculations performed with the Best Combinations were found to differ from the results of existing reaction libraries. In particular, the results of the REACLIB reaction rate library are considerably higher than the reaction rates calculated in this study. Consequently, the utilization of the reaction rates provided in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e is strongly advocated within the context of reaction networks.\u003c/p\u003e\u003cp\u003eThis study, in conjunction with previous similar studies [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e, \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e65\u003c/span\u003e], underscores the imperative for comprehensive investigation of all potential reactions through this method. The Experimental Nuclear Physics group at Kocaeli University has been engaged in two primary research projects. Firstly, the group has been working on the systematic evaluation of reactions that are important for the p-process. Secondly, the group has been working on the creation of a new reaction rate library, which will be available soon.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eThe authors have no competing interests to declare that are relevant to the content of this article.\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eC.Y. and R.T.G. conceived the idea and outlined the main framework of the article. C.Y. and E.T. calculated the theoretical cross sections. C.Y., E.T., T.G., and E.S.D. applied the TLU method and analyzed the results. E.T., T.G., and E.S.D. calculated the reaction rates and prepared all graphs and tables. 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In order to evaluate the experimental results, a total of 432 combinations of eight optical potentials, six level density models and nine gamma strength functions were analyzed. Subsequently, the astrophysical reaction rates were determined by ascertaining the model parameters that optimally represented the experimental outcomes, employing the Threshold Logic Unit (TLU) method. The calculated reaction rates were compared with the existing reaction rate libraries REACLIB and STARLIB, and new reaction rates were proposed for the \u0026sup1;⁰⁶Cd(α,γ)\u0026sup1;\u0026sup1;⁰Sn reaction.\u003c/p\u003e","manuscriptTitle":"Astrophysical Reaction Rates of ¹⁰⁶Cd(α,γ)¹¹⁰Sn","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-08 13:07:55","doi":"10.21203/rs.3.rs-7180952/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-08-18T02:24:47+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-17T03:12:23+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"288338585221960433555007392484740550004","date":"2025-08-06T00:42:18+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-05T12:02:56+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-31T05:05:09+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-07-30T12:40:26+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of the Korean Physical Society","date":"2025-07-21T21:50:31+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-the-korean-physical-society","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Journal of the Korean Physical Society](https://link.springer.com/journal/40042)","snPcode":"40042","submissionUrl":"https://submission.springernature.com/new-submission/40042/3","title":"Journal of the Korean Physical Society","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"fbd1f907-d11e-48bb-8b28-9c888b1567e2","owner":[],"postedDate":"August 8th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-01-13T00:23:19+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-08 13:07:55","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7180952","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7180952","identity":"rs-7180952","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}