Role of consistent parameter sets in an assessment of the alpha-particle optical potential below the Coulomb barrier

TL;DR

This study confirms the reliability of a specific alpha-particle optical potential for predicting reaction cross sections below the Coulomb barrier, using high-precision experimental data and validated statistical-model parameters.

Contribution

The paper demonstrates that a consistent alpha-particle optical potential accurately describes reaction data below the Coulomb barrier without empirical adjustments, based on analysis of independent data.

Findings

Optical potential successfully describes recent reaction data

No empirical rescaling needed for model parameters

High-precision data allows discussion of model uncertainties

Abstract

Background: Further studies of high-precision measurements of alpha-induced reaction data below the Coulomb barrier have still raised questions about the alpha-particle optical model potential (OMP) within various mass ranges, i.e. for 64Zn, 108Cd, 113,115In, 121,123Sb, and 191,193Ir target nuclei. Purpose: The accuracy as well as eventual uncertainties and/or systematic errors of using a previous optical potential are much better considered by analysis of such accurate data. Method: Statistical model (SM) calculations of the (alpha,x) reaction cross sections have been carried out using model parameters which were previously obtained by analyzing independent data, particularly gamma-ray strength functions, and taking into account their uncertainties and questions of extrapolation for nuclei without similar data. Results: Consistent description of the recent alpha-induced reaction data is provided by the above-mentioned optical potential with no empirical rescaling factors of either its own parameters or the gamma and/or nucleon widths. Effects of still uncertain SM parameters on calculated alpha-induced reaction cross sections and conclusions on the alpha-OMP can be now discussed due to unprecedented precision of the new data. Conclusions: The alpha-particle optical potential has been confirmed at incident energies below the Coulomb barrier using statistical-model parameters validated through a former analysis of independent data.

Tables

Table 1. Table 1: Low-lying levels number N d subscript 𝑁 𝑑 N_{d} up to excitation energy E d ∗ subscript superscript 𝐸 𝑑 E^{*}_{d} ens used in cross-section SM calculations, the low-lying levels and s 𝑠 s -wave nucleon-resonance spacings D 0 𝑒𝑥𝑝 superscript subscript 𝐷 0 𝑒𝑥𝑝 D_{0}^{\it exp} (with uncertainties given between parentheses, in units of the last digit) in the energy range Δ Δ \Delta E 𝐸 E above the separation energy S 𝑆 S , for the target-nucleus g.s. spin I 0 subscript 𝐼 0 I_{0} , fitted to obtain the BSFG level-density parameter a 𝑎 a and g.s. shift Δ Δ \Delta , for the given ratio I / I r 𝐼 subscript 𝐼 𝑟 I/I_{r} for excitation energies between g.s. and S 𝑆 S , the average s 𝑠 s -wave radiation widths Γ γ subscript Γ 𝛾 \Gamma_{\gamma} , either measured Capote et al. ( 2009 ) or based on systematics (given between square brackets), and corresponding to the SLO, GLO, and EGLO models, with the parameter T f subscript 𝑇 𝑓 T_{f} of the EGLO model obtained by description of the RSF data OCL .

Nucleus	$N_d$	$E_d^{\ast }$	Fitted low-lying levels and nucleon-resonance data						$a$	$I/I_r$	$\Delta$	$T_f$	${\Gamma }_{\gamma }$
			$N_d$	$E_d^{\ast }$	$S+\frac{\Delta E}{2}$	$I_0$	$D_0^{\mathrm{𝑒𝑥𝑝}}$	${\Gamma }_{\gamma }$					SLO	GLO	EGLO
		(MeV)		(MeV)	(MeV)		(keV)	(meV)	(MeV-1)		(MeV)	(MeV)	(meV)	(meV)	(meV)
67Ga	28	2.282	28	2.282	8.420	0	2.5(2))a		8.20(8)	0.5-0.75	-0.55
67Ge	21	1.747	21	1.747					8.05(8)	0.5-0.75	-0.95(16)
68Ge	16	3.087	16	3.087	12.392	1/2		[550(200)]	8.3(3)	0.5-0.75	0.72	0.5	1890	1700	575
112Sn	23	2.986	23	2.986	10.786	7/2		[140(40)]	13.85(40)	0.5-0.75	1.34	0.46	311	215	106
117Sb	17	1.536	18	1.624	9.889	3		[140(50)]	14.1(4)	0.5-0.75	0.10	0.46	477	344	157
119Sb	23	1.676	23	1.676	9.549	1		[140(50)]	14.4(4)	0.5-0.75	0.08	0.46	433	365	140
124I	50	0.725	51	0.748					15.5(6)	0.5-0.75	-1.10
125I	31	1.392	31	1.392	9.543	2		[140(50)]	14.6(6)	0.5-0.75	-0.32	0.60(14)	419	318	180
126I	26	0.410	30	0.458					14.8(6)	0.5-0.75	-1.36
127I	33	1.480	33	1.480					14.0(6)	0.5-0.75	-0.35
195Au	36	1.443	36	1.443	8.426	1		[128(6)]	18.8(4)	1	-0.12	0.15	390	330	121
198Au	28	0.549	28	0.549	6.515	3/2	0.0155(8)b	128(6)	17.50(9)	1	-1.12	0.15	380	340	128

Full text

Purpose: The accuracy as well as eventual uncertainties and/or systematic errors of using a previous optical potential are much better considered by analysis of such accurate data.

Method: Statistical model (SM) calculations of the $(\alpha,x)$ reaction cross sections have been carried out using model parameters which were previously obtained by analyzing independent data, particularly $\gamma$-ray strength functions, and taking into account their uncertainties and questions of extrapolation for nuclei without similar data.

Results: Consistent description of the recent $\alpha$-induced reaction data is provided by the above-mentioned optical potential with no empirical rescaling factors of either its own parameters or the $\gamma$ and/or nucleon widths. Effects of still uncertain SM parameters on calculated $\alpha$-induced reaction cross sections and conclusions on the $\alpha$-OMP can be now discussed due to unprecedented precision of the new data.

Conclusions: The $\alpha$-particle optical potential has been confirmed at incident energies below the Coulomb barrier using statistical-model parameters validated through a former analysis of independent data.

I INTRODUCTION

More recent high-precision measurements of $\alpha$-particle scattering and induced reaction data below the Coulomb barrier $B$ Ornelas et al. (2016); Mohr et al. (2017); Scholz et al. (2016); Kiss et al. (2016, 2018); Korkulu et al. (2018); Szücs et al. (2018) provide further opportunities to check a previous optical-model potential (OMP) of $\alpha$-particles on nuclei within the mass number range 45\leq$$A$$\leq209 Avrigeanu et al. (2014). It may thus complement the discussion of data provided in the meantime Avrigeanu and Avrigeanu (2016).

Actually, a semi-microscopic double-folding model (DFM) real part and the dispersive contribution of a phenomenological energy-dependent imaginary-potential were firstly involved within an analysis of $\alpha$-particle elastic-scattering angular distributions above $B$ Avrigeanu et al. (2014, 2003, 2009). Subsequently, the phenomenological real potential Avrigeanu et al. (2014) was established using the same data basis. Then, the Hauser-Feshbach statistical model (SM) analysis of $\alpha$-induced reaction cross sections proved the particular energy dependence of the surface imaginary potential at incident energies below $B$ Avrigeanu et al. (2014, 2009); Avrigeanu and Avrigeanu (2009, 2010a, 2010b). The changes corresponding to the correction due to the dispersive relations with an integral over all incident energies, in the real part of the semi-microscopic potential, as well as the phenomenological OMP Avrigeanu et al. (2014) are within uncertainties of the parameter values in the rest of the energy range Avrigeanu et al. (2009). Particular comments have concerned the decreasing side of the volume integral per nucleon of the imaginary surface potential (Fig. 9 of Ref. Avrigeanu et al. (2014)) which is constrained by the elastic-scattering data while the increasing one could be determined by means of the $\alpha$-induced reaction data analysis.

A key point of these analyzes Avrigeanu et al. (2014, 2009); Avrigeanu and Avrigeanu (2009, 2010a, 2010b) has been the use of no empirical rescaling factors of the $\gamma$ and/or neutron widths but consistent parameter sets. The description of all available $\alpha$-induced reaction data, of equal interest for astrophysics and nuclear technology, have thus been obtained. On the other hand, a former OMP Avrigeanu et al. (1994) concerned only the $\alpha$-particle emission in neutron-induced reactions, with distinct predictions from potentials for incident $\alpha$ particles McFadden and Satchler (1966). Thus, this OMP Avrigeanu et al. (1994) was not an earlier version of the above-mentioned ones Avrigeanu et al. (2014, 2003, 2009), while the question on different OMPs for incident and emitted $\alpha$ particles Avrigeanu et al. (1994, 2006); Avrigeanu and Avrigeanu (2015, 2017) is still unanswered.

As the above-mentioned new studies have raised $\alpha$-OMP questions yet open within various mass ranges, similar analyzes of their data become mandatory for the assessment of potential Avrigeanu et al. (2014), before being taken into consideration for the tentative account of $\alpha$-emission as well. Thus, a reasonable description of additional $\alpha$-scattering and induced reaction cross sections on 64Zn at low energies Ornelas et al. (2016) was considered to be provided by several $\alpha$-particle OMPs including Avrigeanu et al. (2014). But then a further $\chi^{2}$-based analysis concluded that neither a better model for calculation of the 64Zn+$\alpha$ reaction cross sections nor better parametrizations of the SM ingredients are available Mohr et al. (2017). The need for further improvement of SM calculations and particularly the $\alpha$-nucleus potential was thus deemed necessary.

Moreover, precise cross sections of the $(\alpha,\gamma)$ and $(\alpha,n)$ reactions on 108Cd, measured for first time close to the astrophysically relevant energies, have been found to provide further support to investigations of the real part of $\alpha$-OMP in order to improve the understanding of reactions involving $\alpha$ particles Scholz et al. (2016). Simultaneous measurement of the $(\alpha,\gamma)$ and $(\alpha,n)$ reactions on 115In, in addition to high-precision elastic scattering Kiss et al. (2016) and together with a best-fit combination of all SM parameters, also concluded that further improvements of the $\alpha$-nucleus potential are still required for a global description of elastic scattering and $\alpha$-induced reactions in a wide range of masses and energies Kiss et al. (2018). It was particularly surprising to find a significant underestimation of isomeric $(\alpha,n)$ and $(\alpha,\gamma)$ cross sections by the OMP Avrigeanu et al. (2014) at once with an excellent description of the elastic scattering data, while the largest deviation from elastic scattering angular distributions is shown by a potential with the best description of the $(\alpha,x)$ data. It was thus concluded that further efforts are needed to establish an OMP which simultaneously describes $\alpha$-particle elastic scattering and reaction data Kiss et al. (2018).

On the other hand, a first measurement on 121Sb close to the astrophysically relevant energy range Korkulu et al. (2018) provided further support to the conclusion that experimental $(\alpha,\gamma)$ data, where they exist, are often strongly overestimated by SM calculations. A similar overestimation of $(\alpha,\gamma)$ measurements with a previously unprecedented sensitivity on 191,193Ir has also been obtained within a SM analysis which reproduced well the $(\alpha,n)$ data Szücs et al. (2018).

The aim of the present work is to analyze, in addition to Avrigeanu and Avrigeanu (2016), the SM results provided by the previous $\alpha$-particle optical potential Avrigeanu et al. (2014) in the case of the new data Ornelas et al. (2016); Scholz et al. (2016); Mohr et al. (2017); Szücs et al. (2018); Korkulu et al. (2018); Kiss et al. (2018), as well as their uncertainties and possible systematic errors. The use of consistent input parameters established or validated by analyzing various independent data (e.g., Ref. Arthur (1980)) constitutes the essential difference between our analysis and the above-mentioned data fit using a range of SM global input parameters.

While detailed presentation of model parameters was given in Refs. Avrigeanu et al. (2014); Avrigeanu and Avrigeanu (2015, 2016); Avrigeanu et al. (2012), latest particular parameter values are given in Sec. II of this work. The SM results obtained using the OMP of Ref. Avrigeanu et al. (2014) are then compared with the above-mentioned measured cross sections Ornelas et al. (2016); Scholz et al. (2016); Mohr et al. (2017); Szücs et al. (2018); Korkulu et al. (2018); Kiss et al. (2018) in Sec. III, followed by conclusions in Sec. IV. Preliminary results were described elsewhere Avrigeanu et al. (2019).

II Statistical model parameters

SM calculations discussed in the following section were carried out within a local approach using an updated version of the computer code STAPRE-H95 Avrigeanu and Avrigeanu (1995), with $\sim$0.1-0.3 MeV equidistant binning for the excitation energy grid. The direct-interaction (DI) distorted-wave Born approximation (DWBA) method and a local version of the code DWUCK4 Kunz (1984) were also used for calculation of the collective inelastic-scattering cross sections using the corresponding deformation parameters Raman et al. (2001); Kibédi and Spear (2002) of the first 2+ and 3- collective states. The collective form of the Coulomb excitation (CE) has been considered in the usual way Kunz (1984), while the comments on CE effects given in Sec. II.A of Refs. Avrigeanu et al. (2014); Avrigeanu and Avrigeanu (2016) apply here as well. The calculated DI cross sections are then involved for the subsequent decrease of the total-reaction cross sections $\sigma_{R}$ that enter SM calculations. Typical DI inelastic-scattering cross sections, e.g. for 64Zn target nucleus, grow up from $\sim$11% to $\sim$18% of $\sigma_{R}$ for $\alpha$-particle energies from 6.6 to 8 MeV, and then decrease to $\sim$3% at the energy of $\sim$16 MeV.

The consistent set of nucleon and $\gamma$-ray transmission coefficients, and back-shifted Fermi gas (BSFG) Vonach et al. (1988) nuclear level densities (NLDs) were established or validated using independently measured data as the neutron total cross sections and $(p,n)$ reaction cross sections exf , $\gamma$-ray strength functions Kopecky et al. (2017); OCL and $(p,\gamma)$ reaction cross sections exf , and low-lying levels ens and resonance data Vonach et al. (1988); Capote et al. (2009), respectively. The details in addition to the ones given formerly Avrigeanu et al. (2014); Avrigeanu and Avrigeanu (2015, 2016); Avrigeanu et al. (2012) as well as particular parameter values are mentioned below in order to provide the reader with all main details and assumptions of the present analysis.

The reaction cross sections calculated within this work are also compared with the content of the evaluated data library TENDL-2017 Koning and Rochman provided by using the code TALYS-1.9 Koning et al. (2008), for an overall excitation function survey.

II.1 Nuclear level densities

The BSFG parameters used to obtain the present SM results, which are either updated or not already provided in Refs. Avrigeanu et al. (2014); Avrigeanu and Avrigeanu (2015, 2016); Avrigeanu et al. (2012), are given in Table 1. They follow the low-lying level numbers and corresponding excitation energies ens used in the SM calculations (the 2nd and 3rd columns) as well as those fitted at once with the available nucleon-resonance data Vonach et al. (1988); Capote et al. (2009) to obtain these parameters. The level-density parameter $a$ and ground state (g.s.) shift $\Delta$ were generally obtained with a spin cutoff factor corresponding to a variable moment of inertia $I$, between half of the rigid-body value $I_{r}$ at g.s., 0.75$I_{r}$ at the separation energy $S$, and the full $I_{r}$ value at the excitation energy of 15 MeV, with a reduced radius $r_{0}$=1.25 fm Avrigeanu et al. (2002). The only different case is that of Au isotopes, for which there is a definite proof for a constant $I_{r}$ value Avrigeanu et al. (2012).

The fit of the error-bar limits of $D_{0}^{\it exp}$ data has also been used to provide limits of the fitted $a$-parameters. Moreover, these limits are used within SM calculations to illustrate the NLD effects on the calculated cross-section uncertainty bands (Sec. III).

On the other hand, the smooth-curve method Johnson (1977) was applied for nuclei without resonance data, using average $a$-values of the neighboring nuclei with resonance data, to obtain only the $\Delta$ values by fit of the low-lying discrete levels. The uncertainties of these averaged $a$-values, following the spread of the fitted $a$ parameters, are also given in Table 1. These uncertainties are obviously larger than those of the $a$-values obtained by fit of $D_{0}^{\it exp}$. Thus, use of their limits in SM calculations leads to increased NLD effects on calculated cross-section uncertainty bands; the same $\Delta$ values have been used within this uncertainty analysis, to take into account an usual uncertainty of 1–2 low-lying levels.

An additional question related to the $\Delta$-parameter value is taken into account in the particular case of 67Ge nucleus, for which only 4 excited levels at mid of the energy range $\sim$1.4–1.9 MeV are currently known. Thus, the NLD change due to this $\Delta$-uncertainty goes over the above-mentioned usual ambiguity of 1–2 low-lying levels and is also considered within the accuracy discussion (Sec. III.1).

II.2 $\gamma$-ray strength functions

The corresponding average $s$-wave radiation widths $\Gamma_{\gamma}$ Capote et al. (2009) including the extrapolated values based on systematics and the $\Gamma_{\gamma}$ distinct $S$-dependence for even-even and odd-$A$ nuclei (e.g., Ref. Toft et al. (2011)) are also provided in Table 1. They have been used together with earlier Nilson et al. (1982); Erlandsson et al. (1980a); Belgya et al. (2006); Erlandsson et al. (1979); Erlandsson et al. (1980b); Nilson et al. (1983); Spyrou et al. (2014); Kopecky et al. (2017) and more-recently OCL measured radiative strength functions (RSF) data for validation of the $\gamma$-ray transmission coefficients by using the former Lorentzian (SLO) Axel (1962), generalized Lorentzian (GLO) Kopecky and Uhl (1990), and enhanced generalized Lorentzian (EGLO) Kopecky et al. (1993) models for the electric-dipole $\gamma$-ray strength functions. The recently-compiled RIP giant dipole resonance (GDR) line-shape parameters were used here. The constant nuclear temperature $T_{f}$ of the final states Larsen and Goriely (2010), which is particularly assumed within the EGLO model, is also given in Table 1 as well as the calculated $\Gamma_{\gamma}$ values corresponding to the three electric-dipole RSF models.

Concerning the $M1$ radiation, the SLO model was mainly used alone, with the GDR parameters derived from photoabsorption data or the global parametrization Capote et al. (2009) for the GDR energy and width, i.e. $E_{0}$=41/$A^{1/3}$ MeV and $\Gamma_{0}$=4 MeV. However, in the particular case of 68Ge nucleus, we also considered the exponential increase of this RSF at decreasing energies approaching zero, predicted by shell-model calculations following the experimental observation of a dipolar RSF upbend (Goriely et al. (2018) and Refs. therein). Thus, the function $f_{up}(E_{\gamma})$=$C$exp$(-{\eta}E_{\gamma})$ has been added (e.g., Refs. Spyrou et al. (2014); Renstrøm et al. (2016)) to the SLO component of the $M$1 strength, with the average parameter values $C$=0.77$\times$10*-8* MeV*-3* and $\eta$=0.578 MeV*-1* found most recently for the $f_{5/2}pg_{9/2}$–shell nuclei Midtbø et al. (2018). The slope thus obtained is not as steep as for 74,76Ge isotopes Spyrou et al. (2014); Renstrøm et al. (2016), in agreement with the results of Midtbø et al. for $N$ mid-shell nuclei. Moreover, the trend of the total $E$1+$M$1 RSF in this case, corresponding to EGLO model and $M$1 upbend to zero energy (Fig. 1), is compatible with measured data of neighboring nuclei especially within the main related uncertainties, as follows.

First, while the GDR parameters of 70Ge RIP were used also for 68Ge, we considered a systematical uncertainty of the EGLO form given by the difference between the GDR peak cross sections $\sigma_{0}$ for 70,72Ge nuclei RIP . Thus, an electric-dipole strength uncertainty band that corresponds to $\sigma_{0}$=(88.4$\pm$16) mb is illustrated by the gray band in Fig. 1. Second, limits of the insight of $M$1-radiation upbend-function $f_{up}(E_{\gamma})$ have been additionally assumed. Therefore, we have considered an upper limit $C$=3$\times$0.77$\times$10*-8* MeV*-3*, using a multiplying factor previously used Goriely et al. (2018). A lower limit, given by the $M$1-upbend disregarding, corresponds to its yet general missing in $(\alpha,\gamma)$ cross-section calculations. The resulting total-uncertainty band shown in Fig. 1, in order to facilitate comparison with various SM calculations, may rather overestimate the uncertainty of adopted RSF than underestimates it.

The $s$-wave average radiation widths $\Gamma_{\gamma}$, either deduced from systematics Capote et al. (2009) or corresponding to $M$1 and each of above-mentioned $E$1 functions, are also given in Table 1 as well as in Figs. 1–6 to provide an immediate comparison of RSF effects on both $\Gamma_{\gamma}$ and $(\alpha,\gamma)$ cross-section calculations. A particular note concerns again 68Ge nucleus. Actually, the $\Gamma_{\gamma}$ error bar that may be estimated in this case by using the measured data for even-even nuclei Capote et al. (2009), versus $S$, is rather large especially due to the greater $S$ value of 68Ge. However, while the EGLO model leads to a calculated $\Gamma_{\gamma}$ close to this inference, the GLO and SLO predictions are higher by more than 5 times its uncertainty. At the same time, the above-assumed uncertainty of the GDR parameters within the EGLO model provides $\Gamma_{\gamma}$ changes of $\sim$17 % while the one including the $M$1 upbend is still only around 30 %. The propagation of these RSF uncertainties on the calculated $(\alpha,\gamma)$ reaction cross sections is discussed next.

III Results and Discussion

III.1 $(\alpha,x)$ reactions on 64Zn

The use of $\alpha$-particle potential Avrigeanu et al. (2014) provided already a suitable description Avrigeanu and Avrigeanu (2015) of the $(\alpha,\gamma)$, $(\alpha,n)$, and $(\alpha,p)$ reaction data provided by Gyürky et al. Gyürky et al. (2012) for 64Zn at energies below $\sim$1.2$B$, as well as of the more recent $(p,\alpha)$ reaction cross sections for the same target nucleus Gyürky et al. (2014). The new high-precision data of Ornelas et al. Ornelas et al. (2016) are particularly worthwhile for the present work as they enlarge the incident-energy range for the three above-mentioned reactions (Fig. 2). This energy extension is particularly useful for the related $\alpha$-capture due to significant spreading of the earlier data [Fig. 2(c)] and existence of the newer data of Gyürky et al. at only three energies. These authors have also performed a careful extrapolation to low energies of these reaction cross sections within their newest analysis of the three reactions Mohr et al. (2017).

The results of our previous calculations Avrigeanu and Avrigeanu (2015) at the higher as well as lower $\alpha$-particle energies, using the same SM parameters and obviously the unchanged $\alpha$-particle potential Avrigeanu et al. (2014), are compared with the measured data in Fig. 2. The nucleon-OMP and RSF dependences shown formerly in Fig. 5 of Ref. Avrigeanu and Avrigeanu (2015) are also included in this figure. The description of the new data Ornelas et al. (2016) is slightly different for the three reactions. First, one may note the good agreement for the major $(\alpha,p)$ reaction, with the changes due to different nucleon-OMP and RSF models within the limits of the experimental error bars [Fig. 2(a)].

Second, the measured $(\alpha,n)$ reaction cross section at the higher incident energy is described in the limit of 2$\sigma$ uncertainty [Fig. 2(b)] while the concurrence provided by the nucleon OMPs of Koning and Delaroche Koning and Delaroche (2003) at lower energies is not confirmed by the proton OMP analysis Avrigeanu and Avrigeanu (2015) as well as the Gyürky et al. data Gyürky et al. (2012). On the other hand, while use of the $a$-values for 67Ga corresponding to the fit of $D_{0}^{\it exp}$ error-bar limits (Table 1) leads to NLD effects within 3.5% of the calculated $(\alpha,p)$ cross sections, a similar statement may concern 67Ge $a$-value but not its $\Delta$ parameter. Thus, a large ambiguity concerning the number of 67Ge low-lying levels to be fitted, between 17 and 22 in the energy range 1.432–1.901 MeV, provides limits of the fitted $\Delta$-value (Table 1) leading to the uncertainty band shown in Fig. 2(b). This band has obviously risen only at the incident energies above the ones corresponding to population of the discrete levels. Nevertheless, it overestimates even more the newest data point at higher energy.

Third and most important, there is an entire agreement of the newly-measured $(\alpha,\gamma)$ cross section and calculated cross sections with the EGLO model for the electric-dipole RSF [Fig. 2(c)], in spite of the other data spreading and large variation of the results corresponding to the SLO and GLO models.

We should also emphasize the increase of the calculated $(\alpha,\gamma)$ cross sections due to inclusion of the $M$1 upbend RSF component, of no more than $\sim$12% around 8 MeV incident energy, and even $<$3% below $\sim$5.5 MeV as well as above 12.5 MeV. However, this change should be compared with the calculated cross-section uncertainties [gray band in Fig. 2(c)] following the above-mentioned ones of the adopted RSF (shown at their turn by light-gray band in Fig. 1), which is increasing from $\sim$2%, at incident energies around 5 MeV, to $<$30% at 8–9 MeV. On the other hand, the accuracy of these calculated cross sections really depends also on NLD parameters. In order to estimate their effects, we carried out SM calculations using the upper and lower limits of the level-density parameter $a$ of 68Ge, with the value (8.3$\pm$0.3) MeV*-1* obtained with the smooth-curve method Johnson (1977). The corresponding uncertainty band is not particularly shown in Fig. 2(c) because it overlaps with the one for the total RSF uncertainty. However, the uncertainty band corresponding to the sum of the above-mentioned RSF and NLD effects, with utmost change from 3%, around the incident energy of 5 MeV, to 48% at 16.5 MeV are displayed too. The calculated cross sections using the $E$1-radiation SLO and GLO models are only at incident energies $<$5 MeV inside this uncertainty band, while their increase over the EGLO results reaches then a factor of $\sim$3.

The lowest-energy region of Fig. 2(c) is expanded in Fig. 2(d), in order to compare the present calculations with the results of the $\chi^{2}$-based assessment of the $(\alpha,\gamma)$ reaction cross sections at two incident energies of particular astrophysical interest, i.e., 3.95 and 5.36 MeV Mohr et al. (2017). This assessment corresponds to the best fit of recently measured data for the above-mentioned reactions using all available options of TALYS-1.8 for the $\alpha$-particle and nucleons OMPs, $\gamma$-ray strength functions, and nuclear level density. While the best fit shows \chi^{2}/F$$\approx7.7 per data point, a reasonable small $\chi^{2}/F$ was considered to satisfy the criterion \chi^{2}/F$$<15 per data point [the error bars in Fig. 2(d)]. The potential Avrigeanu et al. (2014) led to an agreement close to the higher limit of this criterion.

On the other hand, one may note that the smallest \chi^{2}/F$$\approx7.7 per point was derived using the SLO model at obvious variance with the independent analysis of RSF data Avrigeanu and Avrigeanu (2015). At the same time it should be underlined that the lowest $\chi^{2}/F$ values for each reaction channel as well as for all of them correspond to different combinations of the above-mentioned four SM-parameter categories. The case of TENDL-2017 evaluation, which shows the best agreement with the extrapolation below $\sim$0.5$B$ [Fig. 2(d)] while underestimating by at least a factor of 5 [Fig. 2(c)] the $(\alpha,\gamma)$ reaction cross sections Ornelas et al. (2016); Gyürky et al. (2012) recently measured below $\sim$1.5$B$, is also open to discussion.

III.2 $(\alpha,x)$ reactions on 108Cd

The analysis of precise cross sections of $(\alpha,\gamma)$ and $(\alpha,n)$ reactions on 108Cd measured for first time close to astrophysically relevant energies Scholz et al. (2016) has completed a recent similar one for 106Cd including elastic-scattering angular distributions Ornelas et al. (2015). Those data were already discussed Avrigeanu and Avrigeanu (2016) and proved to be well described by the optical potential Avrigeanu et al. (2014) provided that suitable RSF are taken into account. However, the analysis of 108Cd data indicated that additional information about the RSF, for instance, are necessary to additionally test the $\alpha$-particle OMP Scholz et al. (2016).

The previous analysis for 106Cd target nucleus Avrigeanu and Avrigeanu (2016) has been resumed for 108Cd with only one change, due to fact that the excited nucleus 112Sn is closer to 112Cd and 116Sn, with recent RSF data Larsen et al. (2013); Toft et al. (2011) already reviewed in Fig. 3 of Avrigeanu and Avrigeanu (2016). Thus, the calculated $(\alpha,\gamma)$ and $(\alpha,n)$ cross sections shown in Fig. 3(a) and Fig. 3(b), respectively, are obtained using the EGLO parameters for 116Sn Toft et al. (2011) as well as the related SLO and GLO models for $E$1 radiations, along with the SLO one for $M$1 radiation. These results indicate that the $\alpha$-particle OMP Avrigeanu et al. (2014) and only the EGLO model provide a good agreement with the measured $(\alpha,\gamma)$ excitation function Scholz et al. (2016) and $\Gamma_{\gamma}$ value estimated on the basis of RIPL-3 Capote et al. (2009) (also in Table 1). Moreover, the related overpredictions by the GLO and especially SLO models go well beyond the uncertainty band corresponding to the estimated limits of the level-density parameter $a$ (Table 1) for 112Sn. Obviously, the RSF effects on the $(\alpha,n)$ cross sections shown in Fig. 3(b) are within the measured-data errors.

On the other hand, we found this case useful for checking the RSFs obtained for neighboring nuclei. Thus, we used also the EGLO parameters for 112Cd Larsen et al. (2013) that were provided for a couple of $T_{f}$-parameter values. The change shown in Fig. 3(c) is lower than even half of the data-error bars and within or close to the uncertainty band given by the limits of the estimated $\Gamma_{\gamma}$ value (Table 1). Therefore the use of the RSF of neighboring nuclei is supported in the case of no measured data for a given nucleus. At the same time, one may note the rather similar uncertainty bands related to NLD and RSF parameters in Figs. 3(a) and 3(c), respectively.

A particular remark concerns a notable involvement of the ratio between $(\alpha,\gamma)$ and $(\alpha,n)$ cross sections to remove the sensitivity of the adopted $\alpha$-particle OMP Scholz et al. (2016). However, while this ratio was much more sensitive to the adopted RSF model, the SLO model turned out to best fit the measured ratios. As it was argued Scholz et al. (2016), this result does not imply that, unlike the particular combination of SM parameters including it, this model is the best. Moreover, this outcome at variance with the results of RSF data analysis underlines the advantage of using a consistent parameter set established by means of various independent data analysis.

We also considered the $\alpha$-particle OMP of McFadden and Satchler McFadden and Satchler (1966) which was used in Ref. Scholz et al. (2016). First, a good agreement is provided by this OMP at the higher energies of the measured $(\alpha,\gamma)$ and $(\alpha,n)$ cross sections Scholz et al. (2016). Then, at lower energies there is a small overestimation of the $(\alpha,n)$ but a significant one of the $(\alpha,\gamma)$ data. Moreover, while this OMP and above-mentioned RSF effects on calculated $(\alpha,n)$ data are small and in reverse order [Fig. 3(b)], they are quite larger and both overestimating the $(\alpha,\gamma)$ cross sections [Fig. 3(a)]. Therefore, the deviation of an $(\alpha,\gamma)$ evaluation using these parameters is obvious.

The calculated $(\alpha,n)$ cross sections are naturally close by the TENDL-2017 evaluation [Fig. 3(d)] that used the $\alpha$-particle OMP Avrigeanu et al. (2014) as the default option in the TALYS-1.9 code. However, the large TENDL-2017 deviation of $(\alpha,\gamma)$ evaluation [Fig. 3(c)] denotes a quite different case entirely due to the RSF account.

III.3 $(\alpha,x)$ reactions on 113,115In

Particular attention should be paid to the conclusion that further efforts are needed to establish an OMP that simultaneously describes $\alpha$-particle elastic scattering Kiss et al. (2016) and $(\alpha,\gamma)$ and $(\alpha,n)$ reactions on 115In Kiss et al. (2018). Part of this conclusion was due to an excellent description Kiss et al. (2016) of the elastic scattering data with the OMP Avrigeanu et al. (2014), at the same time with a significant underestimation of $(\alpha,\gamma)$ and isomeric $(\alpha,n)$ cross sections at lower energies Kiss et al. (2018). However, a previous analysis of the $\alpha$-induced reaction data on 113,115In below and around $B$ (e.g., Yalç ın et al. (2009)) was carried out Avrigeanu and Avrigeanu (2010b) with a rather good agreement with all data available at that time. This is why we found of interest the inclusion of the newest $(\alpha,x)$ reaction data for 115In within a revision of the former analysis.

An additional aim of this work is to use the EGLO parameters of the RSF model, that were established more recently for the 117Sn excited nucleus Toft et al. (2011). However, in spite of the corresponding $\Gamma_{\gamma}$ values in rather good agreement [Table 1 and Figs. 4(a,c)] with systematics of the measured data Capote et al. (2009), the $(\alpha,\gamma)$ data for both 113,115In are underestimated just above the $(\alpha,n)$ threshold. This underestimation includes the uncertainties corresponding to the limits (Table 1) of both the CN level-density parameter $a$ and the $\Gamma_{\gamma}$ value based on data systematics Capote et al. (2009). The latter limits correspond to large uncertainties which were assumed due to the scarce $\Gamma_{\gamma}$ data Capote et al. (2009) available for odd-even excited nuclei. They were used for an additional RSF normalization that led to the calculated $(\alpha,\gamma)$ cross sections alongside the light-gray uncertainty bands in Fig. 4.

The GLO and SLO models provide a much better agreement only for reaction data while the related $\Gamma_{\gamma}$ values are larger than the systematical estimation by a factor $>$2. As the former analysis of the RSF data Toft et al. (2011) supports only the EGLO model, the questions on these $(\alpha,\gamma)$ excitation functions remain open. Maybe the actual knowledge of the neutron-deficient odd Sb (Z=51) isotopes, with one valence proton, needs further improvement in order to make possible a realistic account of their structure.

On the other hand, the g.s. as well as isomeric $(\alpha,n)$ cross sections are well described in the limit of either the error bars, for 113In [Fig. 4(b)] and half of the data for 115In [Fig. 4(d)], or 2$\sigma$ uncertainty for the rest of 115In data. As the sum of these cross sections is an order of magnitude larger than the $(\alpha,\gamma)$ cross sections shortly above the $(\alpha,n)$ reaction threshold (Fig. 4), the OMP Avrigeanu et al. (2014) is validated also for 113,115In target nuclei. However, it is noteworthy that even the $(\alpha,\gamma)$ reaction cross section for 113In at the lowest energy, just below the $(\alpha,n)$ threshold [Fig. 4(a)], validates the present analysis while the underestimation corresponds to energies where the neutron emission has a sharp increase.

Finally, we address the statement Kiss et al. (2018) regarding the $\alpha$-particle OMP of Ref. Avrigeanu et al. (1994) as an earlier version of the actual potential Avrigeanu et al. (2014), optimized mainly at higher energies. However, this analysis Avrigeanu et al. (1994) addressed the $\alpha$-particle emission in neutron-induced reactions up to E_{n}$$\sim10 MeV, i.e. within several MeV above the corresponding reaction thresholds. Therefore, the above-mentioned emission energies were not higher but below and around $B$.

As a matter of fact, the involvement at energies lower and around $B$ Avrigeanu et al. (2014, 2009); Avrigeanu and Avrigeanu (2010b) of an $\alpha$-particle OMP obtained by elastic-scattering analysis well above $B$ Avrigeanu et al. (2003) is indeed a problem. However, it has already been proved Avrigeanu and Avrigeanu (2009, 2010a, 2010b) that one should take into account the particular $\alpha$-particle surface absorption below $B$, the changes of the related $\sigma_{R}$ being shown in Figs. 1-2 of Refs. Avrigeanu and Avrigeanu (2009, 2010a) and Figs. 3-5 of Ref. Avrigeanu and Avrigeanu (2010b). The proper energy dependence of both the surface and volume components of the $\alpha$-particle imaginary potential has finally been considered Avrigeanu et al. (2014), leading to the suitable account of both the reaction data below $B$ (Fig. 4) and elastic scattering on 113,115In Kiss et al. (2016).

III.4 $(\alpha,x)$ reactions on 121,123Sb

The first measurement of the $(\alpha,\gamma)$ cross sections on 121Sb close to the astrophysically relevant energy range pointed out a strong overestimation by SM calculations Korkulu et al. (2018). Additionally, $(\alpha,n)$ cross sections were obtained for 121,123Sb at lower energies compared to the available data, and especially with much higher precision. Therefore, checking the agreement found earlier for these nuclei (Fig. 3 of Ref. Avrigeanu and Avrigeanu (2010b)) becomes a matter of great interest.

First, we paid closer attention to the RSF account. As recent RSF data of nearby nuclei exist only for 117Sn Toft et al. (2011) and 138,139La Kheswa et al. (2015, 2017), available data of 128I Kopecky et al. (2017) have also been compared with the calculated RSFs of 125I shown in Fig. 5(a). The EGLO parameters for 117Sn Toft et al. (2011) led to a suitable RSF average trend when the $T_{f}$ parameter was increased from 0.46 MeV to 0.6 MeV (Table 1). We considered the related $T_{f}$-difference of 0.14 MeV as an uncertainty estimation of this parameter and found that the corresponding RSF uncertainty band [Fig. 5(a)] covers well even the RSF low-energy upbend of 138La Kheswa et al. (2017). Moreover, while the related RSF change, close to zero energy, is up to 52%, that of the corresponding $\Gamma_{\gamma}$ value is below 27% and rather well within the limits of the data systematics.

The GLO and SLO models using the same GDR parameters led to larger RSF values for $\gamma$-ray energies below 5-6 MeV while $\Gamma_{\gamma}$ values increased by factors close or even above 2 [Table 1 and Figs. 5(a)]. Therefore, despite existing scarce RSF and $\Gamma_{\gamma}$ data, we may consider that a reasonable RSF estimation has finally been obtained particularly with reference to either SLO or GLO models.

Consequently, the use of the EGLO model has led to the agreement with the $(\alpha,\gamma)$ cross sections on 121Sb Korkulu et al. (2018) within the small error bars, except for the two data points at the lowest $\alpha$-particle energies [Fig. 5(b)]. These points are well described by the larger values obtained using the GLO and SLO, which however overestimate the rest of this excitation function. At the same time, the above-mentioned RSF uncertainty band led to an uncertainty band of the calculated $(\alpha,\gamma)$ cross section going from $\sim$ 40% at the lowest energy to less than 20% at the highest one [Fig. 5(b)]. However, the additional consideration of the uncertainty of the level-density parameter $a$ (Table 1) yields a total uncertainty band three to five times larger. This uncertainty estimation covers all measured data as well as the results of using the GLO model, while the SLO results are even larger.

On the other hand, the calculated cross sections using the SLO model of RSF and the $\alpha$-particle OMP of McFadden and Satchler McFadden and Satchler (1966), also shown in Fig. 5(b), seem to be rather close to those obtained by Korkulu et al. (Fig. 6 of Ref. Korkulu et al. (2018)). Moreover, these results indicate that the overestimation by a factor of 2-4 of the measured data has been entirely due to the $\alpha$-particle OMP McFadden and Satchler (1966) only at lowest energies. The disagreement at the higher energies has been caused by the use of the SLO model for the RSF account.

The comparison of the measured and calculated $(\alpha,n)$ cross sections for 121,123Sb [Figs. 5(c,d)] can take the advantage of the recent data unprecedented precision. While one could not differentiate between $\alpha$-particle OMPs Avrigeanu and Avrigeanu (2010b); McFadden and Satchler (1966) by means of the data measured even in the last decade or so, there is now a new case particularly for 123Sb [Fig. 5(c)]. A slight difference between the predictions of the two OMPs Avrigeanu et al. (2014); McFadden and Satchler (1966), much smaller than the error bars of previous measurements, supports now the potential Avrigeanu et al. (2014). On the other hand, the use of the same potential as the default option of TALYS-1.9 led to TENDL-2017 evaluated $(\alpha,n)$ data close to the present calculation at energies where the nuclear level densities and PE effects are not yet playing a significant role.

The calculated and measured data of the $(\alpha,n)$ reaction on 121Sb [Fig. 5(d)] are in a very similar situation to that of $(\alpha,\gamma)$ excitation function. Only the calculated values at the two lowest energies are underestimating the experimental ones Korkulu et al. (2018), yet below a 2$\sigma$ uncertainty. Nevertheless, the case of 121Sb target nucleus is one of the very few in which the analysis of both $(\alpha,n)$ and $(\alpha,\gamma)$ reactions is necessary to validate an $\alpha$-particle OMP, provided that the involved RSF has also been proved in advance.

III.5 $(\alpha,x)$ reactions on 191,193Ir

The similar measurement with a previously unprecedented sensitivity on 191,193Ir by Szücs et al. Szücs et al. (2018) enables an extension of the present work for heavy nuclei. Moreover, a recent measurement of RSF data for 198Au as well as its involvement within the well-known standard 197Au$(n,\gamma)^{198}$Au reaction analysis Giacoppo et al. (2015) provide better conditions for $(\alpha,\gamma)$ reaction suitable account.

Thus, we adopted an RSF energy dependence rather similar to Figs. 5-6 of Ref. Giacoppo et al. (2015) using the EGLO model and (i) the SLO parameters of both $E1$ and $M1$ radiations of Kopecky-Uhl Kopecky and Uhl (1990), (ii) the EGLO parameter $T_{f}$ of Giacoppo et al. Giacoppo et al. (2015) as well as (iii) their pigmy dipole resonance (PDR) parameters with the PDR cross section of 12.2 mb, and (iv) low-energy small resonance (SR) tail (model A in Table II of Ref. Giacoppo et al. (2015)). First, there are less significant SR effects on the suitable EGLO calculated values of RSF [Fig. 6(a)] and $\Gamma_{\gamma}$ (Table 1). Second, the GLO and SLO models led to either RSFs well beyond an uncertainty band corresponding to the average change of 30% (shown in Fig. 3(c) of Ref. Giacoppo et al. (2015) to follow the RSF normalization using various spin-distribution models), or $\Gamma_{\gamma}$ values increased by an average factor of 3. Actually, additional effects due to the minor error bars of $D_{0}$ and $\Gamma_{\gamma}$ experimental values Capote et al. (2009) (Table 1) were not considered anymore because the use of NLD parameters related to the limits of the fitted $D_{0}^{\it exp}$ provides changes of $\Gamma_{\gamma}$ well within its own error bar.

The EGLO model corresponds also to calculated 197Au$(n,\gamma)^{198}$Au reaction cross sections [Fig. 6(b)] in good agreement with the measured values exf that are nearly all within the uncertainty band corresponding to the above-mentioned one for the adopted RSF. The SR contribution is noticeably improving this agreement only at the lowest $\gamma$-ray energies, inside the same uncertainty band. The same excitation function is substantially overestimated by the GLO and SLO models using the same GDR parameters by factors of $\sim$2 and $\sim$3, respectively, even without the SR addition.

The calculated $(\alpha,\gamma)$ excitation function for 191Ir, using the EGLO model for the electric-dipole RSF, is in agreement with the measured data in the limit of 2$\sigma$ uncertainty [Fig. 6(c)]. Moreover, these data are within the uncertainty bands corresponding to the limits of the level-density parameter $a$ for the compound nucleus 195Au (Table 1) and, in addition, an RSF systematic uncertainty of 30% similar to that for 198Au Giacoppo et al. (2015). The GLO and SLO models are leading to larger $(\alpha,\gamma)$ cross sections and $\Gamma_{\gamma}$ values by factors of over 2 and around 3 (also in Table 1]), respectively. On the other hand, it seems that the experimental excitation functions has a faster slope than predicted by our calculations or Ref. Szücs et al. (2018). However, the check of the adopted RSF described above, including the suitable account of the 197Au$(n,\gamma)^{198}$Au cross sections, provide confidence in the $\gamma$-ray and neutron competition we assume.

The comparison between the measured $(\alpha,n)$ cross sections for 191,193Ir and calculated results obtained with the optical potential Avrigeanu et al. (2014) shows a good agreement except the data points measured at the lowest and highest incident energies [Figs. 6(c,d)]. Nevertheless, a continuous increase with energy is shown not only by TENDL-2017 evaluation, which was obtained with the same OMP Avrigeanu et al. (2014), but also by Szücs et al. Szücs et al. (2018) with a modified version of McFadden and Satchler potential McFadden and Satchler (1966). The change of the latter OMP consists in the replacement of the volume imaginary-potential constant depth $W$=25 MeV with a Fermi-type function at an energy 0.9$B$ Avrigeanu and Avrigeanu (2010b) and having a ’diffuseness’ $a_{E}$ used as a free parameter. Szücs et al. found the best description of their $(\alpha,n)$ experimental data Szücs et al. (2018) using a value $a_{E}$=(2$\pm$0.5) MeV corresponding to the limits of the data except that at the highest energy. Our overestimation is rather similar at the lowest energy to that of their best fit, but lower by a factor of 3–4 at the highest energy.

The overestimation of experimental $(\alpha,\gamma)$ cross sections for 191Ir even by the modified OMP corresponding to the lower parameter value $a_{E}$=1.5 MeV Szücs et al. (2018) is also notable. On the other hand, different values $a_{E}$=4–6 MeV were found earlier to provide an excellent reproduction of the experimental cross sections of 187Re$(\alpha,n)^{190}$Ir reaction Scholz et al. (2014). However, with no further change, the OMP Avrigeanu et al. (2014) provides a similar description of the data for 187Re Avrigeanu and Avrigeanu (2016) as well as an improved one for 191,193Ir.

Moreover, we obtained an even better description of the isomeric cross sections of the 193Ir$(\alpha,n)$ reaction in comparison with the related total cross sections [Fig. 6(d)]. Actually, the high-spin second isomeric state of 196Au is the 55th excited state of the residual nucleus at the top of the discrete levels taken into account in SM calculations (Table I of Ref. Avrigeanu et al. (2012)). Therefore its population follows the side feeding and continuum decay, being particularly determined by the $\alpha$-particle OMP, nuclear level density, and RSF. While proved the appropriate assumptions for the latter quantities Avrigeanu, M. et al. (2012), the effective nuclear moment of inertia $I$ which is most important for the isomeric cross section estimation may still be uncertain. Thus, although neutron-induced data analysis suggested a constant $I_{r}$ value for the effective $I$ of 198Au Avrigeanu et al. (2012), more recent RSF and $(n,\gamma)$ reaction data suggest that levels in the quasicontinuum are dominated by lower spins Giacoppo et al. (2015).

Consequently the isomeric cross sections of the 193Ir$(\alpha,n)$ reaction were calculated by using a constant value 0.5$I_{r}$. The corresponding results also shown in Figs. 6(d) prove, however, a lower sensitivity of the calculated $(\alpha,n)$ isomeric cross sections to this quantity, than the neutron activation data (e.g., Fig. 1 Avrigeanu et al. (2012)). Thus, using a 0.5$I_{r}$ value led to underestimated cross sections within the limit of 2$\sigma$ uncertainty. Nevertheless, the agreement of the measured and calculated isomeric data is noteworthy as long as the related TENDL-2017 evaluation, which reproduces the measured total $(\alpha,n)$ excitation function, shows more than one order of magnitude larger isomeric cross sections.

IV SUMMARY AND CONCLUSIONS

Our analysis of recent high-precision measurements of $\alpha$-induced reactions on 64Zn, 108Cd, 113,115In, 121,123Sb, and 191,193Ir, below the Coulomb barrier, points out eventual uncertainties and/or systematic errors of an $\alpha$-particle OMP Avrigeanu et al. (2014) assessment as follows. In any case, independent data were used in advance to establish or validate uncertain statistical-model parameters. Moreover, we took notice of the calculated cross-section uncertainties related to the error-bar limits of level-density parameters and $\gamma$-ray strength functions, that follow the above-mentioned independent data accuracy limits (Table 1). A distinct case is that of the zero-energy upbend of the $M1$-radiation RSF, which was taken into account only for 68Ge nucleus (Figs. 1 and 2), while elsewhere we adopted the usual SLO parameters used within former analyses Toft et al. (2011); Kopecky and Uhl (1990); Larsen et al. (2013); Kheswa et al. (2015, 2017); Giacoppo et al. (2015).

(i) Consistent SM parameters alongside the $\alpha$-particle potential Avrigeanu et al. (2014) can provide a reliable account of all available data Ornelas et al. (2016); Gyürky et al. (2012) and $\chi^{2}$-based predictions Mohr et al. (2017) of $\alpha$-induced reactions on 64Zn. The uncertainty bands corresponding to the adopted RSF and NDL parameter uncertainties cover the recently measured $(\alpha,\gamma)$ cross sections, while the results following the alternative use of SLO and GLO models of electric-dipole RSFs are well above them. The deviations within 2$\sigma$ uncertainty of the measured data at several incident energies, particularly for the $(\alpha,n)$ reaction, are supported by the former trial of independent data.

(ii) The precise cross sections of the $(\alpha,\gamma)$ and $(\alpha,n)$ reactions on 108Cd Scholz et al. (2016) prove not only the $\alpha$-potential Avrigeanu et al. (2014) but also the use of RSFs of neighboring nuclei for a given nucleus with no similar data. On the other hand, the notable involvement of the ratio between $(\alpha,\gamma)$ and $(\alpha,n)$ cross sections to remove the sensitivity of the adopted $\alpha$-particle OMP Scholz et al. (2016), may lead to results at variance with the primary RSF data analysis.

(iii) The total $(\alpha,n)$ as well as g.s. and isomeric $(\alpha,n)$ cross sections for 113,115In Kiss et al. (2018) have also been well described. However, despite of the rather good agreement of the EGLO values with the systematics of measured $\Gamma_{\gamma}$ Capote et al. (2009), the $(\alpha,\gamma)$ reaction cross sections of both 113,115In are underestimated above the $(\alpha,n)$ threshold. The GLO and SLO models provide a much better agreement for reaction data but large overestimate $\Gamma_{\gamma}$. Nevertheless, the $(\alpha,\gamma)$ reaction cross section for 113In at the lowest energy just below the $(\alpha,n)$ threshold validates, however, this potential while the underestimation corresponds to energies where the $\gamma$ channel weakens. Actually, the suitable account of the reaction data below $B$ Kiss et al. (2018) as well as elastic scattering on 113,115In Kiss et al. (2016) do support the proper energy dependence of both surface and volume components of the $\alpha$-particle imaginary potential Avrigeanu et al. (2014). Maybe the actual knowledge of the neutron-deficient odd Sb (Z=51) isotopes may need further improvement in order to enable a realistic account of their structure also involved in SM calculations.

(iv) The reasonable estimation of the RSF of 128I by the EGLO model, especially with reference to either SLO or GLO models, was accompanied by a general agreement with the $(\alpha,\gamma)$ cross sections on 121Sb Korkulu et al. (2018). Moreover, the use of the $\alpha$-particle OMP of McFadden and Satchler McFadden and Satchler (1966) and the SLO/GLO models for RSF has small effects on calculated $(\alpha,n)$ cross sections and even in reverse order, e.g. for 108Cd, but quite larger and both overestimating the $(\alpha,\gamma)$ data. Moreover, the overestimation by a factor of 2-4 of the measured data has been entirely due to the $\alpha$-particle OMP McFadden and Satchler (1966) only at lowest energies. The disagreement at the higher energies has been caused by the use of the SLO model for the RSF account.

(v) The EGLO model for the electric-dipole RSF, formerly proved including the suitable account of the 197Au$(n,\gamma)^{198}$Au cross sections, supports an increase among the measured data in the limit of 2$\sigma$ uncertainty of the calculated $(\alpha,\gamma)$ excitation function for 191Ir. The comparison between the measured $(\alpha,n)$ cross sections for 191,193Ir and calculations using the optical potential Avrigeanu et al. (2014) shows a good agreement except the data points measured at the lowest and highest incident energies. Moreover, we obtained a better account for the isomeric cross section of the 193Ir$(\alpha,n)$ reaction compared to the related total cross section.

A final remark concerns assumption Kiss et al. (2018) of the $\alpha$-particle OMP of Ref. Avrigeanu et al. (1994) as an earlier version of the actual potential Avrigeanu et al. (2014), optimized mainly at higher energies. However, this analysis Avrigeanu et al. (1994) addressed the $\alpha$-particle emission in neutron-induced reactions up to E_{n}$$\sim10 MeV, i.e. within several MeV above the corresponding reaction thresholds. Therefore, the above-mentioned emission energies were not higher but below and around $B$.

At the same time, the involvement at energies lower and around $B$ Avrigeanu et al. (2014, 2009); Avrigeanu and Avrigeanu (2010b) of an $\alpha$-particle OMP obtained by elastic-scattering analysis well above $B$ Avrigeanu et al. (2003) should take into account the particular $\alpha$-particle surface absorption below $B$. Thus, changes of the related $\sigma_{R}$ were shown in Figs. 1-2 of Refs. Avrigeanu and Avrigeanu (2009, 2010a) and Figs. 3-5 of Ref. Avrigeanu and Avrigeanu (2010b). That said, changes of both surface and volume imaginary potentials correspond, through the dispersive relations with an integral over all incident energies, to a change of, e.g., the semi-microscopic real potential. Former $\alpha$-particle elastic-scattering analyses Avrigeanu et al. (2003, 2009) have shown lower sensitivity to the addition of the dispersive correction to DFM real potential. A deeper insight may follow further precise measurements including cross sections of major reaction channels that may not be well described at the moment.

Finally, it seems that the alpha-nucleus OMP Avrigeanu et al. (2014), even though far from perfect, looks like a reasonable compromise when adopted in the extensive model calculations of present interest. Last but not least, the use of the same potential Avrigeanu et al. (2014) as the default option of TALYS-1.9 led to TENDL-2017 evaluated $(\alpha,n)$ data close to our calculation at energies where the nuclear level densities and PE effects are not yet playing a significant role. However, the large TENDL-2017 deviation for $(\alpha,\gamma)$ reactions and the isomeric cross sections of 193Ir$(\alpha,n)$ reaction highlights the importance of a suitable account of all reaction channels for $\alpha$-nucleus optical potential validation.

Acknowledgments

The authors acknowledge the useful discussions and correspondence with Jura Kopecky. This work has been partly supported by Autoritatea Nationala pentru Cercetare Stiintifica (Project PN-18090102), within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training program 2014-2018 and 2019-2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.