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

M. Avrigeanu<EMAIL_ADDRESS>V. Avrigeanu
<EMAIL_ADDRESS>Horia Hulubei National Institute for Physics and
Nuclear Engineering, 077125 Bucharest-Magurele, Romania

###### Abstract

Background: 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 91, 064611 (2015), Paper I]. However, the same was not
the case of neutrons on Zr stable isotopes [Phys. Rev. C 96, 044610 (2017),
Paper II].

Purpose: 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.

Methods: 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.

Results: 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.

Conclusions: An increase of the $\alpha$-emission beyond the statistical
predictions, through consideration of additional reaction channels of the
pickup direct interaction and like–GQR decay, makes possible the description
of both absorption and emission of $\alpha$ particles by the same optical
potential.

###### pacs:

24.10.Eq,24.10.Ht,24.30.Cz,24.60.Dr

##  I Introduction

The reliability of a previous $\alpha$-particle optical-model potential (OMP)
for target nuclei with mass number 45$\leq$$A$$\leq$209 Avrigeanu et al.
(2014) was also proved for $\alpha$-emission in proton–induced reactions on Zn
isotopes Avrigeanu and Avrigeanu (2015) (Paper I). However, its use led to
underestimated predictions of the statistical Hauser-Feshbach (HF) Hauser and
Feshbach (1952) and pre-equilibrium emission (PE) Gadioli and Hodgson (1992)
models for neutrons incident on Zr stable isotopes Avrigeanu and Avrigeanu
(2017) (Paper II). On the other hand, it has recently been shown that this
potential can also describe $\alpha$-emission from excited nuclei in nucleon-
induced reactions within the $A$$\sim$60 range Avrigeanu and Avrigeanu (2021,
2022). This was possible through additional consideration of the pickup direct
reaction (DR) and eventual Giant Quadrupole Resonance (GQR) $\alpha$-emission.
The so-called $\alpha$-potential mystery Rauscher (2013) related to the
account of both $\alpha$-emission and absorption by an OMP, of equal interest
for astrophysics and fusion technology, thus received an alternate solution.
Nevertheless, its support by analysis of more data is imperative, while the
case of neutrons incident on $A$$\sim$90 nuclei Avrigeanu and Avrigeanu (2017)
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 $\alpha$-emission in neutron–induced reactions
on stable Mo isotopes was targeted in an earlier systematic investigation up
to 20 MeV Reimer et al. (2005). However, an $\alpha$-particle OMP Avrigeanu et
al. (1994) describing $A$$\sim$60 compound–nuclei (CN) $\alpha$-particle decay
led to a significant underestimation at incident energies around $\leq$10 MeV
for 92,98Mo. This OMP makes predictions that differ significantly from
potentials for incident $\alpha$ particles McFadden and Satchler (1966)
including a double–folding (DFM) microscopic real potential Avrigeanu et al.
(2014, 2003, 2006, 2009); Avrigeanu and Avrigeanu (2010). The discrepancy in
results corresponding to OMPs Avrigeanu et al. (1994, 2003) led even to the
assumption of a nuclear–density distribution dependent on nuclear temperature
Avrigeanu et al. (2006).

The removal of compensation effects of less accurate model parameters is
essential to establishing an appropriate $\alpha$-particle OMP Avrigeanu et
al. (2014, 2003, 2006, 2009); Avrigeanu and Avrigeanu (2010) or to prove it
Avrigeanu and Avrigeanu (2016, 2019). This aim has been achieved using
”consistent sets of input parameters, determined through analysis of
independent data available in this mass region” Arthur (1980). Moreover, a
proper account of all competitive reaction channels, beyond the
$\alpha$-emission of interest, should also be aimed at the validation of
consistent parameter sets. So, there have been avoided empirical rescaling
factors of the $\gamma$ and/or nucleon widths which however are mandatory
within large-scale nuclear–data evaluations.

This work is an extension of Refs. Avrigeanu and Avrigeanu (2017, 2021, 2022)
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 Thompson (1988). A comparison
of HF+PE results and (i) recent $(\alpha,\gamma)$ and $(\alpha,n)$
cross–sections of Zr and Mo nuclei, for $\alpha$-particle OPM additional
proof, (ii) available data of neutron–induced activation by nucleon–emission
for 93Nb and 92Mo, as for Zr nuclei Avrigeanu and Avrigeanu (2017), and (iii)
$\alpha$-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.

Table 1:  Low-lying levels number $N_{d}$ up to excitation energy $E^{*}_{d}$
ens  used in HF calculations of reaction cross–sections, low-lying levels and
$s$-wave nucleon-resonance spacingsa $D_{0}^{\it exp}$ (with uncertainties
given in units of the last digit in parentheses) in the energy range
$\Delta$$E$ above the separation energy $S$, for the target-nucleus ground
state (g.s.) spin $I_{0}$, fitted to obtain the BSFG level-density parameter a
and g.s. shift $\Delta$ (for a spin cutoff factor calculated with a variable
moment of inertia Avrigeanu et al. (2002) between half and 75% of the rigid-
body value, from g.s. to $S$, and reduced radius $r_{0}$=1.25 fm).

Nucleus | $N_{d}$ | $E^{*}_{d}$ | Fitted level and resonance data | $a$ | $\Delta$
---|---|---|---|---|---
|  |  | $N_{d}$ | $E^{*}_{d}$ | $S+\frac{\Delta E}{2}$ | $I_{0}$ | $D_{0}^{\it exp}$ |  |
|  | (MeV) |  | (MeV) | (MeV) |  | (keV) | (MeV-1) | (MeV)
87Sr | 29 | 2.708 | 53(2) | 3.166 | 8.442 | 0 | 2.6(8) | 9.16(53/-36)(49/-33) | 0.09(19/-14)(15/-10)
88Sr | 47 | 4.801 | 47 | 4.801 | 11.113 | 9/2 | 0.29(8) | 8.70 | 1.63
89Sr | 22 | 3.249 | 22(2) | 3.249 | 6.430 | 0 | 23.7(29) | 9.58(24/-19)(15/-0) | 0.87(7/-5)(-1/4)
90Sr | 15 | 3.039 | 17 | 3.146 |  |  |  | 9.60 | 0.95
91Sr | 15 | 2.237 | 15(3) | 2.237 |  |  |  | 9.7(3) | 0.13(7)(5)
92Sr | 15 | 2.925 | 15 | 2.925 |  |  |  | 10.00 | 0.89
93Sr | 20 | 2.169 | 20(2) | 2.169 |  |  |  | 10.6(3) | 0.09(7)(1)
89 Y | 26 | 3.630 | 26 | 3.630 | 11.478 | 4 | 0.106(35)b | 8.90 | 0.94
90 Y | 30 | 2.366 | 29(2) | 2.327 | 6.857 | 1/2 | 3.7(4) | 9.23(15)(10) | -0.32(5)(2)
88Zr | 22 | 3.094 | 22 | 3.094 |  |  |  | 8.90 | 0.56
89Zr | 23 | 2.300 | 23(3) | 2.300 |  |  |  | 9.2(4) | -0.19(13)(4)
90Zr | 46 | 4.701 | 46 | 4.701 |  |  |  | 9.00 | 1.65
91Zr | 37 | 3.053 | 37 | 3.053 | 7.260 | 0 | 6.0(14) | 9.77 | 0.40
92Zr | 42 | 3.500 | 54(2) | 3.725 | 8.647 | 5/2 | 0.55(10) | 9.67(27)(25) | 0.79(9)(6)
93Zr | 29 | 2.391 | 29 | 2.391 | 6.785 | 0 | 3.5(8) | 10.66 | 0.12
94Zr | 23 | 3.059 | 23 | 3.059 | 8.220 | 5/2 | 0.302(75) | 11.00 | 1.00
95Zr | 14 | 2.022 | 14(2) | 2.022 | 6.507 | 0 | 4.0(8) | 11.04(33)(19) | 0.23(7)(5)
96Zr | 38 | 3.630 | 38 | 3.630 |  |  |  | 11.20 | 1.32
97Zr | 9 | 2.058 | 9(1) | 2.058 | 5.629 | 0 | 13(3) | 11.21(42)(30) | 0.51(7)(2)
91Nb | 29 | 2.660 | 29 | 2.660 |  |  |  | 9.30 | 0.04
92Nb | 41 | 1.851 | 41 | 1.851 |  |  |  | 9.60 | -0.92
93Nb | 35 | 1.784 | 35 | 1.784 |  |  |  | 9.90 | -0.80
94Nb | 48 | 1.281 | 48 | 1.281 | 7.232 | 9/2 | 0.094(10) | 10.75 | -1.29
95Nb | 28 | 1.660 | 28 | 1.660 |  |  |  | 11.30 | -0.45
97Nb | 19 | 2.113 | 19 | 2.113 |  |  |  | 11.60 | 0.26
98Nb | 3 | 0.226 | 3 | 0.226 |  |  |  | 13.50 | -0.52
99Nb | 15 | 1.305 | 15 | 1.305 |  |  |  | 12.40 | -0.31
100Nb | 19 | 0.772 | 19 | 0.772 |  |  |  | 13.50 | -0.48
90Mo | 22 | 3.185 | 22 | 3.185 |  |  |  | 9.00 | 0.67
91Mo | 24 | 2.345 | 29 | 2.716 |  |  |  | 9.30 | 0.09
92Mo | 37 | 4.187 | 37 | 4.187 |  |  |  | 9.20 | 1.36
93Mo | 60 | 2.974 | 77 | 3.161 | 8.092 | 0 | 2.7(5),2.17(25)c | 9.6 | -0.02
94Mo | 60 | 3.456 | 61 | 3.462 | 9.678 | 5/2 | 0.081(24)d | 10.74 | 0.78
95Mo | 25 | 1.692 | 25 | 1.692 | 7.377 | 0 | 1.32(18) | 10.45 | -0.56
96Mo | 38 | 2.875 | 38 | 2.875 | 9.154 | 5/2 | 0.0662(30)e | 11.35 | 0.61
97Mo | 33 | 1.341 | 33 | 1.341 | 6.831 | 0 | 1.05(20).0.661(198)d | 11.16 | -0.91
98Mo | 38 | 2.701 | 38 | 2.701 | 8.664 | 5/2 | 0.06(1),0.047(6)d | 12.20(40)(36) | 0.60(8)(4)
99Mo | 30 | 1.198 | 49 | 1.341 | 5.941 | 0 | 1.0(2) | 12.34(37)(32) | -0.75(8)(4)
100Mo | 32 | 2.464 | 31 | 2.432 |  |  |  | 12.00 | 0.39
101Mo | 23 | 0.626 | 23 | 0.626 | 5.411 | 0 | 0.62(10) | 13.14 | -1.19

aRIPL-3 Capote et al. (2009) if not otherwise mentioned bReference Guttormsen
et al. (2014) cRIPL-1 Beijing file rip  dReference Utsunomiya et al. (2013)
eReference Koehler (2022)

##  II Models and parameters

###  II.1 Compound and pre-equilibrium emission

The same models, codes Koning et al. (2008); Avrigeanu and Avrigeanu (1995);
Kunz (1984), and local approaches Avrigeanu and Avrigeanu (2017, 2021) concern
the HF+PE and collective inelastic–scattering cross–section assessment. Thus,
consistent (i) back-shifted Fermi gas (BSFG) Vonach et al. (1988) nuclear
level density (NLD) parameters, (ii) nucleon and (iii) $\gamma$-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 Koning et al. (2008) and its
default options which include the $\alpha$-particle OMP Avrigeanu et al.
(2014). Furthermore, a correlation with the TALYS–-based evaluated–data
library TENDL-2021 Koning et al. (2019) shows the evaluation progress vs
default options usage.

Figure 1:  Comparison of measured exf  and calculated neutron total
cross–sections of (a) 93Nb, using the global (dashed curve) and local (dash-
dotted curve) OMP parameters of Koning and Delaroche Koning and Delaroche
(2003), and (b) 95Mo (thick, lower curves) and 97Mo (thin, upper curves) with
global (dashed curves) as well as the corresponding modified values of $v_{1}$
parameters given in Table 2 (solid curves).

The low-lying levels and NLD parameters of the BSFG model are given in Table 1
for all nuclei within this work. The limits of the fitted $a$ and $\Delta$
parameters were obtained also by fit of the error-bars of $s$-wave
nucleon–resonance spacings $D_{0}^{\it exp}$. For nuclei without resonance
data, the smooth-curve method Johnson (1977) was used, in which an average of
fitted $a$-values for nearby nuclei is adopted while the $\Delta$ 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 1)
despite better data becoming available in the meantime (e.g. for A=90, 98–100
nuclei ens ).

Figure 2:  As Fig. 1 but for 92,94,96,98,100Mo and modified (Table 2) (solid
curves) local real–potential depth parameter $v_{1}$ Koning and Delaroche
(2003).

The neutron OMP of Koning and Delaroche Koning and Delaroche (2003) has
additionally been reviewed at energies up to $\sim$30 MeV through the SPRT
method Delaroche et al. (1976), i.e. by analysis of the $s$\- and $p$-wave
neutron strength functions $S_{0}$ and $S_{1}$, respectively, the potential
scattering radius $R^{\prime}$ Capote et al. (2009) (Table 2), and the energy
dependence of the neutron total cross–section $\sigma_{T}(E)$ exf . Thus, we
found that the global parameter set Koning and Delaroche (2003) provides a
better agreement between the measured and calculated $\sigma_{T}(E)$ of 93Nb
within this energy range, in comparison to the related local parameter set
[Fig. 1(a)].

The small adjustments shown in Table 2 for the real potential depth parameter
$v_{1}$ Koning and Delaroche (2003) either global, for the odd 95,97Mo [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 rip  except the case of more recent measurements Koehler
(2022); Leinweber et al. (2010); Bahran et al. (2015); Remley et al. (2018).
Nevertheless, these changes have no importance for the $\sigma_{T}(E)$ values
around 1 MeV, which are of obvious importance for the competition between the
neutron and charged–particle decay of excited CN, in heavier Mo nuclei. It is
also within $\sim$4% for the near-spherical 92Mo nucleus as well as for 94Mo.

Because the improvement of the calculated $\sigma_{T}(E)$ at higher energies
is yet within the measured error bars, the present analysis for Nb and Mo
isotopes has also supported the previous use Avrigeanu and Avrigeanu (2017) of
the OMP Koning and Delaroche (2003) for Zr isotopes.

The same OMP parameters have been involved within DWBA assessment of the
collective inelastic scattering, using the deformation parameters Kalbach
(2000); Raman et al. (2001); Kibédi and Spear (2002) of the first 2+ and 3-
collective states. Typical direct inelastic–scattering cross–sections increase
to, e.g., $\sim$7% of $\sigma_{R}$ for neutrons on 93Nb and 98Mo 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 $\sigma_{R}$
that has been taken into account within the PE+HF analysis.

Table 2: Comparison of experimental Capote et al. (2009); Koehler (2022); Leinweber et al. (2010); Bahran et al. (2015); Remley et al. (2018) and calculated $s$\- and $p$-wave neutron strength functions $S_{0}$ and $S_{1}$, respectively, and the potential scattering radius $R^{\prime}$ Capote et al. (2009) of 93Nb and 92,94-98,100Mo isotopes, and the changes of either local or global real–potential depth $v_{1}$ of the even and odd Mo isotopes Koning and Delaroche (2003), respectively (with the use of Koning and Delaroche (2003) notations, the energies are in MeV and geometry parameters in fm) which provide the best results within the SPRT method. Nucleus | Reference Exp./OMP | 104S0 | 104S1 | R’
---|---|---|---|---
93Nb |  Capote et al. (2009) (E$<$0.004 rip ) | 0.45(7) | 5.8(8) | 7.0(2)
|  Koning and Delaroche (2003), global | 0.55 | 6.48 | 6.4
|  Koning and Delaroche (2003), local | 0.44 | 6.28 | 6.5
92Mo |  Capote et al. (2009) (E$<$0.025 rip ) | 0.56(7) | 3.6(6) | 6.4(8)
|  Koning and Delaroche (2003), global | 0.55 | 7.0 | 6.3
|  Koning and Delaroche (2003), local (v1=55.0) | 0.61 | 6.4 | 6.4
|  Koning and Delaroche (2003)\+ v1=53.5, E$<$11.5 | 0.66 | 4.5 | 6.6
|  57.0 |  |  |
94Mo |  Capote et al. (2009) (E$<$0.01 rip ) | 0.44(8) | 7.2(27) | 6.5(13)
|  Koning and Delaroche (2003), global | 0.54 | 7.4 | 6.4
|  Koning and Delaroche (2003), local (v1=54.2) | 0.56 | 7.3 | 6.3
|  Koning and Delaroche (2003)\+ v1=57.5, E$<$0.65 | 0.59 | 8.2 | 5.8
|  53.0 |  |  |
95Mo |  Capote et al. (2009) (E$<$0.0025rip ) | 0.47(17) | 6.9(18) |
|  Leinweber et al. (2010), E$<$0.001 | 0.436(9) |  |
|  Bahran et al. (2015), E$<$0.36 | 0.4(1) | 3.8(1) | 6.97(4)
|  Koehler (2022), E$<$0.005 | 0.47(17) | 3.09(35) |
|  Koning and Delaroche (2003), global (v1=54.6) | 0.50 | 5.1 | 6.4
|  Koning and Delaroche (2003)\+ v1=53.0, E$<$1 | 0.52 | 4.5 | 6.2
|  54.0 |  |  |
96Mo |  Capote et al. (2009) (E$<$0.01 rip ) | 0.62(12) | 7.1(16) | 6.6(13)
|  Koning and Delaroche (2003), global | 0.53 | 8.0 | 6.3
|  Koning and Delaroche (2003), local (v1=55.0) | 0.56 | 7.7 | 6.3
|  Koning and Delaroche (2003)\+ v1=57.2, E$<$3.5 | 0.62 | 7.2 | 5.7
|  52.5 |  |  |
97Mo |  Capote et al. (2009) (E$<$0.002 rip ) | 0.37(7) | 7.85(153) |
|  Leinweber et al. (2010), E$<$0.001 | 0.438(5) |  |
|  Koning and Delaroche (2003), global (v1=54.2) | 0.53 | 8.2 | 6.4
|  Koning and Delaroche (2003)\+ v1=52.5, E$<$0.7 | 0.55 | 5.5 | 6.6
|  53.0, E$>$1.0 |  |  |
98Mo |  Capote et al. (2009) (E$<$0.016 rip ) | 0.48(9) | 6.3(21) | 7.0(14)
|  Remley et al. (2018), E$<$0.005 | 0.7(4) | 6.2(11) |
|  Remley et al. (2018), E$<$0.028 | 0.4(2) | 7.4(10) |
|  Koning and Delaroche (2003), global | 0.51(1) | 8.5 | 6.3
|  Koning and Delaroche (2003), local (v1=54.0) | 0.54 | 8.1 | 6.3
|  Koning and Delaroche (2003)\+ v1=56.0, E$<$3.5 | 0.59(1) | 7.7(2) | 5.8(1)
|  52.0 |  |  |
100Mo |  Capote et al. (2009) (E$<$0.013 rip ) | 0.7(1) | 5.0(8) | 6.5(13)
|  Remley et al. (2018), E$<$0.005 | 0.4(2) |  |
|  Remley et al. (2018), E$<$0.021 | 0.9(2) | 4.2 (6) |
|  Koning and Delaroche (2003), global | 0.50 | 9.0 | 6.3
|  Koning and Delaroche (2003), local (v1=52.0) | 0.52(1) | 8.8 (1) | 6.2 (1)
|  Koning and Delaroche (2003)\+ v1=55.5, E$<$3.5 | 0.58(1) | 7.2(2) | 5.8
|  51.0 |  |  |

The proton OMP of Koning and Delaroche Koning and Delaroche (2003) 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 $(p,n)$ cross–sections at energies higher than 3–4 MeV, where this
reaction channel becomes dominant, with cross–sections close to
optical–potential $\sigma_{R}$.

Figure 3:  Comparison of cross–sections measured exf , evaluated Koning et al.
(2019) ($\ast$), code TALYS-1.96 Koning et al. (2008) default–option results
(short–dotted curves), and calculated in this work using proton global OMP
parameters Koning and Delaroche (2003) and $E$1-radiation EGLO strength
functions(solid curves) for (a) $(p,\gamma)$ reaction on 92Zr, including
results for alternate use of $E$1-radiation GLO (dash-dotted) and SLO (dash-
dot-dotted) models, and (b–d) $(p,n)$ reaction on 92,94,96Zr, respectively.

Moreover, the same is true for the $(p,\gamma)$ reaction below the $(p,n)$
reaction effective threshold, where its cross–sections are, in their turn,
close to $\sigma_{R}$ values. Thus, the good agreement shown in Fig. 3(a)
between the results within this work and recently measured
92Zr$(p,\gamma)^{93}$Nb reaction cross–sections at incident energies of 2–5
MeV Spyrou et al. (2013) supports the global OMP parameters Koning and
Delaroche (2003) even if the earlier $(p,n)$ related data exf  are somehow
underpredicted between 3 and 4 MeV [Fig. 3(b)].

Figure 4:  As Fig. 3 but for $(p,n)$ reaction (upper solid and short–dotted
curves and symbols $\ast$) and $(p,\gamma)$ reaction (lower similar curves and
symbols) on 93Nb exf ; Heim et al. (2020).

The radiative strength functions (RSF) for Nb and especially Mo isotopes do
not yet have a confident parametrization despite the widespread systematics
Kawano et al. (2020); Goriely et al. (2019) and extensive studies (Heim et al.
(2021) and Refs. therein) performed since our former analysis for Zr stable
isotopes Avrigeanu and Avrigeanu (2017). More recent accurate cross–section
measurements of reactions 93Nb$(p,\gamma)^{94}$Mo Heim et al. (2020) and
$(\alpha,\gamma)$ on Zr isotopes Kelmar et al. (2021); Harissopulos (2018), on
the other hand, motivated a further survey of these RSFs.

Figure 5:  Comparison of: (a–c) measured RSFs of 94,95,96Mo nuclei Goriely et
al. (2019); OCL  and calculated sum of $M$1-radiation SLO model and either
$E$1-radiation SLO (dash-dot-dotted), GLO (dash-dotted) RSFs, or the one more
sum of upbend–including $M$1 component (short dotted) and EGLO values of (a)
either lower (solid) or upper (dashed) resonance parameters for 92Mo nucleus
(Table II Tveten et al. (2016)), and (b,c) similar middle resonance parameters
(solid), along with average $s$-wave radiation widths $\Gamma_{\gamma}$ (in
meV) measured Capote et al. (2009); Utsunomiya et al. (2013) and calculated as
well; (d–f) 90-92Zr$(\alpha,\gamma)^{94-96}$Mo reaction cross–sections
measured exf ; Kelmar et al. (2021); Harissopulos (2018), evaluated Koning et
al. (2019) (short dashed), calculated by default options of TALYS-1.96 Koning
et al. (2008) (short-dotted), and with the above–mentioned RSFs (similar
curves), vs $\alpha$-particle laboratory energy (bottom) and ratio of center-
of-mass energy to Coulomb barrier $B$ Nörenberg (1980) (top).

Thus, presently we have adopted the giant dipole resonance (GDR) and
additional $M$1 upbend parameters found recently to describe the RSF data for
92,94Mo nuclei Tveten et al. (2016). The lower and upper resonance parameters
given in Table II of Ref. Tveten et al. (2016)) provide an RSF uncertainty
band for 94Mo, while the related middle resonance parameters were used for the
RSF account for 95,96Mo nuclei. A comparison of the results obtained by using
these parameters within 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 RSF is shown in
Figs. 5(a–c). The EGLO model has led to an enhanced description of both RSF
OCL ; Goriely et al. (2019) and average $s$-wave radiation widths
$\Gamma_{\gamma}$ Capote et al. (2009); Utsunomiya et al. (2013) measured
data. The SLO and GLO models resulted in different RSF energy dependence below
the neutron binding energy, as well as larger $\Gamma_{\gamma}$–values.

Similar results have been obtained for 94Nb nucleus by using the GDR
parameters of Kopecky and Uhl Kopecky and Uhl (1990) but the middle nuclear
temperature and $M$1–upbend parameters of 92,94Mo nuclei Tveten et al. (2016).
The same parameters have been used for 93Nb, 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 93Nb$(p,\gamma)^{94}$Mo and
90Zr$(\alpha,\gamma)^{94}$Mo reactions modeling at low incident energies
concerns also the recent data of Heim et al. Heim et al. (2020) (Fig. 4) and
Kelmar et al. Kelmar et al. (2021) [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 $(p,\gamma)$ cross–sections are described by taking into account
both limits of the resonance parameters of Ref. Tveten et al. (2016), while
only the lower limit is yet close to the $(\alpha,\gamma)$ cross–sections. One
may conclude that different RSFs could describe the two decay channels of the
compound nucleus 94Mo.

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 93Nb and 0+ of the
even–even 90Zr. 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 $(p,\gamma)$ and $(\alpha,\gamma)$
cross–sections.

Moreover, the same is the case of the EGLO model for 95,96Mo nuclei and
$(\alpha,\gamma)$ reaction on 91,92Zr, with increased RSFs due to account of
the middle resonance parameters Tveten et al. (2016) and the more recent data
of Krtička et al. Krtička et al. (2016) shown in Fig. 5(b) for 95Mo. Their
values, which are higher than the former ones OCL , have been taken into
consideration for both 95,96Mo nuclei. Consequently, the corresponding
$(\alpha,\gamma)$ calculated cross–sections [Figs. 5(e,f)] are increased also
for the odd residual nucleus 95Mo.

The PE Geometry-Dependent Hybrid (GDH) model Blann and Vonach (1983) was also
used, which was generalized by including the angular-momentum and parity
conservation Avrigeanu et al. (1990) and knockout $\alpha$-particle emission
based on a pre-formation probability $\varphi$ Gadioli and Hodgson (1992) . It
also includes a revised version of the advanced particle-hole level densities
(PLD) Avrigeanu and Avrigeanu (1998); Harangozo et al. (1998) with a Fermi–gas
energy dependence of the single–particle level (s.p.l) density Kalbach (1985).
The $\alpha$-particle s.p.l. density $g_{\alpha}$=($A$/10.36) MeV-1 Gadioli
and Gadioli Erba (1981), 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/$\pi^{2}$)$a$. Moreover, the above–mentioned OMP
parameters have also been involved in the local density approximation (Blann
and Vonach (1983) and Refs. therein), as also the local–density Fermi energies
for various partial waves, corresponding to the central–well Fermi energy
value $F$=40 MeV.

###  II.2 Direct reaction account

Similar to the previous work on $A$$\sim$60 nuclei Avrigeanu and Avrigeanu
(2021, 2022), the pickup contributions to $(n,\alpha)$ reactions have been
determined within the DWBA method using the code FRESCO Thompson (1988). Thus,
an one–step reaction has also been considered through the pickup of 3He
cluster while the ”spectator model” Smits and Siemssen (1976); Smits et al.
(1979) was involved. The two transferred protons in $(n,\alpha)$ 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 $n$-3He and $p$-$t$ effective
interactions in the $\alpha$ particle assumed to have a Gaussian shape Gadioli
et al. (1986, 1988) set by the fit of the binding energies of 3He and $t$,
respectively. Moreover, the bound states of the three–nucleon transferred
cluster were generated in a Woods–Saxon real potential Smits et al. (1979);
Gadioli et al. (1986, 1988) with the depth adjusted to fit the separation
energies in the target nuclei. The harmonic–oscillator energy conservation
rule Smits and Siemssen (1976); Smits et al. (1979) 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 $\alpha$-particle angular distribution, for
the $(n,\alpha)$ 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. Glendenning (1975)) for the
spectator proton pair Gadioli et al. (1986, 1988); Smits et al. (1979), 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 $\alpha$-particle angular distributions of one–nucleon pickup reactions
$(^{3}He,\alpha)$ or $(t,\alpha)$ toward the residual nucleus of interest, as
shown in the following.

88Sr$(^{3}He,\alpha)^{87}$Sr $\alpha$-particle angular–distribution Fortier et
al. (1989) 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 $\sim$6 MeV excitation energy of 87Sr residual nucleus has
been considered in this respect, as well as for calculation of
90Zr$(n,\alpha)^{87}$Sr pickup excitation function. The spectator proton pair
picked from 2p$\frac{1}{2}$ subshell Gadioli et al. (1986, 1988) has been
considered.

Figure 6:  (Color online) Comparison of the measured (solid circles) Fortier
et al. (1989) and calculated $\alpha$-particle angular distributions (solid
curves) of 88Sr$(^{3}He,\alpha)^{87}$Sr pickup transitions to states shown
with excitation energies in MeV, at the incident energy of 36 MeV.

Figure 7:  As Fig. 6 but for 91Zr$(t,\alpha)^{90}$Y reaction and incident
energy of 17 MeV Decman et al. (1983).

Figure 8:  As Fig. 6 but for 90Zr target nucleus and incident energy of 39 MeV
Galés et al. (1977a).

Figure 9:  As Fig. 6 but for 96Zr target nucleus and incident energy of 39 MeV
Galés et al. (1977b).

91Zr$(t,\alpha)^{90}$Y pickup–reaction analysis and the assumption of a
similarity between the picked–proton SFs and picked–neutron SFs Sharp et al.
(2013) have been involved for calculation of 93Nb$(n,\alpha)^{90}$Y reaction
cross–sections. Thus, 14 levels up to $\sim$ 2.5 MeV excitation energy have
been considered within the analysis of the measured $\alpha$-particle angular
distributions Decman et al. (1983) shown in Fig. 7. The same excited levels
have been concerned with the calculation of the above–mentioned $(n,\alpha)$
pickup excitation function in the following.

90Zr$(^{3}He,\alpha)^{89}$Zr reaction angular–distribution Galés et al.
(1977a) 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$\frac{1}{2}$ subshell has also been considered. 20 levels of
89Zr residual nucleus up to 3.572 excitation energy were considered for this
reaction (Fig. 8) as well as for assessment of 92Mo$(n,\alpha)^{89}$Zr pickup
excitation–function calculation of interest for this work.

96Zr$(^{3}He,\alpha)^{95}$Zr $\alpha$-particle angular–distribution analysis,
of the measured data also at 39 MeV incident energy Galés et al. (1977b),
provided the picked–neutron SFs to be used together with Glendenning’s SF of
the transferred spectator proton pair from 2p$\frac{1}{2}$ subshell. A number
of 23 levels of 95Zr 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 98Mo$(n,\alpha)^{95}$Zr pickup excitation–function calculation.

##  III Results and discussion

###  III.1 Recent $\alpha$-induced reaction data analysis

The recently measured cross–sections of $(\alpha,\gamma)$ Kelmar et al.
(2021); Harissopulos (2018) and $(\alpha,n)$ Kiss et al. (2021); Hamad et al.
(2022); Szegedi et al. (2021); Ong et al. (2022); Choudhary et al. (2022)
reactions, the former being already taken into account for RSF endorsement,
are of particular interest for a customary further validation of the
$\alpha$-particle OMP Avrigeanu et al. (2014), too.

####  III.1.1 90,91,92Zr$(\alpha,\gamma)$94,95,96Mo

The 90Zr$(\alpha,\gamma)^{94}$Mo cross–sections Kelmar et al. (2021) extended
the data energy range below that of the previous experiment Quinn et al.
(2015) taken into account within our former analysis Avrigeanu and Avrigeanu
(2017). This extension has touched even the energies around the $(\alpha,n)$
reaction threshold, where $(\alpha,\gamma)$ cross–section is near the
$\alpha$-particle total reaction cross–section. As a result, the former
becomes a strong constraint on the $\alpha$-particle 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 $\alpha$-particle OMP Avrigeanu et al. (2014) as well.

A similar extension to lower incident energies was performed for $\alpha$
particles incident on 92Zr Harissopulos (2018), the results shown in Fig. 5(f)
providing additional support for the $\alpha$-particle OMP Avrigeanu et al.
(2014). Furthermore, the agreement in Fig. 5(e) of presently calculated and
increased new measured data for the target nucleus 91Zr Harissopulos (2018)
has confirmed this $\alpha$-particle OMP, too.

####  III.1.2 96Zr$(\alpha,n)^{99}$Mo

The first cross–section measurement of 96Zr$(\alpha,n)^{99}$Mo reaction at
incident energies ranging from 6.5 to 13 MeV Kiss et al. (2021), more than six
orders of magnitude, was also a stringent test of the $\alpha$-particle OMP
Avrigeanu et al. (2014). Another recent measurement of the same reaction Hamad
et al. (2022) 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 $\sim$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 $(\alpha,n)$
cross–sections at these energies. Furthermore, the $(\alpha,n)$ reaction
cross–sections account for $\sim$99% of $\sigma_{R}$ below the $(\alpha,2n)$
reaction threshold, implying that this problem seems to be related entirely to
the $\alpha$-particle OMP.

Figure 10:  Comparison of cross–sections measured exf ; Kiss et al. (2021);
Hamad et al. (2022); Szegedi et al. (2021); Ong et al. (2022); Choudhary et
al. (2022); Murata et al. (2019); Villa et al. (2020), evaluated Koning et al.
(2019) (short–dashed), calculated by default options of TALYS-1.96 Koning et
al. (2008) (short–dotted), and in this work for $(\alpha,n)$ reaction on (a)
96Zr (solid curve), and (b) 100Mo (dashed); there are also shown calculated
cross–sections for reactions $(\alpha,2n)$ (dash–dotted), (a)
$(\alpha,\gamma)$ (short-dash-dotted) and $(\alpha,p)$ (dash-dot-dotted), and
(b) $(\alpha,xn)$ (solid).

As a result, the high accuracy of the new data Kiss et al. (2021); Hamad et
al. (2022) 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 of Ref. Avrigeanu et al.
(2014)) due to a significant change in the number of open reaction channels
close to the Coulomb barrier $B$ Avrigeanu et al. (2009); Avrigeanu and
Avrigeanu (2010); Mohr et al. (1997). This $W_{D}$ trend below an energy
$E_{2}$ corresponding to 0.9$B$ Avrigeanu et al. (2009); Avrigeanu and
Avrigeanu (2010) has been proved by reaction cross–section analysis Avrigeanu
et al. (2014, 2009); Avrigeanu and Avrigeanu (2010) while the
elastic–scattering studies have shown that $W_{D}$ decreases at higher
energies.

Because $E_{1}$=8.55 MeV and $E_{2}$=12.15 MeV for $\alpha$ particles incident
on 96Zr, the lower calculated $(\alpha,n)$ cross–sections 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 Avrigeanu et al.
(2009); Avrigeanu and Avrigeanu (2010). An eventual smoothing of the $W_{D}$
abrupt change at $E_{1}$ may significantly improve the data underestimation
around this energy.

####  III.1.3 100Mo$(\alpha,xn)^{102,103}$Ru

A recent measurement of 100Mo$(\alpha,n)^{103}$Ru reaction cross–sections
Szegedi et al. (2021) 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 $(\alpha,xn)$ inclusive
cross–sections between 8.9–13.2 MeV in the center of mass Ong et al. (2022).
Moreover, while new $(\alpha,n)$ cross–sections have been reported in the
energy range 11–32 MeV Choudhary et al. (2022), too, only the lowest energy
point has been considered hereafter to avoid additional PE effects within this
discussion.

Except for the highest energy, the $(\alpha,xn)$ cross–sections calculated in
this work using the $\alpha$-particle OMP Avrigeanu et al. (2014) agree with
the most recent measured data Ong et al. (2022); Choudhary et al. (2022)
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
$(\alpha,n)$ activation cross–sections of Ref. Szegedi et al. (2021) are
significantly underestimated. 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 Avrigeanu et al. (2014). However, there is now a
constant increase in the calculated $(\alpha,n)$ 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 96Zr and 100Mo, 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 $\alpha$-particle energy, should be investigated further.

Nonetheless, the newer data are already well described by the
$\alpha$-particle OMP Avrigeanu et al. (2014) 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. Sauerwein et al. (2011), based on Ref.
Avrigeanu and Avrigeanu (2010), as well as more recently Le and Hung (2022);
Basak, Dipali and Basu, Chinmay (2022). 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 McFadden and Satchler (1966). This may account for the quite
different values obtained for the ’diffuseness’ of this Fermi–type function.

Figure 11:  (Color online) As Fig. 3, of calculated cross–sections in this
work (solid curves), but for (a-c) the $(n,\gamma)$, $(n,2n)$, and $(n,xp)$
reactions on 93Nb, and (d) 14.5 MeV neutron–induced proton spectrum Grimes et
al. (1978); Traxler et al. (1985); Fischer et al. (1988); Ye et al. (1994);
Kokoo et al. (1999) with $(n,p)$ (short-dotted) and $(n,n^{\prime}p)$ (dotted)
CN, and PE (dashed) components.

###  III.2 Nucleon–emission induced by neutrons on 93Nb

There is a large amount of experimental data for neutron interaction with 93Nb
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’
Gruppelaar and Nagel (1985) in nuclear model calculations (e.g., Gadioli and
Hodgson (1992); Kalbach (2000); Watanabe et al. (1995); Demetriou et al.
(1996); Kalbach (1998)). However, the scattered data for the $(n,\alpha)$
reaction on this single Nb natural isotope suggest that more precise
measurements are needed to settle its evaluation (e.g., Refs. Ichihara (2016);
Liang et al. (2017)).

Therefore, a prerequisite for a consistent discussion of the $\alpha$-particle
emission is a suitable account, by using the actual parameter set, of all
competing nucleon-emission data for neutrons incident on 93Nb. The same is the
case for 92Mo, in completion of the earlier study of neutron–induced reactions
on Mo stable isotopes Reimer et al. (2005). Upon due consideration of these
two nuclei, $\alpha$-emission analysis will be feasible for Zr, Nb, and Mo
nuclei altogether.

The $(n,\gamma)$ cross–section analysis should be considered first because of
the significant isomeric-states activation by neutrons on 93Nb. Thus, a former
validation of the $\gamma$-ray 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).
The GLO and especially the SLO models provide calculated cross–sections as
well as $\Gamma_{\gamma}$ values that are much larger.

Figure 12:  (Color online) As Fig. 11(b) but for $(n,xp)$ and $(n,2n)$
reactions on 92Mo exf .

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 92Nb 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 $\sim$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 93Nb 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 cross–sections 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
93Nb.

###  III.3 Nucleon–emission induced by neutrons on 92Mo

The earlier systematic investigation of neutron–induced reactions on Mo stable
isotopes up to 20 MeV Reimer et al. (2005) 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 $\alpha$-emission data correspond merely to
four 92,95,98,100Mo 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 92Mo. 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 Molla and
Qaim (1977).

The $(n,p)$ reaction analysis for the isomeric–state 92Nbm 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. Koning and Delaroche
(2003).

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 energies, 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.

###  III.4  $\alpha$-emission spectra and excitation functions

####  III.4.1 Angle–integrated energy spectra analysis

Figure 13:  (Color online) As Fig. 11(d) but for $\sim$14.5 MeV
neutron–induced $\alpha$ spectra on (a,c) 90Zr and 92Mo Haight et al. (1981),
and (b) 93Nb Kokoo et al. (1999); Grimes et al. (1978); Fischer et al. (1982),
and DR component (dash-dotted).

First, the $\alpha$-particle angle–integrated energy distributions induced by
neutrons on 90Zr Haight et al. (1981), 93Nb Kokoo et al. (1999); Grimes et al.
(1978); Fischer et al. (1982), and 92Mo Haight et al. (1981) 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 $\alpha$-particle PE emission corresponding to (i) the
above–mentioned $\alpha$-particle s.p.l. density $g_{\alpha}$ related to the
level-density parameter $a$, and (ii) $\alpha$-particle pre-formation
probability $\varphi$ 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 $(n,\alpha)$ excitation functions discussed below, suggests a
corresponding $\Delta$$\varphi$$\sim$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 $\alpha$-particle angular distributions corresponding to
one–nucleon pickup reactions $(^{3}He,\alpha)$ or $(t,\alpha)$, and the
spectator proton–pair SF Glendenning (1975) (Sec. II.2).

Figure 14:  (Color online) As Fig. 11 but for $(n,x\alpha)$ reaction on
90,91,92,94,96Zr exf , DR (dash-dotted curve) and PE+CN (dashed) components,
and PE+CN uncertainty bands related to either residual–nuclei LD (gray) or PE
parameter $\varphi$ (light–gray).

The same residual–nucleus levels have been considered within the analysis of
both the $\alpha$-particle angular distributions of the one–nucleon pickup
reactions $(^{3}He,\alpha)$ or $(t,\alpha)$, and the present $(n,\alpha)$
pickup excitation functions. As a result, the most consistent DR component
corresponds to the 43 levels of 87Sr residual nucleus up to $\sim$6 MeV
excitation energy, which is important for calculating the
90Zr$(n,\alpha)^{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 $\sim$ 2.5 MeV excitation energy have
been considered within analysis of the $\alpha$-particle angular distributions
from 91Zr$(t,x\alpha)^{89,90}$Y, as shown in Fig. 7. As a result, a
significant DR component within only $\sim$2 MeV at the end of the spectrum of
93Nb$(n,\alpha)^{90}$Y reaction is shown in Fig. 13(b).

Moreover, the DR component of 20 levels up to 3.572 MeV excitation of 89Zr
residual nucleus, formerly taken into account in 90Zr$(^{3}He,\alpha)^{89}$Zr
angular–distribution analysis (Fig. 8), is also less important for the
$\alpha$-particle energy spectrum of 92Mo$(n,\alpha)^{89}$Zr reaction [Fig.
13(c)].

####  III.4.2 Zr isotopes

90Zr$(n,\alpha)^{87}$Srm 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 $\sim$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 87Sr (Table 1),
or the above–mentioned $\Delta\varphi$=0.02 incertitude of the main PE
parameter. Both of them are minimal at these energies and to $\sim$20 MeV due
to the low spin of 87Srm isomer, leading to a reduced side– and
cascade–feeding. Nonetheless, the absence of other HF+PE uncertainty factors
on calculated cross–sections at low energies indicates that the appropriate
data account supports the $\alpha$-particle OMP Avrigeanu et al. (2014).

Figure 15:  (Color online) As Fig. 14 but for the target nucleus 93Nb exf  and
(a) 93Nb$(n,n^{\prime}\alpha)^{89}$Ym reaction (short dash-dotted curve).

The agreement of the measured and calculated cross–sections at higher
energies, however, does not follow the already lower DR contribution, but
rather an increased PE component in comparison to our previous analysis
Avrigeanu and Avrigeanu (2017). It follows the above–mentioned
$\alpha$-particle s.p.l. density $g_{\alpha}$ value, proved in the present
work by extended energy–spectra analysis and 93Nb target–nucleus case.
Nonetheless, the appropriate account of the excitation–function upper side
supports both this $g_{\alpha}$ value and the parameter $\varphi$–value
firstly suggested by 14–MeV spectra analysis.

91Zr$(n,n\alpha)^{87}$Srm excitation function [Fig. 14(b)] stands as a fine
case of the same nucleus in a different reaction channel and $\alpha$-particle
energy range. The activation of 87Sr in neutron–induced reactions on 91Zr
nucleus, through $(n,$$\alpha$$n)$ reaction is more than 10 times larger in
comparison with that by $(n,n^{\prime}\alpha)$ reaction, at incident energies
of 15-21 MeV. Thus, the suitable agreement of measured Semkova et al. (2010)
and calculated cross–sections of this reaction, in the absence of DR effects,
has supported the $\alpha$-particle OMP Avrigeanu et al. (2014) once more.
Actually this conclusion has already been reachable within the previous
analysis for Zr isotopes Avrigeanu and Avrigeanu (2017), without DR
consideration, which provided a good account only for this reaction.

92Zr$(n,\alpha)^{89}$Sr excitation function analysis has the drawback of no SF
available for the residual nucleus 89Sr 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 92Mo$(n,\alpha)^{89}$Zr reaction,
mentioned in Sec. II.2 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 $(n,\alpha)$ reaction
on 92Zr.

94Zr$(n,\alpha)^{91}$Sr excitation function was measured within several
experiments, as shown in Fig. 14(d), but proper SFs for the residual nucleus
91Sr 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 $\sim$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 $\alpha$-emission on Zr isotopes Avrigeanu and Avrigeanu
(2017) for the $\alpha$-particle OMP Avrigeanu et al. (2014). 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 90Y in the $(n,\alpha)$ reaction on 93Nb (Sec. II.2 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
94Zr$(n,\alpha)^{91}$Sr calculated excitation function remains unchanged. On
the other hand, these DR cross–sections are, as for the target nucleus 90Zr,
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 Avrigeanu and Avrigeanu (2017) by using the $\alpha$-emission OMP
Avrigeanu et al. (1994).

96Zr$(n,\alpha)^{93}$Sr excitation function [Fig. 14(e)] has the same
attributes as those for the target nucleus 92Zr. 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 Avrigeanu and
Avrigeanu (2017) is related again to the above–mentioned PE contribution
change due to the use of the $\alpha$-particle s.p.l. density $g_{\alpha}$
value. No further effect of an eventual DR contribution may be considered, the
expected outcome being the same as for 94Zr.

####  III.4.3 93Nb target nucleus

The DR component, which is shown at the high–energy end of the
$\alpha$-particle energy spectrum in Fig. 13(b), may explain the minor pickup
DR contribution to 93Nb$(n,\alpha)^{90}$Y reaction total cross–sections [Fig.
15(a)]. The same is true for the 7+ isomer of the residual nucleus 90Y 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 90Ym activation, is
fully appropriate to the activation of the 9/2+ isomer via
93Nb$(n,n^{\prime}\alpha)^{89}$Ym 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 $\alpha$-particle OMP Avrigeanu et al.
(2014).

####  III.4.4 Mo isotopes

Figure 16:  (Color online) As Fig. 14 but for the target nuclei 92,95,98,100Mo
exf  and like–GQR (dash-dot-dotted) eventual components.

92Mo$(n,\alpha)^{89}$Zr excitation function analysis reveals a notable
balancing of the DR and PE+CN mechanisms for neutron–induced $\alpha$-emission
on light isotopes of elements. Thus, the former brings about $\sim$1 mb around
14 MeV, for SFs discussed in Sec. II.2, while the measured cross–sections
amount to several tens of mb Molla and Qaim (1977). 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 $\alpha$-emission beyond the
DR+PE+CN cross–sections, in neutron–induced reactions on $A$$\sim$60 nuclei
Avrigeanu and Avrigeanu (2021, 2022), was found around the GQR energies
$E_{GQR}$=$65A^{-1/3}$ MeV Speth and van der Woude (1981) of the related
excited nuclei. Thus, $\alpha$-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 Speth and van der
Woude (1981), we have called them only like–GQR components. On the other hand,
no enhancement beyond the DR+PE+CN cross–sections 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 energy for the target nucleus 92Mo is 6.3 MeV,
so that an eventual like–GQR contribution at the actual extra–yield maximum
around $\sim$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 93Mo, and a width of 2.35 MeV Avrigeanu and
Avrigeanu (2021), 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.

95Mo$(n,\alpha)^{92}$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 Rapp et al. (2003) but also recent data between 4 and 6 MeV Zhang
et al. (2010). 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 96Mo.

First, the present results of the $\alpha$-particle OMP Avrigeanu et al.
(2014) are in obvious agreement with the average trend of the data Rapp et al.
(2003) just above the discrete–resonance energy range exf . The extent of this
agreement could be pointed out by the factor of $\sim$5 between various OMP
predictions formerly considered in this respect (Fig. 3 of Ref. Rapp et al.
(2003)). But more important is that also close to these data were the results
provided by the first version Avrigeanu et al. (2003) of the OMP Avrigeanu et
al. (2014), i.e. based only on the $\alpha$-particle elastic–scattered
analysis (Fig. 1 of Ref. Avrigeanu et al. (2006)). It is also worth noting
that somewhat similar agreement shown in the latter figure by the quite
different $\alpha$-particle OMP Avrigeanu et al. (1994), which is only related
only to neutron–induced $\alpha$-emission from $A$$\sim$60 nuclei. This fact
may suggest that the simple OMP Avrigeanu et al. (1994) 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 92,98Mo using the SFs discussed in Sec. II.2. 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 ($\sigma$), in Fig. 16(b). At the same time, a much larger
underestimation of the more recent data around the incident energy of 5 MeV
Zhang et al. (2010) 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 96Mo. A peak cross–section of 0.6 mb and a width
of 2.35 MeV are given by the fit of the measured data Zhang et al. (2010). It
is worth noting that there are no NLD and PE effects at these energies, the
subsequent uncertainty bands becoming visible only at higher excitation. 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. Zhang et al. (2010) support the assumption of like–GQR
$\alpha$-particle decay.

98Mo$(n,\alpha)^{95}$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.2 is
significantly larger up to $\sim$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 Rahman and Qaim (1985) 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 99Mo, and the peak cross–section of 0.8 mb as
well as the width of 2.35 MeV. The assumption of the like–GQR
$\alpha$-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
Koning et al. (2019) in Fig. 16(c), is leading to a large overestimation of
the measured data at higher energies.

100Mo$(n,\alpha)^{97}$Zr excitation function [Fig. 16(d)] analysis is very
similar to that of 98Mo, with the exception of no measured data below 10 MeV.
The major drawback, however, of the lack of SFs needed for an accurate pickup
DR cross-section estimation, has been overcome similarly to the case of 95Mo
target nucleus. Thus, extrapolating the related components of 92,98Mo nuclei
yielded results that are rather close to those for 98Mo. 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 $\alpha$-emission 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 101Mo. The
width of 2.35 MeV and an assumed peak cross–section of 0.5 mb, close to that
of 98Mo, have provided a possible shape to be confirmed or not by further
measurements. Nevertheless, its proof could be favored by the isotope effect
Molla and Qaim (1977) of significantly lower $(n,\alpha)$ cross–sections for
heavier isotopes, similarly to Ni isotope chain Avrigeanu and Avrigeanu
(2022).

##  IV Conclusions

A recent assessment of a previous $\alpha$-particle OMP Avrigeanu et al.
(2014) also for nucleon–induced $\alpha$-emission on $A$$\sim$60 nuclei,
including pickup DR and eventual GQR $\alpha$-emission Avrigeanu and Avrigeanu
(2021, 2022), 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 $\gamma$ 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 cross–sections of $(\alpha,\gamma)$ reactions on
90,91,92Zr as well as $(\alpha,n)$ on 96Zr and 100Mo nuclei.

On the other hand, the pickup contributions to $(n,\alpha)$ reactions have
been determined within the DWBA method using the code FRESCO Thompson (1988).
The one–step reaction has also been considered through the pickup of 3He
cluster while the ”spectator model” Smits and Siemssen (1976); Smits et al.
(1979) was involved for the two transferred protons in $(n,\alpha)$ reaction.
However, the lack of measured $\alpha$-particle angular distribution for the
$(n,\alpha)$ reactions within this work made possible only straightforward
DWBA calculations of related pickup cross–sections. Thus, the spectroscopic
factors of Glendenning Glendenning (1975) have been used for the spectator
proton pair Gadioli et al. (1986, 1988); Smits et al. (1979), in addition to
SF for the picked neutron that becomes thus responsible for the
angular–momentum transfer. The latter have been obtained through analysis of
$\alpha$-particle angular distributions of one–nucleon pickup reactions
$(^{3}He,\alpha)$ or $(t,\alpha)$ 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 $\alpha$-emission. In
this respect, the previous analyses for Zr Avrigeanu and Avrigeanu (2017) and
Mo Reimer et al. (2005) are completed by a similar work for 93Nb and newer
measured data for the even–even semi–magic nucleus 92Mo. Then, an appropriate
description of the $\alpha$-particle angle–integrated energy distributions
induced by neutrons on 93Nb, 90Zr, and 92Mo has been concerned for validation
of $\alpha$-particle PE and DR account.

Finally, an increase of the $\alpha$-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 $\alpha$
particles by the same potential Avrigeanu et al. (2014), 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 Avrigeanu and Avrigeanu (2017) and Mo Avrigeanu et al. (2006)
before DR and like–GQR decay consideration, due to outstanding cross–section
measurements for 91Zr$(n,n\alpha)^{87}$Srm Semkova et al. (2010) and
95Mo$(n,\alpha)^{92}$Zr Rapp et al. (2003); Zhang et al. (2010) reactions. At
the same time it is suggested that the simple OMP Avrigeanu et al. (1994)
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
$\alpha$-emission.

## Acknowledgments

This work has been partly supported by The Executive Unit for the Financing of
Higher Education, Research, Development and Innovation (UEFISCDI) (Project No.
PN-III-ID-PCE-2021-0642) and carried out within the framework of the
EUROfusion Consortium, funded by the European Union via the Euratom Research
and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and
opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Commission.
Neither the European Union nor the European Commission can be held responsible
for them.

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