Several steps were needed to set up the α-particle OMP of Ref. [4]. First, only the α-particle elastic scattering on nuclei with the mass number A∼100, at energies E<35 MeV, was analyzed above B [19]. Question marks related to the rest of SM parameters needed to calculate cross sections of reactions with either incident or emitted α-particles were thus avoided. Moreover, a double–folding model (DFM), with an explicit treatment of the exchange component, was initially used in a semi–microscopic approach. A dispersive correction to the microscopic DFM real potential was also considered, along with a phenomenological imaginary part that was energy–dependent. A full phenomenological analysis then focused on the same angular distributions, without changing the already determined imaginary part, provided the local OMP parameter sets corresponding to each data set. Lastly, a regional OMP with energy–dependent average mass and charge parameters was derived for further use in SM calculations.

Additional semi–microscopic and phenomenological analyses were carried out for the target nuclei with A between 50 and 120 and α-particle energies of 13–50 MeV [20]. However, an SM assessment of the cross sections of α-induced reactions below B, available for the same A range and energies below 12 MeV, was included. Similar consideration for the heavy nuclei (A between 121 and 197) also resulted in different results for the energy–dependence eventual change below B of the surface imaginary potential depth W _D and the volume imaginary potential depth W _V [21]. Thus, there were no issues in the extrapolation to lower energies of W _V from analysis of the α-particle elastic scattering above B, while W _D presents a distinct case.

The α-particle elastic scattering analysis, above B, provides a decrease of W _D (E) with the energy increase. However, its extrapolation below B would contradict the strong increase in the number of open reaction channels of the compound–nucleus (CN). The related trend was well described by a minimum constant value until an energy limit E ₁, followed by a linear increase of W _D (E) up to a second energy limit E ₂ corresponding to 0.9*B [20]. Only then, a decrease in W _D with increasing energy is in order, until its cancellation at a few tens of MeV [21], along with the W _V continuum increase. Both are considered by the parameters of the OMP, especially the more inelastic channels which are opened by the incident energy increase. These channels are open due to α-particle interactions with the target nucleus on the surface region, firstly, and then within its whole volume. Thus, the data analysis of the α-induced reactions determines the increasing side of W _D while the elastic–scattering data constrain its decrease at higher energies.

Finally, the updated parameter set [4] for A≈45–209 took advantage of the new measured data, including their enlarged accuracy, as well as of an improvement for the well–deformed nuclei (152 < A < 190). This OMP has described successfully, without further parameter changes, most of the recently published (α,x) data [11, 13–17] and elastic–scattering angular distributions [22–24]. Moreover, a similar data account has finally been proven for α-emission from excited nuclei in nucleon-induced reactions on nuclei with A∼60 [16, 17] and A∼90 [11], provided that (i) consistent parameter sets are used, (ii) additional consideration is given to the (n,α) pickup direct reactions to the low–lying states of the residual nuclei, and (iii) all competitive reaction channels and eventually isotopic chains of neighboring elements have been considered.

The above review of the OMP [4] setting up is twofold. The energy limits E ₁ = 13.896 MeV and E ₂ = 16.863 MeV for ¹⁴⁴Sm are are within the incident–energy range ∼10–20 MeV of the recent measurements [1–3]. However, all energies of the (α,γ) data are just below E ₁, whereas the (α,n) data are mainly above it as well as centered on E ₂. Therefore, the former data set constrains the parameters of the OMP [4] triggered by the reaction–data analysis, and the latter is equally related to the setting of the OMP based on the two types of data analysis.

It is noteworthy that, at the time, the former ¹⁴⁴Sm(α,γ)¹⁴⁸Gd data set [6] was the only one available below E ₁ for target nuclei with A ≥130. Based on this analysis, the constant W _D of 4 MeV at energies ≤ E 1 , for A < 130 nuclei [20], was replaced with (2.5 ±1) MeV and a footnote states ” W _D = 3.5 for A<130 while W _D =1.5 fits better the data for A>130” [21]. However, new (α,γ) data on ¹²⁰Te and ¹⁴¹Pr, even fully below E ₁, were driven back to W _D =4 (+0.5/-2.5) MeV [4]. Therefore, an uncertainty band related only to W _D limits at lower energies has covered both (α,γ) data sets [2, 6]. Nevertheless, a parallel assay of the uncertainties of calculated cross sections for the reactions on ¹⁴⁴Sm below and near B, particularly related to the uncertainty of the primary data used to obtain the consistent parameters, remains of interest.

The SM analysis and the assessment of direct–interaction (DI) collective inelastic–scattering cross sections were recently carried out by using the same models and codes [25, 26] within a local approach [4, 27] and excitation energy grid with an equidistant binning ≥0.2 MeV. Therefore, only different or additional aspects are discussed hereafter.

Similar parameters of (i) back-shifted Fermi gas (BSFG) [28], (ii) OMP parameter sets [4, 29], and (iii) radiative strength functions (RSF) [30, 31] were used to account for the nuclear level density (NLD) and particle and γ-ray transmission coefficients, respectively. Particularly, the NLD and RSF consistent parameters were established or validated in advance with distinct data analysis.

The number and maximum excitation energy of the low-lying discrete levels [32] and level density parameters are given in Table 1 for the main SM reaction channels. They are followed by the data used for setting up the corresponding NLD parameters, such as the low-lying levels fitted along with, if known, the s-wave nucleon–resonance spacings D 0 exp [33] in ΔE energy range [34] above the separation energy S, for the ground state (g.s.) spin I ₀ of the target nucleus. The level-density parameter a and g.s. shift Δ thus obtained were then involved within SM calculations at excitation energies above the low-lying discrete levels.

The smooth-curve method has been used for nuclei without resonance data [35]. Thus, an average of the fitted a-values for nearby nuclei [4] was employed to obtain only the Δ value by fitting the low-lying levels. Table 1 also shows the cumulative uncertainty of the NLD parameters corresponding to the incertitude of the above–mentioned data involved within their setting up. Larger uncertainties of the averaged a-values are due to the spreading of the results of D 0 exp fit. The NLD effects on the calculated cross sections are assessed in Section 3 using these uncertainties.

The transmission coefficients were obtained using the global OMPs of either Koning and Delaroche [29], for nucleons, or [4] for α-particles. Moreover, the same parameters provided the DI collective inelastic–scattering cross sections through the distorted-wave Born approximation (DWBA) method and the DWUCK4 code [26]. A more detailed discussion follows in Section 2.3.

The γ -ray transmission coefficients were provided by the recently available widespread RSFs systematics [31, 36]. However, there is no related RSF analysis for the neutron–poor Gd isotopes similar to those in the rare-earth region for the ^{148,149,151,153}Sm isotopes [30, 37]. The comparative survey of the RSF data [36] of lighter Sm and Gd isotopes shown in Figure 1 suggested an apparent mass–and charge–dependency. It may support a prediction for the compound nucleus ¹⁴⁸Gd based on measured RSF data for ^148,149Sm nuclei. Therefore, we employed the giant dipole resonance (GDR) parameters of Kawano et al. for ¹⁵⁰Gd [31] within the former Lorentzian (SLO) [38], generalized Lorentzian (GLO) [39], and enhanced generalized Lorentzian (EGLO) [40] models for the electric-dipole RSF. Moreover, the SLO model and the additional M1 upbend parameters that were recently found to describe the RSF data for ^151,153Sm nuclei [37] were used. A nuclear–temperature EGLO parameter T _f =0.4 MeV, which is between the values found for Sm and Gd isotopes (Table I of Ref. [4]), was considered appropriate.

We obtained a suitable EGLO account of the RSF measured data of ^148,149Sm nuclei [30, 36] as well as the averaged s-wave radiation widths Γ _γ [33]. The dependency of the Γ _γ (S), for the even-even nuclei [33], also provided an estimation of this width and its uncertainty for ¹⁴⁸Gd nucleus as (88 ± 20) meV. Figure 1 shows an RSF uncertainty band corresponding to these Γ _γ estimated limits to match almost all data measured either above or below S. The effect of this uncertainty on the calculated reaction cross sections is also concerned hereinafter. Nevertheless, the SLO and GLO models resulted in different RSF energy dependencies below S, as well as larger Γ _γ –values well beyond the systematic uncertainty (Figure 1).

The α-particle OMP [4] was employed to get the DI collective inelastic scattering with the DWUCK4 code [26] using the corresponding deformation parameters of the first 2⁺ and 3⁻ collective states [41, 42]. The DI cross-section was then subtracted from the σ _R to obtain the CN cross-section for further SM calculations.

A note should be added concerning the CE discussion given in Sec. II of Refs. [4, 13, 15]. Because the inelastic–scattering cross section could be strongly affected by the Coulomb component of the interaction between the projectile and target nucleus, we used a value COUEX = 1.0 in the DWUCK4 calculations [26]. Actually, a previous DWBA analysis pointed out the issue of the collective form factors corresponding to (i) either CE or nuclear excitation (NE) alone, as well as their coherent interference (NE+CE) (e.g., [43]), and (ii) integration radii R _max of either 15 or 30 fm, which are typical to the short–range nuclear interactions [26] and the long-range Coulomb field, respectively [44]. The largest contributor to the CE cross sections were partial waves larger than those contributing to the optical model total–reaction cross section σ _R . Simultaneously, the assessment of α-particle DI inelastic scattering should include the effects of the CE+NE interference corresponding to an integration radius that is typical of the short–range nuclear interactions (∼15 fm).

We obtained α-particle DI inelastic–scattering cross sections up to ∼9% of σ _R for incident energies of ∼14 MeV. Above 20 MeV, these cross sections slightly decreased to 2% of σ _R . The corresponding decrease of σ _R was then considered within the SM analysis of various reaction channels.

Figure 2 provides an initial comparison of the measured cross sections of the reactions (α,γ) [2, 6] and (α,n) [3] on ¹⁴⁴Sm, and calculated results obtained with the α-particle OMP [4] and the above–mentioned model parameters and assumptions. It includes the calculated α-particle σ _R , to highlight the weight of each reaction channel at various energies.

The recent (α,γ) measured data [2], which had higher accuracy than previous ones [6], are described within so small error bars only at incident energies above 12 MeV. Otherwise, there is an overestimation which increases with the energy lowering, until a factor of ∼2 around 10.5 MeV. This trend, at incident energies where the (α,γ) cross section equals σ _R , can be explained neither by the RSF option (Figure 2A) nor the incertitude of the NDL parameters given in Table 1 for the compound nucleus ¹⁴⁸Gd (Figure 2C).

On the other hand, at energies where the α-capture becomes a minor reaction channel, the uncertainty band of the calculated cross section corresponding to the systematic Γ _γ [33] incertitude is similar to the error bar of the highest–energy measured data (Figure 2A). Moreover, a replacement of the EGLO strength function with the GLO and then SLO forms leads each time to additional overestimation of the same size.

A similar uncertainty band is related, at the same energies, to the incertitude of the NLD parameters for the compound nucleus ¹⁴⁸Gd (Figure 2C). Another uncertainty band corresponds to NLD of the residual nucleus ¹⁴⁷Gd at incident energies well above the (α, n) reaction threshold for the continuum population of this nucleus (Figure 2D). Altogether, below the incident energy of ∼12 MeV, no RSF and NLD effects may correspond to the increasing overestimation of the (α,γ) data.

The accurate most recently measured (α, n) cross sections [3] are well described within their small error bars (Figures 2B, D). Only the two lowest-energy points, at 13–14 MeV incident energies, are accounted for at the lowest limit of the experimental errors. However, an interplay of the (α,γ) and (α,n) cross sections for various RSF occurs solely at these energies. Thus, while the EGLO model provides lower cross sections for the former reaction, the reverse occurs with the usual SLO model. On the other hand, the uncertainty band of only ∼10% for (α, n) calculated cross sections at these energies, related to the systematic Γ _γ incertitude, is hardly visible in Figure 2B. Meanwhile, the results of the GLO and SLO models decrease by ∼17% and 36%, respectively.

This case provides a sound motivation for further study of the RSF modeling and competition between the capture and particle emission. At incident energies near 13 MeV, even lower RSF values would provide a better data account for both the (α,γ) and (α,n) calculated cross sections. However, a further decrease of the RSF for the compound nucleus ¹⁴⁸Gd is less probably on the basis of the RSF data available for neighboring nuclei (Figure 1).

The comparison of the σ _R values which were deduced through the elastic–scattering analysis at incident energies between 16 and 20 MeV [1] and the calculated results using the OMP [4] shows an agreement at the highest–energy point but a significant underestimation at the lower energies (Figure 2B). A previous note [1] on this measured and calculated σ _R difference concerned the fact that the setting up of the OMP [4] included the fit of a similar angular distribution measured at the same energy of 20 MeV. Nevertheless, the present suitable account of the more accurate (α, n) cross sections at these incident energies, which amount to 97%–90%σ _R , supports additionally the α-particle OMP [4]. Moreover, a suitable account of elastic–scattering angular distribution by the same OMP was proven for another semi–magic nucleus ¹⁴⁰Ce at a comparable lower energy of 15 MeV [Figure 8(d) of Ref. [4]].

The faster decrease of the (α,γ) cross sections below ∼12 MeV could be associated only with the above–mentioned weighting of the α-particle collective inelastic–scattering cross sections. Their maximum around 9% of σ _R at incident energies near 14 MeV, followed by a decrease to 2% above 20 MeV, is in line with the agreement of the measured and calculated (α,γ) and (α,n) cross sections at these energies. However, following the dependency of the DWBA outcome on the integration radius R _max, we found that the newly measured (α,γ) cross sections, at the incident energies decreasing from 12 to 10.5 MeV, correspond to DI inelastic–scattering cross sections increased by the use of R _max values rising from 15.5 to 17 fm.

Taking into account the agreement already provided by the use of the value R _max=15 fm at the energies ≥13 MeV, it results that a simple linear dependency of the form R _max(fm)=25.4–0.8E is leading to suitable collective DI inelastic–scattering and (α, γ) reaction cross sections also at the incident energies between 10.5 and 13 MeV (Figure 3). Thus, the excitation function of the (α, γ) reaction which formerly overlapped σ _R below ∼12 MeV (Figures 2A, C), has now a distinct trend. However, while R _max=17 fm has currently been used down from the energy of 10.5 MeV, its form below the energy range of the new (α, γ) reaction data [3] should be clarified by further analysis.

The increase of the integration radius beyond the typical 15 fm value to the short–range nuclear interactions, with the decrease of the α-particle energy, may correspond to nuclear effects decreasing [43]. Therefore, it could be a Coulomb excitation effect, as was assumed although in a different manner [7]. Nevertheless, the above R _max(E) dependency is a form used to describe only the recently measured (α,γ) data at α-particle energies above 10.5 MeV [2].