Optical potential for incident and emitted low-energy $\alpha$ particles. III. Non-statistical processes induced by neutrons on Zr, Nb, and Mo nuclei

The reliability of a previous $\alpha$-particle optical-model potential (OMP) on nuclei with mass number 45$\leq$$A$$\leq$209 was proved for emitted $\alpha$ particles as well, for proton--induced reactions on Zn isotopes [Phys. Rev. C {\bf 91}, 064611 (2015), Paper I]. However, the same was not the case of neutrons on Zr stable isotopes [Phys. Rev. C {\bf 96}, 044610 (2017), Paper II]. A recent assessment of this potential also for nucleon--induced $\alpha$-emission on $A$$\sim$60 nuclei, including pickup direct reaction and eventual Giant Quadrupole Resonance (GQR) $\alpha$-emission, is completed for neutrons incident on Zr, Nb, and Mo stable isotopes. Consistent sets of input parameters, determined through analysis of independent data, is involved while no further empirical rescaling factors of the $\gamma$ and nucleon widths have been involved. A suitable account of all competitive reaction channels is confirmed by careful uncertainty analysis, to avoid parameter ambiguities and/or error compensation. Additional validation of this potential is also supported by recently measured $(\alpha,\gamma)$ and $(\alpha,n)$ cross--sections of Zr and Mo nuclei. An increase of the $\alpha$-emission beyond the statistical predictions, through consideration of additional reaction channels of the pickup direct interaction and {\it like}--GQR decay, makes possible the description of both absorption and emission of $\alpha$ particles by the same optical potential.


I. INTRODUCTION
The reliability of a previous α-particle optical-model potential (OMP) for target nuclei with mass number 45≤A≤209 [1] was also proved for α-emission in protoninduced reactions on Zn isotopes [2] (Paper I). However, its use led to underestimated predictions of the statistical Hauser-Feshbach (HF) [3] and pre-equilibrium emission (PE) [4] models for neutrons incident on Zr stable isotopes [5] (Paper II). On the other hand, it has recently been shown that this potential can also describe α-emission from excited nuclei in nucleon-induced reactions within the A∼60 range [6,7]. This was possible through additional consideration of the pickup direct reaction (DR) and eventual Giant Quadrupole Resonance (GQR) α-emission. The so-called α-potential mystery [8] related to the account of both α-emission and absorption by an OMP, of equal interest for astrophysics and fusion technology, thus received an alternate solution. Nevertheless, its support by analysis of more data is imperative, while the case of neutrons incident on A∼90 nuclei [5] is a distinct requisite in this respect.
Thus, the same analysis for the stable isotopes of neighboring elements Zr, Nb, and Mo becomes challenging, with Zr nuclei within both incident and emergent reaction channels. The α-emission in neutron-induced reactions on stable Mo isotopes was targeted in an earlier systematic investigation up to 20 MeV [9]. However, an α-particle OMP [10] describing A∼60 compound-nuclei (CN) α-particle decay led to a significant underestimation at incident energies around ≤10 MeV for 92,98 Mo. This OMP makes predictions that differ significantly from potentials for incident α particles [11] including a double-folding (DFM) microscopic real potential [1, [12][13][14][15]. The discrepancy in results corresponding to OMPs [10,12] led even to the assumption of a nuclear-density distribution dependent on nuclear temperature [13].
The removal of compensation effects of less accurate model parameters is essential to establishing an appropriate α-particle OMP [1, [12][13][14][15] or to prove it [16,17]. This aim has been achieved using "consistent sets of input parameters, determined through analysis of independent data available in this mass region" [18]. Moreover, a proper account of all competitive reaction channels, beyond the α-emission of interest, should also be aimed at the validation of consistent parameter sets. So, there have been avoided empirical rescaling factors of the γ and/or nucleon widths which however are mandatory within large-scale nuclear-data evaluations.
This work is an extension of Refs. [5][6][7] so that only additional HF+PE model parameters for Nb and Mo isotopes are given in Sec. II along with items of the DR analysis by the distorted-wave Born approximation (DWBA) method and code FRESCO [19]. A comparison of HF+PE results and (i) recent (α, γ) and (α, n) crosssections of Zr and Mo nuclei, for α-particle OPM additional proof, (ii) available data of neutron-induced activation by nucleon-emission for 93 Nb and 92 Mo, as for Zr nuclei [5], and (iii) α-emission for all Zr, Nb, and Mo isotopes including eventual excited-nucleus like-GQR decay, are discussed in Sec. III. Conclusions are finally given in Sec. IV.  [20] used in HF calculations of reaction cross-sections, low-lying levels and s-wave nucleon-resonance spacings a D exp 0 (with uncertainties given in units of the last digit in parentheses) in the energy range ∆E above the separation energy S, for the target-nucleus ground state (g.s.) spin I0, fitted to obtain the BSFG level-density parameter a and g.s. shift ∆ (for a spin cutoff factor calculated with a variable moment of inertia [21] between half and 75% of the rigid-body value, from g.s. to S, and reduced radius r0=1. 25

II. MODELS AND PARAMETERS
A. Compound and pre-equilibrium emission The same models, codes [27][28][29], and local approaches [5,6] concern the HF+PE and collective inelasticscattering cross-section assessment. Thus, consistent (i) back-shifted Fermi gas (BSFG) [30] nuclear level density (NLD) parameters, (ii) nucleon and (iii) γ-ray transmission coefficients have also been involved in this work, with the related parameters being established or validated using distinct data. The same OMP and level density parameters have been used in the framework of the HF, PE, and DR models, too.
The excitation functions calculated in this work are also compared with the results of the worldwide used code TALYS-1.96 [27] and its default options which include the α-particle OMP [1]. Furthermore, a correlation with the TALYS-based evaluated-data library TENDL-2021 [31] shows the evaluation progress vs default options usage.
The low-lying levels and NLD parameters of the BSFG model are given in Table I for all nuclei within this work. The limits of the fitted a and ∆ parameters were obtained also by fit of the error-bars of s-wave nucleon-resonance spacings D exp 0 . For nuclei without resonance data, the smooth-curve method [34] was used, in which an average of fitted a-values for nearby nuclei is adopted while the ∆ value is obtained by fit of the low-lying discrete levels. These NLD parameter limits have then been used to illustrate the NLD effects on HF calculated cross-sections (Sec. III). The additional uncertainty of fitted N d led to enlarged NLD parameter uncertainties (second pair of brackets in Table I) despite better data becoming available in the meantime (e.g. for A=90, 98-100 nuclei [20]).
The neutron OMP of Koning and Delaroche [33] has additionally been reviewed at energies up to ∼30 MeV through the SPRT method [35], i.e. by analysis of the sand p-wave neutron strength functions S 0 and S 1 , respectively, the potential scattering radius R ′ [22] (Table II), and the energy dependence of the neutron total crosssection σ T (E) [32]. Thus, we found that the global parameter set [33] provides a better agreement between the measured and calculated σ T (E) of 93 Nb within this energy range, in comparison to the related local parameter set [ Fig. 1(a)].
The small adjustments shown in Table II for the real potential depth parameter v 1 [33] either global, for the odd 95,97 Mo [ Fig. 1(b)], or local for the even Mo isotopes (Fig. 2), have the same results. One may note the neutron energies for this comparison, which are quite different from the usual value of 10 keV. These energies are taken from the earlier RIPL-1 compilation [24] except the case of more recent measurements [26,[36][37][38]. Nevertheless, these changes have no importance for the σ T (E) values around 1 MeV, which are of obvious importance for the competition between the neutron and chargedparticle decay of excited CN, in heavier Mo nuclei. It is also within ∼4% for the near-spherical 92 Mo nucleus as well as for 94 Mo.
Because the improvement of the calculated σ T (E) at higher energies is yet within the measured error bars, the present analysis for Nb and Mo isotopes has also supported the previous use [5] of the OMP [33] for Zr isotopes.
The same OMP parameters have been involved within DWBA assessment of the collective inelastic scattering, using the deformation parameters [39][40][41] of the first 2 + and 3 − collective states. Typical direct inelasticscattering cross-sections increase to, e.g., ∼7% of σ R for neutrons on 93 Nb and 98 Mo at incident energies around 4 MeV and then decrease to slightly less than 5% at energies above 20 MeV. They were then involved in a subsequent decrease of σ R that has been taken into account within the PE+HF analysis.
The proton OMP of Koning and Delaroche [33] was also the first option for the calculation of the proton transmission coefficients for isotopes of Zr and Nb. Nevertheless, it has been checked by analysis of the only available (p, n) reaction data at incident energies of several MeV, as shown for Zr (Figs. 3) and Nb (Fig. 4). The proton OMP fully constrains the calculated (p, n) crosssections at energies higher than 3-4 MeV, where this reaction channel becomes dominant, with cross-sections close to optical-potential σ R .
Moreover, the same is true for the (p, γ) reaction below the (p, n) reaction effective threshold, where its crosssections are, in their turn, close to σ R values. Thus, the good agreement shown in Fig. 3(a) between the results within this work and recently measured 92 Zr(p, γ) 93 Nb reaction cross-sections at incident energies of 2-5 MeV [42] supports the global OMP parameters [33] even if the earlier (p, n) related data [32] are somehow under-    predicted between 3 and 4 MeV [ Fig. 3(b)].
Thus, presently we have adopted the giant dipole resonance (GDR) and additional M 1 upbend parameters found recently to describe the RSF data for 92,94 Mo nuclei [50]. The lower and upper resonance parameters given in Table II of Ref. [50]) provide an RSF uncertainty band for 94 Mo, while the related middle resonance parameters were used for the RSF account for 95,96 Mo nuclei. A comparison of the results obtained by using these parameters within the former Lorentzian (SLO) [52], generalized Lorentzian (GLO) [53], and enhanced generalized Lorentzian (EGLO) [54] models for the electricdipole RSF is shown in Figs. 5(a-c). The EGLO model has led to an enhanced description of both RSF [45,49] and average s-wave radiation widths Γ γ [22,25] measured data. The SLO and GLO models resulted in different RSF energy dependence below the neutron binding energy, as well as larger Γ γ -values.
Similar results have been obtained for 94 Nb nucleus by using the GDR parameters of Kopecky and Uhl [53] but the middle nuclear temperature and M 1-upbend param-  (Table II  [ eters of 92,94 Mo nuclei [50]. The same parameters have been used for 93 Nb, too, with the good agreement shown in Fig. 3(a) for the EGLO model, at once with (p, n) analysis related to the proton OMP setup.
Furthermore, a comparative analysis of 93 Nb(p, γ) 94 Mo and 90 Zr(α, γ) 94 Mo reactions modeling at low incident energies concerns also the recent data of Heim et al. [43] (Fig. 4) and Kelmar et al. [47] [ Fig. 5(d)], respectively. A better agreement with these most accurate measured cross-sections corresponds also to the EGLO model for the electric-dipole RSF, versus the related GLO and SLO models. There are, however, rather distinct details for the two reactions. Thus, the (p, γ) cross-sections are described by taking into account both limits of the resonance parameters of Ref. [50], while only the lower limit is yet close to the (α, γ) cross-sections. One may conclude that different RSFs could describe the two decay channels of the compound nucleus 94 Mo.
On the other hand, there are other distinct issues with these reactions. So, the angular-momentum ranges of the CN initial population are distinct due to the different target-nuclei g.s., i.e. (9/2 + ) of the odd 93 Nb and 0 + of the even-even 90 Zr. More comments that would be well deserved in this respect are not, however, the goal of this work. Therefore we should note only that the alternate SLO and GLO models, already proven in Fig. 5(a) to overestimate the RSF data and average s-wave radiation widths, are also leading to much larger cross-sections of both (p, γ) and (α, γ) cross-sections.
Moreover, the same is the case of the EGLO model for 95,96 Mo nuclei and (α, γ) reaction on 91,92 Zr, with increased RSFs due to account of the middle resonance parameters [50] and the more recent data of Krtička et al. [55] shown in Fig. 5(b) for 95 Mo. Their values, which are higher than the former ones [49], have been taken into consideration for both 95,96 Mo nuclei. Consequently, the corresponding (α, γ) calculated crosssections [Figs. 5(e,f)] are increased also for the odd residual nucleus 95 Mo.
The PE Geometry-Dependent Hybrid (GDH) model [56] was also used, which was generalized by including the angular-momentum and parity conservation [57] and knockout α-particle emission based on a pre-formation probability ϕ [4] . It also includes a revised version of the advanced particle-hole level densities (PLD) [58,59] with a Fermi-gas energy dependence of the single-particle level (s.p.l) density [60]. The α-particle s.p.l. density g α =(A/10.36) MeV −1 [61], on the other hand, has been replaced by the value related to the level-density parameter a via the usual equidistant spacing-model relation g=(6/π 2 )a. Moreover, the above-mentioned OMP parameters have also been involved in the local density approximation ( [56] and Refs. therein), as also the localdensity Fermi energies for various partial waves, corresponding to the central-well Fermi energy value F =40 MeV.

B. Direct reaction account
Similar to the previous work on A∼60 nuclei [6,7], the pickup contributions to (n, α) reactions have been determined within the DWBA method using the code FRESCO [19]. Thus, an one-step reaction has also been considered through the pickup of 3 He cluster while the "spectator model" [62,63] was involved. The two transferred protons in (n, α) reactions are assumed to be coupled to zero angular momentum, acting as spectators, while the transferred orbital (L) and total (J) angular momenta are given by the third unpaired neutron of the transferred cluster.
The prior distorted-wave transition amplitudes and the finite-range interaction were considered, with the n-3 He and p-t effective interactions in the α particle assumed to have a Gaussian shape [64,65] set by the fit of the binding energies of 3 He and t, respectively. Moreover, the bound states of the three-nucleon transferred cluster were generated in a Woods-Saxon real potential [63][64][65] with the depth adjusted to fit the separation energies in the target nuclei. The harmonic-oscillator energy conservation rule [62,63] and the n and l single-particle shell-model state quantum numbers were used to determine the number of N nodes in the radial three-nucleon cluster wave function.
Once more, the lack of measured α-particle angular distribution, for the (n, α) reactions within this work, made possible only DWBA calculations of related pickup cross-sections using (i) the spectroscopic factors (SFs) of Glendenning (Table II of Ref. [66]) for the spectator proton pair [63][64][65], in addition to (ii) SFs for the picked neutron that becomes thus responsible for the angular-momentum transfer. The latter have been obtained through analysis of α-particle angular distributions of one-nucleon pickup reactions ( 3 He, α) or (t, α) toward the residual nucleus of interest, as shown in the following. 88 Sr( 3 He, α) 87 Sr α-particle angular-distribution [67] analysis, at 36 MeV incident energy, provided the SFs of neutrons picked from the 2p, 2d, 1f , 1g shells. The comparison of measured data and the DWBA calculated angular distribution is shown in Fig. 6. A number of 43 levels up to ∼6 MeV excitation energy of 87 Sr residual nucleus has been considered in this respect, as well as for calculation of 90 Zr(n, α) 87 Sr pickup excitation function. The spectator proton pair picked from 2p 1 2 subshell [64,65] has been considered. 91 Zr(t, α) 90 Y pickup-reaction analysis and the assumption of a similarity between the picked-proton SFs and picked-neutron SFs [71] have been involved for calculation of 93 Nb(n, α) 90 Y reaction cross-sections. Thus, 14 levels up to ∼ 2.5 MeV excitation energy have been considered within the analysis of the measured α-particle angular distributions [68] shown in Fig. 7. The same excited levels have been concerned with the calculation of the above-mentioned (n, α) pickup excitation function in the following.     89 Zr reaction angular-distribution [69] analysis, at 39 MeV incident energy, provided the SFs of the picked neutron from the 2p, 2d, 1f , 1g shells, too. The spectator proton pair picked from 2p 1 2 subshell has also been considered. 20 levels of 89 Zr residual nucleus up to 3.572 excitation energy were considered for this reaction (Fig. 8) as well as for assessment of 92 Mo(n, α) 89 Zr pickup excitation-function calculation of interest for this work. 96 Zr( 3 He, α) 95 Zr α-particle angular-distribution analysis, of the measured data also at 39 MeV incident energy [70], provided the picked-neutron SFs to be used together with Glendenning's SF of the transferred spectator proton pair from 2p 1 2 subshell. A number of 23 levels of 95 Zr up to the excitation energy of 4.58 MeV, involving 2p, 2d, 1f and 1g shells have been considered in this respect (Fig. 9) as well as, finally, for 98 Mo(n, α) 95 Zr pickup excitation-function calculation.

III. RESULTS AND DISCUSSION
A. Recent α-induced reaction data analysis The recently measured cross-sections of (α, γ) [47,48] and (α, n) [72][73][74][75][76] reactions, the former being already taken into account for RSF endorsement, are of particular interest for a customary further validation of the α-particle OMP [1], too. The 90 Zr(α, γ) 94 Mo cross-sections [47] extended the data energy range below that of the previous experiment [77] taken into account within our former analysis [5]. This extension has touched even the energies around the (α, n) reaction threshold, where (α, γ) cross-section is near the α-particle total reaction cross-section. As a result, the former becomes a strong constraint on the αparticle OMP, whereas the RSF has no effect on these calculated reaction cross-section. Therefore, the good agreement of calculated and measured data at these energies in Fig. 5(d) supports the α-particle OMP [1] as well.
A similar extension to lower incident energies was performed for α particles incident on 92 Zr [48], the results shown in Fig. 5(f) providing additional support for the α-particle OMP [1]. Furthermore, the agreement in Fig. 5(e) of presently calculated and increased new measured data for the target nucleus 91 Zr [48] has confirmed this α-particle OMP, too. The first cross-section measurement of 96 Zr(α, n) 99 Mo reaction at incident energies ranging from 6.5 to 13 MeV [72], more than six orders of magnitude, was also a stringent test of the α-particle OMP [1]. Another recent measurement of the same reaction [73] supports this issue, with some discrepancies remaining at the lowest incident energy around 8 MeV. The use of the present consistent input parameters led to a suitable agreement, i.e. within measurement error bars, with these new data except for an energy range of ∼1 MeV around the 8.5 MeV incident energy, as shown in Fig. 10(a). Calculated cross-sections of the eventually competitive reaction channels are also shown in the same figure, with none of them being able to motivate the lower calculated (α, n) cross-sections at these energies. Furthermore, the (α, n) reaction crosssections account for ∼99% of σ R below the (α, 2n) reaction threshold, implying that this problem seems to be related entirely to the α-particle OMP.
As a result, the high accuracy of the new data [72,73] at these energies is critical for an additional assessment of the surface imaginary-potential depth W D , which increases between the energies E 1 and E 2 (Table II and Fig. 1 96 Zr, the lower calculated (α, n) crosssections near E 1 have an immediate significance. Thus, even at lower incident energies, the appropriate data account at even lower incident energies has revealed that the related constant minimum W D has a proper value but the sudden change at E 1 is less physical (as expected for a model). The agreement of the measured and calculated cross-section at energies slightly above this limit supports the W D energy dependence within its most important range [14,15]. An eventual smoothing of the W D abrupt change at E 1 may significantly improve the data underestimation around this energy. A recent measurement of 100 Mo(α, n) 103 Ru reaction cross-sections [74] at low energies based on the difference in activation thick target yields at two neighboring energies is also available. Because no agreement was found with previous measurements within their reported uncertainties [ Fig. 10(b)], the same target nucleus was chosen as a proof-of-principle measurement in the inverse kinematics of (α, xn) inclusive cross-sections between 8.9-13.2 MeV in the center of mass [75]. Moreover, while new (α, n) cross-sections have been reported in the en- ergy range 11-32 MeV [76], too, only the lowest energy point has been considered hereafter to avoid additional PE effects within this discussion. Except for the highest energy, the (α, xn) crosssections calculated in this work using the α-particle OMP [1] agree with the most recent measured data [75,76] within their error bars [ Fig. 10(b)]. Except for, once again, the highest energy of this measurement following a sharp change in these data slope, the (α, n) activation cross-sections of Ref. [74] are significantly underes-timated. The disagreement's energy range includes the two above-mentioned energy limits of our OMP, which in this case are E 1 =9.08 MeV and E 2 =12.61 MeV [1]. However, there is now a constant increase in the calculated (α, n) cross-sections over E 1 as well as a suitable agreement between experimental and calculated results around E 2 . One may also note the close similarity between the two neutron-rich nuclei 96 Zr and 100 Mo, as heaviest natural isotopes of their elements and similar nuclear asymmetry (N − Z)/A. Thus, the distinct effects on calculated cross-sections at energies below B, where W D increases with α-particle energy, should be investigated further.
Nonetheless, the newer data are already well described by the α-particle OMP [1] around the energy limit E 2 , i.e. around the surface imaginary-potential depth maximum. The relevance of this energy below which the potential parameters have to be strongly modified was also considered by Sauerwein et al. [81], based on Ref.
[15], as well as more recently [82,83]. Their correction by a further Fermi-type function, on the other hand, did not concern the surface but volume imaginary-potential depth W V of the indeed much simpler OMP of McFadden and Satchler [11]. This may account for the quite different values obtained for the 'diffuseness' of this Fermi-type function.

B. Nucleon-emission induced by neutrons on 93 Nb
There is a large amount of experimental data for neutron interaction with 93 Nb nucleus due to interest in it for structural materials of nuclear reactors, activation monitor in reactor dosimetry, 14 MeV neutron flux determination, and also as an element of superconductor alloys in fusion reactors. It triggered consideration of this interaction even as a 'sample problem' [89] in nuclear model calculations (e.g., [4,39,[90][91][92]). However, the scattered data for the (n, α) reaction on this single Nb natural isotope suggest that more precise measurements are needed to settle its evaluation (e.g., Refs. [93,94]). Therefore, a prerequisite for a consistent discussion of the α-particle emission is a suitable account, by using the actual parameter set, of all competing nucleon-emission data for neutrons incident on 93 Nb. The same is the case for 92 Mo, in completion of the earlier study of neutroninduced reactions on Mo stable isotopes [9]. Upon due consideration of these two nuclei, α-emission analysis will be feasible for Zr, Nb, and Mo nuclei altogether.
The (n, γ) cross-section analysis should be considered first because of the significant isomeric-states activation by neutrons on 93 Nb. Thus, a former validation of the γray transmission coefficients will constrain the isomeric cross-sections to the adopted NLD spin cutoff factors. The particular agreement of the more recent experimental data and the calculated results corresponding to the EGLO model for the electric-dipole RSF is similar to that of the related average s-wave radiation widths likewise shown in Fig. 11(a)  SLO models provide calculated cross-sections as well as Γ γ values that are much larger.
The (n, 2n) and (n, xp) reactions analysis must contend a lack of total cross-section measurements within the latest 40 years. Fortunately, there are recent measurements for the isomeric cross-sections corresponding to the 2 + state of 92 Nb nucleus at 136 keV, as shown in Fig. 11(b). Of particular interest is the agreement of the calculated and recently measured cross-section at the incident energy of ∼14 MeV, i.e., on the flat maximum of this excitation function. In contrast to TALYS-1.96 as well as TENDL-2021 results, this concurrently appropriate account of this isomeric state and total (n, 2n) excitation functions supports both neutron OMP and NLD spin dependence.
The proton emission induced by neutrons on 93 Nb was the subject of several angle-integrated energy distribution studies around the incident energy of 14 MeV, but no measurement of its excitation function has been made. Nevertheless, the total proton-emission crosssections corresponding to these data could be considered at once, with the good agreement shown in Fig. 11(c) for the calculated results of this work. The overall account of the measured energy spectra, in the limit of the error bars [ Fig. 11(d)], may support an appropriate description of nucleon emission in neutron-induced reactions on 93 Nb.

C. Nucleon-emission induced by neutrons on 92 Mo
The earlier systematic investigation of neutroninduced reactions on Mo stable isotopes up to 20 MeV [9] included a local approach and consistent parameter set requirements quite similar to the present analysis. As a result, while different parameters were formerly involved, an analysis of other independent data concerned also their setting up. A definitive account of all available data for competitive reaction channels was obtained, too.
On the other hand, even the actual α-emission data correspond merely to four 92,95,98,100 Mo of the seven stable isotopes of molybdenum. Furthermore, due to the larger amount of available measured data, a reanalysis of the nucleon-emission has now only concerned the lighter and even-even semi-magic nucleus 92 Mo. It matters also the higher charged-particle emission cross-sections owing to the isotopic effect triggered by reaction Q-values, i.e., the CN cross-section decreases with the isotope mass increase [95].
The (n, p) reaction analysis for the isomeric-state 92 Nb m population has been improved as a result of additional data published in the interim, as shown in Fig. 12(a). The better agreement with these data at lower incident energies has thus validated the neutron as well as proton OMPs adopted in the present work in comparison to the local parameter sets of Ref. [33].
The (n, 2n) reaction cross-sections have no additional measurements concerning both ground and metastable states as well as their population sum [ Fig. 12(b-d)]. An effective point is the location of the presently calculated results in between those of TALYS-1.96 default predictions and TENDL-2021 evaluated values. The change from the default results to the final evaluation is obviously approaching the experimental data. However, our calculated cross-sections are, e.g. at higher incident en-ergies, near either the default results for the isomeric cross-sections or the evaluated ones for g.s. population. Nevertheless, in both cases, they are closer to the more recently measured data.

D.
α-emission spectra and excitation functions

Angle-integrated energy spectra analysis
First, the α-particle angle-integrated energy distributions induced by neutrons on 90 Zr [96], 93 Nb [84,88,97], and 92 Mo [96] have been examined to validate the PE component of the energy-spectra above the CN one. Thus, the overall account of the measured spectra, in the limit of the error bars for the more accurate data shown in Fig. 13, may support an appropriate description of α-particle PE emission corresponding to (i) the above-mentioned α-particle s.p.l. density g α related to the level-density parameter a, and (ii) α-particle preformation probability ϕ values of 0.1, 0.14. and 0.11 for Zr, Nb, and Mo isotopes, respectively. Furthermore, the analysis of these spectra as well as the (n, α) excitation functions discussed below, suggests a corresponding ∆ϕ∼0.02 uncertainty.
Second, the high-energy limit of these spectra makes possible the check of the DR pickup cross-sections obtained by using the SF for the picked nucleon from the analysis of α-particle angular distributions corresponding to one-nucleon pickup reactions ( 3 He, α) or (t, α), and the spectator proton-pair SF [66] (Sec. II B).
The same residual-nucleus levels have been considered within the analysis of both the α-particle angular distributions of the one-nucleon pickup reactions ( 3 He, α) or (t, α), and the present (n, α) pickup excitation functions. As a result, the most consistent DR component corresponds to the 43 levels of 87 Sr residual nucleus up to ∼6 MeV excitation energy, which is important for calculating the 90 Zr(n, α) 87 Sr spectrum in Fig. 13(a). Within the highest 2 MeV, it makes a significant contribution to the agreement of calculated and measured spectra. However, there is just a minor increase at lower spectrum energies, i.e., higher excitations of the residual nucleus.
On the other hand, only 14 levels up to ∼ 2.5 MeV excitation energy have been considered within analysis of the α-particle angular distributions from 91 Zr(t, xα) 89,90 Y, as shown in Fig. 7. As a result, a significant DR component within only ∼2 MeV at the end of the spectrum of 93 Nb(n, α) 90 Y reaction is shown in Fig. 13(b).

Zr isotopes
90 Zr(n, α) 87 Sr m excitation function [ Fig. 14(a)] highlights the importance of the pickup DR component, which is higher than that of CN+PE up to the incident energy of ∼12 MeV. Then it decreases to an order of magnitude below the latter around 20 MeV. Their sum is, however, in good agreement with the experimental data along the whole incident-energy range, while several issues should be underlined.
First, there is a relevant data account at lower incident energies, where neither NLD nor PE effects exist on the calculated HF cross-sections. This is shown in Fig. 14(a) by the uncertainty bands corresponding to either the error bars of N d and LD parameter a of the residual nucleus 87 Sr (Table I), or the above-mentioned ∆ϕ=0.02 incertitude of the main PE parameter. Both of them are minimal at these energies and to ∼20 MeV due to the low spin of 87 Sr m isomer, leading to a reduced side-and cascade-feeding. Nonetheless, the absence of other HF+PE uncertainty factors on calculated crosssections at low energies indicates that the appropriate data account supports the α-particle OMP [1].
The agreement of the measured and calculated crosssections at higher energies, however, does not follow the already lower DR contribution, but rather an increased PE component in comparison to our previous analysis [5]. It follows the above-mentioned α-particle s.p.l. density g α value, proved in the present work by extended energyspectra analysis and 93 Nb target-nucleus case. Nonetheless, the appropriate account of the excitation-function upper side supports both this g α value and the parameter ϕ-value firstly suggested by 14-MeV spectra analysis. 91 Zr(n, nα) 87 Sr m excitation function [ Fig. 14(b)] stands as a fine case of the same nucleus in a different reaction channel and α-particle energy range. The activation of 87 Sr in neutron-induced reactions on 91 Zr nucleus, through (n,αn) reaction is more than 10 times larger in comparison with that by (n, n ′ α) reaction, at incident energies of 15-21 MeV. Thus, the suitable agreement of measured [98] and calculated cross-sections of this reaction, in the absence of DR effects, has supported the α-particle OMP [1] once more. Actually this conclusion has already been reachable within the previous analysis for Zr isotopes [5], without DR consideration, which provided a good account only for this reaction. 92 Zr(n, α) 89 Sr excitation function analysis has the drawback of no SF available for the residual nucleus 89 Sr and also no measured data at energies below 14 MeV [ Fig. 14(c)]. Nevertheless, the PE+CN calculated results are just below the more recently measured data, with at least the PE uncertainty band matching their error bars. However, to overcome the former shortcoming, we may assume a DR contribution similar to that of 92 Mo(n, α) 89 Zr reaction, mentioned in Sec. II B and in the following. Its addition to the CN+PE component provides agreement with both the recently measured cross-sections, in the limit of their error bars, and the excitation-function trend. The somehow lower calculated cross-sections are related to the fact that this is merely an attempt to obtain a realistic outline of this (n, α) reaction on 92 Zr. 94 Zr(n, α) 91 Sr excitation function was measured within several experiments, as shown in Fig. 14(d), but proper SFs for the residual nucleus 91 Sr are also lacking. At the same time, the comparison of the CN+PE component and available data shows a quite different look below and above incident energy of ∼14 MeV. Thus, quite recent and accurate data between 14-21 MeV are well described, with error bars just within the calculated NDL and PE uncertainty bands, though below 10 MeV there is an underestimation of several times. This was one of the major flows in our previous analysis of neutron-induced α-emission on Zr isotopes [5] for the α-particle OMP [1]. However, the assumption of a DR contribution. e.g., for a nearby target nucleus, may explain these features.
Thus, a similar DR contribution closer to this case could be that related to the residual nucleus 90 Y in the (n, α) reaction on 93 Nb (Sec. II B and just below). Because its maximum value of around 12 MeV is more than an order of magnitude lower than the CN+PE sum, the upper side of 94 Zr(n, α) 91 Sr calculated excitation function remains unchanged. On the other hand, these DR cross-sections are, as for the target nucleus 90 Zr, higher than the own CN component at the incident energies below 9 MeV. Their sum is quite close to the measured data as well as the results obtained previously [5] by using the α-emission OMP [10]. 96 Zr(n, α) 93 Sr excitation function [ Fig. 14(e)] has the same attributes as those for the target nucleus 92 Zr. The calculated CN+PE component and its PE uncertainty band describe rather well the trend of all measured data as well as cross-section values around 14 MeV except for one disparate data set. The difference from previous analysis [5] is related again to the above-mentioned PE contribution change due to the use of the α-particle s.p.l. density g α value. No further effect of an eventual DR contribution may be considered, the expected outcome being the same as for 94 Zr.

93 Nb target nucleus
The DR component, which is shown at the high-energy end of the α-particle energy spectrum in Fig. 13(b), may explain the minor pickup DR contribution to 93 Nb(n, α) 90 Y reaction total cross-sections [ Fig. 15(a)].
The same is true for the 7 + isomer of the residual nucleus 90 Y which was also recently measured as shown in Fig. 15(b) for energies from the effective threshold to above 20 MeV. Thus, the agreement of measured and calculated cross-sections stands for CN+PE results, with the main PE uncertainty band just across the error bars of the recent data.
The latest comment, concerning the 7+ isomeric state 90 Y m activation, is fully appropriate to the activation of the 9/2 + isomer via 93 Nb(n, n ′ α) 89 Y m reaction. The PE uncertainty band corresponding to this reaction has pointed out no PE effects, too. Therefore the good account of all experimental data does validate entirely the CN+PE parameters that matter, namely the α-particle OMP [1].

Mo isotopes
92 Mo(n, α) 89 Zr excitation function analysis reveals a notable balancing of the DR and PE+CN mechanisms for neutron-induced α-emission on light isotopes of elements. Thus, the former brings about ∼1 mb around 14 MeV, for SFs discussed in Sec. II B, while the measured cross-sections amount to several tens of mb [95]. Hence, the DR addition to the CN+PE component shown in Fig. 16(a) is not significant. Therefore, the comparison of experimental and calculated cross-sections of this reaction is related first and foremost to HF+PE modeling.
However, due to the DR component, there is a clear agreement between calculated and recently measured cross-sections between 12-15 MeV. After that, there are quite different cases at higher and lower energies. Thus, there are only two disparate data sets available at higher energies, even beyond the LD uncertainty band but within the larger similar PE band. On the other hand, at lower energies there is an underestimation up to a factor of two around the incident energy of 8 MeV. Meanwhile, the uncertainty bands, which are yet dropped at these energies in Fig. 16(a), no longer exhibit LD and PE effects.
Previously, an apparent increase in measured αemission beyond the DR+PE+CN cross-sections, in neutron-induced reactions on A∼60 nuclei [6,7], was found around the GQR energies E GQR =65A −1/3 MeV [99] of the related excited nuclei. Thus, α-particle decay of giant resonances populated via neutron capture has been assumed, with Gaussian distributions added in this respect to the DR+PE+CN sum. The widths and peak cross-sections of these distributions were obtained by fitting the extra yields. However, because these widths are lower than the systematic 'best' values [99], we have called them only like-GQR components. On the other hand, no enhancement beyond the DR+PE+CN crosssections has been found in the present work for neutrons incident on Zr and Nb isotopes at energies between 5.8-8.7, and 7.1 MeV, respectively, corresponding to these excited-nuclei GQR energies. Furthermore, the same en-ergy for the target nucleus 92 Mo is 6.3 MeV, so that an eventual like-GQR contribution at the actual extra-yield maximum around ∼8 MeV would be less significant.
Nevertheless, for the sake of discussion, an eventual Gaussian distribution with a peak cross-section of 1 mb at 6.3 MeV related to the GQR energy of 14.35 MeV for the excited nucleus 93 Mo, and a width of 2.35 MeV [6], is shown in Fig. 16(a). The peak cross-section value is chosen to fit the apparent extra-yield at this GQR energy, while the related Gaussian distribution may neither be supported by the available experimental data nor describe the actual extra-yield around the incident energy of 8 MeV. Hence, it has not been included within the finally calculated DR+PE+CN cross-sections. Thus, the underestimation of the measured data below 12 MeV remains an open question at variance with the good agreement around 14 MeV. 95 Mo(n, α) 92 Zr excitation function [ Fig. 16(b)] is a thoroughly different case, not only with an outstanding cross-section set measured from 1 to 500 keV [100] but also recent data between 4 and 6 MeV [101]. The energies of the latter measurement are just around 5.1 MeV incident energy which is linked to the GQR energy of 14.2 MeV for the excited nucleus 96 Mo.
First, the present results of the α-particle OMP [1] are in obvious agreement with the average trend of the data [100] just above the discrete-resonance energy range [32]. The extent of this agreement could be pointed out by the factor of ∼5 between various OMP predictions formerly considered in this respect (Fig. 3 of Ref. [100]). But more important is that also close to these data were the results provided by the first version [12] of the OMP [1], i.e. based only on the α-particle elastic-scattered analysis ( Fig. 1 of Ref. [13]). It is also worth noting that somewhat similar agreement shown in the latter figure by the quite different α-particle OMP [10], which is only related only to neutron-induced α-emission from A∼60 nuclei. This fact may suggest that the simple OMP [10] could include, beyond the CN contribution, the additional ones that only now receive a full consideration.
Second, almost for a complete picture, an eventual pickup DR contribution has been obtained by interpolating between the similar and comparable results for the target nuclei 92,98 Mo using the SFs discussed in Sec. II B. As expected, DR cross-sections of less than 1 mb have a minor addition to the PE+CN sum, which is close to the early data around 14 MeV in the limit of twice the data standard deviation (σ), in Fig. 16(b). At the same time, a much larger underestimation of the more recent data around the incident energy of 5 MeV [101] is obvious.
So, the third and most important is the suitable account of this extra-yield by the addition of a Gaussian distribution corresponding to the GQR energy of 14.2 MeV for excited nucleus 96 Mo. A peak cross-section of 0.6 mb and a width of 2.35 MeV are given by the fit of the measured data [101]. It is worth noting that there are no NLD and PE effects at these energies, the subsequent uncertainty bands becoming visible only at higher exci- tation. The DR contribution becomes also comparable with this like-GQR component only above an incident energy of 7 MeV. Therefore, it may be concluded that the data of Zhang et al. [101] support the assumption of like-GQR α-particle decay. 98 Mo(n, α) 95 Zr excitation function [ Fig. 16(c)] analysis reveals the pickup DR role within the completion of the CN+PE modeling. Thus, while CN+PE uncertainly bands due to the NLD and PE effects are visible from incident energies of 9-10 MeV, the DR contribution of SFs quoted in Sec. II B is significantly larger up to ∼13 MeV, Then, the uncertainty related to PE effects becomes more important above 14 MeV while the same happens for the NLD uncertainty band starting at 17 MeV. Nevertheless, due consideration of the DR has brought the agreement of the calculated cross-sections from the lower limit of the data error bars to their average values.
On the other hand, the only data set [102] available below 10 MeV has shown an extra yield well above the DR+PE+CN cross-sections without either NLD or PE effects at these energies. These data could be described however by adding a Gaussian distribution corresponding to the GQR energy of 14.05 MeV for excited nucleus 99 Mo, and the peak cross-section of 0.8 mb as well as the width of 2.35 MeV. The assumption of the like-GQR α-particle decay appears to be supported once more. On the other hand, a fit of these data below 10 MeV, as seems to be the case for TENDL-2021 [31] in Fig. 16(c), is leading to a large overestimation of the measured data at higher energies. DR cross-section estimation, has been overcome similarly to the case of 95 Mo target nucleus. Thus, extrapolating the related components of 92,98 Mo nuclei yielded results that are rather close to those for 98 Mo. Finally, an unexpected agreement of the experimental data and the calculated CN+PE+DR sum has been found.
To give a complete view of the possible like-GQR αemission for Mo isotopes, it is shown in Fig. 16(d) along with the outline of a Gaussian distribution at the GQR energy of 14.0 MeV for excited nucleus 101 Mo. The width of 2.35 MeV and an assumed peak cross-section of 0.5 mb, close to that of 98 Mo, have provided a possible shape to be confirmed or not by further measurements. Nevertheless, its proof could be favored by the isotope effect [95] of significantly lower (n, α) cross-sections for heavier isotopes, similarly to Ni isotope chain [7].

IV. CONCLUSIONS
A recent assessment of a previous α-particle OMP [1] also for nucleon-induced α-emission on A∼60 nuclei, including pickup DR and eventual GQR α-emission [6,7], is completed for neutrons incident on Zr, Nb, and Mo stable isotopes. Consistent sets of input parameters, determined through analysis of independent data, is involved with no further empirical rescaling factors of the γ and nucleon widths which however are mandatory within large-scale nuclear-data evaluations. Nevertheless, there is an obvious correlation between the accuracy of the independent data, the input parameters determined by their fit, and final uncertainties of the calculated reaction cross sections. Moreover, additional validation of this potential is also supported by recently measured crosssections of (α, γ) reactions on 90,91,92 Zr as well as (α, n) on 96 Zr and 100 Mo nuclei.
On the other hand, the pickup contributions to (n, α) reactions have been determined within the DWBA method using the code FRESCO [19]. The one-step reaction has also been considered through the pickup of 3 He cluster while the "spectator model" [62,63] was involved for the two transferred protons in (n, α) reaction. However, the lack of measured α-particle angular distribution for the (n, α) reactions within this work made possible only straightforward DWBA calculations of related pickup cross-sections. Thus, the spectroscopic factors of Glendenning [66] have been used for the spectator proton pair [63][64][65], in addition to SF for the picked neutron that becomes thus responsible for the angularmomentum transfer. The latter have been obtained through analysis of α-particle angular distributions of one-nucleon pickup reactions ( 3 He, α) or (t, α) toward the residual nucleus of interest.
Nonetheless, a suitable account by actual parameter set of all data for competing nucleon-emission by neutrons on Zr, Nb, and Mo isotopes has been a prerequisite for a consistent discussion of the related α-emission. In this respect, the previous analyses for Zr [5] and Mo [9] are completed by a similar work for 93 Nb and newer measured data for the even-even semi-magic nucleus 92 Mo. Then, an appropriate description of the α-particle angleintegrated energy distributions induced by neutrons on 93 Nb, 90 Zr, and 92 Mo has been concerned for validation of α-particle PE and DR account.
Finally, an increase of the α-emission beyond the CN+PE predictions has been obtained through consideration of additional pickup DR and like-GQR decay of excited nuclei by neutrons on Zr, Nb, and Mo stable isotopes. Thus it becomes possible a description of both the absorption and emission of α particles by the same potential [1], in support of also its use for large-scale nuclear-data evaluations as TALYS corresponding default option. This conclusion had already been reachable within the previous analysis for Zr [5] and Mo [13] before DR and like-GQR decay consideration, due to outstanding cross-section measurements for 91 Zr(n, nα) 87 Sr m [98] and 95 Mo(n, α) 92 Zr [100,101] reactions. At the same time it is suggested that the simple OMP [10] could include, beyond the CN contribution, the additional ones that only now receive full consideration. Nonetheless, further measurements at incident energies corresponding to GQR energies of excited nuclei, as well as heavier isotopes of elements, may shed light on the eventual like-GQR α-emission.