Complete inclusion of parity-dependent level densities in the statistical description of astrophysical reaction rates
Hans Peter Loens, Karlheinz Langanke, Gabriel Martínez-Pinedo, Thomas Rauscher, Friedrich-Karl Thielemann
aa r X i v : . [ nu c l - t h ] A p r Complete inclusion of parity-dependent level densities in thestatistical description of astrophysical reaction rates
H. P. Loens a , b , K. Langanke a , b , G. Mart´ınez-Pinedo a , T. Rauscher c and F.-K. Thielemann c a Gesellschaft f¨ur Schwerionenforschung, Planckstr. 1, 64291 Darmstadt (Germany) b Technische Univerist¨at Darmstadt, Institut f¨ur Kernphysik, Schlossgartenstr. 9, 64289 Darmstadt (Germany) c Universit¨at Basel, Klingelbergstr. 82, 4056 Basel (Switzerland)
Abstract
Microscopic calculations show a strong parity dependence of the nuclear level density at low excitation energy of anucleus. Previously, this dependence has either been neglected or only implemented in the initial and final channelsof Hauser-Feshbach calculations. We present an indirect way to account for a full parity dependence in all steps of areaction, including the one of the compound nucleus formed in a reaction. To illustrate the impact on astrophysicalreaction rates, we present rates for neutron captures in isotopic chains of Ni and Sn. Comparing with the standardassumption of equipartition of both parities, we find noticeable differences in the energy regime of astrophysicalinterest caused by the parity dependence of the nuclear level density found in the compound nucleus even at sizeableexcitation energies.
Key words: parity dependence, Hauser-Feshbach theory, nuclear level density, neutron capture, astrophysical reaction rates,nucleosynthesis
PACS:
1. Introduction
Nuclear reactions in systems with high level den-sity at low and intermediate energies are commonlytreated in the compound mechanism [1,2,3]. This re-action mechanism was first postulated by Bohr in hiswell-known independence hypothesis, stating thatreactions can proceed via formation of a compoundnucleus and that the decay of the compound nucleusis determined entirely by its energy, angular momen-tum, and parity, and not by the way it was formed[4]. This hypothesis remains valid below a projec-tile energy of a few tens of MeV. At higher ener-gies, doorway states, pre-compound, and direct reac-tions become increasingly important. In the follow-ing, we focus on the low energy region and thus on
Email address: [email protected] (H. P. Loens). the pure statistical picture (Hauser-Feshbach the-ory, e.g. [1,2]) because our ultimate goal is to proposean improved description for nuclear astrophysics. Inastrophysical nuclear burning, the relevant energy inthe projectile-target system does not exceed about200 keV for neutron-induced reactions and 10–12MeV for proton- and α -induced reactions [5]. Mostastrophysically important reactions occur at evensignificantly lower energy.Angular momentum conservation is included instandard Hauser-Feshbach theory and thus theBohr hypothesis independently holds for each spin J and parity π of the compound nucleus, formedfrom the interaction of a projectile with a targetnucleus. There are two fundamental assumptions inthe derivation of this theory: 1) There are alwayssufficient compound-nuclear states with J π in therelevant excitation energy range; 2) the wave func- Preprint submitted to Physics Letters B 26 October 2018 ions of the compound nuclear states have randomphases, so that interferences between reactions pro-ceeding through different compound nuclear statesvanish. Due to the strong energy-dependence of thenuclear level density, these assumptions are valid formost reactions (especially at intermediate energy)on stable targets studied in the laboratory. How-ever, it was shown [5] that the level density becomestoo low for the application of the Hauser-Feshbachstatistical model for astrophysically important re-actions involving nuclei far off stability, exhibitingsmall particle separation energies, or even for nucleiclose to stability around closed shells [6] at the lowend of astrophysically relevant energies.For what follows it is important that in appli-cations of the Hauser-Feshbach model one assumesthat both parities are equally present in the com-pound nucleus at the formation energy. This pre-sumption clearly is not valid at very low excitationenergy (e.g. due to pairing effects), but there are alsomany indications from theory as well as some exper-iments that parities may not be equilibrated evenat considerably large excitation energies, in somecases up to 12 MeV [7,8,9]. Similar results are foundin different approaches, e.g., in combinatorial meth-ods using single-particle energies from microscopicHartree-Fock-Bogoliubov calculations [10] as well asin recent Shell Model Monte Carlo (SMMC) calcu-lations [11,12]. We note, however, that very recentexperimental data for Ni and Zr are in accordwith the equal-parity assumption at excitation en-ergies as low as 7 MeV (for Ni in disagreementwith the theoretical predictions). Nevertheless, theassumption of parity-independence of the level den-sity is clearly doubtful for a large number of exoticnuclei in the energy range important in astrophysi-cal environments.In this Letter we incorporate the parity depen-dence of the level density into the statistical de-scription of astrophysically capture reactions in allstages of the reaction. Parity-dependent level densi-ties have been used before in statistical model cal-culations [10,9]. But these concern the distributionof initial and final states, where the first can be pop-ulated due to the finite temperature in the astro-physical environment, while the parity-dependencein the newly formed, excited compound nucleus hasnever been considered before. Speaking in Bohr’sterms, our modification impacts the formation crosssection of the compound state.In the next section we present the details of themodification. This is followed by some restricted ex-
Fig. 1. Schematic sketch of the compound capture reaction.A particle (neutron or proton) is captured in the state µ in the nucleus with mass number A , exciting a state in thecompound/daughter nucleus ( A + 1) at energy E and withangular momentum and parity quantum numbers J and π ,respectively. This state decays by γ emission to the state ν in the same nucleus. amples for application to astrophysical neutron cap-ture which are merely given to discuss the modeland to illustrate the possible implication for astro-physics. The final section gives a summary and anoutlook to future work.
2. Formalism
The Hauser-Feshbach expression for the cross sec-tion of a reaction proceeding from the target state µ with spin J µi and parity π µi to a final state ν withspin J νm and parity π νm in the residual nucleus via acompound state with excitation energy E , spin J ,and parity π (see Fig. 1) is given by σ µν ( E ij ) = π ~ M ij E ij J µi + 1)(2 J j + 1) × X J,π T µj T νo T tot , (1)where E ij , M ij are the center-of-mass energy andthe reduced mass, respectively, in the initial sys-tem, while J j is the ground state spin of theprojectile. The transmission coefficients T µj = T ( E, J, π ; E µi , J µi , π µi ), T νm = T ( E, J, π ; E νm , J νm , π νm )describe the transitions from the compound state tothe initial and final state, respectively. The sum ofthe transmission coefficients of all possible channelsare given by T tot .2s indicated, the transmission coefficients dependon the energy E and the quantum numbers for an-gular momentum J and parity π of the states ex-cited in the compound nucleus. They can in princi-ple be calculated by solving the Schr¨odinger equa-tion for the appropriate degrees of freedom. Such amicroscopic approach will describe simultaneouslyand consistently all the states in the compound nu-cleus including their dependence on parity. Againin principle, this microscopic Schr¨odinger equationcan be mapped onto a complicated optical potentialwhich depends on energy and the other quantumnumbers. There have been first attempts to derivesuch microscopic potentials, which show indeed astrong dependence on parity [13,14]. However, theseattempts are not realistic enough and are restrictedto a few scattering systems. Thus in astrophysi-cal (and other) applications of the Hauser-Feshbachmodel the transmission coefficients are calculatedfrom optical potentials which are expected to give areasonable and global account for the many nucleineeded in nucleosynthesis calculations. For our dis-cussion it is relevant that these global optical poten-tials do not depend on parity and hence also a pos-sible parity dependence of the transmission coeffi-cients is lost. To overcome this shortcoming we pro-pose here an indirect way. It is based on the observa-tion, that for the average transmission coefficients,there is the relation T ∝ h Γ i /D , involving the levelspacing D = 1 /ρ and the average level width h Γ i inthe considered reaction channel. This linear propor-tionality between transmission coefficient and leveldensity ρ leads us to define T ( E, J, π ) = β ( E, J, π ) ˆ T ( E, J ) , (2)where ˆ T ( E, J ) is a transmission coefficient calcu-lated for a global, parity-independent potential (in-cluding centrifugal potential) and the parity depen-dence is introduced by the weighting factor ( π = ± ) β ( E, J, π ) = 2 · ρ ( E, J, π ) ρ ( E, J, +) + ρ ( E, J, − ) . (3)The factor 2 accounts for the proper normalization.This approach assumes that the parity dependenceof the microscopic potential can be fully mappedonto the level density appearing in the standardHauser-Feshbach equations. In the following appli-cations we will use the same ansatz Eq. 2 also for thetransmission coefficients in the final channel. As alltransmission coefficients are evaluated at the sameenergy in Eq. 1, the β factors of the total transmis- sion coefficient (denominator) and one of the β fac-tors of the nominator cancel.As mentioned above, the Hauser-Feshbach ap-proach assumes a “sufficient” number of levels inthe excited compound nucleus so that an averagedtransmission coefficient T is justified and the modelis applicable. For astrophysical application in thedetermination of astrophysical reaction rates theincident energy distribution is given by a Maxwell-Boltzmann distribution giving rise to a relevantenergy window [5,3]. It has been shown that about10 contributing levels (depending on which partialwaves are dominating) within this energy windoware sufficient. This basic conclusion is not affectedby our treatment. However, the parity dependencemay enhance or reduce the number of available rel-evant levels and thus the applicability limits haveto be reevaluated taking into account the spins andparities of the initial states. It should be noted thatwe do not change the total level density but justdistribute it differently between the parities.
3. Results
To explore the possible effects of our modificationwe have performed a series of neutron capture crosssection calculations. At first we have performed con-ventional calculations in which we assumed parityequipartition at all energies in target, compound nu-cleus and residual (calculation a). Secondly we haverestricted the parity dependence to the level densi-ties of the target and residual nucleus (calculationb - these are similar to those of Mocelj et al. [9]).Thirdly, we used a parity-dependent level density inall three steps of the statistical treatment: the target,the compound nucleus and the residual nucleus (cal-culation c). Our calculations have been performedusing the spin and parity dependent level densities ofHilaire and Goriely [10]. To explore how sensitive theresults depend on the set of level densities adoptedwe have repeated our calculations using backshiftedFermi gas level densities with the parametrizationsas derived by Rauscher and Thielemann [5] and theparity dependence as defined by Mocelj et al. [9].In this paper we focus on ( n, γ ) for which the in-fluence of the parity dependence of the level densitycan be discussed considering either the initial neu-tron capture or the final γ decay. In the followingwe have chosen to consider the final γ transitionswhich we assume to be either of parity-conservingM1 or parity-changing E1 multipolarity. The reac-3ion scheme is shown in Fig. 1. Furthermore, we showastrophysical reaction rates which include weightedsums over thermally excited states given by a ther-mal Maxwell-Boltzmann distribution according tothe conditions in a stellar plasma (for the relevantdefinitions see, e.g., Ref. [3]).As a first example we discuss the Ni(n, γ ) Nireaction for which experimental data are availablefor comparison [15].
T [10 K] N A < σ v > [ c m s - m o l e - ] a b c Fig. 2. Stellar reaction rate of Ni( n, γ ) Ni; crosses: rec-ommended values from ’KADoNIS v0.2’ [15], (a) withoutparity dependence, (b) using parity dependence for the finalstates, (c) using parity dependence for the final states and the compound formation using the level densities of [10]
Fig. 2 shows the rate for this reaction as a functionof temperature. One observes basically no differencebetween calculations a) and b); i.e. the considerationof a parity dependence in the level densities of thetarget and residual nucleus has no effect in this case.This is in agreement with the findings of Mocelj etal. [9] and is mainly caused by the fact that both cal-culations use the experimentally known spectrum atlow energies. However, considering the parity depen-dence of the level density in the compound nucleus(calculation c) reduces the rate by about 30% whichis a non-negligible effect. The origin of this reduc-tion becomes clear when one inspects Fig. 3 whichshows the ratio of parity-projected level densities for , Ni defined as ρ π = ρ ( E, π ) = P J ρ ( E, J, π ). Wehave summed over all spins as different values in J have qualitatively the same dependence in the ratio ρ − /ρ + plotted in Fig. 3. For the ( n, γ ) reaction on Ni the γ transitions in the compound nucleus Niare dominated by E1 multipolarity at the relevantenergies. At these energies above the neutron thresh-old negative-parity states dominate the spectrum of Ni due to the negative parity of the unpaired neu- Ni Ni Ni Ni Ni Ni Ni Ni excitation energy [MeV] ρ − / ρ + ρ − / ρ + Fig. 3. Ratios ρ − /ρ + of several nickel nuclides; the par-ity distribution is from Hilaire et al. [10] - we used ρ ( E, π ) = P J ρ ( E, J, π ) here; the arrows mark the neutronseparation energy of the corresponding nickel isotope. tron occupying single particle states in the pf shell.Thus initial states for E1 transitions into these statesmust have positive parity and at stellar conditionsthey have to reside just above the neutron thresh-old energy ( S n = 9 MeV in Ni). At such modestexcitation energies the nuclear models predict stilla dominance of negative parity over positive paritystates in the level density. As we use the same total level density in all calculations a), b), and c), theratio ρ − /ρ + > ρ − /ρ + = 1. We note that this reduction getssmaller with increasing temperature as then higherexcitation energies, at which the ρ − /ρ + gets closerto unity, contribute more to the stellar reaction rate.For M1 transitions the effect is opposite as theserequire negative-parity initial states for this reac-tion. These are enhanced compared to the parity-equipartition assumption and hence the contribu-tion of the M1 transitions relatively increase. This,however, has not much effect on the total cross sec-tion which is dominated by E1 transitions.It is also satisfying that our calculation yieldsa slightly better agreement with the empiricaldata from the KADoNIS compilation when parity-dependent level densities are incorporated into thestatistical model (see Fig. 2).Fig. 4 shows the neutron capture rates for thechain of nickel isotopes as obtained with full paritytreatment (calculation c) relative to the standardtreatment without parity dependence (calculationa). To understand the results one has to consider4hat the importance of parity-dependent level den-sities in the statistical calculation of stellar reactionrates depends on several ingredients: 1) the energydependence of the ratio ρ − /ρ + , 2) the excitation en-ergy of the neutron threshold in the compound, 3)the competition of parity-changing (E1) vs. parity-conserving (M1) γ transitions.
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A of target < σ v > π / < σ v > no r m . T = 10 K
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A of Target < σ v > π / < σ v > no r m . T = 10 K Fig. 4. Ratios of the stellar neutron capture rates of Ni iso-topes with parity influence in target, compound, and resid-ual to the stellar rate without any parity influence - the up-per graph was made by using the level densities from [10]and the lower graph by using the parity distribution from[9] combined with a level density from [5]. The ratios areshown for a temperature T = 1 GK of the stellar plasma. The ratio ρ − /ρ + is shown in Fig. 3 for severalnickel isotopes. Obviously positive parity statesdominate the low-energy spectrum for even-evennuclei, while negative parity-states are more abun-dant at low energies for odd-A nuclei. With increas-ing energy the ratio ρ − /ρ + tends to unity. However,the energy at which parity equipartition is reacheddepends on nuclear structure, i.e. it is basicallydetermined by the energy difference of the Fermienergy to the nearest level with different parity and the occupation of the levels. Thus, the equipar-tition is achieved at increasingly higher energiesfrom Ni to Ni, where the f / and d / orbitalsmatter. By adding more neutrons it becomes en-ergetically easier to excite those to the g / levelto make levels of opposite parity. As the energydifference to this level decreases with increasingneutron number states with opposite parity (com-pared to the ground state parity) can be reachedat lower energies . However, only the excitation ofan odd number of nucleons from or into the nextoscillator shell changes the parity of the state. Thisleads to an oscillatory behavior in the ρ − /ρ + ratiosat energies already below the neutron thresholds(indicated by arrows in Fig. 3). For Ni with theneutron number N = 40 the pf shell is completelyoccupied in the independent particle model. Henceparity-changing transitions appear at quite low en-ergies in the nickel isotopes around Ni. Moving toeven larger neutron numbers, and thus closer to po-tential r-process nuclei around Ni, nickel isotopeshave mainly positive parity states at low energiesas orbitals in the gds shell have positive parity. Wenote further that equipartition is reached at some-what higher energies (about 3 MeV) in even-evennuclei than in odd-A nuclei due to pairing.Fig. 3 also shows the neutron separation energies[16] which obviously decrease with increasing neu-tron number along an isotope chain. The odd-evenstaggering is due to pairing.Usually E1 transitions, which are modelled bya Lorentzian centred around the giant dipole reso-nance in our approach [2], dominate over M1 tran-sitions, which are described by the single-particlemodel which makes the M1 strength energy inde-pendent [2]. However, if the capture occurs at ener-gies significantly below the giant dipole resonance,E1 transitions are strongly suppressed relative toM1 transitions in this model and the latter can dom-inate. This can occur in very neutron-rich isotopeswith very low neutron thresholds.For the nickel isotopes with largest proton excesswe calculate an enhancement in the stellar rates ifthe parity dependence of the level densities is con-sidered (Fig. 4). For these nuclei the neutron thresh-olds are quite high and, due to the excitation of anodd number of nucleons from the ds shell, there is anenhancement of states in the compound with oppo- The protons do not play an important role for determiningthe parity at low energies since nickel is a closed shell nucleusfor protons. − Ni we observe a reductionof the rate, if parity dependence is considered. Theorigin is the same as discussed above for the case of Ni, as E1 transitions are reduced as ρ + /ρ − < ρ − /ρ + <
1) at the energies just above the neutronthreshold for odd-A (even-even) isotopes.For the isotopes − Ni the neutron thresholdsare located at energies where the ratio of parity-dependend level densities is rather close to unity,but still shows some oscillatory behavior. As a con-sequence the rates are slightly enhanced or reduceddepending on the fact whether the ratio is just aboveor below unity at the energies above the neutronthreshold.For the most neutron-rich Ni isotopes, states withthe same parity as the ground state dominate thespectrum at energies around the neutron threshold.This leads to a reduction of the rate if E1 tran-sitions dominate. However, for the capture on thenickel isotopes Ni and Ni the thresholds are solow (0.52 MeV and 0.17 MeV, respectively) thatM1 transitions contribute more in our model thanE1 captures; hence the rate is increased comparedto the case where parity equipartition is assumed.Due to pairing the neutron threshold in Ni is at3.7 MeV and E1 capture dominates. We note thatthese nuclei are close to the r-process path. Henceour discussion clearly shows that the effect of parity-dependend level densities on the neutron capturerate is quite sensitive to the neutron separation en-ergies and the competition of M1 and E1 transitionswhich are both not sufficiently well known yet. Wealso note that the low density of states makes theuse of a statistical model for the very neutron-richisotopes questionable. Furthermore direct neutroncaptures should contribute to the rates for these nu-clei. Here the parity dependence of the optical po-tential should be incorporated into the models andpossible effects studied.We have repeated our calculation of neutron cap-tures on the nickel isotopes using the parity distri-bution of Mocelj et al. [9] combined with the back- shifted Fermi gas model of Rauscher et al. [5]. Asis shown in the lower graph of Fig. 4 the effects arequite similar to those obtained with the Hilaire andGoriely level densities [10].Finally we have performed a series of calculationsfor the neutron capture on the tin isotopes (Fig. 5).Again E1 transitions dominate and for the same rea-sons as explained in details above (e.g. for the caseof Ni) the consideration of parity-projected leveldensities lead to a reduction of the rates. For the tinisotopes − Sn the neutron intruder state h / ,with the opposite parity to the other orbitals in the gds shell, plays a special role which has no equivalentfor the nickel isotopes. It leads to a rather fast par-ity equilibration in the level densities which reachesratios ρ − /ρ + close to unity at energies around theneutron thresholds [10]. As a consequence the ratesfor neutron captures on the mid-shell tin isotopeschange only mildly if a parity dependence of thelevel densities is considered. For the heavier tin iso-topes, the N = 82 neutron shell gap at Sn andthe fact that the two lowest single particle orbitalsbeyond N = 82 ( f / , p / ) have the same parityas the h / intruder level have the effect that theequipartition of parities is reached at larger excita-tion energies than the respective neutron thresholdsfor tin isotopes beyond Sn. For similar reasons asfor the case of Ni, the capture rate is reduced forthese tin isotopes. The odd-even dependence in therates are caused by the pairing effects in the neutronthreshold energies. For the even heavier tin isotopesthe neutron intruder state i / , with opposite par-ity to h / , f / , p / becomes important resultingin a fast parity equipartition of the level densities.As a consequence the capture rates do not changemuch, if parity-dependent level densities are used.
4. Conclusion
We have presented a simple method in the frame-work of the statistical Hauser-Feshbach theory toaccount for a full parity dependence including non-uniformly distributed parities in the nuclear leveldensity of the compound nucleus. This goes beyondprevious work which only accounted for parity-dependent level densities in the initial and finalchannels but not in the compound step of the reac-tion. We applied our method to capture reactionson Ni and Sn nuclei, using a parity dependencein all steps of the compound nucleus reaction. Weconclude that this treatment can have a noticeable6
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A of target < σ v > π / < σ v > no r m . T = 10 K
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A of Target < σ v > π / < σ v > no r m . T = 10 K Fig. 5. Ratios of the stellar neutron capture rates of Sn iso-topes with parity influence in target, compound, and resid-ual to the stellar rate without any parity influence - the up-per graph was made by using the level densities from [10]and the lower graph by using the parity distribution from[9] combined with a level density from [5]. The ratios areshown for a temperature T = 1 GK of the stellar plasma. effect on astrophysical reaction rates for nuclei farfrom stability. In principle, our approach can alsobe extended to include a spin dependence or a moregeneral dependence on the level density of the com-pound nucleus. This will be the focus of future work.This work was performed in the framework ofthe SFB634 of the Deutsche Forschungsgemein-schaft and supported by the Swiss National ScienceFoundation (grant 2000-105328).References [1] E. Gadioli, P. E. Hodgson, Pre-Equilibrium NuclearReactions (Clarendon Press Oxford, 1992).[2] J. J. Cowan, F.-K. Thielemann, J. W. Truran, Phys.Rep. (1991) (4/5). [3] T. Rauscher, F.-K. Thielemann, At. Data Nucl. DataTables (2000) 1.[4] N. Bohr, Nature (1936) 334.[5] T. Rauscher, F.-K. Thielemann, K.-L. Kratz, Phys. Rev.C (1997) (3) 1613.[6] P. Descouvemont, T. Rauscher, Nucl. Phys. A (2006) 137.[7] H. Nakada, Y. Alhassid, Phys. Rev. Lett. (1997) (16)2939.[8] Y. Alhassid, G. F. Bertsch, S. Liu, H. Nakada, Phys.Rev. Lett. (2000) (19) 4313.[9] D. Mocelj, et al. , Phys. Rev. C (2007) (4) 045805.[10] S. Hilaire, S. Goriely, Nucl. Phys. A (2006) 63.[11] Y. Alhassid, G. F. Bertsch, L. Fang, Phys. Rev. C (2003) 044322.[12] C. ¨Ozen, K. Langanke, G. Mart´ınez-Pinedo, D. J. Dean,Phys. Rev. C (2007) (6) 064307.[13] D. Wintgen, H. Friedrich, K. Langanke, Nucl. Phys. A (1983) 239.[14] K. Langanke, H. Friedrich, Adv. Nucl. Phys. , vol. 17(Plenum, New York, 1986).[15] I. Dillmann, M. Heil, F. K¨appeler, R. Plag, T. Rauscher,F.-K. Thielemann, in
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