Solution of the alpha-potential mystery in the gamma-process and its impact on the Nd/Sm ratio in meteorites
aa r X i v : . [ a s t r o - ph . S R ] J u l Solution of the α -potential mystery in the γ -process and its impact on the Nd/Smratio in meteorites Thomas Rauscher ∗ Centre for Astrophysics Research, School of Physics,Astronomy and Mathematics, University of Hertfordshire,College Lane, Hatfield AL10 9AB, United KingdomInstitute for Nuclear Research, Hungarian Academy of Science, 4001 Debrecen, Hungary andDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Dated: August 31, 2018)The
Sm/
Sm ratio in the early solar system has been constrained by Nd/Sm isotope ratiosin meteoritic material. Predictions of
Sm and
Sm production in the γ -process in massivestars are at odds with these constraints and this is partly due to deficiences in the prediction ofthe reaction rates involved. The production ratio depends almost exclusively on the ( γ ,n)/( γ , α )branching at Gd. A measurement of
Sm( α , γ ) Gd at low energy had discovered considerablediscrepancies between cross section predictions and the data. Although this reaction cross sectionmainly depends on the optical α +nucleus potential, no global optical potential has yet been foundwhich can consistently describe the results of this and similar α -induced reactions at the low energiesencountered in astrophysical environments. The untypically large deviation in Sm( α , γ ) and theunusual energy dependence can be explained, however, by low-energy Coulomb excitation whichis competing with compound nucleus formation at very low energies. Considering this additionalreaction channel, the cross sections can be described with the usual optical potential variations,compatible with findings for (n, α ) reactions in this mass range. Low-energy ( α , γ ) and ( α ,n) dataon other nuclei can also be consistently explained in this way. Since Coulomb excitation does notaffect α -emission, the Gd( γ , α ) rate is much higher than previously assumed. This leads to small Sm/
Sm stellar production ratios, in even more pronounced conflict with the meteorite data.
PACS numbers: 26.30.-k, 98.80.Ft, 25.55.-e, 26.30.-k, 26.50.+x, 96.10.+i
The astrophysical γ -process synthesizes proton-richnuclides through sequences of photodisintegrations ofpre-existing seed material. It occurs in explosive Ne/Oburning in core-collapse supernova (ccSN) explosions ofmassive stars [1, 2]. This site was supposed to be themain source of the p-nuclides, i.e., naturally occurring,proton-rich nuclei which cannot be produced in the s-and r-process [3, 4]. A recent investigation has shownthat also type Ia supernovae (SNIa) may be a viable sitefor the γ -process [5, 33], although previous simulationshad not been successful [6, 7].The γ -process produces both Sm and
Sm, theproduction ratio
R ≡ P /P ∝ λ γ n /λ γα = R γ n /R γα depends on the stellar ( γ ,n) and ( γ , α ) rates of Gd,denoted by λ γ n and λ γα , respectively, or alternatively onthe ratios of the reactivities, denoted by R γ n and R γα [8]. This ratio is of particular interest because it wassuggested that surviving Sm may be detected in thesolar system and used for cosmochronometry [9]. No live
Sm has been found to date but at least the signatureof its in-situ decay in meteorites is believed to be seen,from which the isotope ratio at the closure of the solarsystem can be inferred [3, 10–12].There are still large uncertainties involved in determin-ing the production ratio, both from the side of astrophys-ical models and from nuclear physics. To better constrainthe nuclear uncertainties
Sm( α , γ ) Gd was measuredin a pioneering, difficult experiment [13]. Since the stel- lar α -capture reactivity R αγ is dominated by the groundstate (g.s.) transition [8, 14], the laboratory value can beconverted to the stellar ( γ , α ) reactivity R γα by applyingdetailed balance [8, 15]. Although the astrophysicallyrelevant energy range of 9 MeV and below [16] could notbe reached, the lowest datapoint at 10.2 MeV alreadyshowed a strong deviation from predictions. Using anoptical α +nucleus potential with an energy-dependentpart fitted to reproduce the data, a stellar ( γ , α ) ratewas derived which was lower by an order of magnitudethan previous estimates [13] (see Table I). This led to astrongly increased R .This result shed doubts on the prediction of ( γ , α ) ratesat γ -process temperatures and triggered a number of ex-perimental and theoretical studies. Due to the tiny reac-tion cross sections, however, data is still scarce in the rele-vant mass region (at neutron numbers N ≥
82) and closeto astrophysical energies. A comparison of predictionsto data at higher energy often is irrelevant because thecross sections depend not only on the α -widths, as theydo at low energy [14]. To calculate the reaction cross sec-tions in the Hauser-Feshbach model [17], so-called opticalpotentials – describing the effective interaction betweenprojectile and target nucleus – have to be used in the nu-merical solution of the Schr¨odinger equation. Many localand global optical α +nucleus potentials have been de-rived, using elastic scattering at higher energy, reactioncross sections, and theoretical considerations (like fold-ing potentials), e.g., see [18, 19] and references therein.None of the potentials are able to describe the existing( α , γ ) and ( α ,n) data consistently, yet. Rather, a seem-ingly confusing picture arises. Some of the low-energydata are described well, the majority of cases find devi-ations increasing with decreasing energies but never ex-ceeding overprediction factors 2 −
3, and then there is the
Sm( α , γ ) case with its large deviation of more than anorder of magnitude. Also the energy dependence of the Sm( α , γ ) data is peculiar and cannot be reproducedby any prediction (unless fitted to the data). The onlycommon factor seems to be that the predictions usingthe standard optical potential [20] are either close to thedata or considerably higher.Instead of attempting to solve the discrepancy by mod-ifying the nuclear interaction potentials alone, anotherapproach is suggested here. The low-energy deviationsand their variation from one nucleus to another can beexplained by the action of an additional reaction chan-nel which was not considered in the calculations, such asa direct inelastic channel (direct elastic scattering is in-cluded in the usual optical potentials [21]). In the pictureof the optical model, this channel would divert part of theimpinging α -flux away from the compound nucleus for-mation channel and thus lead to fewer compound nucleiat a given projectile flux. In the experiment this is seenas smaller reaction yield. Coulomb excitation (Coulex)is such a reaction mechanism. It has been used exten-sively to study nuclear structure and it is well knownthat Coulex cross section can be comparable to or largerthan compound reaction cross sections at several tens ofMeV. At lower energies they are commonly assumed tobe negligible compared to the compound formation crosssection. It can be shown, however, that for intermediateand heavy nuclei the Coulex cross section σ Coulex declinesmore slowly with decreasing energy than the compoundformation cross section σ form , due to the Coulomb bar-rier. Consequently, the Coulex cross section can becomecomparable to or even exceed the compound formationcross section close to the astrophysical energy range andbelow, even when it has been negligible at intermediateenergy.A straightforward way to include the diversion of α -flux from the compound formation channel in the crosssection calculation is to use a modified compound for-mation cross section (this can simply be implemented byusing modified α -transmission coefficients in the entrance channel) σ form , mod ℓ = f ℓ σ form , with f ℓ = σ form ℓ σ form ℓ + σ Coulex ℓ (1)for each partial wave ℓ . The Coulex cross section can becalculated, e.g., by [22] σ Coulex ℓ ∝
9 9.5 10 10.5 11 11.5 12 12.5 13 S - f a c t o r ( M e V ba r n ) E cm (MeV) ExpMcFS PotMcFS with CoulexMcFS with Coulex, renorm FIG. 1. Experimental S factors for Sm( α , γ ) Gd [13](Exp) are compared to predictions using the standard po-tential [20] (McFS, dashed line), using the same potentialbut corrected for Coulomb excitation (dotted line), and theCoulex corrected prediction with the α -width divided by aconstant factor of 3 (full line). The astrophysically relevantenergy is about 8 − B (E L ) X ℓ f (2 ℓ f + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z F ℓ f ( k f r ) r −L− F ℓ ( kr ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2)using regular Coulomb wave functions F ℓ ( kr ), F ℓ f ( k f r ) atinitial and final α -energies, respectively. The transitionstrengths for electric multipole emission of multipolarity L are given by B (E L ). The results shown here are for thedominant multipolarity L = 2, i.e., E2 transitions andwere obtained using a newly developed Hauser-Feshbachcode, called SMARAGD [23].In astrophysical investigations, often the S-factor S ( E ) = σE exp(2 πη ) is given rather than the reactioncross section σ , with the exponential including the Som-merfeld parameter η accounting for the Coulomb bar-rier penetration. Figure 1 shows how the S-factor of Sm( α , γ ) Gd is changed by inclusion of Coulex whilestill using the standard potential [20]. The energy depen-dence of the data is now accurately reproduced but theabsolute value is still too high. It was assumed in thecalculation, however, that the optical potential used ac-curately describes compound formation in the absence ofCoulex. This does not have to be the case, though, theremay still be an additional energy dependence which hasto be determined independently. The data can be de-scribed well by renormalizing the α -widths obtained withthe standard potential, as also shown in Fig. 1. The re-quired factor of 1/3 is well in line with the typical de-viations found for other reactions involving α particlesat low energy, for which no Coulex occurs (e.g., in (n, α )reactions).The approach outlined above should also remain validwhen applied to other reactions. Due to the scarcity ofsuitable data, there are only few cases to be checked. Al-ready without inclusion of low-energy Coulex, very good σ [ m b ] E c.m. [MeV] SMARAGDExp 0.001 0.01 0.1 1 13 13.5 14 14.5 15 σ [ m b ] E c.m. [MeV]SMARAGDExp FIG. 2. Experimental cross sections [25] of
Yb( α ,n) Hf(top) and
Yb( α , γ ) Hf (bottom) are compared to predic-tions with the code SMARAGD using the standard potential[20]. S - f a c t o r ( M e V ba r n ) E cm (MeV)(McFS,coulex)/1.2ExpMcFS FIG. 3. Experimental S-factors for
Pr( α ,n) Pm [26](Exp) are compared to predictions using the standard po-tential [20] (McFS) and correction for Coulomb excitation(McFS, Coulex), with additional 20% renormalization. Theuncertainty introduced by the B (E2) values is shown by theshaded region. agreement was found between predictions and data for , Ba( α ,n) [24]. Despite the presence of low-lying 2 + states, this remains so when including Coulex becauseits cross section σ Coulex ℓ remains small compared to σ form ℓ in the investigated energy range for the relevant partialwaves. Another example for such a case is shown inFig. 2, where recent data [25] for Yb( α ,n) Hf and
Yb( α , γ ) Hf are compared to predictions. The ex-cellent reproduction of the ( α ,n) data – which mainlydepends on the correct description of the α -widths [14]
11 12 13 14 15 16 17 S - f a c t o r ( M e V ba r n ) E cm (MeV) McFS,coulexExpMcFS FIG. 4. Experimental S-factors for
Tm( α ,n) Lu [27](exp) are compared to predictions using the standard po-tential [20] (McFS) and correction for Coulomb excitation(McFS, Coulex). The uncertainty introduced by the B (E2)values is shown by the shaded region. – shows that the standard potential [20] fares well. Theslight deviations from the data found in the ( α , γ ) re-action must be due to the modeling of the γ - and/orneutron-widths but are not astrophysically relevant, asthe α -capture cross sections depend only on the α -widthat astrophysical energies [14]. Two further cases areshown in Figs. 3 and 4, the reactions Pr( α ,n) Pmand
Tm( α ,n) Lu, respectively. In both cases, theincreasing deviation found for decreasing energy can benicely explained by Coulex. A large uncertainty, how-ever, remains in the B (E2) values which are experimen-tally not well determined for odd nuclei (or nuclei withg.s. other than 0 + ). The prediction for Pr( α ,n) mayneed a small modification of the optical potential, itis 20% too high. But this is considerably lower thanthe usually assumed uncertainties in astrophysical ratepredictions. The large uncertainty stemming from the B (E2) value does not allow to draw a final conclusion on Tm( α ,n) but it seems that it may be feasible to repro-duce the energy dependence of the data without changein the optical potential.To assess the impact on the stellar Gd( γ , α ) rateit should be recalled that Coulex acts in the entrance channel but the α -emission channel should be unaffected.This is also the reason why an optical potential account-ing for compound formation without including Coulex inits absorptive part has to be used. Only such a poten-tial can then be applied to α -emission. (Detailed bal-ance then applies to transitions obtained with such apotential.) This is not the potential that would be ob-tained by α -scattering. If it were possible to perform an α -scattering experiment at such low energy and extractan optical potential without correcting for Coulex, thispotential would include both compound formation andCoulex in its absorptive part but no information on howto distribute the flux across the two possibilities. The TABLE I. Stellar
Sm( α , γ ) Gd reactivities at a plasma temperature 2.5 GK from different sources, obtained with differentcodes and different types of optical α +nucleus potentials. Also shown are the final Sm/
Sm production ratios R obtainedfor different Sm( α , γ ) Gd rates (and their reverse rates) in two models of the ccSN of a 25 M ⊙ star (ccSN-A [13, 31] andccSN-B [2, 34]) and a SNIa model [33]. The values obtained with the optical potential of this work are given on the last line.Type Code Reactivity R (cm s − mole − ) ccSN-A ccSN-B SNIaEquivalent Square Well [28] CRSEC [1] 3 . × − Folding (real), Woods-Saxon (imag.) SMOKER a [29] 1 . × − Woods-Saxon [20] NON-SMOKER a [30] 1 . × − . × − a [13], SMOKER a [29] 1 . × − . × − α -width SMARAGD, this work 1 . × − a The codes SMOKER, NON-SMOKER, MOST used the same routine to calculate Coulomb barrier penetration. result without
Coulex also has to be used for computingthe stellar reactivity R αγ = N A h σv i ∗ αγ for Sm( α , γ ),which then can be converted to the ( γ , α ) reactivity R γα .Since the α -width had to be reduced to reproduce thedata after Coulex was applied (see Fig. 1), it also has tobe reduced in the original result without Coulex.Table I compares the stellar reactivities for Sm( α , γ )obtained with different codes and different potentials, asused in astrophysical applications. It should be notedthat the codes also use different treatments of Coulombbarrier penetration and recently only the implementationin the SMARAGD code has been shown to be adequatefor α -transmission far below the Coulomb barrier. The fi-nal SMARAGD prediction (last line in Table I) is higherthan all previous estimates used in stellar models andin particular higher by two orders of magnitude than thevalue obtained by directly fitting the experimental results[13]. This leads to a reduced isotope ratio R . The ac-tual value varies slightly between stellar models and alsodepends on the Gd( γ ,n) used, as this reaction com-petes with Gd( γ , α ). Table I also shows R obtainedfrom postprocessing of three different models, using dif-ferent Gd( γ , α ) rates (and their reverses). Two modelsuse trajectories from ccSN explosions of 25 solar masses( M ⊙ ) progenitor stars with solar metallicity: one simi-lar to [31] (ccSN-A) and a new model similar to [2] butwith initial solar abundances from [32] (ccSN-B). Thevalue for the SNIa is taken from [33]. All of the cal-culations use the predicted Gd(n, γ ) Gd rate from[30]. Using an open-box model for galactic chemical evo-lution and neglecting any further free-decay and mixingtimescales before inclusion into the early solar system(ESS), a range of 0 . ≤ R ≤ .
23 is permitted by the
Sm/
Sm ratio in the ESS inferred from meteoriticdata [3, 12]. Slightly higher values of R can be accom-modated by making further assumptions on additionaltimescales during which the produced Sm decays be-fore being incorporated into ESS solids. Although the ccSN isotope ratios R vary due to model differences, theyare too low to fall into the permitted range. There is nocalculation available for SNIa with the new potential butif the reduction in the ratio is of the same order as foundfor the ccSN models, then the new ratio could well bewithin the permitted range.The new, low value of R challenges explosive nucle-osynthesis models as well as investigations in galacticchemical evolution and the formation of solids in the ESS.Further studies in both astrophysics and nuclear physics,however, are required to determine the actual value. De-tails in the stellar modelling and the used C( α , γ ) Orate [31] will impact the resulting ratio. Moreover, con-tributions from massive stars with different masses andinitial composition are superposed during galactic evolu-tion. Here, we only showed examples for 25 M ⊙ stars.Finally, the actual seed distribution which is photodis-integrated does not influence R , since both Sm and
Sm originate from the photodisintegration of
Gd.Not only the peak temperature reached in a zone, how-ever, but also the temperature evolution, i.e., how muchtime is spent at a given temperature, impacts the finalratio. A higher temperature favors ( γ ,n) with respect to( γ , α ) and increases Sm production [8]. The produc-tion ratio thus also depends on the expansion timescale,higher explosion temperatures are relevant with shortertimescales. The expansion is different in different ccSNmodels and it may be very different for SNIa. Followingthe expansion of the expanding hot fragments of bothccSN and SNIa – and thus of their actual nucleosynthesis– requires a detailed understanding of the explosion andaccurate, high-resolution hydrodynamic modelling beforefinal conclusions can be drawn.Nuclear experiments can help to test the low-energyCoulex effect introduced here. For improved results, the B (E2) values for odd nuclei have to be determined withhigher precision. In addition, if possible, a simultaneousdetection of the γ -emission from the excited target nu-cleus state while performing a reaction experiment coulddirectly indicate the action of Coulex. Complementarymeasurements of low-energy α -absorption and -emission(difficult for Sm, obviously, but feasible for other testcases) should show a difference in the two directions, notaccountable for by straightforward application of detailedbalance. In this context it is interesting to note that (n, α )experiments on Nd and
Sm find an overpredictionby a factor of 3 [35–38]. This is fully consistent with therequired renormalization found here for
Sm+ α , aftercorrection for Coulex. Finally, the isotope ratio R alsodepends on the Gd(n, γ ) Gd rate which is uncon-strained by experiment.This work is partly supported by the HungarianAcademy of Sciences, the ESF EUROCORES pro-gramme EuroGENESIS, the ENSAR/THEXO collabo-ration within the 7th Framework Programme of the EU,the European Research Council, and the Swiss NSF. ∗ [email protected][1] S. E. Woosley and W. M. Howard, Ap. J. Suppl. , 285(1978).[2] T. Rauscher, A. Heger, R. D. Hoffman, and S. E.Woosley, Ap. J. , 323 (2002).[3] T. Rauscher, N. Dauphas, I. Dillmann, C. Fr¨ohlich, Zs.F¨ul¨op, and Gy. Gy¨urky, Rep. Prog. Phys. , 066201(2013).[4] M. Arnould and S. Goriely, Phys. Rep. , 1 (2003).[5] C. Travaglio, F. R¨opke, R. Gallino, and W. Hillebrandt,Ap. J. , 93 (2011).[6] W. M. Howard, B. S. Meyer, and S. E. Woosley, Ap. J.Lett. , L5 (1991).[7] K. Nomoto, F.-K. Thielemann, and K. Yokoi, Ap. J. ,644 (1984).[8] S. E. Woosley and W. M. Howard, Ap. J. , L21(1990).[9] J. Audouze and D. N. Schramm, Nature , 447 (1972).[10] A. Prinzhofer, D. A. Papanastassiou, and G. A. Wasser-burg, Ap. J. Lett. , L81 (1989). [11] C. L. Harper, Ap. J. , 437 (1996).[12] N. Kinoshita et al. , Science , 1614 (2012).[13] E. Somorjai et al. , Astron. Astrophys. , 1112 (1998).[14] T. Rauscher, Ap. J. Suppl. , 26 (2012).[15] T. Rauscher, Int. J. Mod. Phys. E , 1071 (2011).[16] T. Rauscher, Phys. Rev. C , 045807 (2010).[17] W. Hauser and H. Feshbach, Phys. Rev. , 366 (1952).[18] G. G. Kiss et al. , Phys. Rev. C , 045807 (2009).[19] P. Mohr, G. G. Kiss, Zs. F¨ul¨op, D. Galaviz, Gy. Gyrky,and E. Somorjai, At. Data Nucl. Data Tables, in press;arXiv:1212.2891[20] L. McFadden and G. R. Satchler, Nucl. Phys. , 177(1966).[21] G. R. Satchler, Direct Nuclear Reactions (ClarendonPress, Oxford, 1983).[22] K. Alder, A. Bohr, T. Huus, B. Mottelson, and A.Winther, Rev. Mod. Phys. , 432 (1956).[23] T. Rauscher, computer code SMARAGD, version 0.8.1s(2010).[24] Z. Hal´asz et al. , Phys. Rev. C , 025804 (2012).[25] A. Sauerwein, PhD thesis, University of Cologne, Ger-many (2012).[26] A. Sauerwein et al. , Phys. Rev. C , 045808 (2011).[27] T. Rauscher et al. , Phys. Rev. C , 015804 (2012).[28] J. W. Truran, Astrophys. Space Sci. , 306 (1972).[29] T. Rauscher, F.-K. Thielemann, and H. Oberhummer,Ap. J. Lett. , L37 (1995).[30] T. Rauscher and F.-K. Thielemann, At. Data Nucl. DataTables , 47 (2001).[31] M. Rayet, M. Arnould, M. Hashimoto, N. Prantzos, andK. Nomoto, Astron. Astrophys. , 517 (1995).[32] K. Lodders, in Principles and Perspectives in Cosmo-chemistry, Astrophysics and Space Science Proceedings.ISBN 978-3-642-10351-3. Springer-Verlag Berlin Heidel-berg, 2010, p. 379.[33] C. Travaglio, R. Gallino, W. Hillebrandt, and F. R¨opke,PoS(NIC XII)045[34] A. Heger, private communication (2011).[35] Yu. M. Gledenov P. E. Koehler, J. Andrzejewski, K. H.Guber, and T. Rauscher, Phys. Rev. C , 042801(R)(2000).[36] P. E. Koehler et al. , Nucl. Phys. A688 , 86c (2001).[37] P. E. Koehler, Yu. M. Gledenov, T. Rauscher, and C.Fr¨ohlich, Phys. Rev. C , 015803 (2004).[38] Yu. M. Gledenov et al. , Phys. Rev. C80