Fission fragment distributions and their impact on the r-process nucleosynthesis in neutron star mergers
FFission fragment distributions and their impact on the r-process nucleosynthesis inneutron star mergers
J.-F. Lemaˆıtre ∗ Institut d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles,Campus de la Plaine CP 226, BE-1050 Brussels, Belgium andCEA, DAM, DIF, F-91297 Arpajon, France
S. Goriely
Institut d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles,Campus de la Plaine CP 226, BE-1050 Brussels, Belgium
A. Bauswein
GSI Helmholtzzentrum f¨ur Schwerionenforschung,Planckstraße 1, 64291 Darmstadt, Germany andHelmholtz Forschungsakademie Hessen f¨ur FAIR (HFHF),GSI Helmholtzzentrum f¨ur Schwerionenforschung, Campus Darmstadt, Germany
H.-T. Janka
Max-Planck-Institut f¨ur Astrophysik, Postfach 1317, D-85741 Garching, Germany
Neutron star (NS) merger ejecta offer a viable site for the production of heavy r-process elementswith nuclear mass numbers A (cid:38) I. INTRODUCTION
The r-process, or the rapid neutron-capture process,of stellar nucleosynthesis is invoked to explain the pro-duction of the stable (and some long-lived radioactive)neutron-rich nuclides heavier than iron that are observedin stars of various metallicities, as well as in the solar sys-tem (for a review, see Ref. [1–3]). In recent years, nuclearastrophysicists have developed more and more sophisti-cated r-process models, trying to explain the solar systemcomposition in a satisfactory way by adding new astro-physical or nuclear physics ingredients. The r-process re-mains the most complex nucleosynthetic process to modelfrom the astrophysics as well as nuclear-physics points ofview. Progress in the modeling of type-II supernovae and ∗ [email protected] γ -ray bursts has raised a lot of excitement about the so-called neutrino-driven wind environment. However, untilnow a successful r-process cannot be obtained ab initio without tuning the relevant parameters (neutron excess,entropy, expansion timescale) in a way that is not sup-ported by the most sophisticated existing models [4, 5].Although these scenarios remain promising, especially inview of their potential to contribute to the galactic en-richment significantly, they remain affected by large un-certainties associated mainly with the still incompletelyunderstood mechanism responsible for the supernova ex-plosion and the persistent difficulties to obtain suitable r-process conditions in self-consistent dynamical explosionand neutron-star (NS) cooling models [5–8]. In particu-lar, a subclass of core-collapse supernovae, the so-calledcollapsars corresponding to the fate of rapidly rotatingand highly magnetized massive stars and generally con-sidered to be at the origin of observed long gamma-raybursts, could be a promising r-process site [9]. The pro- a r X i v : . [ nu c l - t h ] F e b duction of r-nuclides in these events may be associatedwith jets predicted to accompany the explosion, or withthe accretion disk forming around a newly born centralBH [10].Since early 2000, special attention has been paid to NSmergers as r-process sites following the confirmation byhydrodynamic simulations that a non-negligible amountof matter could be ejected from the system. Newtonian[11–15], conformally flat general relativistic [16–18], aswell as fully relativistic [19–23] hydrodynamical simula-tions of NS-NS and NS-black hole (BH) mergers withmicrophysical equations of state have demonstrated thattypically some 10 − M (cid:12) up to more than 0.1 M (cid:12) canbecome gravitationally unbound on roughly dynamicaltimescales due to shock acceleration and tidal stripping.Also the relic object (a hot, transiently stable hypermas-sive NS [24] followed by a stable supermassive NS, or aBH-torus system), can lose mass through outflows drivenby a variety of mechanisms [18, 25–30].Simulations of growing sophistication have confirmedthat the ejecta from NS mergers are viable strong r-process sites up to the third abundance peak and theactinides [15, 17, 18, 22, 31]. The r-nuclide enrich-ment is predicted to originate from both the dynamical(prompt) material expelled during the NS-NS or NS-BHmerger phase and from the outflows generated during thepost-merger remnant evolution of the relic BH-torus sys-tem. The resulting abundance distributions are foundto reproduce very well the solar system distribution, aswell as various elemental distributions observed in low-metallicity stars [3]. During the dynamical phase of themerging scenario, the number of free neutrons per seednucleus can reach so high values (typically few hundreds)that heavy fissioning nuclei are produced. Fissioning re-sults in a robust reproduction of the solar-system-likeabundance pattern of the rare-earth elements, as ob-served in metal-poor stars [1, 3, 17, 32]. This supports thepossible production of these elements by fission recyclingin NS merger ejecta. In addition, the ejected mass of r-process material, combined with the predicted astrophys-ical event rate (around 10 My − in the Milky Way [33])can account for the majority of r-material in our Galaxy, e.g. [17, 21, 34, 35]. A further piece of evidence that NSmergers are r-nuclide producers indeed comes from thevery important 2017 gravitational-wave and electromag-netic observation of the kilonova GW170817 7,[36–42].In this specific NS merger scenario, the neutron rich-ness was found to be so high that heavy fissioning nucleican be produced. For this reason, in this astrophysicalsite, fission plays a fundamental role, more particularlyby i) recycling the matter during the neutron irradia-tion (or if not, by allowing the possible production ofsuper-heavy long-lived nuclei, if any), ii) shaping the r-abundance distribution in the 110 ≤ A ≤
170 mass re-gion at the end of the neutron irradiation, iii) definingthe residual production of some specific heavy stable nu-clei, more specifically Pb and Bi, but also the long-livedcosmochronometers Th and U, and iv) heating the envi-ronment through the energy released. More details can be found for instance in Refs. [14, 32, 43–52].Despite the recent success of nucleosynthesis studies forNS mergers, the details of r-processing in these events isstill affected by a variety of uncertainties, both from thenuclear physics and astrophysics point of view. For thisreason, we present here a new effort to further improvethe description of fission fragment distributions (FFD)obtained on the basis of the so-called Scission Point Yield(SPY) model [53]. Sec. II is devoted to the descriptionof such improvements. A phenomenological correction tothe scission distance and a new smoothing procedure forderiving the fission yields are presented in Sec. II.1. Sincethe FFD of relevance for the r-process application cor-respond to the post-neutron-emission one deduced frompre-neutron FFDs, our procedure for estimating the neu-tron evaporation is detailed in Sec. II.2. As r-processis impacted by neutron captures, sensitivity studies areperformed in Sec. II.3 to demonstrate that the neutronemission depends mainly on the primary fission yields.SPY pre-neutron FFDs are compared with experimentaldata and the well-known GEF (“GEneral description ofFission observables”) calculations [54] in Sec. II.4.In addition, it has been shown that weak interactionsmay strongly affect the composition of the dynamicalejecta and thus the efficiency of the r-process [22, 55–63]. In this case, the initial neutron richness of the ejectacan be significantly reduced and consequently the roleof fission, albeit in basically all calculations a significantfraction of neutron-rich ejecta is present. In Sec. III, were-evaluate the role of fission in NS merger in two distinctscenarios, namely the traditional one neglecting weak in-teractions on nucleons and an updated version of such asimulation where neutrino interactions are included phe-nomenologically [56, 62] to reproduce qualitatively theresults found in Ref. [63]. In Sec.III.1, we will presentthe final abundance distributions obtained within bothscenarios as well as the role of the fission recycling andlink them to fundamental initial properties of ejected ma-terial, namely the initial neutron mass fraction, electronfraction and entropy. The relevance of the fission re-cycling and its contribution to the final r-abundances isthoroughly detailed in Secs. III.2–III.4 for both scenarios.Finally, conclusions are drawn in Sec. IV.
II. FISSION FRAGMENTS DISTRIBUTIONSII.1. SPY model
The SPY model is a static and statistical scission pointmodel [53, 64] where a thermodynamic equilibrium atscission is assumed, hence the evolution between the sad-dle and the scission points is neglected. The model isbased on two pillars, namely the absolute available en-ergy balance at the scission configurations and the statis-tical description of the available phase space. The avail-able energy balance is performed for all energetically pos-sible fragmentations of a fissioning system at scission asa function of the deformation of both fragments. Theavailable energy is defined as the difference between thepotential energy of the fissioning system at scission andthe energy of the excited compound nucleus where bothnascent fragments are supposed to be at rest. The po-tential energy of the fissioning system at scission is ob-tained as the sum of the individual binding energies ofthe two fragments and the interaction energy betweenthe fragments composed of the Coulomb repulsion andthe nuclear attraction. The system at scission is treatedas a microcanonical ensemble where all available statesare equiprobable. In this framework, the number of avail-able states of a given fragmentation is the product of thestate densities of the two isolated fragments. The yieldof a fragmentation is the number of available states as-sociated with this fragmentation regardless of the defor-mation of the fragments.All nuclear inputs to the SPY model, i.e. the individ-ual binding energies, proton distributions and proton andneutron single-particle level schemes of each fission frag-ment as a function of its axial deformation are estimatedfor about 7000 nuclei from Z = 20 to Z = 100 withinthe framework of the constrained self-consistent Hartree-Fock-Bogoliubov (HFB) formalism using the SkyrmeBSk27 effective nucleon-nucleon interaction [65]. The nu-clear state density is calculated in the framework of thestatistical model of nuclear level densities [53, 66–69] onthe basis of the discrete single-particle level scheme ob-tained self-consistently within the same HFB calculation.A scission criteria is needed to characterize the scissionconfigurations, more precisely, to define the relative po-sition of the fragments, hence to estimate the Coulomband nuclear energies. This criteria can be based on aninter-surface distance, as in the initial version of the SPYmodel [64] or through the proton density at the scissionneck which defines the scission distance as in our new ver-sion of the SPY model [53]. Presently, with respect to ourlast study [53], a phenomenological modification of thescission distance is introduced in order to better repro-duce experimental fission yields. The corrected scissiondistance ( d sc , corr ) for a fragmentation ( Z , N ) + ( Z , N )is defined through the proton density at the scission neck( d sc ), as in Ref. [53], though with two additional correc-tions, one per fragment d sc , corr = d sc + d corr ( N ) + d corr ( N ) . (1)Each correction term is composed by a positive term anda negative one which are chosen to be dependent on thedifference between the fragment neutron number ( N ) andsome specific neutron numbers ( N corr ), i.e. d corr ( N ) = (cid:88) N corr ∈{ , , , , } . e − ( N − N corr)22 − (cid:88) N corr ∈{ , , , , } . e − ( N − N corr)22 [fm] . (2)Both light and heavy fragments contribute to the dis-tance correction (Fig. 1a) with a maximal correction of ± . d ( N ) with N (cid:54) N CN / d ( N ) with N (cid:62) N CN / d ( N ) + d ( N ) (black dashed curve) for Ufission. The specific neutron numbers N corr (Eq. 2) areindicated by gray arrows. (b) Available energy offragments from thermal neutron-induced fission of Ucalculated with (red curve) or without (green curve) thephenomenological distance correction (Eq. 2). The blueline corresponds to the the difference between correctedand non-corrected energies.for example by around 3 MeV for the neutron-induced fis-sion of
U (Fig. 1b). Such an effect is of the same orderof magnitude as pairing effects and can be regarded asa way to correct the approximate pairing effects whichassumes the pairing gap at zero temperature can be de-scribed by a five-points mass difference [70] and remainsconstant with the nucleus deformation. For nucleosyn-thesis applications, the distance correction is kept con-stant irrespective of the fissioning system, even if a spe-cific adjustment could be performed.This distance correction help to better reproduce theyields of the thermal neutron induced fission of
U(Fig. 2a and d, green and blue curves), more particu-larly its symmetric component is reduced due to the dis-tance correction around N = 64 , , ,
70. As seen inFig. 2c and f, this correction has a negligible impact onthe FFD for the spontaneous fission of
Cf. However,yields of the thermal neutron-induced fission of
Puaround A = 130 becomes underestimated (Fig. 2b and e)due to the same correction as in the U case.As described in Ref. [53], the SPY FFD or yields dis-tribution present strong staggering patterns that are notfound experimentally. Those can be smoothed, as tradi-tionally done, by a double Gaussian function Y smooth ( Z, N ) = (cid:88) i = − (cid:88) j = − Y raw ( Z + i, N + j ) C z e − i σ z C n e − j σ n (3)where C z and C n are normalization factors and σ z = 0 . σ n = 3 are adopted. In the present paper the widthof the Gaussian function is larger in the neutron direc-tion σ n than in the proton direction in order to be si-multaneously consistent with experimental width of theisobaric and isotopic fission yields distributions (Fig. 2).As there is not coupling between Z and N in the dou-ble Gaussian function (Eq. 3), the isotopic yields are notaffected by the smoothing in the neutron direction. Simi-larly to yields, other fragments observables O , such as theavailable energy or the kinetic energy, are also smoothedfollowing the same procedure O smooth ( Z, N ) = (cid:88) i = − (cid:88) j = − O raw ( Z + i, N + j ) C z e − i σ z C n e − j σ n . (4) II.2. Neutron evaporation model
The SPY model allows us to investigate the primaryfragments, i.e. fragments formed at scission, hence theirde-excitation by neutron evaporation. The excitation en-ergy of fragments has two components. First, each frag-ment carries a fraction of the available energy at scis-sion under an intrinsic excitation form. Second, sinceeach fragment can be deformed, a deformation energymay contribute to the excitation energy. No assump-tion about available energy sorting at scission betweenprimary fragments is needed within the SPY framework,contrary to other neutron evaporation models. Indeed,some models require to introduce an energy sorting pa-rameter R T ( A ) [80, 81] or a thermal equilibrium to de-duce the energy sorting on the basis of level density pa-rameters with phenomenological corrections in order toincrease the excitation energy of the light fragment [82].In the SPY model, the energy sorting, denoted by x (Eq. 2 in Ref. [53]), results from the competition betweenthe state density of the light fragment ( ρ ) and the heavyone ( ρ ). The intrinsic excitation energy and deforma-tion energy of each fragment are obtained by averagingover all possible energy sorting and deformations (Eq. 3in Ref. [53]). In the simplest model, the competitionbetween neutron and gamma evaporation during the de-excitation cascade is not taken into account, so that thefission fragments are sequentially de-excited by neutronevaporation as long as it is energetically possible. The ki-netic energy of evaporated neutrons E n can be assumedto follow a Maxwellian distribution φ M ( E n ) = 2 √ πT (cid:112) E n e − E n /T , (5) where T is the nuclear temperature before neutron evap-oration. T can be extracted from the statistical modelof nuclear level densities including BCS pairing interac-tion, denoted as T BCS (see Eqs. 16 and 18 in Ref. [53]and Refs. [66–68]).Following this approximation, the distribution of evap-orated neutrons are calculated for the three well-knownfissioning systems using both raw and smooth pre-neutron yields and shown in Fig. 3. In both cases, thenumber of emitted neutrons is underestimated for theheaviest fragments A = 150 −
160 due to the low avail-able energy, which, in turn, originates from an overes-timate of the kinetic energy of these related fragmenta-tions, as mentioned in Sec. IV of Ref. [53]. However, forheavy fragments with A = 130 − ν is systemati-cally overestimated. This is linked to the available energysorting within a fragmentation which is too favorable forthe heavy fragment.The overestimation the number of neutron emitted bythe heavy fragments leads an overestimation of the to-tal mean number of evaporated neutrons per fission. Asdisplayed in Table I, it is overestimated by 0.77-1.1 neu-tron when smooth yields are used and by 0.6-1 neutronfor raw yields. The overestimation is higher with smoothyields due to a more important contribution of the heavyfragments.TABLE I: Total mean number of evaporated neutronsper fission, ¯ ν tot , from the evaluated data libraryENDF/B-VII.1 [83] and SPY using the raw or smoothyields (see Fig. 3) for the 3 benchmarked nuclei shownin Fig. 3 and obtained either by neutron-induced fission(nif) or spontaneous fission (sf). Nucleus reaction ENDF SPY(raw) SPY(smooth)
U nif 2.44 3.08 3.20
Pu nif 2.88 3.97 3.96
Cf sf 3.77 4.37 4.39
To study the sensitivity of the predicted number ofevaporated neutrons to the various model uncertainties,the impact of the neutron kinetic energy distribution φ ( E n ), the nuclear temperature and the neutron-gammacompetition are analyzed in the next sub-sections for thespecific case of Cf spontaneous fission.FIG. 2: (Color online) Isotopic (top panels) and isobaric (bottom panels) fission yields of U, Pu and
Cf.The black lines with dots represent experimental (pre-neutron-emission) fission yields for fission of U( Q = 6 . Pu ( Q = 6 . Cf ( Q = 0 MeV) (c) [75] and(f) [76]. The green lines (labeled by “raw”) correspond to raw fission yields, the blue lines (labeled by “ d corr ”) toraw yields including distance corrections and the orange lines (labeled by “smooth”) to the final smooth yields afterdistance corrections.FIG. 3: (Color online) Evaporated neutron distributions as a function of the fragment mass number. The blackdotted lines represent experimental evaporated neutron distribution of (a) U ( Q = 6 . Pu( Q = 6 . Cf ( Q = 0 MeV) [79]. The orange/green lines (labeled by “smooth”/“raw”) are SPYevaporated neutron distribution using smooth/raw pre-neutron yields.FIG. 4: (Color online) (a) Evaporated neutrondistribution of spontaneous fission of Cf using theSPY raw yields. (b) Difference with respect to theevaporated neutron distribution computed with theMaxwell distribution (Eq. 5) and temperature from theBCS state density.
II.3. Sensitivity studies of the evaporation model
II.3.1. Impact of the kinetic energy distribution of theneutron
There are various models for the kinetic energy dis-tribution of the neutron φ ( E n ) [84]. The chosen one isrelated to the Maxwell-Boltzmann distribution describ-ing the distribution of speeds of particles in an idealizedgas which yields to the kinetic energy distribution of neu-tron φ M (Eq. 5). Another possible description relies onthe evaporation model of Weisskopf and Ewing [85, 86]yielding to φ W ( E n ) = E n T e − E n /T n (6)for small E n [84] and where T n is the nuclear tempera-ture after the neutron emission. The impact of the neu-tron kinetic energy distribution on the number of evap-orated neutron of Cf spontaneous fission is negligible,as shown in Fig. 4, (orange and light green curves, labeledrespectively by “ φ M , T BCS ” and “ φ W , T BCS ”) where thedifference is seen to rarely exceed 0.1 neutron (Fig. 4b,light green curve, labeled by “ φ W , T BCS ”).
II.3.2. Impact of the nucleus temperature
As an alternative to the BCS theory of nuclear leveldensities, the phenomenological Fermi Gas approxima-tion [87], excluding shell and pairing effects, can also be considered to estimate the nuclear temperature, T FG = (cid:112) U/a (7)where U is the excitation energy and a ∼ A/ φ M , T FG ” and “ φ W , T FG ”), both modelsfor the nuclear temperature ( T BCS or T FG ) give rathersimilar predictions of the evaporated neutron distribu-tion. For the sake of coherence, T BCS is adopted in thesubsequent study.
II.3.3. Impact of neutron-gamma competition
In the previous calculation of the number of emittednumber of neutrons, it was assumed that neutron emis-sion dominates over the electromagnetic de-excitation.To test the impact of such an approximation, TALYS nu-clear reaction code [89] is now used to describe the com-petition between the strong and electromagnetic chan-nels in the de-excitation of the the primary fragments(Fig. 4, green curve, labeled by “TALYS”). Note thatall de-excitation channels are taken into account in theHauser-Feshbach and pre-equilibrium framework. Thedifference between the evaporated neutron distributionfrom TALYS and the one obtained with the simplifiedMaxwellian approach described above is shown in Fig. 4(green curve, labeled by “TALYS”) and does not exceed0.3 neutron.These results indicate that the primary fragments de-excitation can be essentially approximated by a sequenceof neutron emissions, followed finally by gamma emissionwhen no more neutron can be evaporated. This simpli-fied description based on energetic considerations seemsto be sufficient within an accuracy of about 0.1–0.2 neu-tron. This conclusion is only valid for low excited pri-mary fragments up to 20–25 MeV. The crucial point toimprove the description of the emitted neutron distribu-tion concerns the SPY model and particularly the kineticenergy of the fission fragments which directly affect theavailable energy and mean deformation of the fragments.In the following, if not mentioned otherwise, thesmooth yields (Eq. 3) are adopted and neutron evapo-ration are deduced using the Maxwell distribution pre-scription (Eq. 5) with the BCS temperature.
II.4. Systematics
II.4.1. Over all fissionable nuclei
Systematic SPY calculations were performed for iso-topic chains between Z = 70 and 124 from the protonto the neutron driplines with an initial excitation energy Q = 8 MeV (Fig. 5). With Q = 8 MeV, only few nu-clei with Z <
80 can fission. The smoothing procedure(Eq. 3) may impact locally the peak multiplicity (Fig. 5c) (a) Peak multiplicity for the raw pre-neutron isobaric yieldswith corrected distance (Eq. 1). (b) Mean prompt neutron multiplicity per fission according tothe prescription given in Sec. II.2, deduced from the smoothpre-neutron isobaric yields with corrected distance (Eq. 1).(c) Peak multiplicity for the smooth pre-neutron isobaricyields with corrected distance (Eq. 1). (d) Mean available energy release per fission deduced from thesmooth pre-neutron isobaric yields with corrected distance(Eq. 1).(e) Peak multiplicity for the smooth pre-neutron isobaricyields without corrected distance. (f) Isolines of mean available energy release per fission (orangelines) from Fig. 5d and mean prompt neutron multiplicity perfission (green lines) from Fig. 5b.
FIG. 5: (Color online) Systematics in the (
N, Z ) plane of major fission observables for some 3000 nuclei with70 (cid:54) Z (cid:54)
124 for an initial excitation energy of Q = 8 MeV.when compared those obtained with raw yields (Fig. 5a).In particular, peaks can be merged if there are close eachother or if they are small enough. However, the locationof transition between the various fission modes are seennot to be globally affected by the smoothing procedureand does not distort significantly the peaks location ofFFDs and their widths as shown in Fig. 2.Four zones can be identified where the peak multiplic-ity is affected by the distance correction (Eq. 2), as seenwhen comparing Figs. 5c and 5e obtained with or with-out this correction. In general terms, a negative/positivedistance correction makes fragments closer/farther whichinduced a decrease/increase of the available energy, henceof the fission probability. The first one is the shift of thesymmetric/asymmetric fission transition toward neutron- deficient nuclei from N CN ≈
140 to N CN ≈ N frag = 64 , , ,
70 leading to the ap-pearance of the asymmetric mode while for N frag =47 , , , ,
89 the positive distance correction favorsthe asymmetric mode. In the second zone, located at155 (cid:46) N CN (cid:46) i.e. N frag = 64 , , ,
70 and by a positive distance cor-rection for the heavy fragment of the new slightly asym-metric fission mode ( N frag = 85 , (cid:46) N CN (cid:46) N frag = 64 , , ,
70) whichdisfavors the asymmetric mode. The symmetric mode isfurther accentuated by the positive distance correctionfor N frag = 81 , ,
89. Finally, for nuclei in the fourthzone 180 (cid:46) N CN (cid:46) N frag = 85 ,
89 induces a slightly asymmetric fissionmode on top of the symmetric one. Although some fis-sion yields distributions are impacted by the phenomeno-logical correction to the scission distance, globally, SPYpredictions for neutron-rich nuclei remains rather robustwith respect to such a correction, as seen in Figs. 5c –5e.The mean number of neutrons emitted per fission ¯ ν tot ranges from 2 up to 37 neutrons depending on the fission-ing system (Fig. 5b). It depends mostly on the neutronrichness of the fissioning nucleus. Its evolution with pro-ton number is not trivial despite the strong variation ofthe mean available energy of fragmentations (Fig. 5d).For a given fissioning nucleus, an increase of the exci-tation energy ( Q ) enhances the intrinsic excitation en-ergy of fission fragments, hence more neutrons are emit-ted. However, this relation between available energy andemitted neutrons is not sufficient to explain the relationbetween emitted neutrons and mean available energy offragmentations. Along a line of constant available energy(Fig. 5f) the number of evaporated neutron is multipliedby at least five between proton-rich and neutron-rich fis-sioning nuclei. Similarly, along a constant line of emittedneutron number (Fig. 5f), the available energy increasesby a factor up to hundred between light and heavy nuclei.These features are due to the neutron binding energy offission fragments, i.e. the more neutron-rich the fission-ing nucleus, the more-neutron rich the fission fragmentsand the lower their neutron binding energy, hence themore neutrons can be evaporated. Even if the fission ofa heavy nucleus releases more available energy comparedto a light one, the number of evaporated neutrons can besimilar because the neutron binding energies of fragmentsfrom heavy-nucleus fission is higher (less neutron-rich)than the fragments from the lighter one. II.4.2. SPY vs GEF
SPY FFDs with corrected distance are compared inFig. 6 to experimental data or with the evaluation fromthe ENDF/B-VII database [83], if no experimental datais available. Evaluated data (Fig. 6, gray curves) cor-respond to post-neutron FFDs while experimental datacan be pre-neutron or post-neutron emission (Fig. 6,black curves). SPY FFDs (Fig. 6, green curves) are alsocompared to GEF (version 2019/1.2) thermal neutron-induced fission yields [54, 103]. The excitation energiesof experimental or evaluated data may not be exactly thesame as those considered in SPY or GEF calculations buta variation of the excitation energy by a few MeV only impacts moderately the shape of the FFD.In the Po to U region (Fig. 6a), the transition betweensymmetric fission for neutron-deficient nuclei to asym-metric fission for neutron-rich nuclei is rather differentbetween both SPY and GEF predictions. SPY FFDsare in good agreement with experimental data [90] forTh and Pa isotopes where the transition is well repro-duced, in contrast to what is obtained by the GEF model.The transition from symmetric to asymmetric fission pre-dicted by GEF occurs for more neutron-deficient isotopescompared to experimental data.For Th isotopes, GEF transition occurs between Th and Th for which no symmetric peak isfound anymore. From an experimental point of view,the transition seems to start at Th where the sym-metric peak broadens while beyond Th the FFD iscompletely asymmetric. The symmetric to asymmetrictransition of Th predicted by SPY is slightly more com-plex (Fig. 5e and Fig. 6a); there are two peaks in addi-tion to the symmetric component for − Th − which gives a peak multiplicity of two and even fourfor Th . For more neutron-rich Th, this substruc-ture disappears into one symmetric peak which becomesmaller for more neutron-rich isotopes and completelyvanishes for Th (Fig. 5e). This kind of transitionfrom symmetric to asymmetric through a substructureat the top of the vanishing symmetric peak is also visibleon the SPY Pa FFDs.The FFD broadening in the Rn to Ac isotopic chainsis well reproduced by SPY even if the FFDs of neutron-deficient isotopes tend to be narrower than the experi-mental ones. A strong odd-even staggering in GEF FFDis observed for even isotopes of Po and Rn which makesFFD plots completely blue filled (Fig. 6a).For actinides from U to Rf (Fig. 6b), FFDs from SPYand GEF model are rather similar and in good agreementwith experimental data, particularly for U and Es iso-topes. Compared to experimental data for Am and Cmisotopes, the symmetric part of the yields distributionsis underestimated by both models. The peaks are nar-rower with SPY compared to GEF model. However, forneutron-rich nuclei (like Fm), SPY predicts a purelyasymmetric fission, contrary to GEF and evaluated data.This asymmetric splitting predicted by SPY model is dueto the doubly magic nuclei
Sn which is disfavored. In-deed, even if
Sn has more available energy than otherfragmentations, the available states for these fragmenta-tions is significantly lower and consequently their yields islower than other fragmentations leading to an asymmet-ric fission.
Sn affects fissioning systems in the regionaround
Fm (Fig. 5e) where a doubly asymmetric fis-sion is found by SPY model whereas GEF model predictsan symmetric mode. For the same reasons, the symmet-ric fission of
Lr and
Rf are not reproduced with theSPY model contrary to the GEF model. (a) Evaluated isotopic yields (formed by Coulomb excitation) from Ref. [90] converted into isobaric yields using the UCD(Unchanged Charge Density) hypothesis. For
Th post-neutron yields from ENDF/B-VII database [83].(b) Yields of − U are from [90] (converted into isobaric),
U from [91], − U (formed by 0.5 MeV neutron) from [83],
U from [72],
U and
Np and
Cm from [92],
U [76],
Np from[93],
Pu from [94], − Pu and
Es (formedby thermal neutron) from [83],
Am from [95],
Am from [96],
Cm and
Es and − Fm (spontaneous fission) from[83],
Cm from [97],
Cf from [98],
Cf from [76],
Cf and
Fm from [99], − − Fm from [100],
Lr from [101],
Rf from [102].
FIG. 6: (Color online) Isobaric fission yields from SPY model (green curves) compared to those calculated by theGEF model (blue curves) [54] and experimental pre-neutron (black curves) or post-neutron (gray curves) yields. Foreach FFD, the x axis (mass number) ranges from A = 70 to A = 160 by step of 10 and the y axis (FFD) from 0 to15% by step of 2%. The background color refers to the peak multiplicity.0FIG. 7: (Color online) Representation in the ( N , Z ) plan the decay mode with the highest decay rate (a) between β decay followed by 0, 1, 2 or 3 neutron emission ( βkn ), β -delayed fission ( β df), spontaneous fission (sf) or α decay;(b) the corresponding lowest lifetime τ min = 1 /λ max among the various decay mode given in (a). (c) Temperature atwhich fission and radiative neutron capture are equally favorable, above this temperature the neutron inducedfission dominates the neutron capture. (d) Nuclei for which neutron-induced fission is dominant over the ( n, γ )channel for T < . β -delayed fission are depicted bythe gray area (region of efficient fission. Blue hatched areas are the fission seeds areas : the “fission bottleneck”( A <
292 and
N > A (cid:62) A < A (cid:62)
278 and N (cid:54) (cid:54) A <
320 and
N > Z = 82, N = 126 and N = 184 shell closures. See text for more details.1 III. IMPACT OF FISSION ON THE R-PROCESSNUCLEOSYNTHESIS IN NS MERGERS
The NS-NS merger simulation was performed witha general relativistic Smoothed Particle Hydrodynam-ics scheme [16, 31, 104] representing the fluid by a setof particles with constant rest mass, the hydrodynamicalproperties of which were evolved according to Lagrangianhydrodynamics. We only consider dynamical (prompt)ejecta expelled during the binary NS-NS merger in thepresent paper since this ejecta component provides gen-erally the most favorable conditions for fission recyclingdue to the low initial electron fraction. A symmetric1.365 M (cid:12) – 1.365 M (cid:12) binary system compatible with thetotal mass detected in the GW170817 event [36] has beenmodelled on the basis of the so-called SFHO equationof state [105]. R-process calculations are performed on500 ejected “particles” ( i.e. mass elements) representa-tive of the dynamical ejecta. The r-process nucleosyn-thesis is calculated with a reaction network including all5000 species from protons up to Z = 110 lying betweenthe valley of β -stability and the neutron drip line. Allcharged-particle fusion reactions on light and medium-mass elements that play a role when the nuclear statisti-cal equilibrium freezes out are included in addition to ra-diative neutron captures and photodisintegrations. Thereaction rates on light species are taken from the NET-GEN library, which includes all the latest compilationsof experimentally determined reaction rates [106]. Ex-perimentally unknown reactions are estimated with theTALYS code [89, 107] on the basis of the Skyrme HFBnuclear mass model, HFB-21 [108]. On top of these re-actions, β -decays as well as β -delayed neutron emissionprobabilities are also included, the corresponding ratesbeing taken from the relativistic mean-field model ofRef. [109]. Fission processes, including neutron-induced,spontaneous and β -delayed fission, are carefully intro-duced in the network together with the correspondingFFD. Fission barriers obtained with the BSk14 Skyrmeinteraction are adopted here to estimate fission probabil-ities [110, 111]. The new smooth FFDs from SPY model,as described in the previous section, as well as GEF pre-dictions for comparison, are used in the present study.Note that all fission products with a yield larger thantypically 10 − % are linked to the parent fissioning nu-cleus in the network calculation, i.e. about 500 fissionfragments are included for each fissioning nucleus. Theneutron-rich fission fragments located outside the net-work, i.e. across the neutron drip line, are assumed toinstantaneously emit neutrons down to the neutron dripline. The total mass fraction as well as number of nu-cleons is conserved at all time and the emitted neutronsare recaptured consistently by all the existing species.Finally, α -decays are taken into account for all heavyspecies, the rates being extracted from Ref. [112].Despite the recent success of nucleosynthesis studiesfor NS mergers, the details of r-processing in these eventsis still affected by a variety of uncertainties. In particu-lar, the exact impact of neutrino is not yet understood in all details, the main reason of which is the not yetmanageable computational complexity associated withfull neutrino transport in a generically three-dimensional(3D), highly asymmetric environment with nearly rela-tivistic fluid velocities and rapid changes in time. It waslong assumed that neutrino interactions could not, atleast not drastically, affect the initial neutron richnessof the ejecta. However, recent NS-NS merger simulations[22, 55, 57–61, 63] including the effect of weak interac-tions on free nucleons demonstrate that neutrino reac-tions can significantly affect the neutron-to-proton ratioin the merger ejecta, with direct consequences in partic-ular on the amount of synthesized low-mass ( A < i.e. when the temperature hasdropped below 10 GK) amounts to (cid:104) Y e (cid:105) = 0 .
03. This caseis likely to be relevant for NS-BH mergers or highly asym-metric NS-NS mergers, where the lower mass componentdevelops an extended tidal tail, from which considerableamounts of cold, unshocked matter can be centrifugallyejected before neutrino exposure of these ejecta may playan important role [18, 21, 23]. In the second case (sce-nario II), the equilibrium between electrons, positronsand neutrinos is assumed to hold down to the fiducialdensity of ρ eq = 10 g cm − , and from that density on,electron, positron and electron neutrino and antineutrinocaptures on nucleons ν e + n (cid:10) p + e − ¯ ν e + p (cid:10) n + e + (8)are systematically included assuming the (anti)neutrinoluminosities and mean energies remain constant in time,as detailed in Ref. [56], and compatible with the resultsobtained within the Improved Leakage-Equilibration-Absorption Scheme (ILEAS) method [63] (see in partic-ular their Fig. 14). To do so, we adopt the values of L ν e = 0 . × erg/s; L ¯ ν e = 10 erg/s; (cid:104) E ν e (cid:105) = 8 MeV; (cid:104) E ¯ ν e (cid:105) = 12 MeV. The resulting mean electron fraction atthe time of the network calculation reaches (cid:104) Y e (cid:105) = 0 . Y e , of the various mass elements is low and characteristicof the NS inner crust, i.e. less than 0.06; consequently thenumber of free neutrons per seed nuclei can reach a fewhundreds. With such a neutron richness, heavy fissioningnuclei can be efficiently produced during the r-process.2Weak interactions on nucleons (Eq. 8) increase the initialelectron fraction in the ejecta to values ranging in thisscenario II from 0.06 to 0.38 and make the productionof heavy fissioning nuclei and fission recycling during ther-process less favorable, though not impossible.On the basis of the BSk14 fission barriers adopted here,the production of super-heavy neutron-rich nuclei is pos-sible up to Z = 110 if the neutron irradiation is largeenough to overcome the N = 184 shell closure. Beyond Z = 110, the production of elements is unlikely becausethese elements fission spontaneously (Fig. 7a) with a veryshort lifetime (Fig. 7b). This is the main reason why thereaction network has been limited to elements Z (cid:54) Z = 110 and A (cid:62) A (cid:62) i ) β -delayed fission of Pu , ( ii ) spon-taneous fission of Am , ( iii ) β -delayed fission of − Bk − , and ( iv ) spontaneous fission of ele-ments with Z (cid:62)
98 (Fig. 7a). Am and Z (cid:62) − s, which makesneutron capture by these nuclei unlikely. The β -delayedfission lifetime of Pu and − Bk − rangesfrom 10 − to 10 − s. The neutron-induced fission of − Cf − and Am isotopes also happens tobe efficient and contributes to fission recycling (Fig. 7c).The flow through Pu isotopes is negligible due to the fast β -delayed neutron emission of Pu with a lifetime τ βkn = 6 . × − s and Pu with τ βkn = 4 . × − s(Fig. 7b). In addition, the neutron separation energyof Pu equals to 0.28 MeV which makes the neutroncapture by
Pu unfavorable. Consequently, mattermainly flows through the Cm isotopic chain to producesuper-heavy neutron-rich elements. This region formsa “fission bottleneck” (
A <
292 and
N > Cm represents the “crossing point” (Fig. 7d). How-ever, the production of super-heavy neutron-rich nuclei isno more possible for temperatures higher than 2 GK ( i.e. k B T = 0 .
17 MeV) because the neutron-induced fissionsbecome more favorable than the radiative neutron cap-tures of − Cm − and Bk (Fig. 7c) whichclose the “fission bottleneck”.At the end of the neutron irradiation, heavy neutron-rich elements β -decay up to the region of efficient fis-sion. Elements accumulated during the neutron irradia-tion along the N = 184 neutron shell closure β -decay upto reach a region of efficient fission. This can be split intwo zones : the “decay roof 0” ( A < A (cid:62)
278 and N (cid:54) Z <
94 and Z (cid:62)
94, respectively.Elements with 292 (cid:46) A (cid:46)
320 accumulated after the “fis-sion bottleneck” β -decay up to reach another region ofefficient fission named “decay roof 2” (292 (cid:54) A <
III.1. Final abundances
The final abundance distributions of the ejected mate-rial are shown in Fig. 8 for both scenarios I and II. Cal-culations are performed adopting either the SPY or GEFpredictions of the FFDs. While in scenario I, only nucleiwith
A > ∼
120 are essentially produced, weak interactionson nucleons also allow the production of lighter specieswith 80 < ∼ A < ∼
120 in scenario II, but also decrease theproduction of the Pb peak nuclei as well as the long-livedactinides. In this case, the choice of the FFD model (SPYor GEF) is seen to have a negligible impact on the finalabundance distribution. The same conclusions hold ifwe consider our original version of the SPY FFD [53].Without weak interactions, with the low initial electronfractions, the number of neutron per seed is rather high(a few hundreds), so that the production of the heavi-est nuclei is efficient. In this case, nuclear fission plays afundamental role during both the neutron irradiation andat freeze-out [32] and the FFD model directly affects theabundance distribution, particularly around the secondpeak A = 130 where SPY FFDs give higher abundancesthan GEF (see Sec. III.4). This nucleosynthesis hasbeen described in detail in Refs. [17, 18, 31, 44, 56, 62].Note that the corrections introduced in the present study(Sec. II.1) in comparison with our 2019 version [53] have3FIG. 9: (Color online) (a-e) Final abundances of the trajectories computed within scenario II classified into 5groups : “1 st peak”, “between 1 st and 2 nd peak”, “2 nd peak”, “rare earth & no fiss.” and “fiss.”. The green curveswith dots are the final abundances averaged over all trajectories computed within scenario II (a-e). The gray curveswith dots correspond to the mean final abundances within each group. (f) Final abundances of the trajectoriescomputed within scenario I (black curves) and the final abundances averaged over all trajectories (gray curve withdots). (g) Distribution of the trajectories in the ( S , Y e ) plane and iso- X n curves (green curves). (h) Mean value ofthe initial neutron mass fraction X n (green curve) and mean mass (cid:104) A (cid:105) A (cid:62) (gray curve) of each group and theassociated ejecta mass fraction of groups for the scenario II (color bars).a small impact on the final abundance distribution. Theytend to slightly increase the second r-abundance peak at A (cid:39)
130 and decrease its low- A tail. Such differences aresignificantly smaller than the one observed between theSPY and GEF predictions.It is well accepted [114] that the efficiency of the r-process nucleosynthesis can be related to three mainphysical properties of the ejected material, namely theelectron fraction Y e , the entropy per baryon S and theexpansion timescale τ [115]. In scenario I, trajectoriesare characterized by a low initial electron fraction Y e ranging between 0.02 and 0.06 at the beginning of thenucleosynthesis, i.e. when the temperature has droppedbelow 10 GK and the density below the drip density( ρ drip (cid:39) × g/cm ) (Fig. 9g). Initial entropies S reach values up to 170 k B / nucleon, though most of themremain below 50 k B / nucleon. As already mentioned, inscenario II, the initial electron fraction covers a muchlarger range between 0.06 and 0.38 due to the weak in-teractions (Eq. 8).Both the initial electron fraction and entropy affect theinitial mass fraction of free neutrons X n (Fig. 9g, green curves) which is consequently a parameter of first rel-evance in the analysis of the efficiency of the r-processnucleosynthesis. In scenario I, X n ranges between 0.85and 0.98, as seen in Fig. 9g, while in scenario II it variesbetween 0.02 and 0.82. The lower Y e , the larger X n by definition, but also the higher the initial entropy, thelarger X n due to the release of free neutrons throughphotodissociation. Therefore, X n engulfs information onboth the initial electron fraction and entropy. The sud-den variation of X n observed at S = 8 − k B / nucleon,independently of Y e , is linked to the initial abundances oftrans-alpha elements ( A >
4) : (cid:80) A> Y ( A ). The mat-ter is not completely dissociated for trajectories with alow initial entropy S < k B / nucleon, which makesthe initial abundance of trans-alpha elements (with amean mass (cid:104) A (cid:105) A> ranging from 50 to 85) non-negligibleand reduces the initial neutron mass fraction. For S > k B / nucleon, the initial abundances of trans-alpha el-ements is negligible, i.e. the ejected nuclear matter iscompletely dissociated into protons, neutrons and alphas.The low entropy effect is more important for trajecto-ries in scenario II where for 18% of the ejected mass,4 (cid:80) A> Y ( A ) > . (cid:80) A> Y ( A ) > . X n < .
3) and a low initial entropy( S < k B / nucleon); in both cases, the r-process isweak due to the low neutron irradiation.The relevance of the initial neutron mass fraction X n in the analysis of the efficiency of the r-process can beclearly seen in Figs. 9. The various trajectories can beclassified according to their final abundance distributions(Figs. 9a-f). The trajectories computed within scenario Iare classified in the group labeled “no ν ” while thosecomputed within scenario II are divided into five groups :the group labeled “fission” contains trajectories with thehighest fission recycling ( X cumfiss , tot > .
01, more detailsabout the definition of this fission recycling indicatorwill be given in Sec. III.2). Among the trajectories with (cid:80) (cid:54) A (cid:54) Y f ( A ) > .
01, if (cid:80) (cid:54) A (cid:54) Y f ( A ) < . nd peak” else inthe group “rare earth & no fission”. The trajectorieswith (cid:80) (cid:54) A (cid:54) Y f ( A ) > . st peak”. Remaining trajectories correspond to thegroup “between 1 st and 2 nd peak”. The two first groups“1 st peak” and “between 1 st and 2 nd peak” represent each7% and 6%, respectively, of the ejected mass within sce-nario II (see Fig. 9h). The other groups “2 nd peak”,“rare earth & no fission” and “fission” represent 41%,30% and 16%, respectively. The mean initial neutronmass fraction increases with the group (Fig. 9h, greencurve), heavy elements production is possible only witha high initial neutron mass fraction. Similarly, the meanmass of elements heavier than 50 ( (cid:104) A (cid:105) A (cid:62) ) also increaseswith the group (Fig. 9h, gray curve) but reaches a maxi-mum around 170 when fission occurs in groups “fission”and “no ν ” corresponding to the saturation regime in-duced by the fission recycling (see Sec. III.2). In the( S , Y e ) plane (Fig. 9g), the trajectories of the variousgroups are distributed according the iso- X n lines whichclearly demonstrate the relevance of the initial neutronmass fraction X n in the analysis of the r-process effi-ciency. III.2. Fission recycling
Fission recycling for a given trajectory can be quan-tified by the so-called total cumulative fissioning massfraction X cumfiss , tot = (cid:88) Z,A (cid:88) r=fiss (cid:90) dX C dt ( Z, A, t, r ) dt (9)= (cid:88) Z,A (cid:90) X ( Z, A, t ) (cid:104) N n (cid:104) σv (cid:105) Z,A n , f + λ Z,A sf + λ Z,Aβ df (cid:105) dt = (cid:88) Z,A X cumfiss ( Z, A ) FIG. 10: (Color online) Initial neutron mass fraction X n (black curve with dots) and the mass distribution ofthe ejecta (blue curve) depending on the fissionrecycling X cumfiss , tot . The dashed line separatesabundances computed within scenario I ( X cumfiss , tot > X cumfiss , tot < dX C dt ( Z, A, t, r ) corresponds to the reaction flux,expressed in terms of the mass fraction X for a givennucleus ( Z, A ), at a given time t and for a given fissionreaction r . X cumfiss , tot consequently corresponds, at a giventime, to the cumulative mass fraction of fissioning nu-clei that are destroyed by fission reactions and gives aquantitative indicator of the amount of mass recycled byfission processes. Fission can take place by spontaneous(at a rate λ sf ) or β -delayed (at a rate λ β df ) fission orby neutron-induced fission (at a rate N n (cid:104) σv (cid:105) n , f , where (cid:104) σv (cid:105) n , f is the corresponding astrophysical rate and N n the neutron density). Consequently, there are three fac-tors contributing to the fission recycling : ( i ) the quan-tity of fissionable nuclei, ( ii ) the fission rates and ( iii ) theamount of time over which fission reactions may occur.The total fissioning cumulative mass fraction X cumfiss , tot increases with initial neutron mass fraction X n (Fig. 10,black curve with dots). The fission recycling can there-fore be explained in term of initial neutron mass fraction X n , i.e. the more neutron-rich the initial ejecta, thelarger the production of heavy elements, hence the moreefficient the fission. III.2.1. Scenario I
Within scenario I, X cumfiss , tot > X n > . X cumfiss , tot which is found to range between1.19 and 1.82 with a mean value over all trajectories of5 (a) For scenario I where the mean value of the total fissioning cumulative mass fraction (cid:104) X cumfiss , tot (cid:105) traj = 1 . (cid:104) X cumfiss , tot (cid:105) traj = 0 . FIG. 11: (Color online) ( α ) Final abundances obtained with SPY FFDs, averaged for each bin of X cumfiss , tot colored bytheir X cumfiss , tot value and compared to the average final abundances (green solid curve with dots). ( β ) Same finalabundances in the ( A, X cumfiss , tot ) plan colored by their abundances. The mean mass (cid:104) A (cid:105) A (cid:62) for each X cumfiss , tot bins arerepresented by red crosses. (cid:104) X cumfiss , tot (cid:105) traj = 1 .
45. The final abundances distributionfor a given trajectory can be directly linked to its totalfissioning cumulative mass fraction X cumfiss , tot . An overallshift of final abundances towards heavier elements can beexpected with increasing fission recycling but the mean A (cid:62)
50 mass, (cid:104) A (cid:105) A (cid:62) , does not increase with X cumfiss , tot (Fig. 11a β , red crosses), and stays rather constant be-tween 166 and 172. This value remains constant be-cause the final abundances are dominated by the second( A = 130) and third ( A = 195) peaks.6The final abundance of light elements ( A (cid:54) α -particles, is significant with 0 . < (cid:80) A (cid:54) Y f ( A ) < . S , inducing an overall downward shift of thefinal abundances of heavier elements ( A > X n is linked to X cumfiss , tot (Fig. 10), the initial entropy S oftrajectories within a given bin of X cumfiss , tot covers almostthe whole range of possible S (since X n isolines evolve inthe ( S , Y e ) plane, as seen in Fig. 9g). This S admixtureof trajectories within a bin of X cumfiss , tot smooths the overallshift of the final abundances of heavy elements ( A >
A <
195 abundances decrease with X cumfiss , tot , whereasfor elements A >
195 they increase (Fig. 11a α ). Withincreasing values of X cumfiss , tot , the abundances in the massregion 140 < A <
170 drop by an order of magnitude,but around A = 202 they increase from 4 × − up to2 × − , while for A <
80 elements, they drop below10 − (Fig. 11a).Neglecting the effect related to the variation of lightelements abundance, a higher initial neutron mass frac-tion X n induces more fission recycling which implies adecrease of the final A <
80 abundances. A high neu-tron irradiation leads to a significant production of super-heavy elements (
A > N = 184) with 208 < A (cid:54)
278 arethe progenitors of the trans-lead elements. Among theseaccumulated nuclei, those with 260 (cid:46) A (cid:54)
278 (79 (cid:46) Z (cid:54)
94) predominantly β -decay up to reach the “decayroof 0” where they fission. Those with 208 < A < β - and α -decay up to the third abundance peak, whilethe remaining intermediate nuclei with 232 (cid:54) A (cid:46) β - and α -decay up to the third abundance peak or β -decay and reach the “decay roof 0” where they fission ordecay into the long-lived Th and − U nuclei. Fi-nal
A >
204 abundances decrease with X cumfiss , tot becausetheir progenitors are the nuclei accumulated along theneutron shell closure N = 184 during the neutron irra-diation. This accumulation is reduced for high neutronirradiations which enable to overcome the neutron shellclosure N = 184, hence to increase the fission recycling. III.2.2. Scenario II
With the low initial neutron mass fraction X n found inscenario II, about 55% of the ejected mass is not subjectto fission recycling ( X cumfiss , tot < − ) (Fig. 10, blue curve)and the maximum value of X cumfiss , tot = 0 .
6. The fissionrecycling is reduced by a factor 50 with a mean value ofthe total fissioning cumulative mass fraction (cid:104) X cumfiss , tot (cid:105) traj that now amounts to 0.03 to be compared with 1.45 forscenario I. The present scenario is well suited to study lowfission recycling effects on the final abundances. For thisreason, trajectories are sorted in log bins of their total fissioning cumulative mass fraction, as shown in Fig. 11b.The bins, except for the first one, represent each about5% to 10% of the ejected mass (Fig. 10).The final abundance of the light A (cid:54) X cumfiss , tot is linked to the initial neutron mass fraction X n , as observed in Fig. 10 (black curve with dots) where X n is seen to increase linearly with log (cid:16) X cumfiss , tot (cid:17) from0.5 to 0.75. In other words, X cumfiss , tot increases exponen-tially with X n . An increase with X cumfiss , tot , hence with X n ,of the mean mass (cid:104) A (cid:105) A (cid:62) from 105 up to 168 is observedin Fig. 11b β (red crosses), i.e. the higher X n , the heavierthe final nuclei produced.For trajectories with a low initial neutron mass frac-tion, X n ≈ . X cumfiss , tot < − , A >
130 nuclei,including lanthanides, are not significantly produced incomparison with the other trajectories (Fig. 11b) andproduction beyond
A >
180 is negligible. Fission recy-cling does not take place in this case because there are notenough neutrons to overcome significantly the N = 82neutron shell closure. For higher initial neutron massfraction X n ≈ . − < X cumfiss , tot < − ), nuclei in therare earth and third peak regions start to be produced,but those with A >
200 are still highly underproducedand fission recycling remains negligible.Abundances of third peak elements increase in therange 10 − < X cumfiss , tot < − (0 . < X n < .
71) asdoes the mean mass (cid:104) A (cid:105) A (cid:62) which increases linearlywith log (cid:16) X cumfiss , tot (cid:17) up to 168. In this X cumfiss , tot range,abundances of rare earth nuclei (145 < A < − < X cumfiss , tot < − before decreasing for X cumfiss , tot > − . The N = 126 and N = 184 neutron shell closures start to be overcome for X cumfiss , tot > − leading to the synthesis of A >
204 nu-clei in a non-negligible amount. Fission recycling startsto play a role for trajectories with X cumfiss , tot > − (orequivalently with X n > . i.e. . < X cumfiss , tot < .
61 and an averageinitial neutron mass fraction X n = 0 .
76, the third r-abundance peak becomes higher than the second one and
A >
204 nuclei dominate the composition. The meanmass (cid:104) A (cid:105) A (cid:62) does not increase significantly, a satura-tion regime is reached with (cid:104) A (cid:105) A (cid:62) ≈
170 as observedfor scenario I (see Sec. III.2.1); the r-process dynamicsfor such trajectories is similar to the one taking place inscenario I.To complete the discussion of Sec. III.2.1, fission re-cycling plays a dominant role when X cumfiss , tot > . X n > .
75 which concerns 9 .
5% of theejected mass within scenario II. A saturation regime isreached (cid:104) A (cid:105) A (cid:62) ≈ X n induces more fission processes feeding back7FIG. 12: (Color online) Repartition of the fissionrecycling X cumfiss , tot between the 5 fission seed areasdefined in Fig. 7d. The dashed line separates scenario I( X cumfiss , tot >
1) from scenario II ( X cumfiss , tot < < A <
200 mass region.
III.3. Fission seeds
We now turn to the mean contribution of fissioningnuclei, or fission seeds, to the fission recycling and tothe final abundance distribution. The contribution of agiven fissioning nucleus (
Z, A ) to the fission recycling, x FS ( Z, A ), can be defined, for a given set of trajectories,as the fraction of the total fissioning cumulative massfraction (cid:104) X cumfiss , tot (cid:105) traj due to this specific fissioning nu-cleus ( Z, A ), i.e. x FS ( Z, A ) = (cid:104) X cumfiss . ( Z, A ) (cid:105) traj (cid:104) X cumfiss , tot (cid:105) traj . (10)In the specific NS merger cases studied here, (cid:104) X cumfiss , tot (cid:105) traj reaches rather similar values of 1.45 and 1.41 in scenario Iwhen using SPY or GEF FFDs, respectively and 0.03 forboth FFDs within scenario II. This similitude is due tothe fact that SPY and GEF FFDs are not fundamentallydifferent (see Fig. 15) and the properties of the fissionseeds are not related to their FFDs (see Fig. 7). Conse-quently, only results with SPY FFDs are presented here. III.3.1. Location in the ( N, Z ) plane of the contributingfission seeds Due to the specific β -decay and neutron capture prop-erties of nuclei above Th, in particular the “fission bottle-neck”, the “fission roof” and the N = 184 neutron shellclosure, not all the trans-Th nuclei play a role in fission recycling. The contribution of a given fission seed area(see Fig. 7d defining the 5 areas of relevance for the nu-clear physics adopted in the present study) to the fissionrecycling X cumfiss , tot is the sum of x FS ( Z, A ) for all nucleilocated in such a specific fission seed area. Such con-tributions evolve with the fission recycling, as shown inFig. 12. The contribution of the “decay roof 0”, corre-sponding to the fission of nuclei originally accumulatedalong the neutron shell closure N = 184 and Z < β -decay afterthe freeze-out, decreases with increasing X cumfiss , tot since thefission recycling increases with the initial neutron massfraction X n . With increasing X cumfiss , tot and X n , matteraccumulated along N = 184 isotone spreads up to Fmthanks to the successive β -decays and neutron capturesalong this shell closure. More nuclei above Z = 94 areproduced and increase the contribution of the “decayroof 1” and consequently decrease the one of the “decayroof 0”. For X cumfiss , tot > − , the neutron shell closure N = 184 starts to be overcome and nuclei in the “fis-sion bottleneck” area contribute to the fission recycling.In the range 10 − < X cumfiss , tot < − , the contributionof the “fission bottleneck” is roughly constant represent-ing 20% of the total cumulative fissioning mass fraction.The matter starts to flow through the “fission bottle-neck”, via Cm isotopes, and reach the “fission roof” for X cumfiss , tot > − . For high fission recycling ( X cumfiss , tot > III.3.2. Scenario I
Sixteen fission seeds are found to contribute for morethan 1% to the fission recycling (Fig. 13a) within sce-nario I. They are distributed mainly in the “fission bot-tleneck” and the “fission roof” areas, representing respec-tively 57% and 31% of the total cumulative fissioningmass fraction (Fig. 13a).A major part of the flow crossing the N = 184shell closure is concentrated along the Am chain, fed bythe favorable β -decay of Pu , where Am with x FS = 45% is the major contributor to the fission re-cycling. This favorable fission is due to the low fissionbarriers of a few nuclei dropping below 4 MeV accord-ing to the BSk14 Skyrme HFB calculations [111]. Theyare also main contributors in the “fission roof” area, themain one in this area and the second major one of allthe contributors is Mt with x FS = 12%. In thiscase, the primary fission barrier drop below 2 MeV. Asthe main fission seeds are in the 275 < A <
290 and330 < A <
346 ranges, the FFDs of these fission seedsaffect the final production of nuclei in the 80 < A < (a) For scenario I.(b) For scenario II.
FIG. 13: (Color online) Fission seed contribution ( x FS ) depending on the mass of fission seed (the total height).Each bins contains the isotopic distribution of x FS for a given A. III.3.3. Scenario II
As the initial neutron mass fraction is low in this case,the fission recycling remains low too. The contributionof the “fission bottleneck” and the “fission roof” amountto 37% and 6%, respectively, to be compared with 57%and 31% in scenario I. The main fission seeds area is the“decay roof 1” (Fig. 13b) because the low neutron irradi-ation makes it difficult to overcome the N = 184 neutronshell closure. It represents 45% of X cumfiss , tot and the maincontributors are Md and No isotopes. The contributionof the “decay roof 0” increases from 2% in scenario I tothe non-negligible value of 9% in scenario II.The main fission seed is still Am with x FS = 25%with additional ones located in the 276 < A CN < x FS = 2 − X cumfiss , tot . III.4. Fission progenitors
We now turn in the analysis of the so-called fission pro-genitors, i.e. the nuclei for which fission contributes ina non-negligible way to their final abundance. Througha detailed tracking of the fission processes taking placeduring the r-process, the impact of fission on the fi-nal abundances can be quantified by the analysis of thefission fragments progenitors and their related fissionseeds. For each trajectory, the progenitors of a givenfission fragment can be identified and the fission contri-9FIG. 14: (Color online) Contribution of the fission fragments to the final production of nuclei with mass number Ax fFFP ( A, X cumfiss , tot ) obtained with SPY FFDs. Gray curve delimits x fFFP ( A, X cumfiss , tot ) = 0 . x fFFP ( A, X cumfiss , tot ) = 0 . x FFP ( Z, A, t i ) (FFP stands for fis-sion fragments progenitors) to a nucleus ( Z, A ) with themolality Y m ( Z, A, t i ) (in mol/g) at time t i is defined as x FFP ( Z, A, t i ) = x FFP ( Z, A, t i − ) Y m ( Z, A, t i − ) + ∆ Y m, P ∆ t ( Z, A, t i − )∆ t (cid:80) r x FFP ( Z r , A r , t i − ) x RP ( Z, A, r, t i − ) Y m ( Z, A, t i − ) + ∆ Y m, P ∆ t ( Z, A, t i − )∆ t (11)where ∆ t = t i − t i − is an arbitrary time step taken hereto be of the order of ≈ t i /
10 and ∆ Y m, P ∆ t ( Z, A, t i − ) is thetotal production rate (in mol/g/s) of nucleus ( Z, A ) be-tween t i − and t i , expressed in terms of the molality Y m (if normalized to 1, it corresponds to the molar fraction Y ). The reaction fraction x RP ( Z, A, r, t i − ) is the frac-tion of the total production rate due to the reaction ( r ).The reactions considered here are ( n, γ ), ( γ, n ), ( β, γ ),( β, n ), ( β, n ), ( β, n ), ( γ, α ), (sf) ( i.e. spontaneous fis-sion), ( β, f) and ( n, f). In other words, the contributionof fission fragments to the production of a nucleus AZ Xat time t i is the weighted average between the previ-ous contribution x FFP ( Z, A, t i − ) and the contributions x FFP ( Z r , A r , t i − ) due to the production of this elementby a given reaction r : A r Z r Y → AZ X + ... , between t i − and t i , at a rate of ∆ Y m, P ∆ t ( Z, A, t i − ) x RP ( Z, A, r, t i − ). Ifa nucleus is exclusively produced by fission, x FFP = 1.This quantity is well suited to quantify the impact offission fragments on final abundances because it is notaffected by the decay of post-neutron fission fragmentsthanks to the time tracking. To obtain the overall fi-nal x FFP , hereafter x fFFP , (Figs. 16, red dotted curves),the tracking (Eq. 11) is performed up to t = 1 yr whenthe residual fission does not contribute anymore to the production of light nuclei.The contribution of fission fragments to the final pro-duction depends on the fission recycling indicator X cumfiss , tot (Fig. 14, gray curve) and starts to be significant, i.e. max A (cid:2) x fFFP ( A, X cumfiss , tot ) (cid:3) ( X cumfiss , tot ) > . , for X cumfiss , tot > − ( X n > .
65) which corresponds to thevalue given in Sec. III.2.2. For X cumfiss , tot > − , fissioncontributes at least to 50% of the production of the finalnuclei with 108 < A <
120 and 139 < A < X cumfiss , tot ≈ A ≈ x fFFP ( A ) is obviously affected by the FFDs modeladopted, even if the fission recycling does not dependsignificantly on the FFDs model. This sensitivity to theFFDs model is studied for both scenario in the two sub-sections below. In order to clarify the link between x FS and x fFFP , we also introduce here two new quantities,namely the relative contribution of each fission seed tothe production of a final nucleus A by fission x fFS / FFP ( A, A FS ) = x fFFP ( A, A FS ) x fFFP ( A ) (12)0where x fFFP ( A, A FS ) is obtained by tracking only fis-sion fragments produced by the specific fission seed A FS , x RP ( Z, A, t i − ) being replaced by x RP ( Z, A, A FS , t i − ) inEq. 11. and the contribution of a compound nucleus( Z CN , A CN ) to the fission recycling, which can be ex-pressed from Eq. 10 as x CN ( Z CN , A CN ) = (cid:88) ( Z,A ) CN ,r = { sf ,β df , nif } x FS ( Z r , A r , r )(13)where x FS ( Z r , A r , r ) is the contribution to the fission re-cycling (cid:104) X cumfiss , tot (cid:105) traj of the fission seed ( Z r , A r ) fission-ing by a given fission mode r (spontaneous, β -delayedor neutron induced). In Eq. 13, the summation is per-formed in such a way that the mass of the compoundnucleus formed from the fission seed is equal to A CN andthe proton number to Z CN . III.4.1. Scenario I
In scenario I, fission processes contribute significantlyto the production of nuclei with 80 < A <
200 for bothFFDs models (Fig. 16a, red dotted curves) due to thehigh fission recycling. The shape of x fFFP ( A ) is affectedby the FFDs model (Figs. 16a α and γ , red dotted curves).However the link between x fFFP and FFDs is no straight-forward.Two mean fission yields distributions (cid:104) Y FF (cid:105) FS can bedefined, the first one is associated with the “fission bot-tleneck” and the “fission roof”, denoted as pre-freeze-outmean fission yields, corresponds to the mean fission yieldsdistribution during the neutron irradiation, before freeze-out (Figs. 16, green curves). The second one, denoted aspost-freeze-out mean fission yields, is associated with the“decay roofs” (Figs. 16, orange curves), after freeze-out.For both FFDs model, the pre-freeze-out mean fissionyields is dominated by the FFDs of Am ( x CN = 0 . Mt ( x CN = 0 .
12) formed by the neutron capture of
Am and
Mt. With SPY FFDs (see Figs. 15, greencurves), these two main contributors fission asymmetri-cally. The three peaks pattern seen in the pre-freeze-out (cid:104) Y FF (cid:105) FS (Fig. 16a β ) is due to ( i ) the light fragmentsof Am feeding the peak around A = 120 − ii )the heavy fragments of Am plus the light fragmentsof
Mt feeding the peak around A = 145 − iii ) the heavy fragments of Mt responsible for thepeak A = 180 − Am but a more symmetric one for
Mt. As aresult, a double hump pattern is obtained for the pre-freeze-out (cid:104) Y FF (cid:105) FS (Fig. 16a δ ) where the peak around A = 100 −
120 is due to to the light fragments of
Amand the one around A = 150 −
180 to the heavy fragmentsof
Am plus the symmetric fragments of
Mt.The main contributors to the post-freeze-out mean fis-sion yields, it i.e. to the fission recycling in the “decayroofs” areas, are isotopes from Fm to Lr with A CN = 278to 284 where x CN reaches values up to 0.016 for No (see Fig. 15a). Those nuclei are characterized by anadmixture of slightly asymmetric fission and a triple-peak fission where the maximum yields are found around A = 130 for SPY FFDs. GEF FFD is more asymmet-ric leading to post-freeze-out mean fission yields roughlyconstant in the A = 110 −
160 mass range.During the neutron irradiation, fission fragments cap-tures neutrons. The
A <
130 fission fragments accumu-late along the neutron shell closure N = 82 and the rel-ative contribution from the “fission bottleneck” x fFS / FFP increases at A = 120 − N = 82 ( A (cid:38) N = 126( A (cid:46) A > x fFFP and the relative contribution of the“fission bottleneck” x fFS / FFP is higher with GEF thanwith SPY since the pre-freeze-out distribution is moreasymmetric due to the more asymmetric FFD of
Am.At freeze-out, there is no more fission fragments producedby the “fission bottleneck” and the “fission roof”. Thefinal abundance peak A = 130 predicted by SPY model(Fig. 8) is due to the high post-freeze-out mean fissionyields around A = 130 (Fig. 16a, orange curves). x fFFP is locally reduced around A = 130 due to theaccumulation of non-recycled material along the neutronshell closure N = 82. This reduction is limited with SPYmodel since the pre- and post-freeze-out mean fissionyields are higher than in the GEF case around A = 130.A low x fFFP for 80 < A <
100 could be expected since themean fission yields are negligible for both FFD models.However, in this case, there is no matter accumulation inthis mass region, so that the low 80 < A <
100 produc-tion ( Y f < − ) occurring after freeze-out is essentiallydue to the asymmetric fission of seeds located in the “de-cay roof 0”. This leads to a non-negligible value of x fFFP for 80 < A < x fFFP for A >
180 is dueto the accumulation of matter along the neutron shellclosure N = 126. III.4.2. Scenario II
In scenario II, the reduced fission recycling leads toa non-negligible x fFFP that does not exceed 30% onlyaround 130 < A < x CN reaches values up to 0.08 for No. The fissionseed contribution from the “fission roof” is small, so thatthe A = 180 −
190 pre-freeze-out mean fission yields arenegligible with the SPY model. The GEF pre-freeze-outmean fission yields is roughly similar to the one found inscenario I, though the A = 100 −
120 peak is now foundhigher than the A = 160 −
180 one.The low neutron irradiation makes the contributionof the “fission bottleneck” and the “fission roof” rel-atively small within this scenario, leading to a total1 (a) Around the neutron shell closure N = 184, these nuclei are responsible for 64/86% of the fission recycling in scenario I/II.(b) Around the “fission roof”, these nuclei are responsible for 30/5% of the fission recycling in scenario I/II. FIG. 15: (Color online) Post-neutron FFDs from SPY model computed at a constant excitation energy Q = 8 MeV(green curves) and from GEF model for thermal neutron-induced fission (blue curves). For each box, the x axisranges from A = 80 to 210 by step of 10 and the y axis from 0 to 15% by step of 2%. Colored backgroundcorrespond to the contribution of the compound nucleus x CN in scenario I/II. Only contributions x CN > .
001 aredisplayed. FFDs in thick orange boxes correspond to the N CN = 184 neutron shell closure.2 (a) For scenario I. (b) For scenario II. FIG. 16: (Color online) ( α and γ ) Contribution of the fission fragments to the final production ( x fFFP ) (red dottedcurves) and fission seeds contributions to the production of fission fragment progenitors ( x fFS / FFP ) (color scale)computed with SPY( α ) or GEF ( γ ) FFDs . ( β and δ ) Mean fission yields (cid:104) Y FF (cid:105) FS for SPY ( β ) or GEF model ( δ )averaged over fission seeds from the “fission bottleneck” and the “fission roof” (green curves, denoted aspre-freeze-out) or from “decay roofs” (orange curves, denoted as post-freeze-out). x FS < .
25 for 90% the ejected mass ( i.e. X cumfiss , tot < . N = 82and more non-recycled material accumulates along thisshell closure. For this reason, the final abundance around A = 130 is higher than in scenario I with a negligiblefission contribution to the A (cid:54)
130 abundances. As theSPY post-freeze-out mean fission yields is peaked around A = 120 − x fFFP is maximal for A = 135 − A (cid:38)
160 which leads to a higher x fFFP in this mass region. IV. CONCLUSION
In the present study, new improvements have beenbrought to our scission-point model, called SPY, to de-rive the fission fragment distribution for all neutron-richfissioning nuclei of relevance in r-process calculations.These improvements include a phenomenological modifi- cation of the scission distance and an updated smoothingprocedure of the distribution to take into account effectsthat are neglected in the original model. Such correc-tions lead to a better agreement with experimental fissionyields. In particular, SPY FFDs are now in overall goodagreement with the experimental symmetric/asymmetrictransition in the Th and Pa isotopic chains. Those yieldsat scission are also used to estimate the number of neu-trons emitted by the excited fragments on the basis ofdifferent neutron evaporation models. Our new fissionyields significantly differ from to those predicted by thewidely used GEF model.The impact of fission on the r-process nucleosynthesisin neutron mergers is also re-analyzed. Two scenarios areconsidered, the first one with low initial electron fractionis subject to intense fission recycling, in contrast to thesecond one which includes weak interactions on free nu-cleons resulting in less pronounced fission recycling. Theefficiency of the r-process nucleosynthesis and the corre-sponding abundance distribution in a given mass elementhas been discussed essentially in terms of the initial neu-tron mass fraction X n which engulfs information on both3the initial electron fraction and entropy. The contribu-tion fission processes may have to the final abundance dis-tribution has been studied in detail in the light of newlydefined quantitative indicators, namely indicators for thefission recycling ( X cumfiss , tot ), fission seeds ( x FS ) and fissionfragment progenitors ( x fFFP ). The various regions of thenuclear chart responsible for fission recycling during theneutron irradiation as well as after freeze-out have beenidentified for the nuclear physics adopted in the presentcalculations, what we defined as the “decay roofs”, the“fission bottleneck” and the “fission roof”. Within the“fission bottleneck” region, Am isotopes around A = 285play a key role by limiting the nuclear flow to higherelements and strongly contributing to the fission recy-cling, mainly due to the low fission barriers predicted bythe BSk14 interaction in this region. Above Z = 110, theso-called “fission roof” prohibits the production of super-heavy elements. “Decay roofs” essentially recycle heavymaterial after the neutron irradiation freezes out whenthe material accumulated at the N = 184 shell closuredecays back to the valley of β -stability.When considering an efficient r-process (scenario I),fission fragments are found to contribute almost entirelyto the final abundances of nuclei with 100 < ∼ A < ∼ A = 140 −
180 region depending on the FFD modeladopted. Calculations based on the SPY and GEF FFDshave been compared for both r-process scenarios.The present study allows us to have a deep insight onthe role of fission processes in hydrodynamical r-processsimulations on the basis of sound well-defined quantita-tive indicators. Fission remains an important nuclearprocess taking place in binary NS mergers, though ma- jor nuclear as well as astrophysical uncertainties still needto be further addressed before concluding on the definiterole of fission processes. These concern in particular thedetermination of the fission probabilities for all the nu-clei potentially involved during the r-process as well asa detailed description of neutrino absorption and hydro-dynamics in simulations. Much progress remains to bedone along these two directions.
ACKNOWLEDGMENTS
J.-F.L. and S.G. acknowledge financial support fromFNRS (Belgium). This work was supported by the Fondsde la Recherche Scientifique (F.R.S.-FNRS) and theFonds Wetenschappelijk Onderzoek - Vlaanderen (FWO)under the EOS Project nr O022818F. The present re-search benefited from computational resources madeavailable on the Tier-1 supercomputer of the F´ed´erationWallonie-Bruxelles, infrastructure funded by the Wal-loon Region under the grant agreement nr 1117545.H.-T.J. acknowledges financial support from DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) through Sonderforschungsbereich (CollaborativeResearch Center) SFB-1258 “Neutrinos and Dark Mat-ter in Astro- and Particle Physics (NDM)” and underGermany’s Excellence Strategy through Cluster of Excel-lence ORIGINS (EXC-2094)—390783311, and from theEuropean Research Council through Grant ERC-AdGNo. 341157-COCO2CASA. 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