aa r X i v : . [ nu c l - t h ] S e p Alternate Solution of the α -Potential Mystery V. Avrigeanu ∗ and M. Avrigeanu † Horia Hulubei National Institute for Physics and Nuclear Engineering,P.O. Box MG-6, 077125 Bucharest-Magurele, Romania
A consistent set of statistical-model input parameters, validated by analysis of various independentdata, makes possible the assessment of an α -particle optical model potential [Phys. Rev. C ,044612 (2014)] also for nucleon-induced α -emission within A ∼
60 mass-number range. The advantageof recent data for low-lying states feeding is taken as well. Consideration of additional reactionchannels leading to increase of the α -emission beyond the statistical predictions has concerned thepickup direct interaction and Giant Quadrupole Resonance similar features. PACS numbers: 24.10.Eq,24.10.Ht,24.30.Cz,24.60.Dr
Introduction. – The α -particle interaction with nu-clei as well as the corresponding optical model potential(OMP) were of special interest from nuclear-physics ear-liest days. The widely-used phenomenological OMP pa-rameters have been derived from analysis of either elastic-scattering, which is ruled out below the Coulomb bar-rier B , or α -induced reaction data. They are then usedto describe also the α –emission from excited nuclei, interms of the statistical Hauser-Feshbach (HF) [1] and pre-equilibrium emission (PE) [2] models. However, there arealso various assumptions and parameters of these modelsin addition to the α -nucleus potential. Thus, a definiteconclusion on α -potential may become possible only us-ing HF+PE consistent parameter sets already validatedby analysis of other independent data distinct from either α -induced reaction or especially α –emission data.Moreover, there is a so-called α -potential mystery ofthe account at once of both absorption and emission of α -particles in nuclear reactions [3], of equal interest for nu-clear astrophysics and fusion technology. It was referredby Rauscher [3] to competition of the Coulomb excita-tion (CE) with the compound nucleus (CN) formation in α -induced reactions below B , while the former does notaffect the α –emission. Because the corresponding partialwaves and integration radii may provide evidence for dis-tinct account of CE cross section and OM total-reactioncross section [4], an alternate solution may also concernadditional reaction channels but for α –emission. Thus,further consideration should be given to eventual roleof the pickup direct interaction (DI) leading to increaseof the α -emission beyond the HF+PE predictions. Be-sides, a reaction cross-section enhancement that could berelated to the position of a giant quadrupole resonance(GQR) is also worthy of note [5].In fact, a former search for new physics in potentialsto describe nuclear de-excitation assumed that particleevaporation occurs from a transient nuclear stratosphereof the emitter nucleus ([5] and Refs. therein). Recentdefinite conclusion on incident α -particle OMP [6] made ∗ Electronic address: [email protected] † Electronic address: [email protected] feasible the analysis of a possible difference between theOMPs describing either α -particle elastic scattering andreactions or α -emission from excited nuclei [7]. On theother hand, OMP [6] validation for α -emission in low-energy proton-induced reactions on Zn isotopes was re-lated to a surface character [8], at variance with the fast-neutron induced reactions on Zr isotopes [9]. Analysis of α –emission from nuclei excited in reactions induced byeither neutrons or low-energy protons became thus of in-terest while there are quite useful recent data of low-lyingstates feeding [10–15] for A ∼
60 nuclei.Consequently, it is of interest to find out also underthese conditions if the α -emission could be described bythe same OMP [6] which led to a rather good accountof α -induced reaction data [6, 16–19]. It provided in themeantime better results also within large-scale nuclear-data evaluation [20], being adopted as default optionwithin the world-wide used code TALYS [21]. Never-theless, use of a consistent model parameter set, withno empirical rescaling factors of the γ and/or neutronwidths, is essential to avoid compensation effects. Hencethe main HF, PE, and DI assumptions and parametersas well as independent data analysis to support them aregiven in the Supplemental Material [22]. DI pickup in addition to PE+CN α -emission. – Thedue consideration of DI role for α -emission in nucleon–induced reactions is obviously leading to an increase be-yond the PE+HF predictions. Actually, Qaim et al. [23]extended at incident energies ≤
15 MeV the conclusion of’90s that the pickup instead of knockout has the main DIcontribution to low-lying states in ( p, α ) and ( n, α ) reac-tions ([2, 24, 25] and Refs. therein). However, Qaim etal. normalized the semimicroscopic pickup contributionto their N( p, α ) Co data at 15 MeV, with results thatdepend notably on the former PE+HF component [22].Hence, an increased predictive power of the model resultsdemands the primary use of spectroscopic data.In the present work, the pickup contribution to ( p, α )and ( n, α ) reactions was determined within the dis-torted wave Born approximation (DWBA) method usingthe code FRESCO [28] and the same OMPs as withinHF+PE analysis. One–step reaction was consideredthrough the pickup of H and He clusters, respectively. Ni(p, ) Co Levkovski (1991) Qaim+ (1995) PE: (p, x) PE+CN DI (PICKUP) DI+PE+CN (b)a=6.45(35)N d =70(2)6 9 12 15 1810100 ( m b ) E (MeV)61
Ni(p, ) Co Tanaka+ (1972)Sudar+ (1993) AHA (1994) A+(2014)(a) TENDL-2019 a=6.6(2),N d =43(2) =0.05(2) FIG. 1: (Color online) Comparison of (a,b) measured ( p, α ) excitation functions on , Ni [26], evaluated (short-dashed curves)[27], and PE+CN calculated values using the α -particle OMPs of Refs. [6] (dashed curves) and [7] (short-dash-dotted curves),and (b) total PE α -emission (dash-dot-dotted curve), DI pickup (dash-dotted curve), and sum (solid curve); uncertainty bandscorrespond to error bars of NLD parameter a (with uncertainties in units of the last digit [22]) and low-lying level cumulativenumber N d of residual nuclei (gray band), and (a) for GDH α -particle pre-formation probability ϕ (orange band). Moreover, the ”spectator model” [29, 30] was involved,where the two transferred either neutrons or protons in( p, α ) and ( n, α ) reactions, respectively, are coupled tozero angular momentum acting as spectators, while thetransferred orbital ( L ) and total ( J ) angular momentaare given by the third, unpaired nucleon of the trans-ferred cluster. The prior form distorted–wave transitionamplitudes, and the finite–range interaction were consid-ered, with more details in [22]. Finally, 19 states [30–32],until the excitation energy of ∼ p, α ) re-action threshold, the DI pickup component is aroundhalf the CN one but an order of magnitude larger thanPE contribution. The major role of DI pickup onlow-lying residual states, at lower incident energies, isthus proved. The PE component corresponds to theGeometry-Dependent Hybrid (GDH) model [33], gener-alized through inclusion of the angular-momentum andparity conservation [34] and the phenomenological knock-out model of α –emission based on a pre-formation prob-ability ϕ [2]. It increases with the incident energy, be-coming dominant at proton energies >
15 MeV.Second, the DI pickup inclusion is providing a suitabledescription of the measured data at all energies. Thispoint is particularly notable below 10 MeV, where thereis no effect of nuclear level density (NLD) parameters [22]as shown by the related uncertainty bands.Third, there is an obvious data agreement, within sev-eral MeV above the effective ( p, α ) reaction threshold, forthe CN component given by the α -particle OMP of Ref.[7]. This fact may explain the former results found inthis energy range but without DI account [7].The case of the ( p, α ) reaction on Ni is also shown in Fig. 1(a) making use of the CN isotopic effect of cross–section decreasing with the isotope mass increase [35].The lower similar dependence of the DI processes leads toan increased sensitivity of the ( p, α ) reaction on heavierisotope Ni to the pickup contribution. Moreover, thelarger cross sections for this case underlines the differencebetween predictions of the α -particle OMPs [6, 7], andthus support the former one. DI pickup contribution in α –emission spectra. – An ad-ditional check of DI pickup component of the α –emissionin neutron–induced reactions on Co is worthwhile, asfollows. First, there is a large data basis of total α –emission as well as ( n, α ) reaction cross sections overthree decades. Second, Co may be considered as abenchmark because of its medium asymmetry parameter( N − Z ) /A and corresponding isotopic effect. Third, thereare measured at the incident energy of ∼
14 MeV valu-able α –emission angular distributions and spectra [36–38]that are quite important for DI+PE analysis.Actually, only a tentative attempt was possible inthis work, using neutron spectroscopic factors from the Mn( d, p ) Mn reaction analysis [39, 40] and the specta-tor protons pair as picked from 1 f / subshell [22]. Then,27 excited states with well-known J π and transferred L [32, 41], until 2.088 MeV excitation energy, were consid-ered in the pickup cross–section calculation. The goodagreement proved by the analysis of angular distribu-tions for the α -energy bins, within center-of-mass system(c.m.), of 6-10 MeV, 10-12 MeV, and 12-14 MeV (Fig.26 of [22]), is completed hereafter by α -emission spectraand excitation function discussion.Nevertheless, the results in Fig. 2 support the presentattempt of the DI pickup contribution on the lowest-lyingstates of the odd–odd residual nucleus Mn. The data
Fischer+ (1986) [c.m.],E=14.1 Grimes+ (1996)[c.m.],E=14-14.5 Kokooo+ (1999) [lab], E=14.159
Co(n,x ) Mn d / d E ( m b / M e V ) E cm (MeV) CN PE PICKUP E n =14.3 MeV FIG. 2: Comparison of measured α -emission spectra from 14–14.5 MeV neutron–induced reactions on Co [36–38], and cal-culated DI pickup (dash-dotted curve), PE (dashed curves),CN first– (short-dash curve) and second–emission (dottedcurve) components at 14.3 MeV, and their sum (solid curve).
E (MeV) ( m b ) Co(n, ) Mn Liskien+ (1965) Liskien+ (1966) Ghorai+ (1980) Zupranska+(1980) Huang+ (1981) Fischer+ (1986) Meadows+(1987) Ikeda+ (1988) Li Tingyan+(1990) Molla+ (1994) Uno+ (1996) (n, )
TENDL-2019PE+CNDI (n,n’ )=0.04(2) D =2.3(4)N d =49(2) Grimes+(1996): x Mannhart+ (2007) Filatenkov+(2016)
FIG. 3: As Fig. 1 but for ( n, α ) reaction on Co [26] and un-certainty gray band for error bars of s -wave neutron-resonancespacing D exp and N d of residual nucleus Mn (Table I [22]),and additionally calculated cross sections of ( n, n ′ α ) reaction(thin dashed curve) and total α -emission (solid curve). agreement could be increased by taking into account thatdata transformation from laboratory system (lab) to c.m.may provide, as shown for Fe [42], a shift of up to 1MeV to higher emission energies.At the same time, this spectrum analysis provides ev-idence for rather low value of the GDH pre-formationprobability ϕ =0.04. An uncertainty band correspondingto half of it is assumed in the excitation function analysisfor both total α –emission and ( n, α ) reaction cross sec-tion (Fig. 3). The overlap of the related uncertainty bandof calculated cross section and recent data is obvious.On the other hand, Fig. 3 proves also that there are no NLD effects below the incident energy of 10 MeV.The α –particle OMP [6] is confirmed, with the additionalsupport of more data since earlier work [7]. Like-GQR in addition to DI+PE+CN α -emission. –The DI pickup contribution to ( n, α ) reaction on , Fecorresponds to 36 excited states up to 5.943 MeV for theresidual nucleus Cr, and 18 states up to 5.557 MeV for Cr [22]. A first proof of the present approach was givenby calculated angular distributions for the α -energy binsfrom 6 to 14 MeV, compared to data of Fischer et al. [42]for Fe at 14.1 MeV (Fig. 23 of [22]). It is of interestin the following to note the best agreement obtained forthe medium-energy α –emission, i.e. from 10 to 12 MeV.The most significant feature of the comparison of mea-sured and calculated α –emission spectra (Fig. 4) is theDI pickup contribution on lowest states. Then, somespectra overestimation there is at the energies just be-low the DI maxima, at once with an underestimation ofthe low-energy side. However it could be related for Feto a rough attempt to transform data from lab [43] toc.m. by an average shift of 0.5 MeV to higher energies,at variance to the precise one for Fe [42]. There arealso known question marks at α –particle energies below ∼ et al. [42]. At the sametime, the corresponding experimental total α –emissioncross sections are among the lowest data, at their inci-dent energies, within the excitation functions (Fig. 5).Nevertheless, the DI pickup contribution is improvingthe description of these excitation functions except theenergy range below ∼
12 MeV. The uncertainty bandsrelated to number of low-lying levels, NLD, and PE pa-rameters [22] have already proved that none of their ef-fects are in order at these energies. Moreover, the above-mentioned best agreement obtained for the angular dis-tribution for α -energy bin of 6-10 MeV, at 14.1 MeVincident energy on Fe, is additionally supporting boththe CN approach and the involved α –particle OMP [6].Following a former suggestion [5], we have assumedthat such reaction cross-section enhancement could beunderstood as decay from giant resonances populated vianeutron capture. Although generally the decay of theGQR is observed with nucleon emission ([44] and Refs.therein), it has been shown ([45] and Refs. therein) thatan appreciable non-statistical decay through α –emissioncan occur. This assumption has been supported by theposition of the extra yield, beyond DI+PE+CN compo-nent sum, just at the GQR energy E GQR =65 A − / MeV[44], i.e. 17.092 and 16.89 MeV for , Fe excited nu-clei, respectively. Therefore, we have obtained a fit ofthis extra yield by addition of Gaussian distributions at E GQR , with FWHM widths of 2.35 and 3.54 MeV, andpeak cross sections of 8 and 10 mb, respectively (Fig. 5).It is obvious that the widths of these Gaussian distri-butions are much lower than the systematic ’best’ valueΓ=85 A − / MeV (Fig. 14 of Ref. [44]). They are evenlower than the Γ=17 . A − / MeV dependence that seemsto describe better the inelastic hadron scattering datawhich were obtained taking into account also other GRs Fe(n,x ) Cr d / d E ( m b / M e V ) E cm (MeV) Grimes+ (1979) [lab] Grimes+ (1979) [~c.m.] E n =14.8 MeV 4 6 8 10 12 140.1110 Fe(n,x ) Cr CN PE PICKUP Grimes+ (1979) [lab],E=14.8Grimes+ (1979) [c.m.]Fischer+(1984)[c.m.],E=14.1Sterbenz+ (1994) [lab],E=14 E n =14.2 MeV FIG. 4: As Fig. 2 but for , Fe target nuclei [26, 43], at incident energies of 14.8 and 14.2 MeV, respectively.
E (MeV) ( m b ) Fe(n, ) Cr Grimes+ (1979)Paulsen+ (1979)Greenwood (1987)Ikeda+ (1988)Lu+ (1989)Meadows+ (1991)Saraf+ (1991)Meadows+ (1996)Gledenov+ (1997)Mannhart+ (2004)Wang+ (2015)Khromyleva+(2018) Bai+ (2019)
GQR PICKUP PE+CN TALYS-1.9 TENDL-2019 (a) 4 8 12 16 20110100 Fe(n,x ) Cr Grimes+ (1979) Paulsen+ (1981): Fe Fischer+ (1984) Saraf+ (1991) Matsuyama+(1993):Fe Sterbenz+ (1994) Haight+ (1996): Fe Kunieda+(2012): Fe Wang+ (2015) Bai+ (2019)
GQRPICKUPPE+CN (n, )
TALYS-1.9TENDL-2019PE+CN (n,n’ )(b)
FIG. 5: As Fig. 3 but for , Fe target nuclei and additionally calculated ( n, α ) cross sections using TALYS-1.9 and its defaultparameters [21] (short-dotted curves), and like–GQR components (dash-dot-dotted curves) added to DI+PE+CN (solid curves). [44]. Hence we would call these components only like–GQR contributions. Further conclusions on the physicsbehind this empirical addition are not yet evident, whilemore similar cases to be concerned may help. Neverthe-less, a suitable account of the measured ( n, α ) reactionand α -emission cross sections (Fig. 5) is obtained by thisadditional like–GQR contribution which are larger thanthe DI pickup for incident energies <
12 MeV.
Conclusions. – A consistent set of model parameters,validated by analysis of various independent data, makespossible the assessment of an optical model potential[6] also for nucleon-induced α -emission for A ∼
60 nuclei.Particularly, α –emission from , , Fe and Co iso-topes excited by ( n, α ) reaction, and , , , Cu and , , , Zn isotopes excited through both ( n, α ) and ( p, α ) reactions has been analyzed. The advantage ofrather recent data of low-lying states feeding is essential.Further consideration of additional reaction channelsleading to increase of the α -emission cross sectionsbeyond the statistical predictions has concerned the DIpickup and GQR similar features.This work was partly supported by AutoritateaNationala pentru Cercetare Stiintifica (Project PN-19060102) and carried out within the framework of theEUROfusion Consortium and has received funding fromthe Euratom research and training programme 2014-2018and 2019-2020 under grant agreement No 633053. Theviews and opinions expressed herein do not necessarilyreflect those of the European Commission. [1] W. Hauser and H. Feshbach, Phys. Rev. , 366 (1952).[2] E. Gadioli and P. E. Hodgson, Pre-Equilibrium NuclearReactions (Clarendon, Oxford, 1992).[3] T. Rauscher, Phys. Rev. Lett. , 061104 (2013).[4] V. Avrigeanu, M. Avrigeanu, and C. M˘an˘ailescu,arXiv:1605.05455v1 [nucl-th].[5] M. Avrigeanu, W. von Oertzen, and V. Avrigeanu, Nucl.Phys. A , 246 (2006).[6] V. Avrigeanu, M. Avrigeanu, and C. M˘an˘ailescu, Phys.Rev. C , 044612 (2014).[7] V. Avrigeanu, P. E. Hodgson, and M. Avrigeanu, Phys.Rev. C , 2136 (1994).[8] V. Avrigeanu and M. Avrigeanu, Phys. Rev. C ,064611 (2015).[9] V. Avrigeanu and M. Avrigeanu, Phys. Rev. C ,044610 (2017).[10] G. Zhang, H. Wu, J. Zhang, J. Liu, J. Chen, Yu. M.Gledenov, M. V. Sedysheva, G. Khuukhenkhuu, and P.J. Szalanski, Eur. Phys. J. A , 1 (2010).[11] G. Zhang, Y. M. Gledenov, G. Khuukhenkhuu, M. V.Sedysheva, P. J. Szalanski, J. Liu, H. Wu, X. Liu, J.Chen, and V. A. Stolupin, Phys. Rev. C , 054619(2010).[12] Y. M. Gledenov et al. , Phys. Rev. C , 064607 (2014).[13] Z. Wang, X. Fan, L. Zhang, H. Bai, J. Chen, G. Zhang,Y. M. Gledenov, M. V. Sedysheva, L. Krupa, and G.Khuukhenkhuu, Phys. Rev. C , 044601 (2015).[14] T. Khromyleva, I. Bondarenko, A. Gurbich, V.Ketlerov, V. Khryachkov, and P. Prusachenko, Nucl.Sci. Eng. , 282 (2018)[15] H. Bai et al. Phys. Rev. C , 024619 (2019).[16] M. Avrigeanu, A. C. Obreja, F. L. Roman, V.Avrigeanu, and W. von Oertzen, At. Data Nucl. DataTables , 501 (2009).[17] M. Avrigeanu and V. Avrigeanu, Phys. Rev. C ,014606 (2010).[18] V. Avrigeanu and M. Avrigeanu, Phys. Rev. C ,024621 (2016).[19] V. Avrigeanu and M. Avrigeanu, Phys. Rev. C , 733 (1995).[24] E. Gadioli and P. E. Hodgson, Rep. Prog. Phys. , 247(1989).[25] E. Gadioli, S. Mattioli, W. Augustyniak, L. G lowacka,M. Jaskoa, J. Turkiewicz, and A. Chiadli, Phys. Rev. C TENDL-2019:TALYS -based evaluated nuclear data library , https://tendl.web.psi.ch/tendl_2019/tendl2019.html [28] I. J. Thompson, Comput. Phys. Rep. , 167 (1988); v.FRES 2.9 (2011).[29] J. Smits and R. Siemssen, Nucl. Phys. A , 385(1976). [30] J. Smits, R. Siemssen, S. Van Der Werf, and A. Van DerWoude, Nucl. Phys. A , 29 (1979).[31] P. L. Jolivette and C. P. Browne, Phys. Rev. C , 1475(1983); M. Blann, Nucl. Phys. A , 570 (1973).[34] M. Avrigeanu, M. Ivascu, and V. Avrigeanu, Z. Phys.A – Atomic Nuclei , 299 (1990).[35] N. Molla and S. Qaim, Nucl. Phys. A , 269 (1977).[36] R. Fischer, G. Traxler, M. Uhl, H. Vonach, and P.Maier-Komor, Phys. Rev. C , 460 (1986).[37] S. M. Grimes et al. Nucl. Sci. Eng. , 271 (1996).[38] Kokooo, I. Murata, and A. Takahashi, Nucl. Sci. Eng. , 16 (1999).[39] H. Junde, H. Su, and Y. Dong, Nucl. Data Sheets ,1513 (2011).[40] J. R. Comfort, Phys. Rev. , 1573 (1969).[41] R. Capote et al. , Nucl. Data Sheets , 72 (1984).[43] S. M. Grimes, R. C. Haight, K. R. Alvar, H. H.Barschall, and R. R. Borchers, Phys. Rev. C , 2127(1979).[44] J. Speth and A, van der Woude, Rep. Prog. Phys. ,719 (1981).[45] M. Fallot et al. , Nucl. Phys. A , 106 (2007). Supplement to ”Alternate Solution of the α -Potential Mystery” V. Avrigeanu and M. Avrigeanu
Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, 077125Bucharest-Magurele, Romania
Details of the consistent set of statistical-model inputparameters in the main paper, and its validation by ana-lyzing various independent data, are given in this supple-ment. The results obtained using an incident α -particleoptical model potential [Phys. Rev. C , 044612 (2014)]within A ∼
60 mass-number range, by taking the advan-tage of rather recent data of low-lying states feeding, arethen discussed. Further consideration of an additionalreaction channel leading to an increase of the α -emissiondata beyond the statistical model predictions has alsoconcerned the pickup direct interaction (DI). The assess-ment of DI cross sections has been subject to availableinformation on spectroscopic factors related to populatedstates, outgoing particle angular distributions, or at leastdifferential cross–section maximum values. I. INTRODUCTION
The α -particle optical model potential (OMP) param-eters are usually derived from analysis of either elastic-scattering, above the Coulomb barrier B , or α -inducedreaction data (e.g., [1]) in terms of the statistical Hauser-Feshbach (HF) [2] and pre-equilibrium emission (PE) [3]models. Because there are also various assumptions andparameters of the HF+PE models in addition to the α -nucleus potential, only the use of a consistent parameterset (e.g., [4]) may provide a definite conclusion on the α -particle OMP. Details of the consistent set of inputparameters used within the main paper, and its valida-tion by analyzing various independent data, are given inthis supplement.Moreover, the account at once of both absorption andemission of α -particles in nuclear reactions [5] may facethe need for new physics in potentials to describe nuclearde-excitation ([6] and Refs. therein). Therefore, it is ofinterest to see whether the incident α -particle OMP [7] isable to describe also the α -emission from excited nuclei[6, 8] in reactions induced by low-energy protons [9] aswell as fast neutrons [10]. This aim could be particularlyachieved within A ∼
60 mass-number range, by taking theadvantage of rather recent data of low-lying states feeding[11–16] also discussed in this supplement.Further consideration of an additional reaction chan-nel [5, 17] but leading to an increase of the α -emissiondata beyond the HF+PE predictions is also aimed inthe following regarding the pickup direct interaction (DI)[3, 18–20]. An eventual giant quadrupole resonance(GQR) [21, 22] decay through α -particle emission ([23–27] and Refs. therein) is discussed in the main paper.Nevertheless, before proceeding to the above-mentioned points, it should be mentioned that no em- pirical rescaling factors of the γ and/or neutron widthsare used in the present work as well as to establish thisOMP [7, 28, 29] or prove it [30, 31]. First, its use pro-vided a rather good agreement of the calculated and mea-sured α -induced reaction data to date [7, 9, 10, 28–31]. Second, better results obtained also within large-scalenuclear-data evaluation [32] led to its adoption as thelatest default option within the world-wide used codeTALYS [33]. Consequently, conclusion on the suitableuse of this α -particle OMP also for the account of the α -emission will be of a larger interest.Actually, a semi-microscopic double-folding model(DFM) real part and the dispersive contribution of aphenomenological energy-dependent imaginary-potentialwere firstly involved within an analysis of α -particleelastic-scattering angular distributions above B [7, 28,34]. Subsequently, the phenomenological real potential[7] was established using the same data basis. Then,HF statistical model analysis of α -induced reaction crosssections proved the particular energy dependence of thesurface imaginary potential at incident energies below B [7, 28, 29].On the other hand, an earlier OMP [8] concernedonly the α -particle emission in neutron-induced reac-tions, with distinct predictions from potentials for inci-dent α particles [35]. Thus, it was not an earlier version ofthe above-mentioned ones [7, 28, 34], while the questionon different OMPs for incident and emitted α particles[6, 8–10] is still actual.While detailed presentation of model parameters wasgiven in Refs. [7, 9, 30], latest particular values of themare given in Sec. II of this supplement. The PE+HF re-sults obtained using the OMPs of Refs. [7, 8] are thencompared in Sec. III with measured cross sections of low-energy proton– and neutron–induced reactions leadingto excited Fe, Co, Cu, and Zn isotopes, with particu-lar emphasis of above-mentioned Refs. [11–16] in orderto provide more details and support to the additionalwork within the main paper. The main points of the DIanalysis within the distorted-wave Born approximation(DWBA) method and the code FRESCO [36], for calcu-lation of pickup reaction cross sections, are also given inSec. IV. II. STATISTICAL MODEL PARAMETERS
The following HF+PE model calculations were car-ried out within a local approach using an updated ver-sion of the computer code STAPRE-H95 [37], with ∼ + and 3 − collective states. The calculated DI cross sec-tions are then involved for the subsequent decrease of thetotal-reaction cross sections σ R that enter the HF calcula-tions. Typical DI inelastic-scattering cross sections, e.g.for neutrons on Fe, grow up from ∼
7% to ∼
10% of σ R for incident energies from 2 to 7 MeV, and then decreaseto ∼
8% at the energy of ∼
25 MeV.A consistent set of (i) back-shifted Fermi gas (BSFG)[41] nuclear level density (NLD) parameters, (ii) nucleonand (iii) γ -ray transmission coefficients was used alsowithin the present analysis of the fast-neutron inducedreactions on isotopes of Fe, Co, Cu, and Zn. Theseparameters were established or validated using indepen-dently measured data as low-lying levels [42] and s -wavenucleon-resonance spacings, ( p, n ) reaction cross sections[43], radiative strength functions (RSF) [44], average s -wave radiation widths [45], and ( p, γ ) reaction cross sec-tions [43], respectively. The same OMP and level densityparameters have been used in the framework of the DI,PE, and HF models. The details in addition to the onesgiven formerly [7, 9, 10, 30, 31] as well as particular pa-rameter values are mentioned below in order to providethe reader with all main details and assumptions of thepresent analysis.The reaction cross sections calculated within this workare also compared with the TALYS code results corre-sponding to its default options as well as the content ofthe evaluated data library TENDL-2019 [46], for an over-all excitation function survey. A. Nuclear level densities
The BSFG parameters used to obtain the present HFresults are given in Table I. They follow the low-lyinglevel numbers and corresponding excitation energies usedin the HF calculations (the 2nd and 3rd columns) [42] aswell as those fitted at once with the available nucleon-resonance data [41, 45, 47, 48] to obtain these param-eters. The level-density parameter a and ground state(g.s.) shift ∆ were generally obtained with a spin cutofffactor corresponding to a variable moment of inertia I ,between half of the rigid-body value I r at g.s., 0.75 I r atthe separation energy S , and the full I r value at the exci-tation energy of 15 MeV, with a reduced radius r =1.25fm [49].The fit of the error-bar limits of D exp data has alsobeen used to provide limits of the fitted a -parameters.Moreover, these limits are used within HF calculations toillustrate the NLD effects on the calculated cross-sectionuncertainty bands (Sec. III).On the other hand, the smooth-curve method [51] wasapplied for nuclei without resonance data. Thus, average a -values of the neighboring nuclei having these data hsvebeen used to obtain only the ∆ values by fit of the low- lying discrete levels. The uncertainties of these averaged a -values, following the spread of the fitted a parameters,are also given in Table I. These uncertainties are obvi-ously larger than those of the a -values obtained by fit of D exp . Their use in HF calculations leads to increased cal-culated cross-section uncertainty bands. Moreover, even-tual assumption of an additional uncertainty for the fit-ted N d has led to increased NLD parameter uncertaintiesas given in a second pair of brackets in Table I. B. Nucleon optical model potentials
1. Neutron optical model potentials
Fe isotopes’ first option for neutron OMP was obvi-ously the optical potential of Koning and Delaroche [52].However, we paid the due attention to the authors’ re-mark that their global potential does not reproduce theminimum around the neutron energy of 1–2 MeV for theneutron total cross sections of the A ∼
60 nuclei. Follow-ing also their comment on the constant geometry param-eters which may be responsible for this aspect, we appliedthe SPRT method [53] for determination of these param-eters at energies below ∼
20 MeV, through analysis of the s - and p -wave neutron strength functions, the potentialscattering radius R ′ and the energy dependence of neu-tron total cross section σ T ( E ).The RIPL-3 recommendations [45] for the neutron res-onance data and the available measured σ T data [43](Fig. 6) have been used in this respect. One may notethat more accurate data have become available sincethe development of Koning–Delaroche OMP. Moreover,it has been shown that even recent dispersive-coupled-channels OM results describing well the total cross sec-tions above 4 MeV, are still rather similar to [52] predic-tions at lower energies [54]. The model overestimationaround 1 MeV is up to ∼
20 %.Actually, it has been supported the well known be-havior for near–magic nuclei of the iron group that theobserved reduction in the averaged σ T ( E ) requires l -dependent OMPs [55]. We found however that it is pos-sible to describe somehow the σ T ( E ) minimum around1 MeV by adoption of the energy–dependent geometryparameters of either the global ( Fe) or local ( Fe)Koning–Delaroche potentials, at lower energies, given inTable II. This point will be of a particular importance forthe correct account of the neutron evaporation in com-petition with the charged–particle emission.
Cu isotopes have shown a rather likewise case, howeverwith a σ T ( E ) minimum around 2 MeV better approxi-mated by Koning–Delaroche OMP. A similar analysis asabove is given in [56], and their similarly modified localpotentials for , Cu [52] have used in the present workas well.
Zn isotopes proves an even better account of the σ T ( E )minimum above 2 MeV but a minor underestimationof ≤
3% for the maximum around 6 MeV. Both points
TABLE I: Low-lying levels number N d up to excitation energy E ∗ d [42] used in HF calculations, N d ( E ∗ d ) and s -wave nucleon-resonance spacings a D exp (with uncertainty in units of the last digit, in parentheses) in one or more (separated by slash) energyranges ∆ E [50] above separation energy S , for the target–nucleus g.s. spin I , fitted to obtain LD parameter a and g.s. shift∆ (for a spin cutoff factor related to a variable moment of inertia [49] between half and 75% of the rigid-body value, from g.s.to S , and reduced radius r =1.25 fm) with uncertainty related firstly to those of fitted D exp and, in addition, to those of fitted N d (in second pair of brackets). Nucleus N d E ∗ d Fitted level and resonance data a ∆ N d E ∗ d S + ∆ E I D exp (MeV) (MeV) (MeV) (keV) (MeV − ) (MeV) V 25 2.408 25 2.408 9.559 0 10.6(10) b V 43 2.534 43 2.534 5.95 -1.75 V 55 4.053 55 4.053 11.071 / 10.646 6 / 0 2.0(1) / 7.9(6) b V 32 2.591 32 2.591 7.361 7/2 4.0(6) 6.33 -1.13 V 20 2.772 24 2.967 5.75 -0.90 V 17 1.540 17 1.540 6.20 -2.20 Cr 13 2.613 12 2.578 5.50 -0.70 Cr 28 4.367 28 4.367 5.60 0.31 Cr 41 3.376 82(2) 4.214 9.655 0 12.5(12),13.1(12) c b Cr 36 5.139 36 5.139 5.30 0.47 Cr 28 3.435 25(2) 3.262 8.432 / 8.654 0 32.0(35) / 30.1(35) c Cr 38 4.689 38 4.689 9.817 / 10.001 3/2 6.7(6) / 5.81(59) c Cr 32 3.200 32 3.200 6.396 0 50(8) 6.47(24) -0.50(15)6.696 / 6.663 0 54.4(82) b / 52.6(77) c Cr 29 4.349 29 4.349 5.85 0.32 Mn 15 1.956 15 1.956 6.10 -1.20 Mn 33 3.466 42 3.728 5.60 -0.85 Mn 36 2.355 36(3) 2.355 5.7(2) -2.02(9) Mn 58 3.383 58 3.383 10.497 0 7.1(7) b Mn 49 2.118 49(2) 2.118 7.374 5/2 2.3(4) 6.20(42/-3)(37/0) -2.22(32/-3)(24/4) Mn 23 2.341 33 2.758 6.0(2) -1.32 Mn 27 1.470 27 1.470 6.40 -2.25 Fe 9 4.456 5 3.585 5.30 1.15 Fe 24 3.176 24(2) 3.176 5.47(20) -0.90(8) Fe 26 4.782 26(2) 4.782 5.65(20) 0.76(8) Fe 31 3.457 31(2) 3.457 9.548 0 18.0(24) b ,20.5(14) 5.53(14)(10) -0.90(12)(2) Fe 60 5.038 49(2) 4.802 6.0(2) 0.36(17)(12) Fe 30 2.697 30(2) 2.697 8.074 / 8.096 0 19.2(19) b / 25.4(19) 6.00(11)(6) -1.28(8)(3) Fe 60 4.720 60 4.720 10.14 1/2 7.05(70) 5.90 -0.90 Fe 32 2.570 36 2.856 6.756 / 6.696 0 21.6(26) / 25.4(49) b Fe 34 4.053 34 4.053 6.25 0.11 Fe 16 2.143 16 2.143 7.00 -0.72 Co 29 3.980 23 3.775 5.40 -0.33 Co 27 2.789 27 2.789 6.40 -0.88 Co 44 3.553 44 3.553 5.80 -0.97 Co 43 1.979 43(2) 1.979 6.6(2) -2.11(10) Co 60 3.492 61 3.497 10.217 0 4.3(4) b Co 60 2.423 64 2.489 7.542 7/2 1.45(15) 6.99(14) -1.57(10) Co 60 3.417 70(2) 3.575 6.45(35) -0.96(24) Co 17 1.359 16 1.271 7.40 -1.48 Co 11 2.163 8 1.889 7.30 -0.30 Ni 34 4.574 34 4.574 5.90 0.40 Ni 44 3.381 58 3.686 9.405 / 9.324 0 13.4(9) / 12.5(9) b Ni 51 4.613 51 4.613 2.0(7) 6.00 0.06 Ni 36 2.913 36 2.913 8.045 0 13.8(9),13.9(15) b Ni 46 4.455 46 4.455 10.631 3/2 2.10(15) 6.36 0.27 Ni 19 2.353 19 2.353 7.117 / 7.238 0 16(3) / 15(2) b Ni 20 3.849 20 3.849 6.90 0.75 Ni 20 2.520 20 2.520 6.398 0 23.6(30) 7.80 -0.20 Ni 18 3.782 18 3.782 7.50 1.12 Cu 38 3.758 38 3.758 6.30 -0.23 Cu 24 1.505 24 1.505 7.00 -1.75 Cu 36 3.042 38 3.092 6.75 -0.64 Cu 48 1.682 53 1.775 7.20 -2.04 Cu 48 3.043 48 3.043 9.026 0 5.9(7) b Cu 60 2.115 93 2.534 7.993 3/2 0.95(9) d [0.70(9)] 7.46 -1.61 Cu 40 3.132 48 3.278 7.85 -0.10 Cu 20 1.344 20 1.344 7.116 3/2 1.30(11) 7.88 -1.40 Cu 12 2.841 18 3.123 8.20 0.62 Zn 15 1.730 15 1.730 6.70 -1.25 Zn 23 3.730 23 3.730 6.50 0.32 Zn 32 2.403 32 2.403 7.50 -0.79 Zn 34 3.552 68 4.159 11.862 3/2 7.20 0.16 Zn 33 2.138 33 2.138 8.018 0 2.3(3) 8.29 -0.77 Zn 39 3.825 39 3.825 7.70 0.55 Zn 40 2.175 40 2.175 7.278 0 4.62(55) 8.11 -0.97 Zn 51 3.943 51 3.943 10.291 5/2 0.37(2) 8.05 0.59 a Reference [45] if not otherwise mentioned. b Reference [41] c Reference [47] d Reference [48] n + Fe T ( b ) E (MeV)
Guenther+ (1986)Cornelis+ (1995) (a) Koning+(2003): 54Fe 0.1 1 10 60234
TALYS-1.9TENDL-2019 n + Fe Perey+(1991) Cornelis+(1995) Berthold+(1995):Fe Abfalterer+(2001) Kim+(2007): Fe Beyer+(2018): Fe(b)Koning+(2003): 56Fe Koning+(2003) Koning+(2003)-mod.
FIG. 6: (Color online) Comparison of neutron total cross sections for , Fe measured [43] and calculated using either the global(dash-dotted curves) or local (dashed curves) OMP parameters sets of Koning and Delaroche [52], and the energy-dependentchanges of either (a) global or (b) local geometry parameters given in Table II (solid curves). Broad energy–averages over 50,100, and particularly [54] 200 keV of several measured data sets were used for comparison with the OMP results.TABLE II: Comparison of experimental [45] and calculated neutron scattering parameters of , Fe isotopes at neutron energiesof 250 and ∼
400 keV [50], respectively, and (bottom) the changes of the global ( Fe) and local ( Fe) parameters [52] (withthe use of [52] notations, the energies are in MeV and geometry parameters in fm) which provide the best SPRT results. \ Target Fe FeOMP S *10 S *10 R’ S *10 S *10 R’Exp. [45] 6.6(10) 0.42(8) 2.3(6) 0.41(6)global [52] 2.74 0.73 5.2 2.24 0.7 4.7local [52] 2.68 0.82 4.9 2.28 0.7 4.5[52] modified 2.25 0.45 4.1 1.72 0.46 3.5 r V =1.2766-0.04 E , E< r D =1.282+0.014 E , E< E , E< E , E> a V =0.2198+0.15 E , E< a V =0.333+0.066 E , E< E (MeV) ( m b ) Mn(p,n) Fe Johnson+ (1958)Albert (1959)Johnson+ (1964)Viyogi+ (1978)Mitchell+ (1983)Al-Abyad+ (2010)TENDL-201955
Mn(p, ) Fe Fe: a=5.53(10),N d =31(2)(b)0 3 6 9 12 15 18110100 R ad i a t i v e s t r eng t h f un c t i on ( G e V - ) E (MeV) Fe M1OCL (2004): 56Fe:OCL (2004): 57FeOCL (2013): 56Fe
E1 SLO 2480 GLO 2700 EGLO 1470 +M1 up GDR =147.0(54)CM1=[1-3]*10-8 (a)
FIG. 7: (Color online) Comparison of (a) measured [44] and sum of calculated M M Fenucleus, and (b) cross sections for ( p, γ ) and ( p, n ) reactions on Mn, measured [43], evaluated [46], and calculated using theabove-mentioned E1 radiation RSF models and curves for the former; uncertainties correspond to (a) E1-radiation EGLO/GDRparameters (light-gray band) and, in addition, for the M1-radiation upbend (orange band), and (b) the above-mentioned RSFuncertainties (orange band), for ( p, γ ) reaction, and Fe levels and NLD parameter uncertainties (light-gray band) for ( p, n )reaction. The s -wave average radiation widths Γ γ (in meV) are either deduced from systematics [45] or correspond to M1 andeach of above-mentioned E1 models. have been improved through the SPRT method, by usingKoning–Delaroche local parameter set for Ge [52] witha minor change of the real potential radius r V =1.214-0.001 E , for E< Co target nucleus made the object of a detailed studyof neutron-induced reactions [57] that was completedat nearly the same time with Koning–Delaroche OMP.However, a consistent input parameter set was also usedso that an earlier neutron OMP [58] was adopted withchanges according to the SPRT method as well. Becauseof the completeness of that study, only an additionalanalysis of the α -particle emission has been carried on inthe present work, with no further change of the adoptednucleon OMPs [57].
2. Proton optical model potentials
The optical potential of Koning and Delaroche [52] wasconsidered also for the calculation of proton transmissioncoefficients on isotopes of Mn, Fe, Ni, and Cu. A formertrial of this potential concerned the proton reaction crosssections σ R [59] for the stable isotopes of the elementsfrom Mn to Zn, for the lower energies so important instatistical emission from excited nuclei. A comparison ofthese data and results of either the local or global protonOMP [52] was shown in Fig. 2 of Ref. [56]. A goodagreement was found apart from the isotopes of Fe andin particular Ni, for which there is an overestimation of the data by ≥ Fe and Ni isotopes could benefit from the above-mentioned work [56] where the agreement with the corre-sponding σ R data was achieved using energy-dependentreal potential diffusivity. Thus, the constant value a V =0.663 fm [52] was replaced by the energy–dependentform a V =0.563+0.002 E up to 50 MeV, for the target nu-cleus Fe, and a V =0.463+0.01 E up to 20 MeV for Ni.A final validation of the additional energy–dependent a V was obtained for Ni isotopes of interest for this work, byanalysis of the available ( p, γ ) and ( p, n ) reaction data upto E p ∼
12 MeV (Fig. 4 of [56]), the other HF parametersbeing the same as in the rest of the previous as well aspresent work.Consequently the same modified OMP has been usedin this work for Fe isotopes. On the other hand, the en-larged incident-energy range for ( p, n ) reaction on , Niin the present work (Sec. III) led only to a minor changeof the local parameter sets for , Ni [52]. Thus, theenergy-dependent a V =0.463+0.004 E up to 50 MeV hasbeen used within the analysis of the above-mentioned re-action. Discussion of the corresponding calculated crosssections is given in the following section. Cu isotopes made the object of a similar analysis inthe meantime [30], to validate the proton OMP within ananalysis of quite accurate ( α, x ) data at lowest energies,where x stands for γ , n , and p . Following the resultsshown in Fig. 1 of Ref. [30], the same local OMP [60]1has been used also in this work. Mn isotopes have been additionally concerned in thepresent work due to the scarce related σ R data [59] andthe well-known anomalies of proton OMP within theseenergies and mass range [61]. Thus, an analysis of the( p, γ ) and ( p, n ) reaction cross sections was carried outfor Mn target nucleus and incident energies below ∼ p, γ ) re-action below the ( p, n ) reaction effective threshold, whereits cross section is closest to σ R .One may also note that the NLD parameter uncer-tainty related to the limits of the fitted D exp and low-lying levels of Fe (Table I) provides no significant ef-fect on the calculated ( p, n ) cross sections. Therefore theOMP global parameters [52] have been used for protonson Mn isotopes in this work. C. γ -ray strength functions Measured RSF and average s -wave radiation widthsΓ γ [45] data have already been largely used for the suit-able account of γ -ray transmission coefficients for com-pound nuclei (CN) with A ≥
60 [30]. Nevertheless, theisotopes , Fe are most significant for the present work,because of an unexpected RSF upbend to zero energy wasfirstly discovered for them [68]. However, despite moredetailed studies of additionally related data ([69] andRefs. therein), there is not yet an unique parametrizationfor these nuclei. Therefore, firstly, we have adopted therecently-compiled [70, 71] giant dipole resonance (GDR)parameters within the former Lorentzian (SLO) [72], gen-eralized Lorentzian (GLO) [73], and enhanced general-ized Lorentzian (EGLO) [74] models for the electric-dipole RSF. The constant nuclear temperature T f =1.2MeV of the final states [75] was particularly assumedwithin the EGLO model.Then, the SLO model has been used for the M1 ra-diation, with the global [45] GDR energy and width,i.e. E =41/ A / MeV and Γ =4 MeV. The related peakcross section σ =2 mb has been assumed at once with theabove-mentioned T f value in order to describe the RSFdata [44] around the M f up ( E γ )= C exp( − ηE γ ),with the parameter value η =0.8MeV − and limits C =(1–3) × − MeV − [76] [orange band in Fig. 7(a)]. The RSFdata are notably underestimated below ∼ M E R ad i a t i v e s t r eng t h f un c t i on ( G e V - ) Cu =400(150) [syst.] E1 SLO 1150 GLO 731 EGLO+Tf 435 E (MeV) Cu FIG. 8: (Color online) Comparison of the sum of calculated γ -ray strength functions of the E1 and M1 radiations for , Cu nuclei, using to the SLO (dash-dot-dotted curves),GLO (dash-dotted curves), and EGLO (solid curves) mod-els for E1 radiations, and SLO model for M1 radiations; s -wave average radiation widths Γ γ (in meV) are either de-duced from systematics [45] or correspond to M γ -ray strength functions for , Cu [62], Cu[63], Cu [48], Cu [64], , Zn [65], Ga [66], and Genuclei [67]. .
There is shown in the same figure the uncer-tainty propagation of the electric-dipole GDR parameter σ =(147 ± C onthe RSF energy dependence. While the former is impor-tant mainly above the nucleon binding energy but yetwithin the RSF-data errors, the latter is obviously essen-tial below E γ ∼ p, γ ) reac-tion cross sections corresponds to variations well withinthe data spreading as well as the effects due to variouselectric-dipole RSF models.2On the other hand, a comparison of the systematic es-timation [45, 69] and calculated Γ γ values correspondingto the three electric-dipole RSF models, as well as toaddition of the M1 upbend to the EGLO model, is alsoshown in Fig. 7(a). Obviously, it provides an increasedsupport to the latest E1 model.Therefore, the present results of RSF data analysis for Fe and related uncertainty propagation on calculatedreaction cross sections strengthen the previous RSF ap-proach for A ≥
60 (Fig. 2 of Ref. [30]).Use of the above-mentioned electric-dipole RSF modelsis also discussed in Sec. III for proton-induced reactionson , Ni while the corresponding RSFs are shown inFig. 8. There is an obvious difference between the mea-sured RSF data that are smaller for Cu [62] than for Cu [64]. We have described it within the EGLO modelusing different values of the constant nuclear tempera-ture T f . Thus, 0.7 MeV has been considered for Cu, asfor Zn isotopes [30], but 1.2 MeV for Cu. However, theagreement of the measured and our calculated RSF val-ues has been considered at once with that for Γ γ valuesthat are also shown in Fig. 8. Actually, it is of interest tolook for the effects on calculated cross sections in Sec. IIIof the possible RSF uncertainty between the results ofEGLO and GLO models, for Cu, and EGLO and SLOmodels, for Cu.
D. Pre-equilibrium emission modeling
The Geometry-Dependent Hybrid (GDH) model [77],generalized through inclusion of the angular-momentumand parity conservation [78] and knockout α -particleemission based on a pre-formation probability ϕ [3], hasbeen involved within STAPRE-H95 code to provide thePE contribution to the results of the present work. Itincludes also a revised version of the advanced particle-hole level densities (PLD) [79, 80] using the linear en-ergy dependence of the single-particle level density [81].The particular energy dependence of the PE contribu-tion within this approach, that is discussed at large inSec. III.B.5 of Ref. [82] for neutron-induced reactions onMo isotopes, is thus fully appropriate also to this work.The central-well Fermi energy value F =40 MeV has beenused, while the local-density Fermi energies correspond-ing to various partial waves (e.g., Fig. 4 of Ref. [82])were provided within the local density approximation bythe same OMP parameters given in Sec. II B. III.
P E + HF RESULTS AND DISCUSSION
The PE+HF results corresponding to use of the con-sistent parameter set mentioned above (Sec. II) are com-pared in the following with measured cross sections oflow-energy proton- and neutron-induced reactions lead-ing to excited Fe, Co, Cu, and Zn isotopes. A particu-lar attention will be payed to the recent data, e.g., [11– 16]. The aim is to ascertain either the account of the α -particle emission by the α -particle OMP [7] or even-tual questions that may still need further consideration.Nevertheless, the suitable description of all available datafor competitive reaction channels is firstly concerned, inorder to avoid less accurate parameter-error compensa-tion effects. A. Fe isotopes de–excitation
The Fe isotopes and particularly the more abundant , Fe are among the best studied nuclei also for theneutron-induced reactions (e.g., [83] and Refs. therein).There are, however, still open question motivating quiteuseful recent studies as, e.g., [14–16, 54]. Moreover, thereare interesting issues related to the neutron magic num-ber N =28 of Fe as the striking difference between the( n, p ) excitation functions for the target nuclei , Fe.On the other hand, there are recently measured essentialcross sections for populations of the first three discretelevels of the even-even residual nucleus Cr by ( n, α ) re-action on Fe [13]. Their suitable description by modelcalculations is therefore a key test of the model assump-tions and performance. Fe de–excitation
The ( n, p ) reaction large cross sections for this semi-magic target nucleus provides from the beginning an use-ful check of the proton optical potential. Thus, no NLDeffects, including an uncertainty of the low-lying levels,can be seen in Fig. 9(a) until an incident energy of ∼ a of the residual nu-cleus Mn are then leading to an uncertainty of up to ∼
10% that may explain also the variance of the TALYSand TENDL-2019 results. The last comment concernsfirstly the broad plateau of the ( n, p ) excitation function,the NLD effects remaining constant for higher incidentenergies while the PE contribution increases. Neverthe-less, even the PE cross sections depend on the a valuesthrough the related PLDs [79, 80], so that the good agree-ment of our calculated results and the available data doessupport the present approach. The ( n, n ) reaction excitation function shown inFig. 9(b) has firstly been affected by the NLD of theresidual nucleus Fe, along its increasing side. The leveldensity parameter a of this nucleus, with no resonancedata, has also been fixed by the smooth-curve method[51], with an accuracy that leads to uncertainties of cal-culated cross sections close to 10%.Once the continuum of the final residual nucleus Festarts to be populated at higher incident energies, it be-comes visible an effects mixture of the two NLDs. Whilethe NLD uncertainty of only Fe has led to one of ∼ Fe(n,p) Mn Paulsen+ (1971) Smith+ (1975) Chi-Fong Ai+ (1977) Greenwood (1987) Ikeda+ (1988) Lu Han-Lin+ (1989) Ikeda+ (1990) Viennot+ (1991) Meadows+ (1991) Shimizu+ (2004) Mannhart+ (2007) Filatenkov (2016) (a) Mn: a=5.7(2)N d =36(3)16 20 24 280.1110100 ( m b ) Fe(n,2n) Fe Bormann+ (1976)Ryves+ (1978)Greenwood (1985)Viennot+ (1991)Soewarsono+ (1992) Fessler+ (1998)Sakane+ (2001)Wallner+ (2011)
TALYS-1.9 TENDL-2019 PE+CN (b) Fe: a=5.65(20),N d =26(2) Fe: a=5.47(20),N d =24(2)6 9 12 15 180306090120 E (MeV) Fe(n, ) Cr Paulsen+(1979) Grimes+ (1979) Greenwood (1987) Ikeda+ (1988) Lu+ (1989) Meadows+(1991) Saraf+ (1991) Meadows+ (1996) Gledenov+ (1997) Mannhart+(2004) Wang+(2015) Khromyleva+(2018) Bai+ (2019) (c)=0.20(5) Cr : D =13.1(12),N d =25(2) OMP( ): HF / GDH AHA (1994)AHA(1994)/A+(2014) A+(2014)
FIG. 9: (Color online) Comparison of measured [43], eval-uated [46] (short-dashed curves), and calculated TALYS-1.9(short-dotted curves), and this work (solid curves) cross sec-tions of neutron–induced reactions on Fe, with (c) alternateuse of α -particle OMP [8] in HF (dash-dotted curve) and bothHF+GDH (dashed curve) calculations; uncertainty bands cor-respond to error bars of N d , and either LD parameter a or D exp (Table I) of residual nuclei (a) Mn, (b) Fe (orangeband) and, in addition, Fe (gray band), and (c) Cr (orangeband), as well as for GDH α -particle pre-formation probabil-ity ϕ =0.20 ± for the cross sections at the top of the ( n, n ) excita-tion function, the addition of that for Fe increased theone for cross sections up to ∼ The ( n, α ) reaction analysis has taken the advantageof confidence in the above-proved suitable account ofthe main nucleon-emission channels. Unfortunately, theHF+PE results of this work are underestimating notablythe recent accurate data [14–16] as shown in Fig. 9(c)below an incident energy of 11 MeV. On the contrary,there is also an overestimation above ∼
17 MeV but withrespect to only one earlier data set.The residual nucleus Cr has the advantage of knownresonance data [45]. Taking into account the error limitsof a medium value of its s -wave nucleon-resonance spac-ings D exp (Table I) as well as an uncertainty of 2 levelsfor the low-lying level number N d , it results an (orange)uncertainty band of the calculated ( n, α ) cross sectionsthat becomes significant at incident energies higher than12–13 MeV. A larger one corresponds to the assummedlimits of the GDH α -particle pre-formation probability ϕ =0.20 ± n, α ) dataremain truly underpredicted just within the energy rangewhere the α -particle OMP is the main HF parameter.On the other hand, the α -particle OMP [7] that wasused within both HF and PE model calculations, hasfirstly been replaced by [8] for calculation of α -particletransmission coefficients involved in HF calculations.Then the same replacement concerned also the corre-sponding PE intranuclear transition rates within the gen-eralized GDH model [78]. The former replacement pro-vides the agreement already found below ∼
10 MeV [8]but an overestimation over 20% around the incident en-ergy of 16 MeV. Actually this enlargement is twice thatcorresponding to the upper limit of ϕ , becoming similarlywith the PE increase , i.e. above 18 MeV. The latter re-placement had no effect at the lower energies, where PEmechanism is not yet effective and even not consideredin Ref. [8] that concerned incident energies ≤
10 MeV.However, a significant cross-section increase follows athigher energies, becoming even twice the former for theexcitation function maximum at ∼
16 MeVTherefore, a consistent analysis over the whole energyrange of this ( n, α ) excitation function proves substantialunderestimation by α -particle OMP [7] that is removedby the OMP [8] at the price of a large overestimation forthe maximum at ∼
16 MeV. It should be noted that thelatter disagreement can not be compensated by eitherNLD or PE effects within consistent limits.4
E (MeV) Fe(n,p) Mn Ikeda+(1988,1993) Viennot+ (1991) Fessler+ (2000) Coszach+ (2001) Mannhart+ (2007) Mulik+ (2013) Filatenkov (2016) Liskien+ (1965) Smith+ (1975) Mostafa (1976) Ryves+ (1978) Kudo (1984) Fuga+(1991) (c) Mn: D =2.3(4)N d =49(2) 12 15 18 210200400600800 Corcalciuc+ (1978)Frehaut+ (1980,1990) Wallner+ (2010/2011)56 Fe(n,2n) Fe (b)3 6 9 12 15 18050010001500 ( m b ) Negret+ (2014)Beyer+ (2014) TALYS-1.9 TENDL-2019 PE+CN56
Fe(n,n’) Fe (a) Fe: a=5.53(10),N d =31(2) Fe: a=6.0(2),N d =49(2) Fe(n,n’ ) Fe Fe(n,x ) Cr Grimes+ (1979) Paulsen+ (1981): Fe Fischer+ (1984) Saraf+ (1991) Matsuyama+(1993):Fe Sterbenz+ (1994) Haight+ (1996): Fe Kunieda+(2012): Fe Wang+ (2015) Bai+ (2019) (n, )
TALYS-1.9TENDL-2019PE+CN (d) Cr: D =32.0(35)N d =25(2) FIG. 10: As Fig. 9 but for Fe target nucleus, additionally with (a) inelastic scattering, total (solid curve) and for the firstexcited at 847 keV (dashed curve), (d) use of only α -particle OMP [7] within both HF+GDH model calculations of the ( n, α )reaction (short-dash-dotted curve) and total α emission (solid curve). Fe de–excitation
The inelastic scattering of neutrons on Fe was the ob-ject of extensive studies while there are especially morerecent and high-resolution measurements [84–86] whichshould be considered within any further model analysis.This issue is particularly essential for the present workdue to the modified neutron OMP (Sec. II B 1) whereasthe recent studies used the Koning–Delaroche [52] poten-tial.The recent total inelastic neutron-scattering data arecompared in Fig. 10(a) with the latest evaluation [46] andcalcutions with TALYS-1.9 default options and presentwork. There are shown also our results for the popu-lation of the first excited at 847 keV since its measured γ -decay was used together with TALYS predictions to de- rive the total neutron-scattering cross sections [85]. Thecalculated values of this ratio are decreasing from 1, atthe incident energies of 3 MeV, to ∼ ∼
11 MeV, andaround 0.95 above 13 MeV.Actually, the agreement of our calculated results andthe recent data shown in Fig. 12 of Beyer et al. [86] isbetter along the increasing side of this excitation func-tion, due to a correction for the γ -ray angular distributionthat causes a reduction of the available data [84] by upto 30%. On the other hand, the larger calculated crosssections at higher energies are not related to NLD effectsof Fe nucleus, which correspond to an uncertainty bandwithin ≤ The ( n, n ) reaction excitation function shown inFig. 10(b) proves a good agreement with the newest mea-5sured data except the last point just above 20 MeV. Thefast decrease with energy of the two data sets availablein this energy range data seems however unphysical. TheNLD effects for both the residual nucleus Fe and Fe,in addition and also shown for the inelastic scattering onthe first excited state in Fig. 10(a), are below 4%.
The ( n, p ) reaction excitation function is shown inFig. 10(c), with so much changed behavior regarding thesame reaction on Fe. This is due to the magic N num-ber of the lighter isotope as well as the isotopic effectof ( n, p ) and ( n, α ) reaction cross-sections, of decreasingwith the isotope mass increase [87]. On the other hand,it has most recently been proved by Nobre et al. [88]as an ideal mechanism to probe the NLD of Fe and Mn. Their starting point has been that NLD parame-ters which describe reasonably well N d and D exp datado not necessarily lead to consistent calculated cross-section agreements at the precision level required in eval-uations. They finally established additional experimentalconstrains on NLD through quantitative correlations be-tween reaction cross sections and NLD.The uncertainty bands in Fig. 10 (c) go along the sameline as [88]. We have also paid due attention to the suit-able fit of N d and D exp data, however with higher fitted N d (Table I) but also close to numbers of levels consid-ered within complete level schemes in RIPL-3 [45]. Then,while the cross-section uncertainty bands correspondingto the NLD uncertainties of , Fe play a minor role inthe understanding of eventual lower agreement betweenmeasured and modeling ( n, n ′ ) data, the ( n, p ) case isquite different. The NLD uncertainty for Mn had in-deed no effect at incident energies ≤ D value close to the higher limit of D exp . Thus, only arather low NLD, yet compatible with this experimentaldata, has been confirmed by analysis of the measuredreaction cross sections. The α -particle emission excitation function analysis,in the absence of available data only for ( n, α ) reaction,concerns more data sets corresponding to nat Fe target.However, the data for incident energies <
15 MeV standfor ( n, α ) reaction, and include the recent work that haverisen questions yet unanswered [14, 16].While this analysis has followed the above-mentionedcheck of nucleon OMPs and important reaction chan-nels within its energy range, a large underestimation ofthe experimental data at neutron energies of 8–12 MeVis apparent in Fig. 10(d). There is also a lower under-estimation around 14.1 MeV [89], in the limit of twicethe data standard deviation ( σ ). The related α -emissionangular distributions and angle-integrated spectrum arediscussed in Sec. IV and the main paper, respectively. The uncertainty bands are shown firstly with regardto the D exp limits, and then in addition to those of N d ,fitted to obtain the NLD parameters of the residual nu-cleus Cr (Table I). It is thus obvious that effects of N d uncertainty exist just above the neutron energy of 10MeV, and yet minor for other 2–3 MeV. Only then wouldthey be similar to data errors but around calculated crosssections which are still well below the measured values.The effects due to the D exp uncertainty become also vis-ible, and either equal with the former above 15 MeV, ordominant around 20 MeV. Nevertheless, the NLD uncer-tainty band is matching the measured data at incidentenergies >
14 MeV while the case is entirely different atlower energies and needs further attention. Fe de–excitation
The inelastic scattering of neutrons on Fe being re-cently analyzed to a large extent [90], its first discussionis also essential for this work. We have used in this re-spect also the modified neutron OMP (Sec. II B 1) for Fe, with the results shown in Fig. 11(a). The goodagreement of our calculated results and measured dataaround the neutron energy of 1 MeV is most importantfor the rest of the model analysis, as a validation of theneutron competition within CN de-excitation. At higherenergies this agreement is in the 2 σ limit. The ( n, n ) reaction excitation function provides, onthe basis of the same recent and accurate work [90], alsoa good agreement except the latest two points just above16 MeV [Fig. 11(b)]. The overestimation of these dataare obviously smaller than results of TALYS default re-sults and particularly the latest TENDL-2019 evaluation.The NLD effects for the residual nuclei Fe and Fein addition, also shown for the ( n, n ) reaction to thefirst excited-state population, are even lower than for thesame reaction on the even-even nucleus Fe [Fig. 10(b)].
The ( n, p ) reaction excitation function, shown inFig. 11(c), may additionally confirm it as an ideal mech-anism to probe the residual-nuclei NLD [88]. The un-certainty bands in Fig. 11 (c) correspond just to fittedlimits of N d and either D exp data of Fe, or average a -value for Mn. They prove the usefulness of resonancedata availability, making possible an uncertainty bandthree times narrower. On the other hand one may notethat the NLD uncertainties of the two residual nuclei actin opposite ways on the ( n, p ) reaction cross sections.Therefore, possible systematic errors of the NLD param-eters may lead to less wide uncertainty bands of the cal-culated ( n, p ) cross sections, which remain thus half-waybetween the more recent data sets with so large errorbars.
The ( n, α ) reaction analysis may concern only but es-sential measured cross sections for populations of theground state (g.s.) and first two excited states at 0.835and 1.824 MeV, respectively, of the even-even residualnucleus Cr at neutron energies between 5 and 6.5 MeV6 ( b ) Negret+ (2017) Fe(n,inl) (a) 10 15 200.00.51.0 Fe(n,2n) Fe Negret+ (2017): 847 keV (n,2n)
TALYS-1.9 TENDL-2019 57
Fe(n,2n) Fe(2 +1 ) (b) Fe:D =19.2(19),N d =30(2) Fe: a=6.0(2),N d =49(2) (n,2n ) ( m b ) E (MeV) Fe(n,p) Mn Molla+ (1977)Viennot+ (1982)Viennot+ (1991)Kasugai+ (1994)Fessler+ (2000)Filatenkov(2016) Fe: D =19.2(19),N d =30(2) Mn: a=6.0(2)(c) 3 4 5 6 70.020.1110 Fe(n, ) Cr A+(2014) AHA(1994) (n, ) (n, ) (n, ) (n, ) (d) FIG. 11: As Fig. 9 but for Fe target nucleus, additionally with (b) total (thin solid curve) and first 2 + excited-state population(thick solid state) cross sections of ( n, n ) reaction, and (d) total (solid curves) and partial cross sections for population of theground state (dashed curves), first (dash-dotted curves) and second (dash-dot-dotted curves) excited states, calculated usingthe α -particle OMPs of Rev. [7] (thick curves) and Rev. [8] (thin curves), respectively; total ( n, α ) cross sections for Gledenov ∗ et al. [13] correspond to the sum of their measured partial data times our calculated ratio of the total cross section to the sumof the three states. [13]. There are also two additional points added quiterecently to the corresponding total ( n, α ) reaction crosssections between 4 and 5 MeV [15], in addition to theformer energies. Unfortunately, above 5.5 MeV the twodata sets are not consistent while the partial data makepossible at least their excitation-dependence analysis asshown in Fig. 11(d).The great advantage of these data consists in theirmodel analysis free of any NLD as well as PE effects.Once the transmission coefficients for the eventual openreaction channels are already checked, as within presentwork, the calculated cross sections are actually given bythe α -particle OMP. This is proved also by the calculatedfraction of the three partial cross-section sum to the to- tal cross section including the population of the upperstates. It goes from 98%, at the incident energy of 5MeV, to 90% at 6.5 MeV. Actually, we used this ratioto get the total ( n, α ) cross sections corresponding to thesum of the partial data measured by Gledenov et al. [13]and shown in Fig. 11(d). These values are however lowerthan those in Table II of [13] by only 3–7%.Thus we have found that the total as well as partialcross sections are well described up to ∼ n, α )cross sections at these energies are just between the two7 OMP( ): HF / GDH AHA (1994)AHA(1994)/A+(2014) A+(2014)
E (MeV) ( m b ) Co(n, ) Mn Liskien+ (1965) Liskien+ (1966) Ghorai+ (1980) Zupranska+(1980) Huang+ (1981) Meadows+ (1987) Ikeda+ (1988) Molla+ (1994) Uno+ (1996) TENDL-2019 Li Tingyan+(1990) Mannhart+ (2007) Filatenkov+(2016) =0.04(2) D =2.3(4),N d =49(2) FIG. 12: As Fig. 9(c) but for Co target nucleus and reversedcolors of the uncertainty bands. data sets [13, 15]. On the other hand, the calculation re-sults using the OMP [8] are obviously larger by a factorof two, with reference to the measured data and results ofusing the potential [7]. Then, the measured data [13] arejust between the calculated results using the two OMPs[7, 8]. It is thus apparent the additional attention thatshould concern both measurement and analysis works. B. Co de-excitation
Following the above-mentioned detailed study ofneutron-induced reactions on Co [57], only an addi-tional analysis of the α -particle emission has been carriedon in the present work. It has been motivated by the ear-lier need to adjust the α -particle OMP [8] by fit of ( n, α )measured cross sections within several MeV above the ef-fective reaction threshold. No such action has presentlybeen necessary, the use of OMP [7] leading to a gooddescription of entire ( n, α ) excitation function shown inFig. 12. This includes the data measured in the meantimeat energies where there are no NLD effects illustrated bythe corresponding uncertainty band.On the other hand, the use of the NLD parametersgiven in Table I has been of a particular interest becauseof the same residual nucleus for reactions Co( n, α ) Mnand formerly discussed Fe( n, p ) Mn. The cross-sectionuncertainty bands corresponding to the NLD effects havesimilar shapes and the agreement with the measured datais provided by the NLD parameters that correspond toa D value close to the higher limit of its experimen-tal value, i.e. NLD values close to the lowest values yetcompatible with the resonance data.The calculated cross-section uncertainty bands relatedto the assumed uncertainties of the NLD and PE, re- spectively, can be also compared in Fig. 12. The lat-ter has been considered to be given by half of the lowvalue ϕ =0.04 of the α -particle pre-formation probabilitythat correspond to a good agreement of calculated andmeasured cross sections, α -emission angular distributions(Sec. IV), and angle-integrated spectrum [91–93] (see themain paper). It becomes half of the former only aroundthe excitation function maximum, and even larger abovethe neutron energy of 18 MeV. A good agreement there isalso at these energies between the calculated and more re-cently measured cross sections while the others are withinthe uncertainty band due to PE effects.Unlike Fe case, replacement of the α -particle OMP[7] by the earlier one [8], firstly in HF calculations andthen also within PE approach, has led to a notable over-estimation even from the neutron energy of ∼ α -particle pre-formationprobability for the α -emission from the excited odd-oddnucleus Co.Actually, the only data set available for the formeranalysis was overestimated even at that time. On theother hand, that analysis concerned firstly the majorrole of the low-lying discrete levels (Fig. 2 of [8]), whichwas not really the case of the residual odd-odd nucleus Mn. Nevertheless, the good results provided by use ofthe OMP [7] even in these conditions should be kept inmind.
C. Cu isotopes de-excitation
The α -particle emission by de-excitation of the samenucleus excited through neutron– as well as proton–induced reactions would answer entirely the question [10]concerning an eventual difference between the OMPs be-ing able to describe them. Because of no experimentaldata for such case, the next option has been the sameanalysis for various isotopes populated by the two dis-tinct reactions. Incident either neutrons on Cu or protonson Ni isotopes provides the first object of this compara-tive study. ( n, α ) reactions: , Cu de–excitation
The above-mentioned detailed study of neutron-induced reactions on Cu isotopes [56] makes useful in thefollowing only an additional analysis of the α -emission.In a similar way to the involvement [57] of the earlierOMP [8] within ( n, α ) reaction on Co, a decrease ofthe real well diffuseness [8] was used in order to obtainthe agreement with data even at lower incident energies.On the contrary, no change of OMP [7] parametershas been necessary to obtain a good agreement with thedata available for both target nuclei , Cu (Fig. 13).This agreement concerns firstly the feeding of the low-lying levels, within the several MeV above the effective8
E (MeV) Cu(n, ) Co ( m b ) Garuska+ (1980) Winkler+ (1980) Greenwood (1987) Hanlin+ (1990) Yongchang+(1990) Csikai+ (1991) Konno+ (1993) Ykeda+ (1996) Meadows+ (1999) Semkova+ (2004) Gledenov+ (2014) Filatenkov+(2016) (a) D =1.45(15) Cu(n, ) Co Artem’ev+ (1980) Cserpak+ (1994)
OMP( ): HF / GDH AHA (1994)AHA(1994)/A+(2014) A+(2014) TENDL-2019 (b) =0.06(3) a =7.4(2)
FIG. 13: As Fig. 12 but for , Cu target nuclei. reaction thresholds. Then, a similar agreement has alsobeen found around the cross-section maxima and withthe data available for Cu at higher energies.Thus, an uncertainty band corresponding to the errorbar of the available D exp value for the residual nucleus Co (Table I) has risen up to ∼
10% at incident ener-gies of 13–14 MeV. It decreases for higher energies butyet in contact with especially recent measured data. Onthe other hand, there is only one measured data set for Cu target nucleus, below the excitation function top,and even somehow scattered. The assumed accuracy forthe level density parameter a , given by the smooth-curvemethod, has led to a larger uncertainty band of the cal-culated cross sections, that seems to cover all these data.The above-mentioned agreement has corresponded to alow value ϕ =0.06 of the α -particle pre-formation proba-bility, so close to that found also for Co target nucleus.We have taken into account again its 50% variation inorder to estimate the calculated data uncertainty bandrelated to PE effects. It resulted to be rather similar tothose related to the NLD effects for both target nucleiup to ∼
15 MeV, while the PE effects become dominantat higher energies. Nevertheless, the α -particle OMP [7]provides an accurate account of data available for bothCu isotopes, additional data for Cu being eventuallyquite useful. ( p, α ) reactions: , Cu de–excitation
The key analysis of this section has been that of thereaction Ni( p, α ) Co, following the pioneering work ofQaim et al. [18]. However, the consistent analysis of allavailable data for various reaction channels and isotopeshas been an essential condition, i.e. including the target nucleus Ni. Cu de–excitationThe ( p, γ ) reaction analysis makes possible a confirma-tion of nucleon OMPs mentioned in Sec. II B. Thus, onemay see in Fig. 14(a) that the proton-capture cross sec-tions at the lowest proton energies ≤ σ R , are smaller by even a factor of 2 if themodified proton OMP [52] is taken into account. Then,there are rather equal proton-OMP and GLO-model RSFeffects, at incident energies just below 3 MeV, where theneutron-emission is not yet open. While these effectshave led to a cross-section increase higher than 50%, theuse of the SLO-model RSF may add another similar in-crease, in close relation to the RSFs behavior in Fig. 8.Fortunately there are rather recent data which supportwell the present results. The ( p, n ) reaction measured data are most importantfor validation of the proton OMP. A better agreement es-pecially with the more recent data in Fig. 14(b) is givenby a 10–20% decrease of the calculated cross sections fol-lowing the use of modified OMP [52]. It is thus increasedthe confidence in the presently calculated cross sections. The ( p, α ) reaction is characterized by the two mea-sured data sets shown Fig. 14(c). They agree in the limitsof rather large error bars while the calculated cross sec-tions using the α -particle OMP [7] are in between themup to the proton energy of ∼
12 MeV. There are, yet mi-nor, changes due to the proton OMP, with no effect onthe comparison of calculated and measured data.At higher energies the agreement remains only withthe larger measured data even taking into account the9 ( m b ) Ni(p,n) Cu Johnson+ (1964)Tanaka+ (1972)Antropov+ (1992)Tingwell+(1988) Szelecsenyi+ (1993)Szelecsenyi+ (1993): (p,2n) Singh+ (2006) (p,2n) R (b) TENDL-2019 10 15 2010100 E (MeV)61
Ni(p, ) Co Tanaka+ (1972)Sudar+ (1993) OMP( ): HF / GDH AHA (1994) AHA(1994)/A+(2014) A+(2014) (c) a=6.6(2),N d =43(2) =0.05(2)1 2 3 40.0010.010.11 Ni(p, ) Cu Krivonosov+ (1974) Krivonosov+ (1977)Tingwell+(1988) Simon+ (2013) R (a) OMP/n/p: Koning+ (2003) This work (EGLO) GLO SLO FIG. 14: As Fig. 12 but for proton-induced reactions on Ni,with additional calculated cross sections corresponding to thealternate use of (a) the SLO (dash-dot-dotted curves) andGLO (dash-dotted curves) RSF models for E1 radiations,and (a,b) Koning-Delaroche nucleon OMPs [52] (short dash-dotted curves). uncertainty band due to the assumed accuracy of N d and a value obtained by the smooth-curve method for theresidual nucleus Co. This band has a width up to ∼ α -particle pre-formation probability ϕ =0.05. The so low ϕ value, actually just in between those for the neutron-induced reactions on Co and , Cu, shows that thelarger calculated cross sections at proton energies ≥ ∼
10 MeV, the validationof the α -particle OMP [7] is obvious. An alternate con-sideration of the OMP [8] increases the calculated crosssections by ∼
50% at above-mentioned proton energy of ∼
10 MeV and yet around 35% at the excitation-functionmaximum. On the other hand, one may note that this α -particle OMP effect match at higher energies the un-certainty band due to NLD accuracy. So, a real mixtureof parameter uncertainties may occur above the incidentenergy of 20 MeV, making possible meaningful conclu-sions on them only at lower energies. Cu de–excitationThe ( p, γ ) reaction analysis presents interesting simi-larities as well as differences from the same reaction on Ni. First, the same is the role of the modified protonOMP [52] on proton-capture cross sections at incident en-ergies just below ∼ , Cu provide a fully changedcase for higher energies as shown in Fig. 15(a).The RSF values given by the EGLO model in betweenthe lower GLO and higher SLO ones, above the γ -rayenergy of ∼ ∼ The ( p, n ) reaction comparative analysis of the mea-sured and calculated data in Fig. 15(b) is again quitesimilar to that for Ni target nucleus, taking addition-ally the advantage of larger reaction cross section. The0 Ni(p, ) Cu OMP/n/p: Koning+ (2003) This work (EGLO) (a)Sevior+ (1983) Simon+ (2013) TENDL-2019 GLO SLO R ( m b ) Ni(p,n) Cu Guzhovskij+ (1969)Tanaka+ (1972)Sevior+ (1983)Levkovski (1991)Antropov+ (1992)Szelecsenyi+ (1993)Avila-Rodriguez+(2007) Adam Rebeles+ (2009)Uddin+ (2016) (b) TENDL-2019 R E (MeV)64
Ni(p, ) Co Levkovski (1991) Qaim+ (1995) OMP( ): HF / GDH AHA (1994) AHA(1994)/A+(2014) A+(2014)(c) OMP( ): McFadden+ (1966)OMP( /n): McFadden+(1966)/Uhl(1991)a=6.45(35),N d =70(2) FIG. 15: As Fig. 14 but for Ni target nucleus, with (c) addi-tional calculated cross sections corresponding to the alternateuse of the α -particle OMP of Ref. [35] (dotted curve), and inaddition the neutron OMP [94] (short-dotted curve). more recent data at proton energies above 20 MeV pro-vide also an increased validation of the proton OMP aswell as the PE account. The ( p, α ) reaction is also characterized by the twomeasured data sets, shown in Fig. 15(c), of which onlyone has proper incident-energy error bars. It made how-ever the object of the deeper analysis by Qaim et al. [18],whose review has been challenging in the light of progressin the field, in meantime.First, by using the α -particle OMP [7], we found anobvious underestimation also at lower incident energiesbut particularly of 20–25% above 11 MeV. On the otherhand, these calculated cross sections are yet higher by ∼
50% than Qaim et al.
HF+PE results. In order tounderstand this difference, we have looked to eventualeffects of the OMPs concerned within Ref. [18]. Thus,use of the α -particle OMP of Ref. [35] led to decrease ofour results by less than 10%. However, the additional useof the neutron OMP [94] provides an excitation functionthat is indeed rather close to the HF+PE results of Qaim et al. [18].The eventually remaining dissimilarity between theHF+PE results of Ref. [18] and ours, using their OMPs,can be well motivated by that of the NLDs taken intoaccount. Despite the use of the same BSFG model, theNLD parameters for the residual nucleus Co have beenbased on systematics, in the absence of correspondingresonance data. Thus, in Fig. 15(c) is also shown the un-certainty band of our calculated results, due to the limitsassumed for the N d and a value of Co (Table I). Itproves that there is no NLD effect up to ∼
12 MeV, whilethen its width is close to 40% of the excitation-functionmaximum.Nevertheless, even if this uncertainty band comes nearthe largest cross sections, there is still an apparent un-derestimation. At the same time, similarly to the caseof Ni target nucleus, PE effects become visible around ∼
12 MeV. They become only around half of the NLDones at the cross-sections maximum, and rather equalaround 20 MeV.Moreover, an alternate consideration of the OMP [8]increases the question marks. Thus, it provides calcu-lated cross sections larger than the measured data [18]in the 2 σ limit. Therefore, further consideration of ad-ditional pickup direct interaction leading to an increaseof the α -emission data beyond the HF+PE predictions[18] should be carefully taken into account in order toestablish the correct α -particle OMP. D. Zn isotopes de–excitation
The analysis of α -particle emission in neutron– as wellas proton–induced reactions has the advantage, for de-excitation of Zn isotopes, of accurate data at low incidentenergies [11, 12, 15, 95–97]. The α -particle OMP studybecomes thus less dependent by NLD and PE effects.Therefore, incident either neutrons on Zn or protons on1Cu isotopes provides the second object of this compara-tive study.Moreover, the recent analysis of α -particle induced re-actions on Ni isotopes around the Coulomb barrier [30]has just proved their suitable description by means ofthe α -particle OMP [7]. Therefore it is so motivating tocheck the accuracy of the same potential in the inversereactions. At the same time, there are useful also in thepresent work the corresponding consistent parameter setas well as several competing reactions being already dis-cussed therein Ref. [30]. ( n, α ) reactions: , Zn de–excitation
The rather recent measured cross sections of the ( n, α )reactions on , Zn, beyond the largest usefulness ofquite low incident energies, have the advantage of dataavailable for other reaction channels. The former anal-ysis of these data makes possible a valuable validationof their common model parameters, to be firstly pointedout in the following. Zn de–excitationThe ( n, p ) reaction comparative analysis of the mea-sured and calculated data in Fig. 16(a), making use ofmore recent measurements, provide again an increasedvalidation of the proton OMP as well as the PE account.It seems that the worse description of more recent dataabove the incident energy of 18 MeV is well compensatedby the accurate acoount of the the whole rest of the ex-citation function. The ( n, n ) reaction analysis shown in Fig. 16(b) val-idates firstly the neutron OMP and then also the PEaccount, within incident–energy range below 20 MeV.While these energies are of first interest for the presentwork, more data at higher energies would be quite usefulfor the modeling support. The ( n, α ) reaction is the first one with recent experi-mental cross sections [Fig. 16(c)] corresponding to feed-ing of only low-lying states. An agreement provided bythe use of α -particle OMP [7] has been found at the low-est incident energy of ∼ σ . This iscertainly not due to NLD effects, the calculated cross-section uncertainty band corresponding to the measured D limits for the residual nucleus Ni becoming visibleonly above 9 MeV. A similar case is that of the PE takeninto account using an α -particle pre-formation proba-bility ϕ =0.20 ± Zn(n,p) Cu Smith+ (1975) Casanova+(1976) King+ (1979) Husain+ (1983) Ikeda+ (1990) Ikeda+ (1991) Konno+ (1993) Ghorai+ (1995) Kielan+ (1995) Huang+ (1999) Mannhart+(2007) Furuta+ (2008)
TENDL-2019 TALYS-1.9(a)
12 15 20 25 30 35 40210100 ( m b ) Zn(n,2n) Zn Soewarsono+ (1992) Uwamino+ (1992)Konno+ (1993)Ghorai+ (1995)Mannhart+ (2007)Vrzalova+ (2013)Bhatia+ (2013)Filatenkov (2016) (b)
E (MeV) Zn(n, ) Ni Casanova+ (1976)Yuan+ (2003)Zhang+ (2007)Zhang+ (2008)Khromyleva+(2018) (c) OMP( ): HF / GDH AHA (1994) AHA(1994)/A+(2014) A+(2014) D =13.8(9)=0.20(5) FIG. 16: As Fig. 9 but for Zn target nucleus. ( m b ) Zn(n,p) Cu Konno+ (1993)Kielan+ (1995)Nesaraja+ (1999) Shimizu+ (2004)Furuta+ (2008)Bhike+ (2009) (a)0 2 4 60.1110
E (MeV) Zn(n, ) Ni Zhang+(2010) A+(2014) AHA(1994) (n, ) (n, ) (n, ) (n, .. ) (b) FIG. 17: As Fig. 11 but for ( n, p ) and ( n, α ) reactions on Zn. point at ∼ ∼
50% from 4MeV. Zn de–excitationThe ( n, p ) reaction has fortunately more recent dataat incident energies between 1.6 and 6 MeV, as shownin Fig. 17(a). This is the energy range where there areavailable the α -emission data, various model parametersbeing involved within analysis of both reaction channels.The agreement of the present model calculations withthese data as well as the data available at energies ≤ The ( n, α ) reaction partial cross sections for the g.s.,first excited state at 1.346 MeV, and higher states from2.277 MeV, were firstly measured at the neutron energy of 6 MeV [11] and, in addition, at 4 and 5 MeV [12].The usefulness of these data for the assessment of the α -particle OMP is that already underlined for the samereaction on Fe [13, 15].The analysis results are similar however only for theg.s. partial cross sections. The measured data trend iswell described by calculations using both OMPs [7, 8],the former potential leading to an underestimation of ∼ α -particleOMPs [7, 8] at variance with the experimental one. Thesame is true also for the TENDL-2019 evaluation, whichactually used the former potential. Therefore, additionalwork should concern this issue, maybe also experimen-tally. ( p, α ) reactions: , Zn de–excitation
The analysis of the ( p, α ) reaction on , Cu has takenthe advantage of the recent analysis of α -particle inducedreactions on Ni isotopes around the Coulomb barrier[30]. Thus, the corresponding consistent parameter setas well as the ( p, γ ) and ( p, n ) competing reactions, ofinterest also in the present work, have already been dis-cussed. That is why in the following we have taken intoaccount their suitable description, formerly proved, andhave straightforwardly proceeded to ( p, α ) analysis.Moreover, there are partial cross sections measuredearlier [98] for g.s. population of the residual even-evennuclei , Ni and first excited state of Ni, at protonenergies below 4.8 MeV (Fig. 18). The original study ofSwitkowski et al. looked for ( p, n ) threshold effects thatare quite different due to a lower threshold of only 2.167MeV for Cu, vs. 4.215 MeV for Cu. They found in-deed a severe fall of the ( p, γ ) and ( p, α ) cross sectionsfor Cu, at variance to the case of higher threshold for Cu.The previous analysis of proton-induced reactions on , Cu [30] describe well the ( p, n ) threshold effect forthe ( p, γ ) reaction on Cu, as well as the importanceof using improved proton OMP (Fig. 1 of [30]). In thepresent work we found firstly worthy of note the effect ofthe alternate use of the proton OMP of Koning-Delaroche[52] and its modified version for low energies. Thus, thiseffect is rather normal only for Cu, the former potentialleading to larger ( p, α ) cross sections above the ( p, n )threshold, while it is absent or even inverted just abovethe ( p, α ) threshold for Cu.A different effect is also shown in Fig. 18 by the un-certainty band corresponding to the error bars of s -waveaverage radiation widths Γ γ deduced from systematics[45]. It is minor for Cu but similar to the data er-3 ( m b ) E (MeV)63
Cu(p, ) Ni Switkowski+(1978): 1st. (a) [ =510+230] A+ (2014): 1st. 2 30.0010.010.11 Cu(p, ) Ni AHA (1994): g.s. A+ (2014): g.s. p’s OMP: Koning+(2003) Switkowski+(1978): g.s. (b)[ =450+180]
FIG. 18: (Color online) Comparison of measured [98] and calculated partial cross sections of ( p, α ) reaction on , Cu andpopulation of g.s. (solid curves) and first excited state (dash-dotted curve), using the α -particle OMP of Ref. [7], and theOMPs alternate use of either Ref. [8] for α -particles (dashed curves), or [52] for protons (dash-dot-dotted curves); uncertaintybands correspond to the error bars of s -wave average radiation widths Γ γ (in meV) deduced from systematics [45]. ror bars for Cu below the ( p, n ) threshold. However,a contribution in this respect has also the isotopic effectleading to ( p, α ) cross sections lower by even an order ofmagnitude for the heavier isotope.We paid due attention to various effects as above-mentioned in order to avoid any misinterpretation of therather questionable comparation of the presently calcu-lated cross sections using the α -particle OMP [7], and themeasured data. The best agreement has been obtainedfor the excitation functions of first excited state of Nias well as , Ni g.s. within ∼ p, α )threshold. There is also a somewhat good account of thesound ( p, n ) threshold effect for Cu, as well as of theweak one for Cu just above 4.2 MeV. Otherwise, the( p, n ) threshold effect for Cu is followed by a ratherconstant underestimation.On the other hand, the calculated cross sections us-ing the earlier α -particle OMP [8], larger by ∼
30% thanthe former results, show distinct cases for the two nuclei.Thus, except several data points around the proton en-ergy of 3 MeV, they are overestimating the data for Cubut match the data for Cu above the ( p, n ) thresholdeffect. Unfortunately, even Switkowski et al. [98] consid-ered the resolution of their data to be insufficient for amore detailed analysis.Nevertheless, this analysis of ( p, α ) reaction on , Cuis pointing out that the measured data for the heavierisotope could be also well described by [7] if an addi-tional contribution would exist. This eventual additionwould be negligible for Cu, due to the isotopic effect, ifit is rather similar for both nuclei and as large as the dif-ference between the measured and calculated values for Cu. These issues suggest that a DI contribution is theone missing within modeling work.
IV. PICKUP DI MODELING
The due consideration of DI role within the α -emissionin both neutron– and proton–induced reactions is obvi-ously leading to an increase beyond the PE+HF predic-tions. Moreover, since the beginning of ’90s it is con-cluded that the pickup instead of knockout has the mainDI contribution to the low-lying levels in ( p, α ) and ( n, α )reactions ([3, 19, 20] and Refs. therein). While this con-clusion was achieved at incident energies above 20 MeV,there were Qaim et al. [18] extending it even around 10MeV However a better connection of their phenomeno-logical results to available spectroscopic data would beuseful for an increased predictive power as needed in thepresent work.In the present work, the pickup contributions to ( p, α )and ( n, α ) reactions have been determined within the dis-torted wave Born approximation (DWBA) formalism us-ing the code FRESCO [36] as well as the same above-mentioned particle OMPs. One–step reaction has beenconsidered through the pickup of H and He clusters,respectively. Moreover, the ”spectator model” [99, 100]was involved, where the two transferred either neutronsor protons in ( p, α ) and ( n, α ) reactions, respectively, arecoupled to zero angular momentum acting as spectators,while the transferred orbital ( L ) and total ( J ) angularmomenta are given by the third, unpaired nucleon of thetransferred cluster. The prior form distorted–wave tran-sition amplitudes, and the finite–range interaction havebeen considered. The p- H as well as n- He effective in-teraction in the α particle are assumed to have a Gaussianshape [20, 101, 102]: V r = − V e − ( r/r ) , (1)4where r =2 fm, and V is determined by fit of the bindingenergies of H and He, respectively.The three–nucleon transferred cluster bound stateswere generated in a Woods–Saxon real potential [20, 100–102] with the depth adjusted to fit the separation ener-gies in the target nuclei. The number of nodes ( N ) inthe radial three–nucleon cluster wave function was de-termined by the harmonic–oscillator energy conservationrule [99, 100]:2 N + L = Σ i =1 [2( n i −
1) + l i ] , (2)where n i and l i are the single–particle shell–model statequantum numbers. If the three nucleons are picked from1 f p shell, in the present work on A ∼
60 target nuclei,then 2 N + L =9. Otherwise, if the unpaired nucleon ispicked from 2 s d shell, then 2 N + L =8, while if it is from1 g subshell, it follows that 2 N + L =10.Nevertheless, the assessment of DI cross sections is sub-ject to available information on spectroscopic factors re-lated to populated states, outgoing particle angular dis-tributions, or at least differential cross–section maximumvalues. A. Ni ( p, α ) Co Our starting point on the pickup contribution to α -emission in nucleon–induced reactions at low energieswas the pioneering work of Qaim et al. [18]. In or-der to describe N( p, α ) Co reaction, they normalizedthe formerly calculated semimicroscopic pickup contri-bution to account for the measured data at 15 MeV.However, this normalization depends notably on the pre-ceding PE+HF calculated results. Since these are quitedifferent as shown in Sec. III C 2, we have looked for abso-lute values by DWBA analysis using spectroscopic factorscorresponding to the outgoing α -particle angular distri-butions reported at 15 MeV by Jolivette and Browne[103], and at 30 MeV by Smits et al. [100].Thus, for description of picked α -particle angular dis-tributions (Figs. 19-20) within ”spectator model” [99,100], the 0 + g.s. of Ni target nucleus led to the trans-ferred angular momentum L being fully set by the resid-ual Co final–state spin and parity.A particular note should concern the angular distri-butions corresponding to 2.238 and 2.558 excited statesof Co residual nucleus. The spectroscopic factors ob-tained through their analysis at 15 MeV incident energy(Fig. 19) did not describe the data at 30 MeV (Fig. 20).It became yet possible to describe the data at 30 MeV bytaking into account the excitation of the doublets shownin Fig. 20. Finally, 19 states [42], until the excitationenergy of ∼ d / d ( m b / s r) g.s. c.m. [deg] Ep=15 MeV Ni(p, ) Co FIG. 19: (Color online) Comparison of measured (solid cir-cles) [103] and calculated (solid curves) α -particle angular dis-tributions of Ni( p, α ) Co pickup transitions to states, withexcitation energies in MeV, at incident energy of 15 MeV. d / d ( m b / s r) g.s. .001 Ni(p, ) Co .0010.010.1 .0010.01 L= .0010.010.1 .0010.01 E p =30 MeV c.m. (deg) FIG. 20: As Fig. 19 but for incident energy of 30 MeV [100],and additional sum (solid curves) for doublets at 2.238, 2.558,1.664 MeV (dashed curves), and 2.230, 2.571, 1.674 MeV(dash-dotted curves), respectively. B. ( n, α ) reactions There are scarce information with regard to the anal-ysis of pickup ( n, α ) reactions for A ∼
60 target nuclei.Thus no measured angular distribution of α particlesfrom pickup processes has been found for the ( n, α ) re-actions within the present work.Consequently we carried out the pickup ( n, α ) cross-sections calculations standing on the spectator role ofthe picked proton pair [101, 102], with the spectroscopicfactors given by Glendenning (Table II of Ref. [105]). Atthe same time, the spectroscopic factors for the pickedneutron, that becomes thus responsible for the angular-momentum transfer, have been obtained by angular-distribution analysis for neutron pickup processes, as( He, α ), ( d, t ), and ( p, d ), populating the same residualnuclei.5 c.m. [deg] d / d ( m b / s r) g.s 0.11 Cr( He, ) Cr E He =25 MeV FIG. 21: As Fig. 19 but for α -particle angular distributions of Cr( He, α ) Cr pickup transitions, at incident energy of 25MeV [104]. Fe ( n, α ) Cr The pickup Fe( n, α ) Cr cross–section calculationwas carried out using the assumed spectator proton pair.Thus, the transferred angular momentum L was uniquelydetermined by the residual–nucleus final state. The Glen-denning spectroscopic factor [105] corresponding to thetransferred spectator proton pair from the 1 f / subshellwas involved in the DWBA analysis too.The picked-neutron spectroscopic factors were ob-tained from the comparison of the measured α -particleangular distributions [104] and the DWBA analysis of Fe( He, α ) Cr reaction, at 25 MeV incident energy,(Fig. 21). Thus, 36 excited states up to the excitationenergy of 5.943 MeV [104, 106] have been considered forcalculation of Fe( n, α ) Cr pickup excitation functionin the main paper. Fe ( n, α ) Cr The above-described approach concerned also thepickup Fe( n, α ) Cr cross–section calculation, usingthe assumed spectator proton pair and the transferredangular momentum L determined by the residual state.Thus, the Glendenning spectroscopic factor [105] corre- sponding to the transferred spectator proton pair fromthe 1 f / subshell was also involved in the DWBA anal-ysis too.The picked-neutron spectroscopic factors were ob-tained from the comparison of the measured α -particleangular distributions [107] and the DWBA results for Cr( He, α ) Cr reaction, at 30 MeV incident energy,(Fig. 22). Then, 18 excited states up to the excitationenergy of 5.557 MeV [107–109] were involved in calcu-lation of Fe( n, α ) Cr pickup excitation function (seethe main paper). The same calculations concerned alsothe analysis of α -emission angular distributions and anangle-integrated spectrum at 14.1 MeV [89] in the mainpaper.Actually, Fischer et al. [89] measured double-differential α –emission spectra for 16 reaction anglesranging between 22 o –165 o , at the incident energy of 14.1MeV. Because of rather large statistical errors, resultsintegrated over either energy or angle were presented,as well as the corresponding total α -emission cross sec-tion of the Fe( n, α ) reaction also shown in Fig. 10(d).Comparison of calculated angular distributions for the α -energy bins, within c.m. system, of 6-10 MeV, 10-12MeV, and 12-14 MeV, and the Fischer et al. data isshown in Fig. 23.The above–mentioned pickup results were added to theequivalent forms a + b · cos θ [110, 111] corresponding to6 c.m. (deg) c.m. (deg) d / d ( m b / s r) g.s 0.010.1 E He =18 MeV Cr( He, ) Cr FIG. 22: As Fig. 19 but for α -particle angular distributionsof Cr( He, α ) Cr pickup transitions, at incident energy of18 MeV [107]. the PE+CN isotropic component, as shown in Fig. 23.First, the isotropic PE+CN component goes from a goodagreement with data at lower α -particle energies (6–10MeV), to some overestimation at higher energies (10–12 and 12–14 MeV). Then, the relevant point is thatthe anisotropy of the measured distributions is well ac-counted by the present DI pickup approach. Fe ( n, α ) Cr The lack of α -particle angular-distribution data orspectroscopic factors for picked-neutron reactions, e.g.,( p, d ), ( d, t ) or ( He, α ) on Fe, makes possible only aqualitative estimation of the related pickup ( n, α ) crosssections. We rely on the likeness sequence of the firstthree low-lying sates of Cr and Fe nuclei, having thesame number 30 of neutrons, and 24 and 26, respec-tively, protons in the 1 f / subshell. Thus, we used forthis specific reaction the neutron spectroscopic factorsreported by Daehnick et al. [112, 113] from analysis of Fe( d, t ) Fe pickup reaction, and Glendenning spectro-scopic factor [105] corresponding to the transferred spec-tator proton pair from 1 f / subshell.The consequently pickup cross sections obtained forthe three low-lying sates of Cr by ( n, α ) reaction on Fe are shown in Fig. 24 in addition to the CN compo-nents given formerly in Fig. 11(d). While there is indeedan order of magnitude between the two mechanism con-tributions, a slightly improved trend is provided by thepickup consideration for the g.s. and first excited state.Moreover, an agreement seems to become possible in the
E =6-10 MeV Fe(n, x)
Cr,
En=14.1
PE+CN 0.48+0.22*cos (a)0.00.20.4 E =10-12 MeV PE+CN 0.36+0.07*cos Fischer+ (1984) DI (PICKUP) DI+PE+CN (b)30 60 90 120 1500.00.10.20.30.4 c.m. (deg) d / d ( m b / s r M e V ) E =12-14 MeV PE+CN 0.15+0.045*cos (c) FIG. 23: (Color online) Comparison of measured [89] an-gular distributions for Fe( n, α ) reaction and different α -energy bins in the c.m. system, of (a) 6-10 MeV, (b) 10-12MeV, and (c) 12-14 MeV, and calculated values of the pickupDI (dash-dotted curves), PE+CN (dashed curves) equivalentforms a + b · cos θ (dotted curves), and sum of the DI+PE+CNcomponents (solid curves). case of the larger cross sections for 2 +1 excited state evenat incident energies of 6 and 6.5 MeV. It remains thus asignificant underestimation at these energies only for the0 + g.s. while a good agreement is already provided forthe 4 +1 excited state by CN mechanism.Following the discussion in the main paper on like–GQR contributions in ( n, α ) reaction on , Fe, a simi-lar attempt has concerned also the analysis of these data.Thus, their description can be obtained by inclusion of a7
E (MeV) ( m b ) Fe(n, ) Cr g.s.2nd g.s. FIG. 24: As Fig. 11(d) but only for g.s. (firstly upper, thenmiddle curves), first (middle, then upper curves) and second(lowest curves) excited states of Cr, and CN cross sectionscalculated using the α -particle OMPs of Rev. [7] (dashedcurves), DI pickup contributions (dash-dotted curves), andCN+DI sum (solid curves). ( m b ) E (MeV)
Gledenov+(2014) Khromyleva+(2018)
TALYS-1.9TENDL-2019 CN+DI+GQR CN DI (PICKUP) GQR Fe(n, ) Cr FIG. 25: As Fig. 11(d) but only for total ( n, α ) reaction crosssections [13, 15], and additional like–GQR component (dash-dot-dotted curve).
Gaussian distribution at E GQR =16.792 MeV [22] for theexcited nucleus Fe, with a FWHM width of 2.35 MeVand a peak cross section of 1 mb (Fig. 25). It results againa like-GQR component larger than the DI pickup crosssections along the yet increasing side of the former. Bothof them are still minor but provide an increased agree-ment with one of the two data sets. Nevertheless thesedata sets, that are not consistent above the incident en-ergy of 6 MeV, make less certain any definite conclusionon a possible like–GQR component within α -emission. Co ( n, α ) Mn Unfortunately, there is no spectroscopic informationconcerning Mn population through pickup reactions asalso ( p, d ), ( d, t ), or ( He, α ). Hence Mn excited statesand spectroscopic factors for transitions to neutron-hole
E =7-10 MeV Co(n, ) Mn, E n =14.1 Fischer+ (1986) PE+CN 0.43+0.2*cos (a)0.10.20.30.40.5 E =10-12 MeV PE+CN 0.20+0.26*cos (b)30 60 90 120 1500.020.050.1 c.m. (deg) d / d ( m b / s r M e V ) E =12-14 MeV PE+CN 0.06+0.074*cos DI (PICKUP) DI+PE+CN(c)
FIG. 26: As Fig. 23 but for Co target nucleus [91] and (a)the first c.m. α -energy bin of 7-10 MeV. states corresponding to the coupling of the 2 p / and1 f / neutron outside of the magic N=28 shell had to besomehow covered.Our tentative attempt in this respect concerned the useof neutron spectroscopic factors from the Mn( d, p ) Mnreaction analysis [112, 114] and the spectator protonspair as picked from 1 f / subshell. And lastly, 27 ex-cited states with well-known J π and transferred orbitalangular momentum [42, 45], until 2.088 MeV excitationenergy, were considered in the pickup assessment in thefollowing as well as in the main paper.A quite similar measurement as the above-mentionedfor Fe( n, α ) reaction [89] was performed on Co targetnucleus [91] as well. Comparison of calculated angulardistributions for the c.m. α -energy bins of 7-10 MeV,810-12 MeV, and 12-14 MeV, and the Fischer et al. dataat 14.1 MeV is shown in Fig. 26, while comments onthe angle-integrated spectrum and total α -emission crosssection are in the main paper.The above–mentioned pickup results were added to theequivalent form a + b · cos θ corresponding to the PE+CNisotropic component only for the higher α -particle ener-gies (12–14 MeV) in Fig. 26(c). The lower limit of 2.088MeV excitation energy for the above-mentioned 27 statesof the odd–odd residual nucleus Mn was the origin ofthis matter, that is yet in agreement with the lowest-lyingstates being populated through the pickup mechanism.The anisotropy of the corresponding measured distribu-tion has been better described, with a slight overestima-tion but yet within 2 σ at backward angles. The largestdistribution, at the α -particle energies 7-10 MeV, is alsoin good agreement with measured data, providing thus asupport for the PE+CN present account. Zn ( n, α ) Ni The analysis of the pickup contribution to Zn( n, α ) Ni reaction has taken into account theassumption that the spectator protons pair is pickedfrom 2 p / subshell. Then, the transferred orbitalmomenta L to Ni excited states in ( d, t ) and ( He, α )reactions that were previously analyzed [115, 116]were considered as well. The spectroscopic factorsfor the population of the g.s and two excited statesin Ni( d, t ) Ni reaction were obtained by analysis oftriton angular distributions [117] (Fig. 27) leading tocalculated excitation function in Fig. 6 of Ref. [115].Next, the spectroscopic factors reported for neutron–removing reaction Ni( p, d ) by Schiffer et al. [118] wereused for other 13 excited states until 2.64 MeV excitationenergy. Most important are however the g.s. and excitedstates at 0.067 and 0.283 MeV, within a first group, andat 0.656 and 0.909 MeV, within a second group, for whichZhang et al. [96, 97] measured α -particle angular distri-butions at neutron incident energies from 2.54 to 5.95MeV. The same analysis as for Fe and Co is shownin Fig. 28 for these data, with major conclusions for theinvolved reaction mechanisms.First, the experimental remark [97] of the anisotropicfirst–group and almost isotropic second–group angulardistributions is confirmed by the pickup contributions to d / d ( m b / s r) g.s. c.m. [deg] Ni(d,t) Ni E d =15 MeV FIG. 27: As Fig. 19 but for triton angular distributions of Ni( d, t ) Ni pickup transitions [117]. E n =2.54 Zn(n, ) Ni Zhang+ (2008)
DI (PICKUP) DI+CN CN 0.63+0.3*cos CN 1.3+1.9*cos CN 0.94+0.26*cos E n =4 CN 2.83+0.4*cos E n =5.5 Zhang+ (2008) d / d ( m b / s r) CN 1.38+1.8*cos
30 60 90 120 1501234 CN 0.96+1.45*cos c.m. (deg)
123 CN 2.11+0.6*cos E n =5.03 Zhang+ (2007)
30 60 90 120 150123456 CN 3.2+0.7*cos E n =5.95 Zhang+ (2007)
FIG. 28: As Fig. 23 but for Zn target nucleus [96, 97],neutron energies from 2.54 to 5.95 MeV, and populated g.s.and excited states at 0.067 and 0.283 MeV, within a firstgroup (left side), and 0.656 and 0.909 MeV, in a second group(right side) of Ni. the calculated data. It is true even for the incident ener-gies of 4 and 5.03 MeV, where there is an agreement ofcalculated and measured values only for the data trend,but the same data underestimation as for the total ( n, α )cross sections in Fig. 16(c). On the other hand, the sogood agreement at the lowest incident energy of 2.54 MeVis a definite support of the α -particle OMP [7], while theearlier one [8] leads to values that are twice the data.This angular–distribution analysis is completed bythat of the excitation function for the total ( n, α ) re-action as well for the above–mentioned first and secondgroup of states (Fig. 29). Thus, underestimation for thetotal ( n, α ) cross sections at the incident energies of 49 Zn(n, ) Ni Yuan+ (2003)Zhang+ (2007) Zhang+ (2008) Khromyleva+(2018) TALYS-1.9 TENDL-2019 (a)1040 ( m b ) Zn(n, ) Ni (b)3 4 5 6 70.211050 E (MeV) Zn(n, ) Ni PE+CN DI (PICKUP) DI+PE+CN (c)
FIG. 29: As Fig. 25 but for (a) Zn( n, α ) Ni [15, 43, 96, 97],(b) g.s. and excited states at 67, 283, (c) 656, and 909 keV. and 5.03 MeV follows the same feature of the first group,with no improvement due to the additional DI contribu-tion. However, the good agreement at the lowest incidentenergy of 2.54 MeV is extended above 5 MeV following the DI inclusion. Finally the best agreement proved forthe second group at all energies provides a definite sup-port of the α -particle OMP [7], the earlier one [8] leadingto values greater than twice the data [Fig. 16(c)].A final remark may concern the rather opposite nu-clear asymmetries of Fe and Zn. Thus the isotopiceffect led to so different cross sections, the DI compo-nent becoming significant only for the neutron-richer Fe(Fig. 25).
V. CONCLUSIONS
A consistent set of statistical-model input parameters,validated by analysis of various independent data, makespossible the assessment of an optical model potential [7]also for nucleon-induced α -emission within A ∼
60 range.Particularly, α –emission from , , Fe and Co iso-topes excited by ( n, α ) reaction, and , , , Cu and , , , Zn isotopes excited through both ( n, α ) and( p, α ) reactions has been analyzed. The advantage ofrather recent data of low-lying states feeding is essen-tial. Further consideration of additional reaction chan-nels leading to increase of the α -emission cross sectionsbeyond the statistical predictions has also concerned theDI pickup. The assessment of DI cross sections has beensubject to available information on spectroscopic factorsrelated to populated states, outgoing particle angular dis-tributions, or at least differential cross–section maximumvalues. Acknowledgments
This work has been partly supported by Autori-tatea Nationala pentru Cercetare Stiintifica (Project PN-19060102) and carried out within the framework of theEUROfusion Consortium and has received funding fromthe Euratom research and training programme 2014-2018and 2019-2020 under grant agreement No 633053. Theviews and opinions expressed herein do not necessarilyreflect those of the European Commission. [1] P. Demetriou, C. Grama, and S. Goriely, NuclearPhysics A , 253 (2002), ISSN 0375-9474, URL .[2] W. Hauser and H. Feshbach,Phys. Rev. , 366 (1952), URL https://link.aps.org/doi/10.1103/PhysRev.87.366 .[3] E. Gadioli and P. E. Hodgson, Pre-Equilibrium NuclearReactions (Clarendon, Oxford, 1992).[4] E. D. Arthur, Nuclear Science and Engineering , 137(1980), https://doi.org/10.13182/NSE80-A19446, URL https://doi.org/10.13182/NSE80-A19446 .[5] T. Rauscher, Phys. Rev. Lett. , 061104 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.111.061104 .[6] M. 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