EElectron capture in stars
K Langanke , , G Mart´ınez-Pinedo , , and R.G.T. Zegers , , GSI Helmholtzzentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany Institut f¨ur Kernphysik (Theoriezentrum), Department of Physics, TechnischeUniversit¨at Darmstadt, D-64298 Darmstadt, Germany Helmholtz Forschungsakademie Hessen f¨ur FAIR, GSI Helmholtzzentrum f¨urSchwerionenforschung, D-64291 Darmstadt, Germany National Superconducting Cyclotron Laboratory, Michigan State University, EastLansing, Michigan 48824, USA Joint Institute for Nuclear Astrophysics: Center for the Evolution of the Elements,Michigan State University, East Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing,Michigan 48824, USAE-mail: [email protected] , [email protected] , [email protected] Abstract.
Electron captures on nuclei play an essential role for the dynamics ofseveral astrophysical objects, including core-collapse and thermonuclear supernovae,the crust of accreting neutron stars in binary systems and the final core evolution ofintermediate mass stars. In these astrophysical objects, the capture occurs at finitetemperatures and at densities at which the electrons form a degenerate relativisticelectron gas.The capture rates can be derived in perturbation theory where allowed nucleartransitions (Gamow-Teller transitions) dominate, except at the higher temperaturesachieved in core-collapse supernovae where also forbidden transitions contributesignificantly to the rates. There has been decisive progress in recent years in measuringGamow-Teller (GT) strength distributions using novel experimental techniques basedon charge-exchange reactions. These measurements provide not only data for the GTdistributions of ground states for many relevant nuclei, but also serve as valuableconstraints for nuclear models which are needed to derive the capture rates for themany nuclei, for which no data exist yet. In particular models are needed to evaluatethe stellar capture rates at finite temperatures, where the capture can also occur onexcited nuclear states.There has also been significant progress in recent years in the modelling of stellarcapture rates. This has been made possible by advances in nuclear many-body modelsas well as in computer soft- and hardware. Specifically to derive reliable capturerates for core-collapse supernovae a dedicated strategy has been developed basedon a hierarchy of nuclear models specifically adapted to the abundant nuclei andastrophysically conditions present at the various collapse conditions. In particularat the challenging conditions where the electron chemical potential and the nuclear Q values are of the same order, large-scale diagonalization shell model calculations havebeen proven as an appropriate tool to derive stellar capture rates, often validated byexperimental data. Such situations are relevant in the early stage of the core collapseof massive stars, for the nucleosynthesis of thermonuclear supernovae as well for thefinal evolution of the core of intermediate-mass stars, involving nuclei in the mass range A ∼ − a r X i v : . [ nu c l - t h ] S e p lectron capture in stars This manuscript reviews the experimental and theoretical progress achievedrecently in deriving stellar electron capture rates. It also discusses the impact theseimproved rates have on the various astrophysical objects.
Submitted to:
Rep. Prog. Phys. lectron capture in stars
1. Introduction
Electron capture is one of the fundamental nuclear processes mediated by the weakinteraction. In this reaction, a free proton or one bound inside a nucleus is transformedinto a neutron by capture of an electron producing an electron neutrino. Electroncaptures on nuclei play an important role in various dense astrophysical environmentsand all three properties which characterize this process (change of the nuclear charge,reduction of the number of electrons and energy release by neutrinos) have importantconsequences in these astrophysical environments [1]. Stellar electron captures, however,differ significantly from those which can be studied in the laboratory. In the latter,the decay occurs within an atom (or ion) by capturing an electron from the atomiccloud where electrons in tightly bound orbitals are strongly preferred due to their largerprobability density at the nucleus. However, in the high-density, high-temperatureenvironments of stars the atoms are strongly (like in our Sun) or completely ionized(in advanced stellar burning stages or supernovae). Hence stellar capture rates differfrom laboratory rates and are, unfortunately, yet not directly experimentally accessibleand have to be modelled.Supernovae are arguably the most important astrophysical sites in which electroncaptures on nuclei play a decisive role. This includes the core collapse of massive stars[2, 3, 4], the final evolution of the ONeMg cores in intermediate-mass stars [5, 6], thecrust evolution of neutron stars in binaries [7, 8] as well as the nucleosynthesis occurringin thermonuclear (Type 1a) supernovae [9, 10]. In all scenarios the densities at whichelectron captures play a role are in excess of about 10 g cm − and at finite temperatureswhich range from 10 K in electron-capture supernovae to above 10 K which areencountered in the advanced core collapse of massive stars. Under these conditionselectrons are characterized by a relativistic Fermi gas with Fermi energies of MeV totens of MeV. As a consequence, electron captures can occur under these conditions alsoon nuclei which, under laboratory conditions, are stable [2].At the relatively low electron energies the capture is dominated by allowedGamow-Teller (GT) transitions, with forbidden transitions contributing at the higherdensities/temperatures or in exceptional cases [2, 3, 11, 12, 13]. This observationhas been the basis to recognize the importance of electron captures in core-collapsesupernova, but also of the decisive progress that has been achieved in recent years toderive reliable stellar capture rates. The pioneering work of Bethe, Brown, Applegate,and Lattimer [2] derived capture rates on the basis of a single GT transition transformingan f / proton into an f / neutron. This assumption was motivated by the IndependentParticle Model (IPM) structure of Fe which is quite abundant during the early collapsephase. The important insight into the collapse dynamics drawn in their pioneering workwas that electron capture is a very efficient cooling mechanism and that the entropystays low during the entire collapse. As a consequence the composition of the core ispredominantly made by heavy nuclei rather than being dissociated into free nucleons.The challenge of deriving an improved set of stellar capture rates was taken up by Fuller, lectron capture in stars A = 21 −
60 at appropriate temperature and density conditions in the core [14, 3, 15, 16].These derivations were based again on the IPM, but considered experimental datawherever available. Fuller noticed that, within the IPM, GT transitions from pf protonorbitals are Pauli blocked for nuclei with N ≥
40 for which the pf shell for neutrons iscompletely filled [17]. Based on this observation Bruenn derived a parametric descriptionfor stellar electron capture rates which assumed vanishing capture rates for all nucleiwith neutron numbers N >
40 [18]. Although Cooperstein and Wambach pointedout that the Pauli blocking could be overcome at high temperatures and by forbiddentransitions [11], the Bruenn prescription has been the default for electron capturesin supernova simulations until the early 2000s (e.g. [1]). On this basis, simulationspredicted that during the advanced collapse phase for densities in excess of 10 g cm − electron capture proceeds on free protons rather than on nuclei. As free protons aresignificantly less abundant than nuclei during the collapse, electron capture and theassociated core cooling was drastically throttled once capture on nuclei was blocked.During the last two decades the role played by electron captures for the supernovadynamics has been decisively revised. This was made possible by new theoreticalinsights, improved models and not the least by the development of novel experimentaltechniques to determine nuclear GT strength distributions. This breakthrough wasmade possible by the observation that strongly forward-peaked cross sections in charge-exchange reactions, mediated by the strong interaction, are dominated by the spin-isospin operator needed to derive weak-interaction GT transitions [19, 20]. Thepioneering GT measurements were performed at TRIUMF using the ( n, p ) charge-exchange reaction [21, 22, 23]. Despite of its moderate energy resolution of aboutan MeV, these measurements clearly showed that the nuclear GT strength is significantlymore fragmented and also reduced in total strength compared to the predictions of theIPM. These findings were subsequently confirmed by measurements performed at KVI,Groningen using the ( d, He) [24, 25, 26] and at NSCL, Michigan State University byexploiting ( t, He) charge-exchange reactions, respectively [27]. With both techniques,experimenters were successful to measure GT strength distributions for many pf shellnuclei with an energy resolution nearly an order of magnitude better than being possiblein the pioneering TRIUMF experiments. These measurements became an indispensableconstraint for nuclear models, which were developed in parallel to the experimentalprogress.Due to the strong energy dependence of phase space electron capture rates arequite sensitive to the detailed fragmentation of the GT strength if the Fermi energiesof the electron reservoir and the nuclear Q value are of the same magnitude [12, 28].This is the case during hydrostatic silicon burning and at the onset of the collapseat core densities up to about 10 g cm − . Under these conditions the core consistsmainly of nuclei in the Fe-Ni mass range, while sd shell nuclei are also present duringsilicon burning [29]. The method of choice to describe the properties of these nuclei lectron capture in stars sd shell and for pf shell nuclei at a truncation level that guaranteed sufficientlyconverged results for the nuclear quantities needed to derive reliable electron capturerates. This in particular includes detailed description of the GT strength distributionswhich, except for a slightly shell-dependent constant factor, reproduced the total GTstrength and its fragmentation quite well [31, 32]. This success was first used by Odaand collaborators to derive shell model electron capture rates for sd shell nuclei [33].This was followed by the calculation of individual capture rates for nuclei in the massrange A = 45–65 based on GT strength distributions derived in large-scale shell modelcalculations [12, 34]. Due to the finite temperature of the astrophysical environment theshell model calculations also include GT transitions occurring from thermally excitednuclear states. The shell model rates became the new standards in supernova simulationsfor intermediate-mass nuclei. It turned out to be quite relevant that the shell modelrates for pf shell nuclei are systematically and significantly smaller compared to theprior rates based on the IPM [12]. As a consequence, simulations with the shell-modelrates showed a noticeably slower deleptonization and resulted in different Fe-core massesat the end of the presupernova phase when the collapse sets in [29, 35].The Pauli blocking of the GT strength at the N = 40 shell closure exists in theIPM [17], but can be overcome by correlations which move protons or neutrons into thenext major shell (the sdg shell) [36]. To describe such cross-shell correlations within thediagonalization shell model requires usually model spaces with dimensions which are notfeasible with today’s computers. However, such studies exist for Se (the intermediatenucleus in the double-beta decay of Ge) showing that its GT strength is small, butnon-vanishing, even for the ground state [37]. This finding is in good agreement withthe experimental determination of the GT strength by the ( d, He) technique [38]. Asa consequence the stellar electron capture rate on Se is sizable, showing that theassumption of neglecting the capture on nuclei with
N >
40 is not justified. To deriveat the stellar capture rate for such nuclei a hybrid model had already been proposedand applied prior to the shell model studies of Se. This model is based on two steps[36, 39]: At first, the crucial cross-shell correlations are studied using the Shell ModelMonte Carlo approach [40, 41], which is a stochastical approach to the shell modelallowing to calculate nuclear properties at finite temperature considering correlations inunprecedentedly large model spaces. These calculations have been applied to determinepartial occupation numbers for protons and neutrons in the combined pf - sdg shells andat finite temperature. In the second step, these partial occupation numbers servedas input in RPA calculations of the GT and forbidden strength distributions andsubsequently the stellar capture rates. The use of the RPA for these nuclei is justified asthey dominate the core abundance only at higher densities and temperatures where theFermi energy of the electron gas is noticeable larger than the Q value of the respectivenuclei requiring only a reasonable reproduction of the total strength and its centroid for lectron capture in stars
6a reasonable estimate of the rate. The hybrid model has been applied to about 200 nucleiin the mass range A = 65 −
110 [28]. The studies clearly implied that Pauli blockingof the GT strength is overcome by cross-shell correlations at the temperature/densityconditions at which these nuclei are abundant [28, 42]. The SMMC calculations also yieldrather smooth trends in the partial occupation numbers at the relevant temperature ofabout 1 MeV. Based on observation a simple parametrization of the occupation numbershas been derived which was the basis of RPA calculations of stellar capture rates foranother 2700 nuclei [13].On the basis of the shell model calculations for sd and pf shell nuclei, of the hybridmodel for cross-shell N = 40 nuclei and the parametric study for the heavier nucleian electron capture rate table has been derived for core-collapse conditions [13]. Thenuclear composition of the core has been assumed to be in nuclear statistical equilibrium(NSE) [43]. When incorporated into supernova simulations these rates had significantconsequences for the collapse dynamics. In particular, the simulations show that captureon nuclei dominate over capture on free protons during the entire collapse. Furthermore,the dominating capture on nuclei leads to a stronger deleptonization of the core and tosmaller temperatures and entropies, in comparison to the previous belief that captureon nuclei would vanish due to Pauli blocking [28, 42].As an alternative to the hybrid model, the temperature-dependent QuasiparticleRPA model has been developed and applied to stellar electron capture for selectednuclei by Dzhioev and coworkers [44]. This approach formally improves the hybridmodel as it describes correlations and strength function calculations consistently withinthe same framework. In contrast to the hybrid model it restricts correlations to the 2p-2h level which due to the diagonalization shell model studies is not completely sufficientto recover all cross-shell correlations. This shortcoming is relevant for ground statestrength functions, but gets diminished with increasing temperatures. Satisfyingly bothquite different approaches yield similar capture rates in the density/temperature regimeswhere nuclei with neutron gaps at N = 40 and N = 50 matter during the collapse [45].Electron capture also plays a role for the final fate of the O-Ne-Mg cores ofintermediate-mass stars [46, 47] and for the nucleosynthesis occurring behind the burningflame during a thermonuclear supernova [48, 49]. In these scenarios only sd - and pf -shell nuclei are relevant and hence the respective diagonalization shell model rates canbe applied. For the dynamics of the O-Ne-Mg cores, however, also beta decays are quitedecisive for selected nuclei. The relevant rates can also be calculated quite reliable withinthe shell model (e.g. [33, 50]. It has been pointed out that the electron capture on Neconstitutes a very unusual case as its rate is dominated by a second-forbidden ground-state-ground-state transition in the relevant density/temperature regime [51]. As anexperimental milestone this transition has very recently been experimentally determinedwith quite considerable consequences for the fate of intermediate-mass stars [52].Electron captures on selected nuclei play also a role during hydrostatic stellarburning or during s-process nucleosynthesis. In these environments, ions are notcompletely stripped from electrons so that the capture predominantly occurs from bound lectron capture in stars Be whichis a source of high-energy solar neutrinos. The respective solar rate is derived in[53, 54, 55, 56, 57]. Electron capture on Be is also important in evolved stars as it affectsthe abundance of Li in red giant branch and asymptotic giant branch stars [58]. Durings-process nucleosynthesis certain pairs of nuclei (like
Rh-
Os,
Tl-
Pb) serve aspotential cosmochronometers [59]. These pairs are characterized by very small Q valuesagainst electron captures so that, in the inverse direction, β decay with an electron boundin the ionic K-shell (or higher shells) becomes possible and even dominates the decay.Such bound-state β decay strongly depends on the degree of ionization and of correctionsdue to plasma screening, while the competing electron capture process is often modifiedby contributions due to thermally excited nuclear levels. The formalism to describe therelevant electron capture, β and bound-state β -decay rates for the appropriate s-processtemperature and density conditions is derived in [60]; detailed rate tables can be foundin [61]. Application of these rates in s-process simulations are discussed in [62]. Forreviews of s-process nucleosynthesis the reader is referred to [63, 64].In this review we will summarize the theoretical and experimental progress achievedduring the last two decades in describing stellar electron captures on nuclei. Section 2 isdevoted to the experimental advances describing the various techniques to measureGamow-Teller strength distributions. Section 3 starts with some general remarksdefining the strategy how to derive the rates at the relevant conditions, followed by somebrief discussions of the adopted models and the rates derived within these approaches.InSection 4 we summarize the consequences of modern electron capture rates in core-collapse supernovae, for the fate of O-Ne-Mg cores in intermediate-mass stars and forthe nucleosynthesis in thermonuclear supernovae.
2. Experimental techniques and progress
To accurately estimate electron-capture rates on nuclei present in stellar environments,it is key to have accurate Gamow-Teller strength distributions from which theelectron-capture rates can be derived. Direct information about the Gamow-Tellerstrength distribution can in principle be obtained from β -decay and electron-capturemeasurements, but this provides only information about transition strengths betweenground states and a limited number of final states. Moreover, since in most astrophysicalphenomena electron captures near the valley of stability and/or on neutron-rich isotopesare most important, the Q value for β + /EC decay is often negative and directinformation is available only on the Gamow-Teller transition strength from the groundstate of the mother to the ground state of the daughter that is derived from β − decayin the inverse direction, and only if the ground-state to ground-state decay is associated lectron capture in stars E b (cid:38)
100 MeV) haveserved as that indirect method, as it is possible to extract the Gamow-Teller strengthdistribution up to sufficiently high excitation energies to perform detailed assessments ofthe validity of the theoretical models employed. The remarkable feature of this methodis that detailed information about transitions mediated by the weak nuclear force canbe extracted from reactions with hadronic probes mediated by the strong nuclear force.The methods and associated experimental techniques are described in this section.It is important to note that only a limited number of charge-exchange experimentscan be carried out and that these experiments only provide data on transitions from theground state of the mother nucleus. Since in many astrophysical scenarios a relativelylarge number of nuclei play a significant role and, if the stellar environment is at hightemperature, transitions from excited states also play a role, it is rarely possible to relyon experimental data of Gamow-Teller strength distributions only. To make accurateestimates for electron-capture rates in stars, theoretical nuclear models are necessary,which can be tested against charge-exchange data where available. Another importantconsideration is that electron captures in stars take place on stable and unstable nuclei.Hence, the obtain information about the latter, charge-exchange experiments withunstable nuclei are needed. As described below, such experiments are challenging andrelevant techniques for performing charge-exchange experiments in inverse kinematicsare still in development, although good progress have been made over the past decade.For the purpose of extracting Gamow-Teller strength distribution of relevance forelectron captures in stars, charge-exchange experiments in the β + /EC direction or ( n, p )direction are necessary and the primary focus in this section. These experiments probeproton-hole, neutron-particle excitations. However, charge-exchange data in the β − or ( p, n ) direction (neutron-hole, proton-particle excitations) are important as well.Firstly, the development of the techniques to extract Gamow-Teller strengths has beenprimarily developed by using charge-exchange reaction in the β − direction, starting withthe pioneering work by [68]. Many detailed studies have been performed by using the( He, t ) reaction. benefiting in part from the fact that for mirror nuclei the β + decayof the neutron-deficient nucleus and the ( p, n )-type reaction on the mirror neutron-richnucleus populate states with the same isospin. This allows for detailed comparisons ofGamow-Teller strengths through β decay and charge-exchange reactions [69].Secondly, for certain astrophysical phenomena, detailed information about theGamow-Teller strengths in the β − direction are needed. Thirdly, by assuming isospin-symmetry, information about Gamow-Teller strengths in the β + /EC direction cansometimes be derived from data in the β − direction. Finally, the theoretical modelsused to calculate Gamow-Teller strength distributions in the β + direction usuallyrely on the same parameters of the nuclear interaction as those calculated in the β − direction. Hence, by comparing the results of models against data from charge- lectron capture in stars β − direction, additional information about the strengthsand weaknesses of those models is obtained. The summed Gamow-Teller strengths inthe β + and β − directions are connected through a sumrule, first developed by Ikeda,Fujii and Fujita [70]: S β − ( GT ) − S β + ( GT ) = 3( N − Z ) (1)Although experimentally, only about 50–60% of the sum-rule strength is observed in theGamow-Teller resonance at excitation energies below ∼
20 MeV [71, 72], referred to asthe “quenching” phenomenon [20], it allows one to obtain information about the strengthin the β + /EC direction from the measurement in the β − direction. However, as theelectron-capture rates that are derived from the Gamow-Teller strength strongly dependson the strength distribution and not just the magnitude of the strength, measurement ofthe strength in the β − direction is of limited use for detailed evaluations of the strengthdistribution. This is especially true for nuclei with increasing neutron number for fixedatomic number as S β − becomes increasingly larger than S β + . On the other hand, theIkeda sum rule is a very useful constraint for the total GT strength for the cross sectioncalculation of charged-current ( ν e , e − ) reactions for neutron-rich nuclei [73, 74]. The extraction of the Gamow-Teller strength distribution from charge-exchange reactiondata at intermediate beam energies is based on the proportionality between the Gamow-Teller transition strength B (GT) for small linear momentum transfer, q ≈
0, expressedthrough the following relationship [68]: (cid:20) dσd
Ω ( q, ω ) (cid:21) GT = F ( q, ω )ˆ σB (GT) , (2)in which dσd Ω ( q, ω ) is the measured differential cross section for a transition associatedwith energy transfer ω = Q gs − E x and linear momentum transfer q . Q gs is the ground-state reaction Q value that is negative for a transition that requires energy. E x is theexcitation energy of the final nucleus. B (GT) is the Gamow-Teller transition strengthand represents the same matrix elements as probed in β and EC decay transitionsbetween the same initial and final states. The condition that q = 0 requires that thecross section is extracted at or close to a center-of-mass scattering angle of zero degreesand that an extrapolation is required based on a calculation to correct for the finitereaction Q value. This extrapolation is represented by the factor F ( q, ω ). The factor ˆ σ is the so-called unit cross section, which depends on the reaction kinematics, the nucleiinvolved in the interaction and the properties of the nucleon-nucleon ( N N ) interaction.In the Eikonal approximation [68], these components are factorized:ˆ σ = KN | J στ | . (3)In this factorization, K is a calculable kinematic factor, N is a distortion factor,and J στ is the volume integral of the spin-transfer, isospin-transfer στ componentof the N N interaction [75]. The distortion factor accounts for the distortion of the lectron capture in stars B (GT)is known from β -decay experiments. Once established for one or a few transitionsfor given nucleus and charge-exchange reaction, the proportionality can be applied toall transitions identified as being associated with ∆ L = 0 and ∆ S = 1, except forthe extrapolation to q = 0 through the factor F ( q, ω ) of Eq. 2. This extrapolationcarries a relatively small uncertainty. Calibrations against known transitions from β decay are not always possible. Therefore, mass-dependent parametrizations ofthe unit cross sections have been successfully developed for the ( p, n )/( n, p ) [76] and( He , t )/( t, He) [77, 78] reactions, which provide a convenient way to extract Gamow-Teller strength distributions for such cases.In order to use Eq. 2 and extract the Gamow-Teller strength distribution frommeasured differential cross sections, one must first identify the contributions to theexperimental spectra that are associated with monopole (∆ L = 0) and spin-transfer(∆ S = 1). This is performed by investigating the differential cross sections as afunction of scattering angle, since excitations that are associated with increasing unitsof angular momentum transfers have angular distributions that peak at larger scatteringangle. Therefore, through a process called multipole decomposition analysis (MDA) [79],in which the measured differential cross sections of a particular peak or data in anexcitation-energy bin is fitted by a linear combination of calculated angular distributionsfor different units of ∆ L , the ∆ L = 0 contribution to the cross section is extracted. Anexample for the Ti( t, He) reaction is shown in Fig. 1. Since the ∆ L = 0, ∆ S = 0contribution is almost completely associated with the excitation of the isobaric analogstate, it does not contribute to the ∆ L = 0 yield for ( n, p )-type reactions for nuclei with N ≥ Z , as the isospin of states in the final nucleus always exceed that of the target.The Fermi sum rule of S − − S + = ( N − Z ) is nearly fully exhausted by the excitationof the isobaric analog state in the β − ( p, n ) direction.For ( n, p )-type reactions, at excitation energies (cid:38)
10 MeV, contributions to the∆ L = 0 yield arise from the excitation of the isovector giant monopole resonances(IVGMR) and isovector spin giant monopole resonance (IVSGMR) [65]. In charge-exchange reactions with beam energies of (cid:38)
100 MeV, the IVSGMR dominates. Sincethe angular distribution of the IVSGMR is very similar to that of Gamow-Tellerexcitations, the two are not easily separable experimentally. Only through a comparisonbetween ( n, p ) and ( t, He) data it is possible to disentangle the two contributions [81].Since the transition density for the IV(S)GMR has a node near the nuclear surface,a cancellation occurs for the ( n, p ) probe that penetrates relatively deeply into thenuclear interior, whereas such a cancellation does not occur for the peripheral ( t, He)reaction [82, 83]. Hence, the excitation of the IV(S)GMR is enhanced for the latter lectron capture in stars d / dd E x ( m b s r M e V ) Ti( t , He) Sc E = 115 MeV/ u cm = 0.67 E x = 3.1 MeV E x = 10.0 MeV DataSum L = 0 L = 1 Data L = 0 L = 1 L
20 1 3 4 50 5 10 15 0.110.110 E x ( Sc) (MeV) cm (deg) L Figure 1.
An example of the MDA for the Ti( t, He) reaction at 115 MeV/ u . Onthe left-hand side, differential cross sections at 4 scattering angles are shown. Onthe right-hand side, the MDA analyses for two excitation energy bins (at 3.1 MeVand 10 MeV) in Sc are shown. At 3.1 MeV (10.0 MeV), the ∆ L = 0 (∆ L = 1)contribution is strongest. The stacked colored histogram on the left-hand side indicatethe contributions from the different angular momentum transfers based on the MDA.(from Ref. [80]). probe. As this comparison between probes is generally not available, the extraction ofGamow-Teller strengths for the purpose of estimating electron-capture rates and benchmarking the theory is usually limited to excitation energies up to about 10 MeV.Since the extracted Gamow-Teller strengths from the charge-exchange data arecalibrated against known weak transitions strengths, the uncertainties introduced by theneed to extract absolute cross sections through careful beam intensity normalizationsand target thickness measurements are absent. If phenomenological relationshipsbetween the unit cross section and mass number are utilized [77] to determine theunit cross section, usually a measurement with a target for which the unit cross sectionhas been well established is included in an experiment, so that a relative normalizationcan be performed, rather than relying on an absolute cross section measurement that isusually more uncertain. This helps to reduce experimental systematic uncertainties toabout 10% [77].The main remaining uncertainties in the extraction of Gamow-Teller strengths arise lectron capture in stars L = 0, ∆ S = 1 and ∆ L = 2, ∆ S = 1 amplitudes that both contribute to∆ J = 1 transitions in which the parity does not change. This interference is mediatedby the tensor- τ component of the N N force [75, 20]. The uncertainty introduced bythis effect depends on the magnitude of the Gamow-Teller transition strength and wasestimated [27] to be ≈ . − .
035 ln( B (GT)), which amounts to an uncertainty ofabout 20% for B (GT)=0.01. The results of this study are shown in Fig. 2. A B (GT) of0.01 is close to the detection limit in charge-exchange experiments. It has been shownthat this uncertainty estimate is not strongly dependent on the nucleus studied [84]. Itis also clear that the systematic deviation fluctuates around 0, and after integrating overmany states, the uncertainty in the summed or average transitions strength is small. -0.500.510 -3 -2 -1 B(GT) R e l a t i v e s y s t e m a t i c e rr o r i n B ( G T ) belowdetection limit T=0T=1T=2 Mg( He,t) 140 MeV/u
Isospin final state
Figure 2.
Results from a theoretical study to estimate the magnitude of theuncertainty in the proportionality between Gamow-Teller strength and differentialcross sections for the Mg( He , t ) reaction at 140 MeV/ u due to the effects of thetensor- τ component of the N N interaction. Transitions to final states in Al withisospin T = 0 , , and 2 are included. The uncertainty increases with decreasing B (GT).The detection limit of 0.01 is indicated. (from Ref. [27]). The extraction of Gamow-Teller strengths from charge-exchange reactions in the β + direction for the purpose of constraining electron-capture rates has primarily beenperformed with three probes: the ( n, p ), ( d, He), and ( t, He) reactions. In thissubsection, a brief overview of these three probes and experimental methods will beprovided. lectron capture in stars ( n, p ) reaction Although ( n, p ) charge-exchange reactions have been performedat a variety of facilities, the pioneering work at TRIUMF has been particularlyimpactful for the purpose of testing theoretical models used to estimate electron-capture rates for astrophysical simulations. The nucleon charge-exchange facility atTRIUMF [85, 86] utilized the ( p, n ) reaction on a Li target to produce neutrons ofabout 200 MeV associated with transitions to the ground and first excited state of Be that were subsequently impinged on the reaction target of interest. The setupused a segmented target chamber, which allowed for the insertion of several targetssimultaneously. Events induced by reactions on different targets were disentangledthrough the analysis of hitpatterns in multi-wire proportional chambers placed in-between the targets. Usually, one of the targets was a CH target, so that the well-known H( n, p ) reaction could be utilized to perform absolute normalizations of the neutronbeam intensity. Protons produced in the ( n, p ) reaction were momentum analyzed inthe medium-resolution spectrometer (MRS). Measurements at different scattering angleswere utilized to determine the differential cross sections as a function of center-of-massangles, facilitating the multipole decomposition analysis and extraction of Gamow-Teller strength from the proportionality between strength and differential cross sectiondiscussed above. A wide variety of experiments were performed for the purpose ofextracting Gamow-Teller strengths for astrophysical purposes, primarily on stable nucleiin the pf shell (see e.g. [87, 88, 89, 90, 91, 92, 93]). The excitation energy resolutionsachieved varied between 750 keV and 2 MeV, depending on the experiment. In Fig. 3,three examples of the extracted ∆ L = 0 contributions for the , , Ni( n, p ) reactions areshown, displaying a concentration of Gamow-Teller strength at low excitation energies,with a long tail up to higher excitation energies. ( d, He) reaction
The ( d, He) reaction has become one of the most powerfulprobes to study the Gamow-Teller strengths in the β + direction. This probe was firstdeveloped for the purpose of extracting Gamow-Teller strengths at RIKEN by usinga 260 MeV deuteron beam [94], followed by the development of this probe at TexasA&M [95] by using a 125 MeV deuteron beam. In these experiments, a resolution of 500–700 keV could be achieved, and the beam intensities were limited due to the backgroundfrom deuteron break-up reactions. The method was perfected in experiments withthe Big-Bite Spectrometer at KVI in combination with the EuroSuperNovae (ESN)detector [96] and using deuteron beams of ∼
170 MeV. Owing to the use of data signalprocessing, two-proton coincidence events could be selected online, strongly reducing thebackground from deuteron break-up reactions and making it feasible to run at higherincident beam rates. In addition, the excitation energy resolution was improved tovalues of typically 150 keV. A recent overview of the ( d, He) program at KVI can befound in Ref. [26].The use of the ( d, He) probe requires that the momentum vectors of the two protonsfrom the unbound He must be measured with high accuracy in order to reconstructthe momentum of the He particle created in the ( d, He) reaction, as well as the lectron capture in stars E (MeV) E x ( )VeM())cS E (MeV) , Sc) E x ( = 0.991 MeV + + + + G Sc SS pp EE SS nn EE xx G a m o w - T e ll e r s t r e n g t h ( M e V - ) data (statistical error)GXPF1A KB3G Excitation Energy in Cu (MeV) systematic errorshell model theory Ni(p,n) 110 MeV/uinverse kinematics
Honma et al. Poves et al. (a) (c)(d)(b)
Figure 3. (a) Differential cross sections associated with ∆ L = 0 for the , , Ni( n, p )reaction at 198 MeV (from Ref. [93]). (b) Differential cross sections at forwardscattering angles for the V( d , He) reaction at 170 MeV. Owing to the high-resolution,individual transitions are well resolved (from Ref. [97]). (c) left: γ energy versusexcitation energy for the Ti( t , He+ γ ) reaction (see also Fig. 1). right: by gatingon the Sc excitation-energy range around 0.991 MeV, the decay by a very weak 1 + state can be identified, sufficient for estimating the Gamow-Teller transition strengthto this state (from Ref. [80]). (d) Extracted Gamow-Teller strength distribution fromthe Ni( p, n ) reaction at 110 MeV/ u , performed in inverse kinematics. Two sets ofshell model-calculations with different interactions are super imposed (from Ref. [98]). lectron capture in stars (cid:15) between the two protons. On the other hand, the determination ofthe relative energy makes it possible to enhance the spin-transfer nature of the probe.As the incident deuterons are primarily in the S state, a spin-transfer ∆ S of 1 isensured if the outgoing protons couple to the S state, which can be accomplishedby reconstructing the relative energy between the protons and selecting events thathave small relative energies (typically smaller than 1 MeV). This removes transitionsassociated with ∆ S = 0 from the spectra and makes it easier to isolate the ∆ S = 1Gamow-Teller transitions.A variety of ( d, He) experiments were performed at KVI with the goal to extractGamow-Teller strengths for testing theoretical models used to estimate electron-capturerates of interest for astrophysical simulations, see e.g. Refs. [99, 97, 100, 101, 102,103, 104]. Because of the high resolution achieved, detailed studies of the Gamow-Tellerstrength distribution could be performed, including for nuclei for which it was difficult toobtain the targets, such as , V [97, 100], as shown in Fig. 3(b). Clearly, the excellentresolution achieved makes it possible to extract very detailed information about theGamow-Teller strength distribution. ( t, He) reaction
The use of the ( t, He) reaction has the disadvantage thatit is complicated to generate tritium beams. Although tritium has been used toproduce primary beams (see e.g. Ref. [105]), experiments performed for the purposeof extracting Gamow-Teller strength distributions for astrophysical purposes utilizedsecondary tritium beams. These experiments are performed at NSCL with the S800Spectrometer [106]. A primary O beam is impinged on a thick Beryllium productiontarget to produce a secondary tritium beam of 345 MeV [84]. Because the momentumspread of the secondary beam is large (typically 0.5%), the dispersion-matchingtechnique [107] is utilized to achieve excitation-energy resolutions ranging from 200 − particles per second, butwith the completion of the Facility for Rare Isotope Beam (FRIB), the beam intensitieswill increase significantly.In addition to the good excitation-energy resolution that can be achieved with the( t, He) reaction, it has the advantage that the inverse ( He , t ) reaction is studied in greatdetail and with excellent resolution at comparable beam energies [67, 78, 26]. This makesit possible to utilize the dependence of unit cross section on mass number determinedfrom ( He , t ) data for extracting Gamow-Teller strengths from ( t, He) experiments[27, 77, 78].As for the ( n, p ) and ( d, He) reactions, the ( t, He) reaction has been used to studya variety of nuclei to test theoretical models used in the estimation of electron-capturerates in astrophysical scenario, e.g. Refs. [108, 109, 110, 111, 112, 80, 113, 114]. Sincethe electron-capture rate is very sensitive to the transitions to the lowest-lying finalstates in the daughter nucleus, especially at low stellar densities, the ( t, He) probewas combined with the high-resolution detection of γ -rays in the Gamma-Ray EnergyTracking In-beam Nuclear Array (GRETINA) [115]. This has made it possible to extract lectron capture in stars Ti( t, He + γ ) reaction, for which the B (GT) for the transition to the first 1 + state at0.991 MeV could only be determined due to the measurement of the decay γ rays.In recent years, the focus of the experiments has shifted from nuclei in the pf -shellto nuclei near N = 50 [108, 109, 110] given their relevance for electron capture ratesduring the collapse of massive stars (see section 4.1). ( p, n ) reaction and isospin symmetry For nuclei with
N > Z , the Gamow-Tellertransition strength in the β + direction can also be extracted from ( p, n ) reactions underthe reasonable assumption that isospin-symmetry breaking effects are small. Hence,states with isospin T +1 populated from a ( n, p )-type reaction for a nucleus with ground-state isospin of T , have analogs in the ( p, n )-type reaction on that same nucleus. Bymeasuring the ( p, n )-type reaction and identifying the T + 1 states in the spectrum, theGamow-Teller transition strengths of relevance for estimating electron-capture rates canbe extracted. Unfortunately the excitation of states with higher isospin is suppressedcompared to states with lower isospin [116], and the T + 1 states sit on a strongbackground of states with isospin T − T , which are also excited in a ( p, n )-type reaction on a nucleus with isospin T . Still for nuclei near N = Z Gamow-Tellerstrengths have been extracted from ( p, n ) data for the purpose of testing theoreticalmodels used to estimate electron-capture rates in nuclei [117, 118].For nuclei with N = Z and assuming isospin symmetry, the Gamow-Teller strengthdistribution in the β + and β − directions are identical and a ( p, n )-type measurementscan be used to directly obtain the Gamow-Teller strength distribution of relevance forthe electron-capture rates. This feature was used to extract the Gamow-Teller strengthdistribution from Ni. By using a novel method to perform a ( p, n ) experiment in inversekinematics [98, 119], the Gamow-Teller strength distribution in Cu was extracted (seeFig. 3(d), which is the same as the Gamow-Teller strength distribution from Ni to Co. In this experiment, the excitation-energy spectrum in Cu was reconstructed bymeasuring the recoil neutron from the ( p, n ) reaction when the Ni beam was impingedon a liquid hydrogen target. Since it is important to measure the reactions at smalllinear momentum transfer to main the proportionality of Eq. 2, the relevant recoilneutrons have very low energies and were detected in a neutron-detector array developedespecially for that purpose [120]. With this method, it became possible to measure ( p, n )reaction in inverse kinematics on any unstable nucleus and it was recently used to study
Sn [121]. ( n, p ) -type charge-exchange reactions on unstable isotopes Since many of the nuclei that undergo electron captures in stellar environments areunstable, it is important to develop experimental techniques to perform ( n, p )-typecharge-exchange experiments in inverse kinematics. This poses a significant challenge.A neutron target is not available and all candidate reactions have a light low-energy lectron capture in stars p, n ) reaction, the recoil particle is notreadily available for the precise reconstruction of the excitation energy and scatteringangle of the reaction. If the excitation-energy of the residual nucleus after the charge-exchange reaction is below the nucleon separation energy, a precise measurement ofthe momentum and scattering angle of the residual can be sufficient to reconstruct theevent kinematics. This method was used to extract the low-lying β + Gamow-Tellerstrength distributions from unstable nuclei B and Si through the ( Li , Be) reactionin inverse kinematics [122, 123]. Unfortunately, this probe is very difficult to use forstudying Gamow-Teller strength distributions in unstable nuclei heavier than Si. If theexcitation energy of the residual exceeds the nucleon separation energy, it is necessaryto measure the decay nucleon in addition to measuring the residual and achieving thenecessary energy and angular resolutions to reconstruct the event kinematics becomeschallenging because of the strong forward kinematic boost of the laboratory referenceframe [122].Most recently, efforts have been initiated to utilize the ( d, He) reaction in inversekinematics to study ( n, p )-type charge-exchange reactions on unstable isotopes. In suchexperiments, the rare-isotope beam is impinged on an active-target time projectionchamber in which deuteron gas serves both as the target and the detector medium [124].The tracks from the two protons originating from the unbound He particle can be usedto reconstruct the momentum of the He particle, from which the excitation energy andscattering angle of the charge-exchange reaction can be determined. The unique two-proton event signature is also very helpful to separate the ( d, He) reaction from othertypes of reactions that occur in the time projection chamber and that have much highercross sections. If successful, the method will be equally powerful for the extraction ofGamow-Teller strengths in the β + direction as the ( p, n ) reaction in inverse kinematicsis for the extraction of Gamow-Teller strengths in the β − direction.
3. Strategy and model to calculate stellar electron capture rates
During their long lasting lives stars balance gravitational contraction thanks to theenergy gained from nuclear fusion reactions in their interior. Massive stars developa sequence of core burning stages (started by hydrogen burning via the CNO cycle,then followed by helium, carbon, neon, oxygen and the finally silicon burning). Duringthis evolution the density ρ and temperature T in the core increases gradually and hasreached values in excess of 10 g cm − and 10 K, respectively, at the end of siliconcore burning. At these high temperatures nuclear reactions mediated by the strong andelectromagnetic force are in equilibrium with their inverse reactions. This situation iscalled Nuclear Statistical Equilibrium (NSE) and determines the nuclear composition forgiven values of temperature, density and the proton-to-neutron ratio (usually defined bythe proton-to-nucleon ratio Y e ). Once NSE is reached the star cannot generate energyfrom nuclear fusion reactions anymore. Hence the core looses an important source of lectron capture in stars Y e value only slightly smaller than 0.5. However, the electrons, present in thecore to balance the charges of protons, form a highly degenerate relativistic gas and canbalance the gravitational contraction of a stellar mass up to the famous Chandrasekharlimit M Ch = 1 . Y e ) M (cid:12) with the solar mass denoted by M (cid:12) . Once this limitingmass is exceeded to continued silicon burning or, as we will see below, due to electroncaptures, the electron gas cannot stabilize the core anymore. The core collapses underits own gravity. 𝜌 (g cm − ) ( M e V ) h 𝑄 i = 𝜇 𝑛 − 𝜇 𝑝 𝜇 𝑒 𝑄 𝑝 × T Figure 4.
Various energy scales related to electron captures on nuclei and protons asfunction of density during the collapse. Shown are the temperature, T , the chemicalpotential of electrons, µ e , the Q value for electron capture on protons (constant) andthe average Q value for electron capture on nuclei approximated as the difference inchemical potential of neutrons and protons (adapted from [25]). It is important to note that Y e can be modified by charge-changing reactions which,however, can only be mediated by the weak interaction. Such reactions (electron capture,beta decay) are not in equilibrium under the early collapse conditions (as for examplethe neutrinos produced by the processes can leave the star and hence are not availableto initiate inverse reactions) and can change the nuclear composition. It is also veryimportant to note that under core-collapse supernova conditions, i.e. at sufficiently highdensities, electron capture and beta decay do not balance each other. The reason forthis unbalance lies in the fact that the electron Fermi energy, which scales like ρ / ,grows noticeably faster than the Q values of the nuclei present in the core (see Fig. 4).As a consequence, the electron capture rates are accelerated, while beta decays on theopposite are throttled due to Pauli blocking of the final electron states. Hence electroncaptures win over beta decays with three very important consequences. First, electroncaptures reduce the number of electrons and hence the degeneracy pressure which the lectron capture in stars g cm − where the diffusion time scale due to coherentscattering on nuclei becomes longer than the collapse time of the core. Neutrinos arethen effectively trapped in the core which until bounce collapses as a homologous unit.Third, electron capture reduces Y e and makes the core composition more neutron-rich.The NSE composition is driven to heavier nuclei with larger neutron excess (see Fig.5). This effect is the reason why nuclei with valence protons and neutrons in differentmajor shells become relevant for electron captures, introducing the Pauli unblocking asmentioned above. −5 −4 −3 −2 Z ( P r o t on N u m be r) T= 9.01 GK, ρ = 6.80e+09 g/cm , Y e =.0.433 Log (Mass Fraction) −5 −4 −3 −2 Z ( P r o t on N u m be r) T= 17.84 GK, ρ = 3.39e+11 g/cm , Y e =.0.379 Log (Mass Fraction) Figure 5.
Mass fraction of nuclei in Nuclear Statistical Equilibrium at conditionswhich resemble the presupernova stage (top) and the neutrino trapping phase (bottom)of core-collapse simulations (courtesy of W.R. Hix).
Electron capture plays an important role for the dynamics of the core collapse ofmassive stars for core densities between 10 g cm − and 10 g cm − . Fig. 4 shows theevolution of crucial energy scales for this density regime. The strongest growing quantity lectron capture in stars µ e which increases from 6 MeV to about 40 MeV.As nuclei get increasingly more neutron rich due to continuous electron captures, theaverage electron capture (cid:104) Q (cid:105) value of the nuclear composition present at the variousstages of collapse grows too, but this increase is noticeably smaller from about 4 MeVto 12 MeV. At all stages the average nuclear (cid:104) Q (cid:105) value is larger than for free protons(1.29 MeV). Finally the temperature in the core also grows during the collapse, fromabout T = 0 . T = 2 . (Z−1,A)051015 (Z,A) + σ τ distributionelectron (Z,A) transitionsIndividual Supernova (Z−1,A)051015
Supernova
Gamow−Tellerresonance distributionelectron (Z,A)(Z−1,A)051015 (early collapse phase) (late collapse phase)Low−lyingstrength
Laboratory
Figure 6.
Sketch of electron capture conditions at different conditions: a) in thelaboratory (left) where the electron is captured from an atomic orbital, b) in theearly collapse phase (middle) where the electron is captured from a Fermi-Dirac (FD)distribution with an electron chemical potential of order the nuclear Q value, andc) later in the collapse at higher densities where the electron is captured from a FDdistribution with a chemical potential which are noticeably larger than the nuclear Q values. It is important that, with increasing core density, the electron chemicalpotential grows faster than the average nuclear Q value. Electron captures in the star(middle and left) can also proceed from thermally excited states where the temperature,respectively average nuclear excitation energy, is increasing during the collapse. Fig. 6 depicts the consequences which the different behavior of the energy scaleshas for the electron capture process. We schematically compare the situation in thelaboratory with the one in the early stage of the collapse where µ e ≈ (cid:104) Q (cid:105) and at anadvanced stage with µ e (cid:29) (cid:104) Q (cid:105) . In the laboratory the daughter nucleus must be morebound than the decaying nucleus ( Q < Q value are similar (for example the Q value of the abundant Fe is 4.20 MeV) whichmakes the calculation of the rate quite sensitive to the reproduction of the low-lyingGT strength distribution. An additional complication arises from the fact that thestellar environment has a finite temperature. Hence the capture can also occur fromthermally excited nuclear states which can have different GT strength distributions thanthe ground state. The nuclear composition at this stage of the collapse is dominated by lectron capture in stars pf shell nuclei are now feasible and have been proven to reproduce GTstrength distributions and energy levels quite well. Thus, diagonalization shell model isthe method of choice to determine the capture rates for pf shell (and sd shell) nuclei.Due to continuous electron captures the nuclei abundant in the core compositionbecome more neutron rich and heavier. The right panel of Fig. 5 shows the NSEdistribution for the conditions reached around the onset of neutrino trapping. Themost abundant nuclei correspond to nuclei with valence protons in the pf shell, whilethe valence neutrons occupy orbitals in the sdg shell. Hence the description of cross-shell correlations is the challenge to determine capture rates for these nuclei. We alsonote that at the higher densities more nuclei contribute to the NSE abundances. Thisis an effect of the slight increase of core entropy as neutrino-trapping sets in and of thedecrease of the relative differences of nuclear binding energies as the composition movesto heavier neutron-rich nuclei. The right part of Fig. 6 describes the energy situationencountered at higher densities in the collapse (a few 10 g cm − and above). At first,the electron chemical potential is now significantly larger than the average nuclear Q -value. For example, the neutron-rich nuclei Fe (with N = 40) and Ge ( N = 50) haveQ-values of 13.8 MeV and 13.0 MeV, respectively. Furthermore the temperature hasgrown to about T = 1 MeV. At such temperatures the average nuclear excitation energy,estimated in the Fermi gas model as E ∗ = A T / Fe and 10.2 MeV for Ge and the capture, on average, occurs from excited states, making it even easier forelectron capture to overcome the Q value. Under these conditions calculations of stellarcapture rates for the abundant nuclei on the basis of the diagonalization shell model arenot appropriate nor possible. At first, diagonalization shell model studies of nuclear GTstrength distributions for the relevant cross-shell nuclei is not feasible due to model spacerestrictions yet. Moreover, there are simply too many thermally excited nuclear states inthe mother nucleus which can contribute to the capture process. However, the detailedreproduction of the GT strength distribution — as required at lower densities where pf shell nuclei dominate — is not needed at the advanced conditions of the collapse. At first,the fact that the electron Fermi energy and the average nuclear excitation energy aretogether noticeably larger than the average nuclear Q value makes the capture rate lesssensitive to the detailed reproduction of the GT strength distribution. Thus it sufficesif the total GT strength and its centroid are well described. This is possible within theRandom Phase approximation (RPA). Second, due to the exponential increase of thelevel density with excitation energy, there will be many states which contribute to thecapture so that some averaging is expected over the GT strength functions. However,there are two further demands which have to be considered. The Pauli unblocking of theGT strength requires the consideration of multi-particle-multi-hole correlations. Thesecorrelations are not expected to be the same at the higher excitation energies than for theground state. A many-body method which accounts for both of these effects is the ShellModel Monte Carlo approach which allows the calculation of average nuclear propertiesat finite temperature considering all many-body correlations in unprecedentedly large lectron capture in stars A < E i to final states in the daughternucleus at E f . This formalism explicitly considers that the stellar interior has finitetemperature T Thus beta decays and electron captures can occur from excited nuclearlevels, where the thermal nuclear ensemble is described by a Boltzmann distribution.Beta-decay λ β and electron capture rates λ ec can be derived in perturbation theoryand the respective formulas and derivations are presented in [14, 34]. Analyticalapproximations are provided in [51]. In the derivation of the weak-interaction ratesonly Gamow-Teller transitions are included (with an important exception for Ne, asdiscussed below). pf shell nuclei The first derivation of stellar weak interaction rates for the pf -shell nuclei relevant for core-collapse supernovae has been presented in Ref. [12]. Thecalculations are based on diagonalization shell model calculations considering either allcorrelations in the complete pf shell or at a truncation level which basically guaranteedconvergence of the low-energy spectra and the GT strength distributions which arethe essential quantities to calculate electron capture and beta decay rates. The GTstrength functions were determined using the Lanczos method. Hence it represents thestrength for individual states at low energies, while at moderate excitation energiesthe GT strength is not completely converged and gives the average value for a rathersmall energy interval. We note that the shell model gives in general a good account ofnuclear properties in the pf shell if appropriate residual interactions including monopolecorrections are used (see Ref. [30] and references therein). Ref. [32] presented detailedstudies of the GT strength distributions and validated the method by comparison tothe charge-exchange data available at that time. In fact, good agreement with data wasfound, if the shell model GT distributions were reduced by a constant factor (0 . ([127, 128]). The origin of this renormalization (often called quenching of GT strength) lectron capture in stars −2 0 2 4 6 8 10 12 E (MeV) G T S t r eng t h E (MeV) Fe Fe Ni V Mn CoFFNExpSM
Figure 7.
Comparison of experimental and shell model GT strength distributionsfor several pf shell nuclei. The data are derived from ( n, p ) charge-exchangeexperiments [91, 92, 89]. The shell model results are given as histograms and foldedwith the experimental energy resolution. The energies at which the FFN evaluationplaced the GT strengths are shown as arrows. Fig. 7 compares the shell model GT + strength distributions with the experimentaldata derived from ( n, p ) charge-exchange reactions and the energy position at which theFFN rates assumed the total GT + strength to reside. The fragmentation of the GTstrength is quite obvious. It is even more visible in high-resolution data determinedby the ( d, He) and ( t, He) techniques, e.g. see the data for V( d, He) in panel b offig. 3. The data for nickel isotopes showed that the KB3 residual interaction, used inRefs. [32, 12], had some shortcomings in describing low-energy details of the GT strength lectron capture in stars Sc Ti Ti V V Fe Ni Ni Ni Ni Zn l E C / l E C r Y e ~10 g/cm T=10x10 K shell-model GXPF1aKB3GQRPA { (d, He)(t, He)(p,n) x x x x xx x x x x xxxKVINSCLIUCF t heo r y e x pe r i m en t Figure 8.
Comparison of electron capture rates for pf shell nuclei calculated fromGT + distributions derived experimentally and from shell model calculations with twodifferent residual interactions and within the QRPA approach. The astrophysicalconditions represent a situation at which pf shell nuclei dominate the core composition.The rates presented are originally from [118] and used existing data from ( d, He),( t, He), and ( p, n ) experiments, as discussed in Section 2. For the purpose of thisreview, they are supplemented with later results from the ( t, He) reactions on Sc[111], Ti [80], and Fe [112].
Fig. 8 compares electron capture rates calculated for all pf shell nuclei, for whichexperimental GT + distributions have been measured, with the predictions from the shellmodel on the basis of two residual interactions (KB3G [134] and GXFP1a [135]). Thechosen astrophysical conditions correspond to the presupernova stage of the collapseat which the pf shell nuclei dominate the abundance distribution. The GXPF1a ratesgiving a nearly perfect reproduction, except for Sc. The KB3G rates are slightlyworse than those based on the GXPF1a interaction, but still very good, except for Scand Zn. On the other hand, the rates based on the QRPA calculations, with theirrestricted account of correlations, can deviate from the data and shell model rates byup to a factor of 10 for light pf nuclei, although for the heavier pf shell nuclei the ratesbased on the QRPA calculations do well at this stellar density.The rates presented in Fig. 8 have been calculated solely from the ground stateGT distribution. This assumes that the GT distributions of excited mother states isthe same as for the ground state, shifted only by the respective excitation energy. Thisassumption often is called Brink-Axel hypothesis [136, 137] It cannot be strictly validas it does not allow for deexcitations. As we will see below it is also not appropriate lectron capture in stars pf shell nuclei in the mass range A = 45–64. These calculations approximated the state-by-state formalism discussed above by considering the low-energy states and their GTdistributions explicitly. These contributions were supplemented by the considerationsof ‘back-resonances’. These are GT transitions calculated for the inverse reaction andthen inverted by detailed balance [3, 15]. The calculated energies and GT transitionstrengths had been replaced by experimental data whenever available. A detailed tableof the weak interaction rates for the individual nuclei and for a fine grid of astrophysicalconditions at which pf shell nuclei are relevant have been published in [34]. The ratetable is publicly available and is incorporated in several leading supernova codes. Aprocedure how to interpolate between the grid points in temperature, density and Y e value is discussed in [34], based on the work of [16].Fig. 9 compares the shell model and FFN electron capture rates for several nuclei.The chosen nuclei represent the most abundant even-even, odd- A and odd-odd nucleifor electron captures as been identified by simulations on the basis of the FFN rates atthe respective astrophysical conditions during early collapse (presupernova phase). Theshell model rates are systematically smaller than the FFN rates with quite significantconsequences for the presupernova evolution, as discussed below. The reasons for thesedifferences is mainly due to the treatment of nuclear pairing which had been empiricallyconsidered in the FFN calculations. This leads in particular to the drastic changesobserved for odd-odd nuclei. The shell model rates also considered experimental datawhich were not available at the time when the FFN rates were derived. The differencesbetween the FFN and shell model beta decay rates are smaller than for electron captureand do not show a systematic trend [142, 12]. sd shell nuclei Beta decays and electron capture on sd shell nuclei (massnumbers A = 17–39) can occur during silicon burning in massive stars [29]. Theprocesses are, however, of essential importance for the fate of the O-Ne-Mg core whichdevelops at the end of hydrostatic burning in intermediate mass stars. This was themotivation for Oda et al. [33] to derive at stellar beta decay and electron capture rates for sd shell nuclei covering the relevant astrophysical conditions (temperatures 10 –10 Kand densities 10 –10 g cm − ). The rate evaluations used the state-by-state formalismdefined above. The spectra of the nuclei and the respective Gamow-Teller strengthdistributions for ground states and excited states were determined by diagonalization inthe sd shell using the Brown-Wildenthal USD residual interaction which had been provenbefore to give a quite reliable account of nuclear properties for sd shell nuclei. Like forthe nuclei in the pf shell the Gamow-Teller strength distributions were renormalizedby a constant factor. The rates have been made available in table form for a grid of lectron capture in stars T −4 −3 −2 −1 −4 −3 −2 −1 λ e c ( s − ) −16 −13 −10 −7 −4 −1 LMPFFNAufderheide T −6 −4 −2 −6 −5 −4 −3 −2 −1 −3 −2 −1 ρ =10.7 Fe Ni ρ =4.32 ρ =4.32 Co Co ρ =33 Mn ρ =10.7 ρ =33 Co Figure 9.
Comparison of the FFN and shell model rates for selected nuclei as functionof temperature (in 10 K) and at densities (in 10 g cm − ) at which the nuclei arerelevant to the capture process in simulations which used the FFN rates. The trianglesrefer to shell model estimates derived on the basis of rather strong truncations [141]. temperature-density- Y e points.More recently, an updated rate table has been published by Suzuki which is based onthe USDB residual interaction (a modified version of the USD interaction) and additionalexperimental information [50]. These modern shell model rates differ not too much fromthose of Ref. [33]. However, they are given on a finer mesh of temperature and density.This finer grid is particularly required for the study of the core evolution of stars in themass regime 8–10 M (cid:12) . Of particular importance are the URCA pairs ( Ne- Na, Na- Mg and potentially Mg- Al) which have Q values against electron captures whichare reached during core contraction at densities around 10 g cm − . As the environmentalso has a finite temperature of order 10 –10 K, which smears the electron chemical lectron capture in stars Figure 10. (left) Cumulative GT strength for Mg calculated in sd shell modelstudies with two different interactions. The ground state strength is knownexperimentally. (right) Beta decay and electron capture rates for the URCA pair Na and Mg as function of density and a specific temperature. The curve labelled’product’ is given by the sum of the two rates and identifies the density at which theURCA pair operates most efficiently. The rates are given with and without screeningcorrections. (from [50]) potential and implies the presence of thermally excited states, it is possible that bothelectron captures and beta decays occur between the pairs of nuclei. The neutrinosproduced in both processes carry energy away making the URCA pairs an efficientcooling mechanism. The operation of URCA pairs is restricted to a relatively narrowdensity range requiring the knowledge of weak interaction rates on a rather fine density-temperature grids. Such rates have been provided in [143, 50]. Fig. 10 compares theGT + strength for the Mg ground state as calculated with the USD [33] and USDB [50]interactions. The transition to the Na ground state is known experimentally. We notethe rather close agreement between the two calculations. 10 shows the beta decay andelectron capture rates calculated on the basis of the sd shell model. With increasingdensity, the electron chemical potential grows which reduces the beta decay rates due toPauli final state blocking and increases the electron capture rates. At the URCA densitylog( ρY e ) = 8.81 both rates match. The product of beta decay and electron capture ratesindicates the density range at which the URCA pair operates. Screening effects inducedby the astrophysical environment shift the URCA density to slightly larger values (seebelow).While the URCA pairs cool the core, electron capture on the two abundant nuclei Mg ( Q = 6 .
03 MeV) and Ne ( Q = 7 .
54 MeV) heat it. (The third abundant nucleus O has such a high Q -value that electron capture does not occur at the densitiesachieved during the evolution of the ONeMg core). Electron captures on these nuclei setin once the core density is large enough for the electron chemical potential to overcomethe respective Q value. (Due to its lower Q value this occurs first on Mg.) At these lectron capture in stars Na and F, respectively, occursthen instantaneously with noticeably larger capture rates, as the odd-odd daughternuclei have significantly smaller Q -values against electron capture due to pairing effects.As µ e > Q , the capture often leads to excited states in the final nuclei Ne and Owhich de-excite via gamma emission heating the environment. log [ ρ Y e (g cm −3 )] −35−30−25−20−15−10−50 l og [ λ ec ( s − )] Takahara et al. + → + + → + + → + + → + Total
Figure 11.
Electron capture rate for Ne as function of density and for a specifictemperature. The rate labelled ’Takahara et al’ was evaluated from GT distributionscalculated within the shell model [47]. The rate is broken down into the individualstate-by-state contributions where the energies and transition strengths are all takenfrom experiment. The label ‘0 + → + ’ identifies the contribution from the secondforbidden ground-state-to-ground-state transition whose strength has been measuredby Kirsebom et al. . [52]. This transition dominates the capture rate at the densitiesmost relevant for the core evolution of intermediate-mass stars. (from [52]). The electron capture rates for Ne and Mg and their daughters have beendetermined on the basis of shell model calculations by Takahara et al. [47] and Odaet al. [33]. These rates have been the default values until recently in studies of thecore evolution of intermediate mass stars. The Mg capture rate has been updated inRef. [51] using experimental data which became available in the meantime leading torather small modifications. This is different for the electron capture rate on Ne whichcan be considered a milestone and an exception. At first, Ref. [51] showed that allrelevant Gamow-Teller contributions to the rate could be derived from experiment usingdata from ( p, n ) charge-exchange measurements [144] (applying isospin symmetry) andfrom beta decays of F (see Fig. 11). Furthermore, the authors noticed that, due to therelatively low temperatures of a few 10 K, the, at the time unknown, Ne- F ground-state-to-ground-state might contribute to the capture rate just at the relevant densities,despite that it is highly suppressed due to angular momentum mismatch. The strength ofthis second forbidden transition has recently been measured in a dedicated experiment atthe IGISOL facility in Jyv¨askyl¨a [52, 145] and it was found large enough to increase the Ne capture rate by several orders of magnitude as is shown in Fig. 11. We emphasize lectron capture in stars Ne in the temperature-density range important forintermediate mass stars is now completely determined by experiment. This is quitean achievements and shows the great opportunities offered by modern Radioactive IonBeam (RIB) facilities. That a second forbidden transition essentially contributes to anastrophysical electron capture rate is exceptional and due to the low temperature ofthe environment and the peculiar structure of Ne. In core-collapse supernovae thetemperatures are an order of magnitude higher at the same densities making allowedGamow-Teller transitions the dominating contributor to electron capture rates.In the astrophysical environment the weak interaction processes are modified dueto screening effects. The screening corrections for electron capture have been developedin [13], the extension to beta decays is given in [51]. There are two important effectsinduced by the astrophysical environment. At first, screening enlarges (reduces) theenergy threshold for electron captures (beta decays). Second, it reduces the electronchemical potential. Both effects together reduce the electron capture rates, while theyenhance beta decay rates. Rate modifications due to screening are are relatively mildof order a factor of 2. The effects for the URCA pair Na- Mg are shown in Fig. 10.Modifications of the electron capture rates during the collapse of a massive star arediscussed in [13] and exemplified in their Fig. 10. Many tabulations of electron capturerates (e.g. Refs. [14, 3, 15, 16, 34, 33, 146, 147] do not include screening corrections. Ref.[13] presents a formalism how these rates can be approximately corrected for screeningeffects.Weak-interaction rates based on diagonalization shell model exist for nuclei in themass range A = 17–65, with the exception of A = 39–44. Studies of these nucleirequire the inclusion of correlations across the Z, N = 20 shell closures and hence largemodel spaces enabling allowed Gamow-Teller and also forbidden transitions. Steps inperforming such demanding calculations have been taken so that a shell model evaluationalso for this mass range appears to be in reach. Weak-interaction rates for A = 39–44 were provided by Fuller et al in their seminal work based on the IPM, but also byNabi and Klapdor-Kleingrothaus within the framework of the QRPA [146]. The latterreference gives electron capture rates for a wider range of nuclei. A > A ≥
65 require an accurate descriptionof cross-shell correlations. The associated model spaces make diagonalization shellmodel calculations in general unfeasible. It is fortunate that by the time nuclei with A ≥
65 dominate the core composition the density, and accordingly the electron chemicalpotential, has grown sufficiently that the capture rates are mainly sensitive to the totalGT strength and its centroid. For these nuclei a hybrid model [36, 39] has been proposedto evaluate the stellar capture rates. In this model the rates are calculated within anRPA approach in appropriately large model spaces using partial occupation numbers.These occupation numbers are calculated within the Shell Model Monte Carlo (SMMC) lectron capture in stars pf - sdg for nuclei with neutron numbers N ≤
61 and pf / - sdg - h / for even heavier nuclei) using adjusted pairing+quadrupole interaction to avoidthe infamous sign problem [41]. The hybrid model has been validated in [13] and appliedto about 250 nuclei in the mass range A = 66–120 [39, 13]. B ( G T + ) Excitation energy (MeV)
T=2 T=4 T=6 T=8 Full Exp.(d, He)
Figure 12.
Comparison of experimental GT strength distribution for Se (shownas running sum) with results obtained by shell model diagonalization using the RGresidual interaction and different levels of truncations (from [148]).
In this context, a special nucleus is Se with Z = 34 and N = 42. Thus,its GT + strength vanishes in the Independent Particle Model (and in the Bruennparametrization used in supernova simulations prior to 2003 [18]). The GT strengthhas been experimentally determined using the ( d, He) charge-exchange technique atGroningen [38] proving that cross-shell correlations indeed unblock the GT strength (seeFig. 12). Diagonalization shell model calculations, performed in different model spacesand with different residual interactions, are able to describe the low-energy spectra of Ge and Se and also the GT strength (Fig. 12). These shell model calculationsshowed that cross-shell correlations are a relatively slowly converging process requiringthe inclusion of multi-particle-multi-hole configurations. For example, the consideration lectron capture in stars e c ( s - ) Temperature (GK)
Exp.(d, He) SM-NS SM-RG SM-JUN45 SMMC+RPA = 10 g cm -3 Ye = 0.45 e c ( s - ) Temperature (GK)
Exp.(d, He) SM-NS SM-RG SM-JUN45 SMMC+RPA = 10 g cm -3 Ye = 0.45
Figure 13.
Electron capture rates on Se at ρ = 10 g cm − (left) and ρ = 10 g cm − (right) as function of temperatures. The rates have been calculatedfrom the experimental ground state data [38] and within diagonalization shell modelapproaches using different residual interactions. The results labelled ‘SMMC+RPA’have been obtained within the hybrid model. (from [148]). We note that Se, being an odd-odd nucleus, is never very abundant duringcore collapse. Nevertheless, Fig. 13 compares the electron capture es calculated fromthe experimental and diagonalization shell model (for different interactions and modelspaces) GT distributions with those obtained in the hybrid model for two differentcore densities and for various temperatures [148]. The lower density corresponds topresupernova conditions, where electron capture is dominated by pf shell nuclei. Therates calculated from the data and the shell model GT strength distributions agree quitewell. The hybrid model rates agree with the other rates within a factor of 3 for the rangeof temperatures given, but they show a distinct different T -dependence. This is relatedto the fact that the hybrid model does not resolve the fragmentation of the GT strength,which is particularly important at low temperatures and densities. In fact, at the higherdensity, the agreement between all rates is quite satisfactory. Under these conditionsthe electron chemical potential is noticeable larger than the capture Q -value, makingthe rate less sensitive to details of the GT distribution. The hybrid model calculationconsiders also forbidden multipoles whose contributions increase with temperature, butare relatively small [148]. We note that the shell model and experimental rates are solelydetermined from the ground state GT distribution, while the hybrid model considersfinite temperature effects in the calculation of the occupation numbers. These turn outto be not so relevant as the N = 40 gap is already strongly overcome by correlations inthe ground state. This will be different for the N = 50 gap, discussed below.The Thermal Quasiparticle RPA (TQRPA) model [149] is an alternative approachproposed to calculate electron capture (and neutrino-nucleus reaction) rates at finitetemperatures [44, 150]. Like the SMMC, also the TQRPA is based on an equilibrium lectron capture in stars TQRPA HM Ge Y e = 5 10 Y e = 10 Y e = 10 Y e = 5 10 l og ( ec ) TQRPA HM Ge TQRPA HM Ge T (MeV) 1.0 1.2 1.4 1.61.52.02.53.0
T (MeV) 1.0 1.2 1.4 1.62.02.53.03.5
T (MeV) 1.0 1.2 1.4 1.64.04.55.05.5
T (MeV)
Figure 14.
Comparison of electron capture rates for , , Ge for different densities ρY e and as function of temperature, calculated within the TQRPA and hybrid model(HM). (from [44]). The TQRPA approach has been used to calculate electron capture at finitetemperatures for selected Fe and Ge isotopes [44] and for nuclei at the N = 50 shellclosure [150, 45]. The differences between the two models become illustrative in Fig. 14which compares the electron capture rates calculated in both approaches for various lectron capture in stars Q value increases,but also the occupation of the g / neutron orbital grows decreasing the unblocking of pf shell neutron orbitals. Neutron-rich Ge isotopes appear in the core composition attemperatures ≥ ρY e ≥ g cm − and both models predict quitesizable capture rates for these conditions. There are, however, differences between thetwo models. In general, the hybrid model capture rates are larger than those obtained inthe TQRPA, most evidently at lower densities. Furthermore, the TQRPA model showsa steeper rise of the capture rates with temperature than the hybrid model. These factsare foremost related to the increased unblocking probabilities in the hybrid model due tomany-body correlations which result in larger GT strength at lower excitation energies.The differences in the rates become smaller with increasing density and temperature.This is mainly due to the growing electron chemical potential which makes the rate lesssensitive to the details of the GT strength distribution. Secondly, forbidden transitionscontribute increasingly with growing density and temperature. These contributions arenot subject to blocking effects.We have seen that many-body correlations overcome the N = 40 shell closurealready in the ground state and unblock the GT contribution to the capture rate.But what happens at the magic number N = 50? In fact, measurements of the GT + distribution for Kr ( Z = 36 and N = 50) [109] and for Sr ( Z = 38 , N = 50) showsonly very little strength, mainly located at excitation energies between 8–10 MeV [110].This points to a rather strong blocking of GT transitions at N = 50. Electron capturerates, calculated from the experimental ground state data, are indeed significantly lowerthan expected from systematics [110]. The results for Kr and Sr are surprising,given that a significant amount of GT strength ( ∼ . Zr [153]even though, based on transfer reaction experiments [154], the proton 0 g / occupationnumber for Sr and Zr are comparable: 0.7 and 1.0, respectively. A high-resolutionexperiment for Zr will be necessary to better understand these results.In the collapsing core, N = 50 nuclei (e.g. Ge and Ni with Y e values of 0.39 and0.34, respectively) are very abundant at densities in excess of about 10 g cm − [4, 155]and at temperatures T > (cid:104) E (cid:105) = 10 MeV, which is larger than the Z = 28 proton gapand the N = 50 neutron gap. This implies that the capture at the stellar temperaturesoccurs on average on states with important many-body correlations across the two gaps,in this way unblocking the GT contribution to the capture rate. This is indeed bornout in TQRPA calculations performed for N = 50 nuclei between Ni and Sr. Theobtained capture rates are shown in Fig. 15. Satisfyingly the TQRPA calculations findsno GT strength in the Kr and Sr ground states at low energies, in agreement withobservation. In fact the TQRPA capture rates, calculated solely from the T = 0 GTdistributions, agree with those obtained from the experimental GT distributions for lectron capture in stars -4 -2 Total1 + GT + (g.s.)SkM* Ni E C ( s - ) Kr e (MeV) Approximation9 10 11 1210 -4 -2 SkO’ Total1 + GT + (g.s.) log ( Y e , g/cm ) Ge Sr Figure 15.
TQRPA electron capture rates for selected N = 50 nuclei calculated at T = 0 and at T = 1 MeV and as function of density. The upper axis shows thecorresponding electron chemical potential. The calculations have been performed fortwo Skyrme interactions: SkM ∗ (blue lines) and SkO’ (red lines). The calculatedtotal capture rates include also contributions from forbidden transitions; the GTcontribution is presented individually. The shaded area is the rate obtained from theexperimental ground state GT distribution (taken from [110]. The thick line labelled‘Approximation’ represents the ’single state approximation’ adopted from [39]. (from[45]). both nuclei (see Fig. 15). The TQRPA calculation shows, however, a strong thermalunblocking of the GT strength as protons are moved into the g / orbital and neutronsout of the pf shell. This leads to a strong increase in the capture rate for all nuclei(see Fig. 15). Thermal unblocking of the GT strength has the largest effect at smallelectron chemical potentials µ e (low densities), while its relative importance decreaseswith growing µ e . With increasing density contributions from forbidden transitionsbecome more important and dominate the rate for densities of order ρY e > g cm − ,hence at the conditions where N = 50 nuclei are abundant in the collapse. The capture lectron capture in stars Ni to Sr. This hastwo reasons: the growing Q value with neutron excess and the increased promotion ofprotons into the g / orbital.In summary, GT measurements for nuclei which become relevant in the highdensity/temperature environment during supernova collapse are indispensable toconstrain nuclear models and to create trust in them. However, they cannot directlybeen used to determine the stellar capture rate as thermal unblocking effects modifythe rates under such conditions noticeably. This is in particular true at shell closures,i.e. for N = 50 nuclei. For these nuclei forbidden transitions might be as relevant asGT transitions and should be experimentally constrained as well.Fig. 15 also shows the rate estimated by a parametrization put forward in Ref. [39].This simple parametrization assumes that the capture proceeds through a singletransition from an excited state in the parent nucleus at E i to a state in the daughternucleus at E f with ∆ E = E f − E i (single-state approximation). Then the capture ratecan be written as [16] λ = ln(2) BK (cid:18) Tm e c (cid:19) (cid:2) F ( η ) + 2 χF ( η ) + χ F ( η ) (cid:3) (4)where χ = ( Q + ∆ E ) /T , η = ( µ e − Q − ∆ E ) /T , K = 6146 s and B represents a typical(Gamow-Teller plus forbidden) matrix element. The quantities F k are the relativisticFermi integrals of order k . Q is the ground state ground state Q -value that is positive forcapture in protons and neutron-rich nuclei. This approximation was used in Refs. [39, 42]to estimate the rates of the many heavy nuclei which are abundant at larger densitiesand for which no rates existed at that time. The two parameters (energy position andGT strength) were fitted to the rates of about 200 nuclei for which individual pf shellmodel and hybrid model rates were available. Fig. 16 compares the shell model rateswith the single-state approximation (4) using B = 4 . E = 2 . g cm − . In this density regime the nuclear composition is largely dominated bynuclei for which shell model rates exist. The general trend seen in Fig. 16 is also borneout in Fig. 15 where the approximation badly fails at low densities, but gives reasonableagreement at ρY e > g cm − . Ref. [155] compares the shell model and single-staterates at slightly different astrophysical conditions.The single-state parametrization has been adopted for heavy nuclei in supernovasimulations which systematically studied the influence of nuclear ingredients (electroncapture rates, Equation of State, mass models) on the collapse dynamics [155, 156] (see lectron capture in stars
15 10 5 0
Q (MeV) −2 −1 λ ec ( s − ) ρ Y e =0.07, µ e =9.62, T=0.93 ρ Y e =0.62, µ e =20.2, T=1.32 ρ Y e =4.05, µ e =37.8, T=2.08 Figure 16.
Electron capture rates on nuclei, for which individual shell model ratesexist, as function of Q value for 3 different stellar conditions. Temperatures aremeasured in MeV, density in 10 g cm − . The solid lines represent the rates obtainedfrom the single-state approximation 4. (from [39]). below), where Ref. [156] used the improved single-state parametrization of [157] (seebelow). Most supernova codes now use the rate table as provided by Juodagalvis et al. [13].This table defines electron capture rates on a grid of the three important parameterscharacterizing the astrophysical conditions during collapse: temperature, density, Y e value. The rate evaluation assumes the core composition to be given by nuclearstatistical equilibrium, hence it does not provide rates for individual nuclei.The rate table is based on the hierarchical strategy defined above. For the nucleiwith A <
65 the shell model rates of Oda et al. [33] ( sd shell) and of Langanke andMartinez-Pinedo [34] ( pf shell) have been adopted. This guarantees a reliable anddetailed reproduction of the GT strengths for the important nuclei at collapse conditionswhere µ e ∼ Q . The rates for nuclei in the range A = 39 −
44 have been taken fromFuller, Fowler and Newman [3]. For the heavier nuclei the table adopts the rates fromhybrid model calculations. For about 200 nuclei in the mass range A = 65 −
110 thesewere calculated by using SMMC partial occupation numbers in RPA calculations. Fora few nuclei in this mass regime and for even heavier nuclei, in total about 2700 nuclei,the rates were evaluated on the basis of a parametrization of the occupation numbers,derived in accordance with the SMMC studies, and RPA response calculations. Inthis way the most relevant nuclear structure input, like shell gaps, are accounted for.Screening corrections due to the astrophysical environment have been incorporated intothe rates. lectron capture in stars sd shell nuclei are important for the core evolutionof intermediate mass stars. Rates for individual nuclei for the relevant density andtemperature regime are given in [33] and updated in [50].Nuclei in the mass range A = 45–65 are essential for the early phase of core collapsesupernovae and for the nucleosynthesis in thermonuclear (Type Ia) supernovae. Theweak-interaction rates for these pf shell nuclei are individually given in [34]. The ratesare not corrected for screening, which, however can be accounted for using the formalismdeveloped in [13].We note that at specific astrophysical conditions (e.g. during silicon burning), atwhich the sd and pf shell nuclei are relevant, the temperature is in general not highenough to establish an NSE composition. Hence the knowledge of individual rates isessential.To make it easier to incorporate complete sets of electron-capture rates inastrophysical simulations, a library of rates was created [155, 158, 159] based on the ratetables for specific mass regions described above and on the single-state approximationfor nuclei where rates based on microscopic calculations are not available. This libraryis incorporated in the weak-rate library NuLib [160].
4. Electron captures in astrophysical applications
Simulations of the evolution of massive stars distinguish two distinct phases motivatedby their specific needs and requirements. 1) During hydrostatic burning energy releasedby nuclear reactions in the star’s interior are essential to balance gravity. The densitiesare low enough that neutrinos, produced in weak interactions, can leave the starunhindered transporting energy away. This loss has to be considered in the energybalance, but a detailed treatment of neutrino transport is not required. However, thesimulations have to incorporate a detailed network of nuclear reactions to follow thenuclear energy production and the change in composition. This stellar evolution periodlasts to the so-called presupernova phase when the core density has reached values ofabout 10 g cm − and the inner part of the iron core collapses with velocities in excessof 1000 km s − [161, 29].The final models obtained by the stellar evolution codes become the input forthe supernova codes in which the gravitational collapse of the iron core and theexplosion are simulated. The astrophysical conditions relevant during these simulationlead to two important changes compared to stellar evolution. The temperaturesare sufficiently high ( T > a few GK) so that the nuclear composition can be wellapproximated by an NSE distribution, without the need to follow a complicated networkof nuclear reactions. On the other hand, the involved densities require a detailedbookkeeping of neutrinos. This is achieved by Boltzmann transport. An additionalcomplication arises from the fact that the assumption of spherical symmetry, which lectron capture in stars
10 15 20 25 30 35 40
Star Mass (M ⊙ ) ∆ Y e Y e WWLMP
10 15 20 25 30 35 40
Star Mass (M ⊙ ) −0.2−0.10.00.1 ∆ M F e ( M ⊙ ) M F e ( M ⊙ ) WWLMP
10 15 20 25 30 35 40
Star Mass (M ⊙ ) −0.2−0.10.00.1 ∆ S ( k B ) ce n t r a l e n t r opy / b a r yon ( k B ) WWLMP
Figure 17.
Comparison of the center values of Y e (left), the iron core sizes (middle)and the central entropy (right) for 11 − M (cid:12) stars between the models using the FFNrates (WW models [161]) and models which used the shell model weak interaction rates(LMP [35]). The lower panels show the changes in the 3 quantities between the WWand LMP models. Heger et al. have investigated which effect the diagonalization shell model rateshave on the presupernova evolution for stars in the mass range M = 13–40 M (cid:12) [29, 35].To this end they repeated calculations of Weaver and Woosley [161], keeping the stellarphysics as much as possible, but replacing the weak interaction rates for pf shell nucleiby those of Ref. [34] (LMP rates). Fig. 17 summarizes which consequences the shellmodel rates have on three quantities which are relevant for the following collapse. Thecentral Y e value is larger by ∆ Y e = 0 . β decays can compete withelectron captures. Although this does not occur by specific URCA pairs, but rather byan ensemble of nuclei, the effect is the same: the star is additional cooled, while the Y e is kept constant. The study confirmed that β decays become increasingly Pauli blockedwith growing density and can be safely neglected during collapse. Fig. 17 also indicatesthat the iron core masses are generally smaller with the LMP rates. However, this isnot a continuous effect and shows variations among the models with different stellarmasses. Finally, the LMP rates lead to presupernova models with lower core entropy forstars with M <
20 M (cid:12) . For the more massive stars, the effect is not unique; stars with M = 30–40 M (cid:12) show an increased core entropy. We mention that lower (larger) core lectron capture in stars g cm − the composition is dominated by nuclei with Z <
40 and
N >
40 for which, for a longtime, it was assumed that electron captures vanish (e.g. [18]) due to Pauli blocking ofthe GT strength. As a consequence the capture process at the later of the collapsecontinued solely on free protons, which are, however, less abundant than heavy nucleiby orders of magnitude. As we have discussed above, the GT strength at the N = 40shell gap is unblocked by multi-nucleon correlations. Furthermore, the blocking at the N = 50 shell closure, which results in a strong reduction in the experimental groundstate GT strength, is overcome at the finite-temperature core conditions by thermalexcitations.Arguable the most important result reported in Refs. [39, 42] is the fact thatelectron capture proceeds on nuclei rather than on free protons during the entire collapse,in contrast to previous belief (e.g. [1]. These findings are based on supernova simulationsperformed independently by the Garching and Oak Ridge groups which both adoptedthe hybrid model capture rates for more than 100 nuclei in the mass range A = 65–110,supplemented by the shell model rates for pf shell nuclei. For the heavy nuclei, thecapture rates were estimated by the single-state approximation. The capture rate onfree protons was taken from [18]. ρ c (g cm −3 ) Y e , c , Y l e p , c BruennLMSH Y lep Y e ρ c (g cm −3 ) s c ( k B ) , T c ( M e V ) BruennLMSH
T s ρ c (g cm −3 ) −1 d E d t ( M e V s − nu c l e on − ) BruennLMSH ρ c (g cm −3 ) 〈 E ν 〉 ( M e V ) Figure 18.
Comparison of a supernova simulation for a 15 M (cid:12) star using theshell model weak interaction rates from Ref. [28] (labelled LMSH) and the Bruennparametrization which neglects capture on nuclei for N >
40 [18] (labelled Bruenn).The figure shows the central core values for Y e and Y l ep (electrons plus neutrinos)(left), the entropy and temperature (middle) and the neutrino emission rate (right) asfunction of core density. The insert in the right figure shows the average energy of theemitted neutrinos (courtesy of Hans-Thomas Janka). lectron capture in stars Y e at neutrinotrapping at densities around 10 g cm − . At higher densities the total lepton fraction Y lep becomes constant, while the electron fraction Y e still decreases. This is relatedto neutrino tapping and the formation of the homologous core [1]. In this regime,continuous electron captures reduce the electron abundance, but the neutrinos generatedby this process interact with matter mainly by coherent scattering on nuclei with a ratelarge enough that their diffusion time scale is longer than the core collapse time scale.Neutrinos are trapped and add to the total lepton fraction in the core. But beforetrapping, the neutrinos can still leave the star and are an additional cooling mechanismleading to smaller core entropies than obtained in previous calculations. Lower entropiesreduce the abundance of free protons in the NSE composition, which increases theimportance of capture on nuclei due to their increased abundances. Neutrinos producedby capture on nuclei have smaller average energies due to the higher Q -value thanneutrinos produced by capture on free protons. Hence the luminosity of electronneutrinos is increased due to more captures, but their average energies are shifted tolower values. We stress that the rate for capture on individual nuclei is noticeablysmaller than the capture rate on free protons. The dominance of capture on nucleiresults for the overwhelmingly higher abundance of nuclei compared to free protons andare a result of the low entropy, i.e. of the capture process.The fact that electron capture on nuclei proceeds until neutrino trapping is reachedreflects itself also in the core dynamics and profiles. In the simulations with the improvedrates, as shown in Fig. 19 the shock forms with significantly less mass included (smaller‘homologous core’ size) and a smaller velocity difference across the shock. Despite thismass reduction, the radius from which the shock is launched is actually slightly pushedoutwards due to changes in the density profile. Despite these significant alterationsalso one-dimensional supernova models employing the new electron capture rates failto explode. No noticeable differences in the simulations are observed if the rate setof Juodagalvis et al. [13] is used which replaces the rates for nuclei, for which in [42]the single-state approximation was used, by rates estimated in the spirit of the hybridmodel. Multidimensional supernova simulations describe electron capture now by therates of Ref. [13]. However, no dedicated investigation of the role of electron capture(i.e. in comparison to the case where capture on heavy nuclei is neglected) has beenperformed.In a recent supernova simulation [166] electron capture on nuclei has been identifiedas the dominating weak-interaction process and the main source of electron neutrinosduring collapse. However, it was shown that pair-deexcitation of thermally excitednuclear states is an important source of the other neutrino types (electron anti-neutrinos,muon and tau neutrinos and their antiparticles).The contribution of a particular nucleus to the reduction of Y e during collapse, lectron capture in stars Y e ( E l e c t r on F r a c t i on ) E n t r op y pe r ba r y on V e l o c i t y ( k m / s ) Enclosed MassBruenn prescriptionLMP+hybrid rates
Figure 19.
Comparison of Y e (upper), entropy (middle) and velocity profiles (lowerpanel) at bounce obtained in supernova simulations with the shell model rates for nucleiin the mass range A = 45–110 ([34, 39], thick line) and the Bruenn rate parametrization([18], thin line). (from [42]). depends on the product of its abundance and of its capture rate. Both quantities aretime-dependent and have to be integrated over the duration of the collapse. This studyhas been performed by Sullivan et al. [155] using rates calculated based on microscopicnuclear models where available. For the heavy nuclei, for which such rates are notindividually available, they adopted the single-state approximation of Eq. (4).Fig. 20 shows in the upper panel which nuclear ranges contribute to the changeof Y e with time, ˙ Y e . The top axis shows the time until bounce. The correspondingdensities are 1 . × , 4 . × , 1 . × g cm − at t − t b = − , − , − Y e grows with time during collapse and reaches its maximumafter trapping has already set in. The increase reflects the fact that the electron chemicalpotential grows faster than other scales, in particular the average nuclear Q value of thecomposition,resulting in strong increases of the capture rates. The change in the capturerate is mainly driven by nuclei in the mass range A = 65-105. Rates calculated within lectron capture in stars Figure 20. upper panel (a): The contribution of nuclear electron capture to thechange of Y e as function of Y e which continuously reduces with time. As reference theupper axis vindicates the time until bounce. lower panel (b) The top 500 nuclei whichcontribute strongest to electron capture. (from [155]). lectron capture in stars pf shell nuclei, for which accurate diagonalization shellmodel rates exist, dominate in the early collapse. Here capture rates are, however,smaller due to the smaller electron chemical potentials involved. Nuclei heavier than A = 105 contribute or dominate just before and during trapping.The lower panel of Fig. 20 identifies how individual nuclei contribute to ˙ Y e ,determined by integrating the respective contributions during collapse until trappingoccurs. Due to this study, the relevant nuclei are those around the N = 50 shell closurecentred in this range from Ni to Ge.Sullivan et al. [155] also investigated which effect a systematic modification of theelectron capture rates has on the supernova dynamics. When scaling the capture ratesfor all nuclei by factors between 0.1 and 10, they observed significant modifications.A systematic reduction of the rates throttles the effects which captures on nuclei haveduring collapse, as outlined above, driving the results back towards those where captureon nuclei were neglected. A systematic rate reduction by a factor 10 indeed increases theenclosed mass at bounce by about 16%, which is a similar effect as reported in Ref. [42].Sullivan et al. [158] and Pascal et al. [156] argue that the single-state approximationmight overestimate the rates for nuclei close to the N = 50 shell gap. A similarconclusion was drawn from the measurements of Gamow-Teller distributions for theground states of the N = 50 nuclei Kr [109] and Sr [110]. As discussed above,the single-state approximation in fact does not consider nuclear structure effects whichshould be quite relevant in particular at shell closures. We note that structure effectsare considered in the shell model rates used in Ref. [13] to set up a rate table forelectron capture under collapse conditions, assuming, however, NSE for the nuclearcomposition. It has been shown that the use of alternative and improved Equations ofState has rather small effects on the supernova dynamics [155, 156]. The dependenceof the core composition on different equation of states and its indirect impact on stellarelectron capture rates has been investigated in Refs. [167, 168]. An improved versionof the single-state approximation is presented in [157]. The impact of a reduction ofthe N = 50 shell gap has been explored on ref. [169]. We also mention again that theTQRPA calculations and the hybrid model indicate that the blocking of the GT strengtharound N = 50 is largely overcome at stellar conditions due to thermal unblocking.Furthermore, both models predict sizable contributions from forbidden transitions atthe astrophysical conditions at which N = 50 nuclei are abundant during the collapse. Thermonuclear, or Type Ia, supernovae are a class of supernovae which are distinct fromthe core-collapse version by their explosion mechanism and also due to their spectralcomposition (Type Ia spectra do no exhibit hydrogen lines, in contrast to spectra of core-collapse or Type II supernovae). In the currently favored model Type Ia supernovaecorrespond to the explosion of a White Dwarf in a binary star system triggered by mass lectron capture in stars N = Z nuclei, i.e. C, O, Ne. The accretion adds to the WhiteDwarf mass bringing it towards the Chandrasekhar limit and increases the density in itsinterior to the point where carbon burning can be ignited. As the burning occurs in ahighly degenerate environment, the energy set free cannot lead to expansion, but ratherheats the surrounding. This results in a self-reinforcing acceleration of the burninguntil degeneracy can be lifted and the entire White Dwarfs is disrupted. The explosionmechanism - complete disruption of a White Dwarf in a thermonuclear runaway - leadsto similarity among Type Ia events. For example, the observed peak magnitude andwidth of the lightcurves obey a simple scaling law (Philipps relation [170]) which makesType Ia supernovae to standard candles for cosmological distances. This fact has beenexploited to deduce the current acceleration of our Universe. -3 -2 -1
10 20 30 40 50 60 70
O NeNa SiAlMg PS ClAr KCa ScTi VCr MnFe CoNi CuZn ( Y i / Y i , S un ) / ( Y F e / Y F e , S un ) Mass Number
Figure 21.
Influence of electron capture rates on type Ia nucleosynthesis. The two leftpanels show yields calculated for the WS15 progenitor model of Ref. [171] calculatedwith the FFN (left) and LMP (middle) electron capture rates (courtesy of F. Brachwitz,from [28]). The right panel shows the yields calculated for the W7 progenitor model ofRef. [171] replacing the LMP rates with improved shell model rates for selected nucleiin the Ni-Fe region (from [172]). All yields are relative to the solar abundances. Theordinate is normalized to Fe).
After the burning flame has moved through the matter, the inner material behindthe front, with a mass of about 1 M (cid:12) , has reached temperatures sufficiently high todrive the nuclear composition into nuclear statistical equilibrium. As the White Dwarfwas composed of N = Z nuclei, mainly Ni is produced. Deviations towards nucleiwith neutron excess occur due to electron captures in the hot and dense matter behindthe front. The impact of these captures depend, besides the astrophysical conditionsof density (about 10 g cm − and temperature ( T ∼ K), on the speed of the flame(i.e. the time for electron captures before the star is disrupted) and obviously on therates themselves. As discussed above, the diagonalization shell model (LMP) ratesare systematically lower than the FFN rates for pf shell nuclei, in particular for thenuclei in the Ni-Fe mass range which are of importance for captures behind the typeIa burning front. Brachwitz et al. have performed nucleosynthesis studies in a one-dimensional supernova simulation based on the well known W15 progenitor model ofRef. [171] which starts from a 1.38 M (cid:12) C-O White Dwarf [10, 173]. The faster FFN lectron capture in stars (cid:12) core mass, reachingvalues down to Y e = 0 .
44 in the center, while the slower LMP rates produce Y e = 0 . Ti or Cr, which are strongly overproduced comparedto the solar abundances (left panel of Fig. 21). This overproduction constituted aserious problem [9] as roughly half of the Fe content of the solar abundances aresynthesized in type Ia supernovae and hence all nuclides, produced in type Ia, shouldnot have overproduction factors larger than 2 as otherwise their relative abundancesare in conflict with observation. As is shown in the middle panel of Fig. 21, theoverproduction is removed when the slower LMP shell model rates are used. Suzukihas recently confirmed this finding in a study which replaced the LMP shell modelrates for selected nuclei in the Ni-Fe mass range by those obtained with the GXPF1residual interaction which gives better agreement with the measured GT strength in Niisotopes [172]. Suzuki used a different progenitor model than Ref. [173] (the W7 modelof [171]). But also his study shows that the overproduction of neutron-rich nuclei isremoved if modern diagonalization shell model capture rates are used rather than theFFN rates (right panel in Fig. 21). Satisfyingly Suzuki only observes a small differenceof 4% in the calculated abundances based on his shell model rates and on the LMPrates. Detailed studies of the sensitivity of nucleosynthesis in type Ia supernova can befound in refs [48, 49].Electron captures and beta-decays, operating via URCA pairs (see section 3.1.2),are also important during the accretion and simmering phases of the evolution of COWDs before the type Ia supernova explosion as they determine the neutron excess andthe density at which the thermal runaway occurs [174]. Particularly important duringthese phases is the N( e − , ν e ) C rate whose value is determined by beta-decay andcharge-exchange data [175].
An old isolated neutron star can be described in beta equilibrium. However, such anequilibrium is broken in the crust if the star accretes mass from the interstellar medium(ISM) or from a binary star. For an old neutron star traversing the ISM, a mass oforder 10 − M (cid:12) per year will be accreted as a layer on the neutron star surface. Thetemperature of the layer is low and is usually approximated as T = 0 [176]. In abinary system the mass flow can be higher leading to repeating burning of a surfacelayer with characteristic emission of X-rays with typical durations up to ∼
100 s (X-rayburster [177, 178]). Due to the re-occurrence of the break-outs, with typical periods oforder a year, the ashes of previous events are pushed to higher densities and temperaturesof order a few 10 K can be reached. These binary systems can also exhibit rare day-long X-ray bursts (so-called superbursts). Here carbon flashes, triggered by the fusionof two C nuclei, heat the neutron star envelope so that hydrogen and helium burningbecomes stable, suppressing the usual shorter x-ray bursts. These can only occur after lectron capture in stars Si. Particular challenging is the evaluation of the pycnonuclear triple-alpha reactionrate as neither the Be intermediate state nor the Hole state in C can be thermallyreached [182, 183, 184]. The fresh material produced by pycnonuclear reactions rests onoriginal neutron star crust material, i.e Fe or Ni.Electron captures can occur once the density reaches a value at which the electronchemical potential can overcome the nuclear Q value. For O, which is the main productproduced by pycnonuclear helium reactions, this happens at ρ ∼ × g cm − . Asbeta decay of the daughter nucleus N is prohibited due to complete filling of theelectron phase space at T = 0, the daughter nucleus immediately undergoes a secondelectron capture to C as the required density is less than for O. At the densityrequired for the double electron capture on O the underlying material of Fe and Ni has already undergone double electron captures to Cr, followed to Ti, and Fe,respectively (see below). As shown in Ref. [176] this leads to several unstable situationswhere a denser layer (i.e. C) rests on less denser layers (i.e. Ti or Fe), resulting inan overturn of the unstable interfaces. This scenario had been proposed as a possibleexplanation for gamma-ray bursts before these were identified as extra-galactical eventswith luminosities larger than observed for supernovae.Double electron captures are expected also to occur in the crust of an accretingneutron star in a binary system. When accretion pushes the original surface layer,made mainly of Fe, to higher densities, electron captures will transform Fe to Cronce ρ > . × g cm − . Haensel and Zdunik have studied the consequences forthe accreted neutron star crust, build on a single-nucleus ( Fe) approach [7] (seealso [185, 186]. Upon pushing the matter to even higher densities, further doubleelectron captures proceed ( Cr → Ti at ρ = 1 . × g cm − , Ti → Ca at ρ = 8 × g cm − , Ca → Ar at ρ = 2 . × g cm − ), before the density isreached at which neutrons are emitted from the nucleus ( ρ = 4 . × g cm − , neutrondrip). Thus, the double electron capture of Ar is accompanied by the emission of freeneutrons, Ar → S + 4 n . The successive electron captures lowers the charge of thenuclei so that pycnonuclear fusion reactions, induced by zero-point motion fluctuationsin the Coulomb lattice become possible. The double electron captures, but in particularpycnonuclear fusion reactions are considerable heat source, as is discussed in [7, 187].The crust composition, containing other nuclei than Fe, complicates the situation.This is also true for the ashes of X-ray burst events which, due to repeating outbursts,are also successively pushed to higher densities and run through a similar sequence,to the one described above, of double electron captures, neutron deliberation and lectron capture in stars
10 20 30 40 50 60 70 80 90 1000123456 [ M e V ] superburst ignition Figure 22.
Depth at which URCA pairs of mass number A operate in neutron stars.The size of the data points corresponds to the neutrino luminosity of the pair, settingits mass fraction to X = 1. (The top 5 are colored in red.) The grey band indicatesconstraints for superburst ignition assuming an ignition at a column depth between0.5–3 × g cm − . (from [188]). pycnonuclear fusion reactions [189]. As pointed out by Schatz et al. [8] the ashesof former burst events have finite temperatures (a few 10 K) which, although smallcompared to typical electron capture Q values, open up a small energy window at whichbeta decays of electron capture daughters can occur. For such an URCA process tooccur the electron capture process has to satisfy two conditions: it must be mediatedby an allowed transition to a state at excitation energies E x < T and the beta-decayingnucleus must no have a strong electron capture branch. On general grounds even-evennuclei, which are the most abundant nuclei in the crust, do not form URCA pairs butrather perform double electron captures [188]. On the other hand, nearly all odd-Anuclei can form URCA pairs. The authors of Ref. [188] have identified about 85 URCApairs. Fig. 22 shows the neutrino luminosities of these pairs (setting the mass fractionof the nucleus on which an electron is captured to X = 1) and at which depth in theneutron star they operate. As pointed out in [8] cooling by URCA pairs in the crustreduces the heat transport from the crust into the region of the x-ray burst or superburstashes which reside at less dense regions (this region is often called the ocean). Thislowers the steady-state temperature in the ocean. This puts constrains on the ignitionof the C + C fusion reaction to start the next burst cycle. This ignition has nowto occur at higher densities [188]. URCA pairs can also directly operate in the ocean.However, due to the lower densities nuclei are less neutron-rich with smaller Q valuesfor electron captures than those in the crust. As the neutrino luminosity scales with Q , This strongly reduces the effectiveness of URCA pairs in the ocean [188].Due to the simultaneous observation of the gravitational wave and theelectromagnetic signal from GW170817, the merger of two neutron stars in a binarysystem has been identified as one of the astrophysical sites, see e.g. [190], wherethe r-process [191, 192, 193, 194] operates. Particularly important to determinenucleosynthesis in mergers is the Y e value of the ejected material that is determinedby weak processes. The initial Y e profile of the neutron stars can be determined frombeta-equilibrium. However, as the neutron stars approach each other and finally mergethe temperature increases and neutrino emission becomes very important. An accurateprediction of the neutrino luminosities requires a description of high density neutrino- lectron capture in stars ν e and ¯ ν e together with electron and positron captures leads to substantial changes on the Y e ofthe ejected material particularly in the polar regions [197, 198, 199, 200]. These processesoccur when the material is hot and constitutes mainly of neutrons and protons.Another important source of material is the so-called secular ejecta originatingfrom the accretion disk that surrounds the central remnant produced by the merger.If this is a long-lived neutron star, the neutrino luminosities are large enough toaffect the neutron-to-proton ratio of the ejected material [201]. If the central objectis a black-hole, the neutron-to-proton ratio is determined by a dynamical beta-equilibrium [202] between electron, positron captures and beta-decays operating in theaccretion disk [203, 204, 205] on hot material that mainly is made of neutrons andprotons. Due to the current understanding, electron capture on nuclei does not play animportant role for r-process nucleosynthesis in neutron-star merger events. The final fate of stars depend on their masses at birth. Stars with masses less thanabout 8 M (cid:12) advance through hydrogen and helium burning. As they suffer significantmass losses by stellar winds their masses at the end of helium burning is not sufficientto ignite further burning stages. They end their lives as White Dwarfs, which arecompact objects with a mass limit of 1.44 M (cid:12) (Chandrasekhar mass), stabilized byelectron degeneracy pressure. Stars with masses in excess of about 11 M (cid:12) develop acore at the end of each burning phase which exceeds the Chandrasekhar mass. As aconsequence they can ignite the full cycle of hydrostatic burning and end their lives ascore-collapse supernovae, leaving either neutron stars or black holes as remnants. Thefate of intermediate-mass stars (8–11 M (cid:12) ) balances on a knife edge between collapsinginto a neutron star or ending in a thermonuclear runaway which disrupts most of thecore [206]. Simulations of such stars are quite sensitive to astrophysical uncertaintieslike convective mixing or mass loss rates [206]. On the other hand the major nuclearuncertainty, related to electron capture on Ne, has recently been removed as this rate,as we have described above, is now known experimentally at the relevant astrophysicalconditions. We briefly summarize the consequences which this nuclear milestone has forthe fate of intermediate mass stars.Intermediate mass stars go through hydrostatic hydrogen, helium and core carbonburning, but are not massive enough to ignite further advanced burning stages. Inthe center of the star a core develops which mainly consists of O and Ne, withsmaller amounts of Na and , Mg. Due to its position on the Hertzsprung-Russelldiagram stars with such an ONe core are referred to as Super Asymptotic Giant Branch(AGB) stars. It is important to note that cores of Super-AGB stars are more densethan their counterparts after helium burning in more massive stars. As nuclear burninghas ceased in the ONe core its gravitational collapse is counteracted by the electron lectron capture in stars ρ c ( g cm − )0.00.20.40.60.81.0 T c ( GK ) Mg Na Mg Na Na Ne ˙ M = 10 − M fl yr − With forbiddenNo forbidden
Figure 23.
Temperature-density evolution of the ONeMg core of an intermediatemass star. The labels indicate at which densities the URCA pairs and the electroncaptures on Mg and Ne operate. The red (blue) lines show the evolution with(without) inclusion of the forbidden ground-state-to-ground-state contribution to the Ne electron capture rate.The calculation assumes that the ONe core accretes a massof 10 − M (cid:12) per year from ongoing hydrostatic burning. (from [52]). degeneracy pressure. However, the densities achieved in the core result in electronchemical potentials large enough to initiate electron capture reactions, which reducethe pressure against collapse. Here two distinct processes play the essential role for thedevelopment of the core. This is shown in Fig. 23 which displays the final temperature-density evolution of the core center. First, several URCA pairs ( Mg- Na, Na- Ne, Na- Ne) operate at various phases of this final evolution. These pairs are efficientcooling mechanism. Second, electron captures also occur on the abundant α -nuclei Mg and Ne once the electron chemical potentials overcome the capture Q values(the Q value of O is too high for electron captures at these densities). But for these N = Z nuclei, the electron capture daughters ( Na and F) also capture electronwhich due to their smaller Q values than in the first captures occur significantly fasterthan competing β -decays. Furthermore the second capture proceeds often to excitedstates in their daughters which then decay by γ emission to the ground states and heatthe environment (see Fig. 23). Due to its lower initial Q -value the double electroncapture on Mg proceeds at lower densities than the one on Ne. Recognizing the lowtemperature at the onset of electron capture on Ne (about 300 keV), basically due tothe efficient URCA cooling [207], the authors of Ref. [51] pointed out that the transitionfrom the Ne 0 + ground state to the F 2 + ground state, although second forbidden,could dominate the rate at core conditions as all other transitions were exponentiallysuppressed by either the tail of the electron distribution or a Boltzmann factor due to lectron capture in stars -1 Radius (km)0.00.51.0 T i g n ( K ) ˙M = 10 −7 M ⊙ yr −1 ρ i g n ( g c m − ) Figure 24.
Temperature (blue) and density (red) profiles of an ONe core at ignition ofoxygen fusion, calculated with (solid lines) and without (dashed lines) consideration ofthe forbidden ground-state-to-ground-state transition in the Ne electron capture rate.The calculations were performed using the spherical MESA code [208] and assumingthat the ONe core accretes a mass of 10 − M (cid:12) per year from ongoing hydrostaticburning. (from [52]). The inclusion of the ‘forbidden’ ground-state-to-ground-state transition in theelectron capture rate on Ne shifts the onset of this capture to lower densities beforethe last epoch of URCA cooling by the Na- Ne pair (see Fig. 23). This shift in densityhas, however, important impact on the fate of the star ending either as gravitationalcollapse or thermonuclear explosion. This fate is determined by the competition betweenelectron capture and nuclear energy generation by oxygen fusion [206]. If the ignitionof oxygen (requiring temperatures in excess of 10 K) occurs at low enough densities,the fusion generates sufficient energy to reverse the collapse and to disrupt the star ina thermonuclear explosion. At higher densities, the deleptonization behind the burningfront is so rapid that the loss of pressure cannot be recovered by nuclear burning. In thiscase the core continues to collapse, ending as a neutron star. The increase of the Neelectron capture rate due to the contribution of the forbidden transition seems to shiftthe fate towards thermonuclear explosions [52]. This is demonstrated in Fig. 24 whichis based on a spherical simulation of the core evolution. In the calculation, performedwithout the inclusion of the forbidden contribution to the rate, oxygen is ignited in thecenter, while in the case using the experimental Ne capture rate with inclusion of theforbidden transition, the star develops an isothermal core with temperatures below theignition value. In this inner core double electron capture on Ne continues generatingheating which leads to an off-center ignition of oxygen burning (at a radius of 58 km forthe case shown in Fig. 24). We also note that, without the forbidden contribution, the lectron capture in stars (cid:12) more mass than gravitational collapse and intermediate mass stars are much moreabundant than heavier stars. A first exploration shows that the ejecta of thermonuclearexplosions are particularly rich in certain neutron-rich Ca, Ti, Cr isotopes and in trans-iron elements Zn, Se and Kr [52]. This might have interesting implications for theunderstanding of the early chemical evolution of our galaxy [209].
Subluminous B stars are core-helium burning stars with thin hydrogen envelopes andmasses of about 0.5 M (cid:12) [210]. Often these stars exist in tight binaries with WhiteDwarfs [211]. When the White Dwarf accretes matter from the unburned outer layersof its companion star, also some amount of N is present depending on the initialmetallicity of the donor star. Electron capture on N is then a decisive factor for thefate of the accreted material.Due to its low Q value of 0.667 MeV, electrons are captured on N once thedensity of the accreted matter on the WD surface exceeds a threshold value of about1 . × g cm − [212]. As the respective temperatures are rather low (T less thana few 10 K), the capture solely proceeds by the allowed transition between the Nand C ground states. The respective transition matrix element is known from the Cbeta decay. Coulomb corrections due to environment effects are relatively minor, butare considered in recent astrophysical applications [212].For the temperatures involved and for densities larger than about 10 g cm − ,the electron capture rate is larger than the competing β decay and, in the helium-richenvironment, the electron capture is followed by an α capture on C [213]. The energygeneration by this so-called NCO reaction ( N( e − , ν e ) C( α, γ ) O) [214] — despitesome uncertainties in the α -capture rate on C [215] — is larger than by the triple-alpha reaction in the relevant temperature-density range. Thus, it is the NCO reactionwhich triggers a rather steep rise of the temperature in the environment so that asa second step also the triple-alpha reaction will be ignited. This finally leads to athermonuclear instability which is observed as helium flashes.The evolution of these flashes depend crucially on the accretion mass flow [213].If the mass flow is large (10 − M (cid:12) yr − ), the energy released from the gravitationalcontraction leads to heating of the environment enabling the C nucleus to capture an α particle fast. The electron capture process controls the NCO reaction sequence andno significant amount of C is being built up. For smaller mass flows (10 − M (cid:12) yr − ) lectron capture in stars α captures on C. This occurs at conditions with higherdensities and after N has been completely converted to C. Simulations also show thatfor smaller accretion rates, the core becomes convectively unstable. The time scale onwhich the flashes develop depend also on the accretion rate and are significantly shorterfor smaller rates (a few 10 yr for 10 − M (cid:12) yr − )
5. Summary
In his authoritative review on core-collapse supernovae, Hans Bethe stated in 1990 [1]:“The theory of electron capture has gone a full circle and a half.” He was referring tothe fact that in early models, capture was assumed to occur on free protons. This wasput into question by BBAL [2] who noted that the concentration of free protons duringcollapse is very low and that the capture takes place on nuclei with mass numbers A = 60–80, changing a proton in the f / shell to a neutron in the f / orbital byallowed Gamow-Teller transition. Following Bethe, the third semi-circle is due to Fuller’sobservation that the neutron f / orbitals are occupied at neutron number N = 38 [17],blocking Gamow-Teller transitions within the pf shell. Hence at the time when Bethewrote his famous article, electron capture in supernovae was assumed to occur on freeprotons and capture on nuclei was switched off for nuclei with N >
38 [18].As we have summarized in this manuscript, the experimental and theoretical workof the last two decades implies that this picture is too simple. Experimental techniquesto measure Gamow-Teller strength distributions based on charge-exchange reactionswith progressively better energy resolutions — advancing from the pioneering ( n, p )reactions to much more refined ( d, He) and ( t, He) reactions — give clear evidencethat nuclear correlations play a decisive role in the total strength, and even more, forthe fragmentation of the GT distribution, thus invalidating the Independent ParticleModel on which the early electron capture work, which was discussed and reviewed byBethe in 1990 [1], were based. In parallel, many-body models became available whichwere capable to account for the relevant nuclear correlations and which describe theexperimental GT data quite well. Importantly, these models, and also experimentaldata, imply that the GT strength is not blocked at the shell gap between the pf and g / orbitals caused by strong nuclear cross-shell correlations. As a major consequence,electron capture takes place on nuclei during the entire collapse. With this result, thetheory of electron capture has gone now two complete circles.The evaluation of stellar electron capture for core-collapse supernovae rests onthe fact that, with progressing core density, the electron chemical potential growssignificantly faster than the average Q value which dominate the core composition.As a consequence, detailed description of the nuclear strength functions (i.e. Gamow-Teller) are only needed for nuclei in the Fe-Ni mass range ( A = 45–65), which aremost abundant during the early collapse phase, while for the neutron-rich, heavy nuclei, lectron capture in stars pf - and sd -shell nuclei, where the latteroccur in burning stages prior to collapse), diagonalization shell model calculations canbe performed which in general reproduce the measured GT strength functions quite well.In fact, if the capture rates are calculated solely from the ground state distributions,the rates obtained from data and from shell model agree within better than a factor of2 at the relevant astrophysical conditions. The theoretical capture rates (i.e. [34, 13])consider from excited states also from shell model calculations, accounting for the factthat each nuclear state has its own individual strength distribution. There are noindications that the shell model results for excited states might be less reliable than forthe ground states. However, there is concern that the procedure applied in [34] mightslightly underestimate the partition function at higher temperatures [139].The fact that cross-shell correlations unblock Gamow-Teller transitions even inthe ground states of nuclei with proton numbers Z <
40) and neutron numbers
N >
40 has been experimentally proven by experimental data for Gamow-Teller strengthdistributions and also from spectroscopic information obtained from transfer reactions.Thus, the assumption that GT transitions are Pauli blocked for nuclei with
N >
38 hasbeen disproven by experiment. Modern many-body models like the diagonalization shellmodel (for selected nuclei) and the Shell Model Monte Carlo approach can reproducesuch cross-shell correlations. The latter approach has been adopted to determine partialoccupation numbers in large model spaces including the shell gaps at N = 40 and 50.The capture rates were then calculated within a ‘hybrid model’ from these occupationnumbers within the framework of the Random Phase Approximation, exploiting thefact that these heavier nuclei become abundant during the collapse at sufficientlyhigh densities requiring only the overall, but not the detailed reproduction of theGT strength functions. Contributions from forbidden transitions were included, whichbecome progressively important with increasing density. The hybrid model indicatesthat the gaps at N = 40 and 50 lead to some reduction of the capture rates, but therares are clearly large enough so that captures on nuclei dominate the one on free protonsduring the entire collapse. This is clearly borne out in modern supernova simulations,thus closing the second circle as referred to by Hans Bethe.The unblocking of the GT strength at the neutron numbers N = 40 and 50 hasalso been confirmed by calculations performed within the Thermal Quasiparticle RPAapproach, which consistently considers correlations up to the 2p-2h level. As cross-shellcorrelations require in general correlations higher than 2p-2h, GT strength, in particularat low excitation energies, can be missed. This translates into the observation that atmodest temperatures and densities capture rates obtained within the Thermal QRPAare somewhat smaller than in the hybrid model. At higher temperatures and densitiesthe two models give very similar results, including the neutron-rich nuclei with N = 50, lectron capture in stars T = 1MeV, configurations from higher shells, which are strongly reduced in the ground state,are present in the thermally excited nuclear states and significantly unblock the GTstrength. This observation is quite important as the ground state GT distribution forsuch nuclei has been experimentally observed to have nearly vanishing strength andthe electron capture rate would nearly be blocked if calculated from the ground statedistribution. While the unblocking appears to be quite solid on theoretical ground,experimental verification is desirable.Although core-collapse supernovae are arguably the most important astrophysicalapplication, electron captures play also a role in other astrophysical environments.In thermonuclear supernovae the rate of electron captures on nuclei determine theproduction yield of neutron-rich nuclei. As the relevant nuclei are those in theFe-Ni mass range, the experimental and theoretical (by diagonalization shell modelcalculations) progress have constrained the relevant capture rates significantly up to adegree that improved description of details of the GT strength distribution changed thenucleosynthesis yields by only a few percent. The description of capture rates for sd -shellnuclei, again based on shell model calculations and data, has reached a similar degreeof accuracy which appears to be sufficient for the simulation of this process for the coreevolution of intermediate mass stars. However, attention has been drawn recently tothe fact that in the low-temperature-low-density environment of such stellar cores onlya few transitions dominate the capture rates and that in exceptional situations also aforbidden transition can noticeably contribute to the rate. Such a situation happens forthe capture on Ne where the second forbidden transition from the Ne ground-stateto the F ground state enhances the capture rate just at the most crucial conditionsfor the core evolution. The transition strength has now been measured so that theentire electron capture rate on Ne is now experimentally determined in the relevanttemperature-density regime.Double electron captures, initiated on abundant even-even nuclei, are relevant forthe crust evolution of accreting neutron stars. The process is triggered once the electronchemical potential (i.e. the core density) is high enough for electrons to overcome the Q value between the even-even mother nucleus and the odd-odd daughter. As the Q valueof the second capture step on the odd-odd nucleus is smaller due to nuclear pairing, thisenergy gain can be transferred into crust heating. For simulations of the crust evolution,generally one is not so much interested in the capture rate (which is often fasted thancompeting time scales), but in the portion of the energy gain which is translated intoheat. As this can involve quite exotic neutron-rich nuclei a detailed determination ofthis energy portion is a formidable nuclear structure challenge and current models arelikely too uncertain.Despite the progress which has been achieved in recent years in the determinationof stellar electron capture rates, further improvements are certainly desirable and, inspecific cases, needed. Additional precision measurements of Gamow-Teller strength lectron capture in stars sd - and pf -shell nuclei will lead to further improvements and torefinements of the shell model calculations, however, it is not expected that theseimprovements will have significant impact on supernova dynamics or nucleosynthesis.It is, however, desirable that the gap of nuclei (with mass numbers A = 38–45), forwhich no shell model electron capture rates exist, should be filled. Such calculationsare challenging as they require an accurate description of cross-shell correlations. Theywould certainly benefit from some detailed experimental GT distribution measurements.A particularly interesting and important case is Ar, which serves as material forneutrino detectors like ICARUS [216, 217, 218], which holds potential for the detectionof supernova neutrinos. Detailed GT − data from ( p, n ) [219] and ( He , t ) charge-exchange data [220] and M1 data from ( γ, γ (cid:48) ) photon scattering reactions [221] canserve as experimental constraints for the determination of charged-current ( ν e , e − ) andneutral-current ( ν, ν (cid:48) ) cross sections on Ar. However, GT + data, which are relevantfor electron capture and charged-current (¯ ν e , e + ) cross sections do not exist yet. Inprinciple, forbidden transitions, not considered in the shell model electron capture ratesfor sd and pf -shell nuclei, can contribute to the rates. But such contributions willonly be relevant in core-collapse supernovae at higher temperatures than those at whichthese nuclei dominate the core composition. The case of Ne, for which a secondforbidden transition dominates the capture rate at the relevant conditions during thecore evolution of intermediate mass stars, shows, however, that such exceptional casescan occur in cases of rather low temperatures where only a few transitions contribute tothe capture rate. No other case has yet been identified, however, caution is asked for.The shell gaps at neutron numbers N = 40 and 50 do not block electron captureon nuclei in current supernova models. In both cases, this is based on modern many-body models which at N = 40 overcome the gap by nucleon correlations, while for N = 50 thermal excitations are the main unblocking mechanism (plus contributionsfrom other multipoles than Gamow-Teller). For N = 40, the finding is supportedby experimental data, although yet quite limited. It would be desirable if the datapool could be enlarged. It is particularly tempting that recent developments openup the measurements of GT distributions for unstable neutron-rich nuclei, based oncharge-exchange reactions performed in inverse kinematics. Such additional data wouldcertainly be welcome to further constrain models. At N = 50, theoretical modelsimply that cross-shell correlations induced by thermal excitation render ground state GTdistributions not applicable for the calculation of capture rates at the finite temperatureswhich exist in the astrophysical environment when these heavier and very neutron-richnuclei dominate the capture process. Although the two models which have been appliedto N = 50 nuclei agree rather well in their rate predictions, improvements of the modelsare conceivable. On one hand, the finite-temperature QRPA model should be extendedto non-spherical nuclei and, in the midterm, also to include higher correlations like insecond QRPA approach. On the other hand, the Shell Model Monte Carlo approachis uniquely suited to study nuclear properties at the finite temperatures of relevance.It might be interesting to calculate the GT strength function at those temperatures lectron capture in stars Acknowledgements
The authors gratefully acknowledge support and assistance by their colleagues SamM. Austin, B. A. Brown, E. Caurier, S. Couch, D. J. Dean, J. Engel, T. Fischer,Y. Fujita, A. Heger, H.-T. Janka, A. Juodagalvis, R. W. Hix, E. Kolbe, O. Kirsebom,A. Mezzacappa, F. Nowacki, E. O’Connor, A. Poves, J. Sampaio, H. Schatz, A. Spyrou,T. Suzuki, F.-K. Thielemann, S. E. Woosley, Y. Zhi, and A. P. Zuker. G.M.P.acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) – Project-ID 279384907 – SFB 1245 “Nuclei: From FundamentalInteractions to Structure and Stars” and the “ChETEC” COST Action (CA16117),funded by COST (European Cooperation in Science and Technology). R.Z. gratefullyacknowledges support by the US National Science Foundation under Grants PHY-1913554 (Windows on the Universe: Nuclear Astrophysics at the NSCL), PHY-1430152(JINA Center for the Evolution of the Elements), and PHY-1927130 (AccelNet-WOU:International Research Network for Nuclear Astrophysics [IReNA])
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