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EPJ manuscript No. (will be inserted by the editor) Selected topics in Nuclear Astrophysics Gabriel Martı́nez-Pinedo Gesellschaft für Schwerionenforschung Darmstadt, Planckstr. 1, D-64259 Darmstadt, Germany Abstract. In this lectures after a brief introduction to stellar reaction rates and its implementation in nuclear networks I discuss the nuclear aspects of the collapse of the inner core of massive stars once it has run out of its nuclear energy source and of the star’s explosion as a type II supernova and the explosive nucleosynthesis occurring during this explosion which leads to the production of heavy elements by the rapid neutron capture process and potentially also by the recently discovered νp process. 1 Introduction Nuclear astrophysics aims at describing the origin of the chemical elements in the Universe as well as of the various nuclear processes, occurring in and powering astrophysical objects, which lead to the production of elements. It is impossible to cover the large width of this truely interdisciplinary field in these lectures. Hence I have made a biased choice and concentrate in the following topics: i) the nuclear aspects of the collapse of the inner core of massive stars once it has run out of its nuclear energy source and of the star’s explosion as a type II supernova; ii) the explosive nucleosynthesis occuring during this explosion which leads to the production of heavy elements by the rapid neutron capture process and potentially also by the recently discovered νp process. Due to this selection I will omit several other fascinating and timely topics of nuclear astrophysics. These include the slow neutron capture process, which produces about half of the nuclei heavier than iron, the p process, which makes the neutron deficient heavy nuclei, explosive hydrogen burning which occurs in novae and x-ray bursts, nucleosynthesis during the Big Bang, in cosmic rays and by neutrinos during a core-collapse supernova, Gamma Ray Bursts, and also thermonuclear (type Ia) supernova. Recent developments on these subjects are reviewed in a special issue on Nuclear Astrophysics edited by K. Langanke, F.K. Thielemann, M. Wiescher [1]. Of course, it is still very much recommended to read the two pioneering papers of Burbidge, Burbidge, Fowler and F. Hoyle [2] and A.G.W. Cameron [3]. 2 Cross sections and stellar reaction rates Nuclear astrophysics is concerned with the description of reactions taking place in astrophysical plasmas that change the composition and are responsible for the energy generation and nucleosynthesis. These reactions include fusion reactions during the various phases of stellar evolution or explosive burning, photodisintegration of nuclei and weak interaction reactions like electron capture, beta decay and neutrino absorption and scattering. In general, target and projectiles follow some distribution that normally is isotropic and depends only on the momentum of the particles and if the particles are in thermal equilibrium can be characterized by temperature and chemical potential of the particles. If the targets, a and projectiles b follow 2 Eur. Phys. J. Special Topics 156, 123 (2008) a specific distributions, na (p) and nb (p), the number of reactions per cm3 and per second and per pair of reactants for the process a + b → c + d, (1) is given by: rab = Z σ(v)vna (pa )nb (pb )dpa dpb (2) where v = |va − vb | is the relative velocity of particles a and b. The evaluation of this integral depends of the type of particles and distributions involved. For the typical conditions found in astrophysical plasma nuclei follow a Maxwell-Boltzmann distribution that is normalized to the total number of particles: Z G(T ) E−µ n= 4πp2 dp (3) exp (2π~)3 kT where G(T ) is the partition function that measures the internal degrees of freedom of the nuclei and it is defined: X G(T ) = (2Ji + 1)e−Ei /(kT ) (4) i where the sum runs over excited states [4,5] with excitation energy Ei and angular momenta Ji . As nuclei are non-relativistic we can use for the energy and chemical potential the expressions: E = mc2 + p2 , 2m µ = mc2 + kT η where η is the degeneracy parameter, so that equation (3) can be integrated to give: r G(T )eη 2π~2 n= , Λ = Λ3 mkT (5) (6) with Λ the thermal de Broglie wavelength. This equation provides a way of checking the validity of the Maxwell-Boltzmann statistics (η ≪ −1) for given temperature and density conditions: nΛ3 ≪ G(T ). 2.1 Reactions with particles of similar mass When the mass of the particles that partitipate in the reaction are similar we can use equations (3) and (6) and the integral (2) over the distributions yields: rab = hσvia,b na nb ≡ ha, bina nb (7) where hσvi is the distribution averaged σv that is related to the energy dependent cross section, σ(E) by: hσvi = 8 πµ 1/2 1 (kT )3/2 Z 0 ∞ E σ(E)E exp − dE kT (8) where µ is the reduced mass. For charged particle interactions, the reaction cross section depends critically in the Coulomb barrier and for low energies the reaction is only possible via the tunnel effect, the quantum mechanical penetration through a barrier at a classically forbidden energy. At low energies, the cross section σ(E) is dominated by the penetration factor, which for point-like particles and in the absence of a centrifugal barrier, s-wave, is well approximated by [6] Eur. Phys. J. Special Topics 156, 123 (2008) 2πZa Zb e2 P (E) = exp − ~v ≡ exp(−2πη(E)), where η(E) is often called the Sommerfeld parameter numerically equal to: r b A , 2πη(E) = 1/2 = 31.29Za Zb E E 3 (9) (10) with the energy, E = 12 µv 2 , defined in keV and A the reduced mass in atomic mass units. It is convenient and customary to define the cross section in terms of the astrophysical S-factor by factoring out the known energy dependence of the penetration factor and the de Broglie factor S(E) −2πη(E) e . (11) E Using the S-factor instead of the cross section the astrophysical reaction rate can be written: σ(E) = hσvi = 8 πµ 1/2 1 (kT )3/2 Z 0 ∞ E b S(E) exp − − 1/2 dE. kT E (12) The product of the two exponentials peaks at an energy denoted Gamow-energy, EG , with a width denoted Gamow-window, ∆EG , that determines the effective energy range at which the reaction takes place at a given temperature 2/3 bkT EG = = 1.22(Z12 Z22 AT62 )1/3 keV, 2 3 p EG kT = 0.749(Z12 Z22 AT65 )1/6 keV, ∆EG = √ 3 (13a) (13b) with T6 the temperature in units of 106 K. Typical reaction cross sections at Gamow energies during hydrostatic burning phases are in the sub-picobarn range and therefore extremely difficult to measure. For this reason, most of the available reaction rates are based on extrapolations of measurements at relatively high energies down to the stellar energy range. For non-resonant reactions the extrapolation can be safely performed in terms of the astrophysical S-factor as this has a weak energy dependence that reflects effects arising from the strong interaction, from the antisymmetrization, from small contributions from partial waves with l > 0, and for the finite size of the nuclei that can be described theoretically [7]. In addition to a non-resonant contribution, the total reaction rate can include contributions from resonances. For a reliable extrapolation of the resonant reaction component, the number, energies and strengths of low energy resonances and even subthreshold resonances must be known with great accuracy. This has been achieved for several reactions involving light nuclei [8] and medium mass nuclei for nova nucleosynthesis [9–11]. In many cases, however, no information or at best incomplete information is available about the energy dependence of the S-factor and the possible contributions of low energy resonances, like for example for the 12 C(α, γ) reaction [12]. This introduces significant uncertainties into the rates, which result in deviations of several orders of magnitude. Nuclear reaction rates are enhanced in stellar environments as the repulsive Coulomb barrier is somewhat shielded by the electrons present in the plasma [13]. Such shielding effects are also encountered in laboratory measurements of nuclear cross sections at low energies as the electrons present in the target (and perhaps also in the projectile) effectively reduce the Coulomb barrier [14]. We stress, however, that laboratory and plasma screening are different. Hence, the first has to be removed from the data to determine the cross sections for bare nuclei which then, in astrophysical applications, has to be modified due to plasma effects. As the temperature and/or charge of the particles increases the Gamow-energy defined in equation (13) is shifted to higher excitation energies in the compound nucleus. At some point 4 Eur. Phys. J. Special Topics 156, 123 (2008) the average resonance width becomes larger than the average level spacing and the compound reaction mechanism becomes applicable. In this case the reaction rate can be computed using statistical model using average resonance properties, also called Hauser-Feshbach approach [15, 7]. This approach is also applicable for neutron capture reactions when the reaction Q-value or the level density is sufficiently large. This is true for most intermediate and heavy nuclei close to the stability with non-magic neutron numbers. The relevant quantity is the number of available levels in the Gamow peak. An estimate of the applicability range of the statistical model is given in ref [16]. Many of the (n, γ) reactions relevant for s-process nucleosynthesis have been determined with accuracies reaching the 1% level [17]. However, experimentally only the capture on the ground state can be measured while in the astrophysical environment excited states can be thermally populated enhancing the reaction rate. This stellar enhancement have to be determined theoretically by Hauser-Feschbach calculations[18,19]. For neutron-magic nuclei and near the neutron drip line in the case of r-process the level densities at the neutron separation energies are so low that the statistical model is not applicable anymore and the capture reaction proceeds via the direct radioactive capture mechanism [20,21] For recent surveys of experimentally determined rates see[22,8,18,11] 2.2 Reactions with light or massless particles and decays If one of the particles that participate in the reaction is massless or is much lighter than the other particle that we denote target (for example the electron is 2000 times less massive than the nucleon). The relative velocity in equation (1) can be approximatted by the velocity of the light particles that for massless particles (photons and neutrinos) is just the speed of light, c. In this case the integral over the target nucleus gives just the density of targets and the remaining integral over the projectile leads to an effective decay rate of the target nucleus that depends on thermodynamical properties like density and temperature and the chemical potential of the projectile: Z rab = λb (T, ρ, µa )nb , λb = σva n(pa )dpa (14) The distribution of projectiles depends of the type of particles being considered. Photons follow a Bose distribution with chemical potential zero: n(p) = 1 p2 π 2 ~3 exp(pc/kT ) − 1 (15) The photodisintegration rate is therefore: λγ (T ) = 1 π 2 ~3 c3 Z 0 ∞ σ(E)cE 2 dE. exp(E/kT ) − 1 (16) There exist several recent attempts to evaluate experimental photodisintegration cross sections and determine the photodisintegration rates (see ref.[23]). Typically, photodisintegration rates are determined from the inverse capture reaction via detailed balance. The relationship between both rates can be obtained easily assuming that an equilibrium has been achieved between the capture reaction (a + b → c) and the photodisintegration (c → a + b). Equilibrium implies that the number of reactions occurring in both directions is the same: na nb ha, bi = λγ nc . 1 + δab (17) Moreover, in equilibrium the sum of chemical potentials for the particles in the right side of the reaction and the left side has to be the same: µa + µb = µc (18) Eur. Phys. J. Special Topics 156, 123 (2008) 5 Using equations (5) and (6) we can relate the chemical potentials to the number densities of particles obtaining: 3/2 3/2 2π~2 Gc Ac nc = nb na exp(Q/kT ), (19) mu kT Ga Gb Aa Ab where Q = ma c2 + mb c2 − mc c2 is the reaction Q-value and Ai is the atomic weight of species i. After substitution in equation (16) we finally obtain 3/2 3/2 mu kT Ga Gb Aa Ab ha, bi λγ = . (20) exp(−Q/kT ) 2π~2 Gc Ac 1 + δab While equation (19) is only valid in equilibrium, equation (20) is a consequence of detailed balance and is valid whenever the population of states in parent and daughter nucleus follows a thermal distribution. Electrons in an astrophysical plasma follow a Fermi-Dirac distribution and can exists with any state of degeneracy and being relativistic or non-relativistic. Moreover, at high temperatures it is necessary to account for the possibility of creating electron-positron pairs. To determine the chemical potential one defines the net number of electrons present in the system, ne , that has to be equal to the number of protons due to charge neutrality, substracting from the number of electrons, ne− the number of positrons, ne+ : Z ∞ 1 1 1 2 (21) ne = ne− − ne+ = 2 3 − dpp π ~ 0 e(E(p)−µ)/kT + 1 e(E(p)+µ)/kT + 1 p with µ the chemical potential of the electrons and E(p) = m2 c4 + p2 c2 . The rate of electron captures is then determined by integrating in equation (14) over the electron Fermi-Dirac distribution: Z 1 σ(E)p2 dE. (22) λec = 2 3 π ~ e(E−µ)/kT + 1 In the same way one can compute positron capture rates integrating over the positron distribution. Normally, neutrinos are not in thermal equilibrium and its distribution is explicitly computed solving the Boltzmann transport equation[24,25]. In nucleosynthesis applications neutrino spectra are typically fitted assuming a Fermi-Dirac distribution with temperature Tν and degeneracy parameter η: n E2 , (23) F2 (η)(kTν )3 exp(E/kTν − η) + 1 with E = pc, n the neutrino number density of neutrinos and the relativistic Fermi integral defined by Z ∞ xn Fn (η) = . (24) exp(x − η) + 1 0 An improved fit can be obtained using what is called “alpha fit”[26]: 1+α 1+α (1 + α)E n (25) E α exp − nα (E) = Γ (1 + α) hEi hEi n(E) = hEi its average energy and α a parameter that is adjusted to reproduce the average hE 2 i for the neutrinos: α+2 hEi2 (26) α+1 Finally, for normal decays with a half-life t1/2 , the decay rate is defined by λ = ln 2/t1/2 . At high temperatures and densities it is necessary to include the contribution of excited states and the blocking of the final electron phase space as will be discussed in section 3.2 hE 2 i = 6 Eur. Phys. J. Special Topics 156, 123 (2008) 2.3 Nuclear networks Energy generation and nucleosynthesis in stellar burning processes can be simulated by largescale nuclear reaction network calculations for the temperatures and densities of the particular stellar environment. In addition to nuclear reactions, expansion and contraction of the plasma can also produce changes in the number densities, ni . To separate the nuclear changes in composition from these hydrodynamic effects, the nuclear abundance, Yi , is defined as the ratio between the numberPdensity of species i and the total number density of nucleons (or baryons) in the plasma (n = i ni Ai , with Ai the number of nucleons of species i, and n ≈ ρ/mu 1 ). The mass fraction, Xi , of a nucleus is related to the abundance by Xi = Ai Yi . The conservation of P the number P of baryons reduces to i Yi Ai = 1. Likewise, the equation for charge conservation becomes i Yi Zi = Ye , where Ye (= ne /n) is the number of electrons per nucleon (also known as electron fraction or electron abundance). The rate of change for isotopic abundances can be expressed in the form: Ẏi = X j Nji λj Yj + X j,k i Nj,k X ρ i hj, kiYj Yk + Nj,k,l mu j,k,l ρ mu 2 hj, k, liYj Yk Yl , (27) where the three sums are over reactions which produce or destroy a nucleus of species i with one (decays, photodisintegrations, electron captures or neutrino-nucleus interactions), two (twoparticle fusion reactions) and three (three-particle reactions) reactant nuclei respectively. They are expressed in terms of the quantities λj and hj, ki as discussed above, while the tree body terms hj, k, li typically describe two successive captures with an intermediate particle-unstable nucleus. The N s provide a proper accounting of the number of nuclei participating in the i i reaction and are given by: Nji = Ni , Nj,k = Ni /(|Nj |!|Nk |!) and Nj,k,l = Ni /(|Nj |!|Nk |!|Nl |!). The Ni ’s represent positive or negative numbers specifying how many particles i are created or destroyed in the reaction. The denominators avoid double counting of the number of reactions when identical particles react (for example in the 12 C + 12 C or in the triple-α reactions, for details see[27]). Numerical methods for the solution of the system of differential equations (27) and the coupling to hydrodynamics are discussed in ref. [28]. During most of stellar evolution the nuclear composition is determined by a network of nuclear reactions between the nuclei present in the stellar environment. Electromagnetic reactions of the type (p, γ), (α, γ), and once free neutrons are produced, also (n, γ), play a particularly important role to fuse nuclides to successively larger nuclei. As the stellar environment has a finite temperature T , these reactions are in competition with the inverse dissociation reactions ((γ, p), (γ, α), (γ, n)) and for temperatures exceeding T ≈ Q/30, where Q is the threshold energy (Q-value) for the dissociation process to occur, a capture reaction and its inverse get into equilibrium. Similarly also nuclear reactions mediated by the strong interaction get into equilibrium with their inverse once the temperature is high enough for the charged particles to effectively penetrate the Coulomb barrier. These conditions are achieved in the core of massive stars before the supernova explosion when all reactions mediated by the electromagnetic and strong interaction are in equilibrium with their inverse and the nuclear composition becomes independent of the rates for these reactions. This state of matter is denoted nuclear statistical equilibrium. As the weak interaction is normally not in equilibrium the set of equations (27) is reduced to an equation determining the change of Ye due to weak interactions: X X Ẏe = − (λi,ec + λi,β + )Yi + (28) (λi,pc + λi,β − )Yi , i i where the sum runs over all nuclei present and includes all weak reactions that either increase Ye (β − and positron capture) or decrease Ye (β − and electron capture). The nuclear abundances can be uniquely determined that the total number P for a given (T, ρ, Ye ) with the constrains P of nucleons is conserved ( i Yi Ai = 1) and charge neutrality ( i Yi Zi = Ye ). Due to the two conserved quantities there exist two independent chemical potentials which are conventionally 1 Notice that in cgs units mu is numerically equal to 1/NA , with NA the Avogadro number Eur. Phys. J. Special Topics 156, 123 (2008) 7 chosen as µn and µp for neutrons and protons, respectively. For a nucleus of charge number Z and mass number A in equilibrium with free nucleons, the chemical potential is related to the chemical potentials of free neutrons and protons by: µ(Z, A) = Zµp + (A − Z)µn . (29) As nuclei obey Boltzmann statistics we can use (5,6) to obtain a expression for the abundance of every nuclear species in terms of neutron, Yn and proton, Yp , abundances. GZ,A (T )A3/2 Y (Z, A) = 2A ρ mu A−1 YpZ YnA−Z 2π~2 mu kT 3(A−1)/2 eB(Z,A)/kT (30) with B(Z, A) = (A−Z)mn +Zmp −M (Z, A)c2 the nuclear binding energy. At high temperatures NSE favors free nucleons, for intermediate temperatures α particles dominate while for low temperatures the most bound nuclei for which Z/A ∼ Ye is favoured. 3 45 T= 9.01 GK, l= 6.80e+09 g/cm , Y =.0.433 e 40 Z (Proton Number) 35 30 25 20 15 10 Log (Mass Fraction) 5 ï5 ï4 ï3 ï2 0 0 10 20 30 40 50 60 N (Neutron Number) 70 80 90 3 45 T= 17.84 GK, l= 3.39e+11 g/cm , Y =.0.379 e 40 Z (Proton Number) 35 30 25 20 15 10 Log (Mass Fraction) 5 ï5 ï4 ï3 ï2 0 0 10 20 30 40 50 60 N (Neutron Number) 70 80 90 Fig. 1. Abundances of nuclei for two sets of conditions during the core-collapse of a massive star. The upper panel represents typical conditions during the early collapse while the lower panel shows conditions near to neutrino trapping. A NSE code has been used in the calculation of the abundances [29] (Adapted from [30]). The fact that temperatures are high enough in the late stage of stellar evolution to drive matter into NSE facilitates simulations significantly as the matter composition becomes in- 8 Eur. Phys. J. Special Topics 156, 123 (2008) dependent of the rates for reactions mediated by the electromagnetic and strong interaction. Nevertheless the nuclear composition changes quite drastically during the collapse as temperature and density increase as the collapse progresses and, importantly, the weak interaction is initially not in equilibrium. As we will discuss in the next section, electron captures on nuclei and on free protons occur which reduce Ye and the nuclear composition is shifted to nuclei with larger neutron excess (smaller proton-to-nucleon ratio) which favors heavier nuclei. The increasing temperature has the consequence that the number of nuclei present in the composition with sizable abundances grows. This effect and the shift to heavier and more neutron-rich nuclei is confirmed in Fig. 1 which shows the NSE abundance distribution for two typical conditions during the collapse. Note that with increasing density, correlations among the nuclei and effects of the surrounding plasma become increasingly relevant [31,32]. 3 Core Collapse Supernovae ν M Progenitor (~ 15 M ) H 13 ~10 00000 11111 00000 11111 Dense 00000 11111 7 m 00000 11111 Core 00000 10 c11111 11111 00000 00000 11111 He ν ν ν ν 1−1 ν Early "Protoneutron" Star c. Se ~0. M 0000 1111 0000 Dense 0000 1111 10 cm 1111 1111 0000 Core 0000 1111 6 M Hot Extended Mantle ν Fe ν ν − Sphere M O/Si cm M ν 7 (Lifetime: 1 ~ 2 10 y) 1111 0000 0000 1111 ν ν ν rn pe Su νe k e +p n + νe and Photodisintegration of Fe Nuclei oc 3 10 8 cm νe Sh ov a Late Protoneutron Star (R ~ 20 km) ~1 Sec. Collapse of Core (~1.5 M ) "White Dwarf" (Fe−Core) 30000 − 60000 km/s (R ~ 10000 km) Fig. 2. Final phases of the evolution of a massive star showing the collapse of the core that results in the supernova explosion and the formation of a neutron star (Adapted from [33]). Massive stars end their lifes as type II supernovae, triggered by a collapse of their central iron core with a mass of more than 1M⊙ . The general picture of a core-collapse supernova is probably well understood and has been confirmed by various observations from supernova 1987A. It can be briefly summarized as follows: At the end of its hydrostatic burning stages (Fig. 2), a massive star has an onion-like structure with various shells where nuclear burning still proceeds (hydrogen, helium, carbon, neon, oxygen and silicon shell burning). As nuclei in the iron/nickel range have the highest binding energy per nucleon, the iron core in the star’s center has no nuclear energy source to support itself against gravitational collapse. As mass is added to the core, its density and temperature raises, finally enabling the core to reduce its free energy by electron captures of the protons in the nuclei. This reduces the electron degeneracy pressure and the core temperature as the Eur. Phys. J. Special Topics 156, 123 (2008) 9 neutrinos produced by the capture can initially leave the star unhindered. Both effects accelerate the collapse of the star. With increasing density, neutrino interactions with matter become decisively important and neutrinos have to be treated by Boltzmann transport. Nevertheless the collapse proceeds until the core composition is transformed into neutronrich nuclear matter. Its finite compressibilty brings the collapse to a halt, a shock wave is created which traverses outwards through the infalling matter of the core’e envelope. This matter is strongly heated and dissociated into free nucleons. Due to current models the shock has not sufficient energy to explode the star directly. It stalls, but is shortly after revived by energy transfer from the neutrinos which are produced by the cooling of the neutron star born in the center of the core. The neutrinos carry away most of the energy generated by the gravitational collapse and a fraction of the neutrinos are absorbed by the free nucleons behind the stalled shock. The revived shock can then explode the star and the stellar matter outside of a certain mass cut is ejected into the Interstellar Medium. Due to the high temperatures associated with the shock passage, nuclear reactions can proceed rather fast giving rise to explosive nucleosynthesis which is particularly important in the deepest layers of the ejected matter. Reviews on core-collapse supernovae can be found in [34–36]. Nevertheless, the most sophisticated spherical supernova simulations, including detailed neutrino transport [37–39], currently fail to explode indicating that improved input and/or numerical treatment is required. Among these microscopic inputs are nuclear processes mediated by the weak interaction, where recent progress has been made possible by improved manybody models and better computational facilities, as is summarized in [40]. Here we focus on the electron capture on nuclei, which strongly influences the dynamics of the collapse and produces the neutrinos present during the collapse, and on recent developments in explosive nucleosynthesis, where again neutrino-induced reactions are essential. Recent two-dimensional simulations stress also the importance of plasma instabilities, which in fact initiated successful numerical explosions [41,42]. 3.1 Electron captures in core-collapse supernovae - the general picture Late-stage stellar evolution is described in two steps. In the presupernova models the evolution is studied through the various hydrostatic core and shell burning phases until the central core density reaches values up to 1010 g/cm3 [43,44]. The models consider a large nuclear reaction network. However, the densities involved are small enough to treat neutrinos solely as an energy loss source. For even higher densities this is no longer true as neutrino-matter interactions become increasingly important. In modern core-collapse codes neutrino transport is described self-consistently by multigroup Boltzmann simulations [25,24]. While this is computationally very challenging, collapse models have the advantage that the matter composition can be derived from Nuclear Statistical Equilibrium (NSE) as the core temperature and density are high enough to keep reactions mediated by the strong and electromagnetic interactions in equilibrium (see section 2.3). This means that for sufficiently low entropies, the matter composition is dominated by the nuclei with the highest binding energies for a given Ye . The presupernova models are the input for the collapse simulations which follow the evolution through trapping, bounce and hopefully explosion. The collapse is a competition of the two weakest forces in nature: gravity versus weak interaction, where electron captures on nuclei and protons and, during a period of silicon burning, also β-decay play the crucial roles [45]. The weak-interaction processes become important when nuclei with masses A ∼ 55 − 60 (pf -shell nuclei) are most abundant in the core (although capture on sd shell nuclei has to be considered as well). As weak interactions changes Ye and electron capture dominates, the Ye value is successively reduced from its initial value ∼ 0.5. As a consequence, the abundant nuclei become more neutron-rich and heavier, as nuclei with decreasing Z/A ratios are more bound in heavier nuclei. Two further general remarks are useful. There are many nuclei with appreciable abundances in the cores of massive stars during their final evolution. Neither the nucleus with the largest capture rate nor the most abundant one are necessarily the most relevant for the dynamical evolution: What makes a nucleus relevant is the product of rate times abundance. 10 Eur. Phys. J. Special Topics 156, 123 (2008) For densities ρ < 1011 g/cm3 , stellar weak-interaction processes are dominated by GamowTeller (GT) and, if applicable, by Fermi transitions. At higher densities forbidden transitions have to be included as well. In addition thermal population of excited nuclear states needs to be considered. The formalism for the calculation of stellar weak-interaction rates is described in refs [46–48]. Here we discusse a usefull approximation that assumes that the total electron capture rate can be described by a unique transition from an initial state to a final state. This approximation is valid for capture in protons and for high temperatures when the capture is dominated by transitions to the Gamow-Teller resonance [49] and allows to understand the requirements of nuclear models to describe electron capture processes. 106 40 105 104 30 102 10 1 1 H Ni 69 Ni 76 Ga 79 Ge 89 Br 68 100 10−1 10−2 10−3 10−4 (MeV) λec (s−1) 103 10 10 10 11 ρ (g cm−3) 10 µe 20 〈Q〉 = µn−µp 10 Qp 12 0 10 10 10 ρ (g cm−3) 11 1012 Fig. 3. (left panel) Comparison of the electron capture rates on free protons and selected nuclei as function of density along a stellar collapse trajectory taken from [50]. (right panel) energy scales relevant for the determination of the electron capture rates. µe is the electron chemical potential. hQi and Qp are the average Q value for electron capture on nuclei and protons respectively. The cross section for the capture of an electron with energy Ee from a nuclear state in the initial nucleus to a final nuclear state whose Gamow-Teller transition is described by a Gamow-Teller matrix element, B(GT ) is given by: σ(Ee ) = 2 G2F Vud Ee F (Z, Ee )B(GT )(Ee − Q) 4 2π~ c5 pe (31) where GF is the Fermi coupling constant, Vud is the up-down element in the Cabibbo-KobayashiMaskawa quark-mixing matrix. F (Z, Ee ) is the Fermi function that takes in account the Coulomb distortion of the electron being capture in a nucleus of charge Z. Q is the effective Q-value for the capture and is assumed to be possitive for capture in protons and neutron rich nuclei. The stellar electron capture rate is obtained integrating over the thermal electron spectrum using equation (22): Z ∞ 2 F (Z, Ee )Ee pe (Ee − Q)2 G2F Vud dEe . (32) λec = B(GT ) 2π 3 ~7 c5 e(Ee −µe )/kT + 1 Q For relativistic electrons we can use the approximation F (Z, Ee )pe c ≈ Ee [51] and use the definition of Fermi integrals to rewrite the above equation 5 (ln 2)B(GT ) T λec = F4 (η) + 2χF3 (η) + χ2 F2 (η) , (33) 2 K me c 2 where χ = Q/(kT ), η = (µe −Q)/(kT ), and we have introduced the constant K = 2π 3 (ln 2)~7 /(G2F Vud m5e c4 ) + + whose value is K = 6147 ± 2.4 s as measured in superallowed 0 → 0 decays [52]. Figure 3 compares the electron capture rates for free protons and selected nuclei along a stellar trajectory taken from [50]. These nuclei are abundant at different stages of the collapse. For all the nuclei, the rates are dominated by GT transitions at low densities, while forbidden B(GT+) Eur. Phys. J. Special Topics 156, 123 (2008) 11 0.4 51V(d,2He)51Ti 0.3 0.2 0.1 0.1 B(GT+) 0.2 0.3 large shell model calculation 0.4 0 1 2 3 4 5 6 7 Ex [MeV] Fig. 4. Comparison of the measured spectrum GT+ strength in the one computed in a large-scale shell model calculation [53] 51 V(d,2 He)51 Ti reaction with the transitions contribute sizably for ρ & 1011 g cm−3 . The electron chemical potential µe and the reaction Q value are the two important energy scales of the capture process. (They are shown on the right panel of figure 3.). Further, µe grows much faster than the Q values of the abundant nuclei. For the low densities (. 1010 g cm−3 ) present during the presupernova phase, µe ≈ Q and the term F2 dominates in equation (33). The rate is then larger for the nuclei with smaller Q values and is very sensitive to the phase space requiring an accurate description of the details of the GT+ distribution of the involved nuclei. It has been demonstrated [54,47] that modern shell model calculations are capable to describe nuclear properties relevant to derive stellar electron capture rates (spectra and GT+ distributions) rather well (an example is shown in Figure 4) and are therefore the appropriate tool to calculate the weak-interaction rates for those nuclei (A ∼ 50 − 65) which are relevant at such densities. For intermediate densities (1010 –1011 g cm−3 ), the term F3 dominates and the rate still has some dependence on the Q-value. For high densities (& 1011 g cm−3 ), µe ≫ Q so that the term F4 dominates and the rate becomes independent of the Q-value, depending only on the total GT strength, but not its detailed distribution. Thus, less sophisticated nuclear models might be sufficient. However, one is facing a nuclear structure problem which has been overcome only very recently. Once the matter has become sufficiently neutronrich, nuclei with proton numbers Z < 40 and neutron numbers N > 40 will be quite abundant in the core. For such nuclei, Gamow-Teller transitions would be Pauli forbidden (GT+ transitions change a proton into a neutron in the same harmonic oscillator shell) were it not for nuclear correlation and finite temperature effects which move nucleons from the pf shell into the sdg shell (see figure 5). To describe such effects in an appropriately large model space (e.g. the complete pf sdg shell) is currently only possible by means of the Shell Model Monte Carlo approach (SMMC) [55]. In [56] SMMC-based electron capture rates have been calculated for many nuclei which are present during the collapse phase. 12 Eur. Phys. J. Special Topics 156, 123 (2008) g9/2 g9/2 N=40 Blocked GT Unblocked Correlations Finite T f5/2 GT p1/2 p1/2 f5/2 p3/2 p3/2 f7/2 f7/2 neutrons 1111111111111 0000000000000 0000000000000 1111111111111 Core 0000000000000 1111111111111 0000000000000 1111111111111 neutrons protons protons 111111111111 000000000000 Core 000000000000 111111111111 000000000000 111111111111 Fig. 5. In the independent particle model GT transitions are blocked at neutron number N = 40. 3.2 Weak-interaction rates and presupernova evolution 0.450 1.2 0.445 0.440 MFe (M⊙) 1.8 Ye 0.435 1.6 0.430 0.425 1.4 WW LMP 0.420 0.1 0.0 0.010 0.005 10 ∆S (kB) 0.015 -0.1 15 20 25 30 Star Mass (M⊙) 35 40 -0.2 10 1.1 1.0 0.9 0.8 0.7 WW LMP 0.6 0.1 ∆MFe (M⊙) ∆Ye 0.020 WW LMP central entropy / baryon (kB) 2.0 15 20 25 30 Star Mass (M⊙) 35 40 0.0 -0.1 -0.2 10 15 20 25 30 35 40 Star Mass (M⊙) Fig. 6. Comparison of the center values of Ye (left), the iron core sizes (middle) and the central entropy (right) for 11–40 M⊙ stars between the WW models and the ones using the shell model weak interaction rates (LMP) [45]. The lower parts define the changes in the 3 quantities between the LMP and WW models. Up to densities of a few 1010 g/cm3 electron capture is still dominated by capture on nuclei in the A ∼ 45−65 mass range, for which capture rates have been derived on the basis of large-scale shell model diagonalization studies. Importantly, the shell model rates are noticeably smaller than those derived previously on the basis of the independent particle model [47]. To study the influence of these slower shell model rates on presupernova models Heger et al. [45,44] have repeated the calculations of Weaver and Woosley [43] keeping the stellar physics, except for the weak rates, as close to the original studies as possible. Fig. 6 examplifies the consequences of the shell model weak interaction rates for presupernova models in terms of the three decisive quantities: the central Ye value and entropy and the iron core mass. The central values of Ye at Eur. Phys. J. Special Topics 156, 123 (2008) 13 the onset of core collapse increased by 0.01–0.015 for the new rates. This is a significant effect. We note that the new models also result in lower core entropies for stars with M ≤ 20M⊙ , while for M ≥ 20M⊙ , the new models actually have a slightly larger entropy. The iron core masses are generally smaller in the new models where the effect is larger for more massive stars (M ≥ 20M⊙ ), while for the most common supernovae (M ≤ 20M⊙ ) the reduction is by about 0.05 M⊙ . Electron capture dominates the weak-interaction processes during presupernova evolution. However, during silicon burning, β decay (which increases Ye ) can compete and adds to the further cooling of the star. With increasing densities, β-decays are hindered as the increasing Fermi energy of the electrons blocks the available phase space for the decay. Thus, during collapse β-decays can be neglected. We note that the shell model weak interaction rates predict the presupernova evolution to proceed along a temperature-density-Ye trajectory where the weak processes are dominated by nuclei rather close to stability. Thus it will be possible, after radioactive ion-beam facilities become operational, to further constrain the shell model calculations by measuring relevant beta decays and GT distributions for unstable nuclei. Ref. [45,44] identify those nuclei which dominate (defined by the product of abundance times rate) the electron capture and beta decay during various stages of the final evolution of a 15M⊙ , 25M⊙ and 40M⊙ star. 3.3 The role of electron capture during collapse Until recently core-collapse simulations assumed that electron capture on nuclei are prohibited by the Pauli blocking mechanism [57]. However, based on the SMMC calculations it has been shown in [56,58] that capture on nuclei dominates over capture on free protons (the later was evaluated in [59] and has always been included in the simulations). Even if the electron capture rate on a proton is larger than that for individual nuclei (see figure 3) the collapse proceeds with low entropy keeping the protons significantly less abundant than heavy nuclei. Once P the abundances are considered the reaction rate for electron capture on heavy nuclei (Rh = i Yi λi , where the sum runs over all the nuclei present and Yi denotes the number abundance of species i) dominates over the one of protons (Rp = Yp λp ) by roughly an order of magnitude throughout the collapse [56,58]. The effects of this more realistic implementation of electron capture on heavy nuclei have been evaluated in independent self-consistent neutrino radiation hydrodynamics simulations by the Oak Ridge and Garching collaborations [58,60,61]. The changes compared to the previous simulations, which basically ignored electron capture on nuclei, are significant: In denser regions, the additional electron capture on heavy nuclei results in more electron capture in the new models. In lower density regions, where nuclei with A < 65 dominate, the shell model rates [47] result in less electron capture. The results of these competing effects can be seen in the first panel of Figure 7, which shows the center value of Ye and Ylep (the lepton-to-baryon ratio) during the collapse using the standard treatment of Bruenn [59] and the new rates for heavy nuclei (denoted LMSH). At densities above 1012 g/cm3 , Ylep is (nearly) constant indicating that the neutrinos are trapped and a Fermi sea of neutrinos is being built up. Weakinteraction processes are now in equilibrium with their inverse. The changes in electron captures strongly reduce the temperatures and entropies (middle panel) in the inner core and hence affect its composition. The panel on the right shows the energy per baryon and second emitted in neutrinos at the center of the star. The inset shows the mean energy of the produced neutrinos. With the LMSH treatment, more neutrinos are produced with lower mean energy than with the Bruenn treatment due to the fact that nuclei have larger Q-values for electron capture. Weak-interactions also strongly influence the shock evolution as discussed in ref. [58]. Astrophysics simulations have demonstrated that electron capture rates on nuclei have a strong impact on the core collapse trajectory and the properties of the core at bounce. The evaluation of the rates has to rely on theory as a direct experimental determination of the rates for the relevant stellar conditions (i.e. rather high temperatures) is currently impossible. Nevertheless it is important to experimentally explore the configuration mixing between pf and sdg shell in extremely neutron-rich nuclei as such understanding will guide and severely Eur. Phys. J. Special Topics 156, 123 (2008) 0.45 0.30 0.25 105 2.0 s 1.5 1.0 1011 1012 1013 ρc (g cm ) 1014 0.0 10 10 104 103 50 102 40 101 100 Bruenn LMSH −3 Bruenn LMSH 30 20 10 0.5 Bruenn Y e LMSH dE (MeV s−1 baryon−1) dt 0.35 sc (kB), Tc (MeV) Ylep Ye,c, Ylep,c T 2.5 0.40 0.20 10 10 106 3.0 〈Eν〉 (MeV) 14 0 1010 1012 1014 ρc (g cm−3) −1 1011 1012 1013 ρc (g cm ) −3 1014 10 1010 1011 1012 1013 ρc (g cm−3) 1014 Fig. 7. Comparison of the evolution of several quantities at the center of a 15 M⊙ star: Ye is the number of electrons per baryon, Ylep is the number of leptons per baryon, s is the entropy per baryon, T is the temperature, dE/dt is the neutrino energy emission rate per baryon, hEν i is the average energy of the emitted neutrinos. The initial presupernova model was taken from [45]. The thin line is a simulation using the Bruenn parametrization [59] while the thick line uses the LMSH rate set (see text). The LMSH rate set considers electron capture rates on nuclei for densities below 1013 g cm−3 . Above this density electron capture is only possible on protons explaining the kink in the energy emission rates in the right panel. Notice that this has no effect in other properties of the core as at that density the core is in full weak equilibrium. The models were calculated by the Garching collaboration (Courtesy of M. Rampp and H.-Th. Janka). constrain nuclear models. Such guidance is expected from future radioactive ion-beam facilities like FAIR, RIBF, and SPIRAL 2. 4 Nucleosynthesis of heavy elements in neutrino heated ejecta When in an successful explosion the shock passes through the outer shells, its high temperature induces an explosive nuclear burning on short time-scales. This explosive nucleosynthesis can alter the elemental abundance distributions in the inner (silicon, oxygen) shells (see ref. [62] for an in-depth review). More interestingly the deepest ejected layers reach sufficiently high temperatures that matter is fully dissociated in free protons and neutrons. In addition the hot protoneutron star born during the explosion cools emitting large amounts of neutrinos during a period of tens of seconds. This neutrinos heat the matter in the neutron star surface and produce an outflow of baryonic matter that is commonly denoted as neutrino-driven wind [63,64]. Once the matter reaches its maximum temperature expands and cools adiabatically, under normal conditions this means constant entropy, with nuclei being resembled once the temperatures become sufficiently low. The nucleosynthesis in this ejecta and the final elemental abundances depends on outflow parameters like the expansion timescale, entropy and Ye value of the ejected matter [65]. Recently explosive nucleosynthesis has been investigated consistently within supernova simulations, where a successful explosion has been enforced by slightly increasing the neutrino absorption cross section on nucleons or reducing the neutrino mean-free path. Both effects increase the efficiency of the energy transport to the stalled shock. The results presented in [66,67] showed that in an early phase after the bounce the ejected matter is actually proton rich as already anticipated by Qian and Woosley [64] while matter ejected latter may become neutron-rich, potentially leading to r-process nucleosynthesis [68,69]. Eur. Phys. J. Special Topics 156, 123 (2008) Proton rich (νp-process) ν̄e + p → n + e+ 64 Ge + n → 64Ga + p 64 Ga + p → 65Ge; . . . Neutron rich (r-process) neutrons + seeds → heavy nuclei (A ∼ 100–??) 4 4 T≈ ( 0.25 MeV 3 GK T≈ ( 0.75 MeV 9 GK T≈ ( 0.9 MeV 10 GK ..... seeds (A ∼ 50–100) ..... seeds (N = Z ∼ 28–32) 15 He(αα, γ)12C 4 He(αα, γ)12C He(αn, γ)9Be 2p + 2n → 4He Alpha formation Weak interaction freeze−out ion He on ati ng i reg ng reg ati He νe + n ⇄ p + e− ν̄e + p ⇄ n + e+ νe , ν̄e , νµ , ν̄µ , ντ , ν̄τ Proto−neutron Star Fig. 8. Evolution of matter outflows from the protoneutron star surface. Figure 8 shows the evolution of matter ejected from the proto-neutron star surface. Near to the neutron star the matter is composed of neutrons and protons under extreme neutrino and antineutrino fluxes. Due to the large temperatures electrons and positrons are also created. Under this conditions, Ye , is determined by a competition between electron capture and antineutrino capture (decrease Ye ) and positron capture and neutrino capture (increase Ye ) [67]. As the matter moves to larger radii and cools, the electron and positron capture rates decreases much faster than neutrino and antineutrino absorptions, due to the strong temperature dependence of the former (see eq. 33). Under, this conditions the evolution of Ye is governed by: Ẏe = λνe n Yn − λν̄e p Yp (34) with λνe n the rate for the reaction νe + n → p + e− and λνe n for the reaction ν̄e + p → n + e+ and Yn and Yp are the neutron and proton abundances. If matter is exposed long time enough to the neutrino fluxes Ye will try to reach its equilibrium value obtained from the condition Ẏe = 0. If the composition at this time is given by neutrons and protons then we have Ye = Yp and Yn = 1 − Ye and for Ye we get the estimate: Ye = λ νe n . λνe n + λν̄e p (35) If neutrino interactions continue when a substantial amount of alpha particles is present, the equation governing the change of Ye becomes [70] Xα − (λν̄e p + λνe n )Ye (36) 2 which is obtained using Ye = Yp + Xα /2 and Yn + Yp + Xα = 1 with Xα the mass fraction of alpha particles that are assumed to be inert to neutrino interactions. In this case, Ye tries to reach: Ẏe = λνe n + (λν̄e p − λνe n ) 16 Eur. Phys. J. Special Topics 156, 123 (2008) Ye ≈ λνe n λν̄ p − λνe n Xα + e λνe n + λν̄e p λνe n + λν̄e p 2 (37) which is larger (smaller) than the value in equation (35) for λν̄e p > λνe n (λνe n > λν̄e p ). This is the so called α-effect that drives the composition to Ye ≈ 0.5 hindering the occurrence of the r-process in neutron rich ejecta. If we neglect weak magnetism corrections [71] and the existence of a threshold for antineutrino absorption in protons the rates for neutrino and antineutrino absorption at a distance r can be determined by: ∆2 Lν ǫ + 2∆ + (38a) σ λνn = ν 0 4πr2 (me c2 )2 hEν i Lν̄ ∆2 λν̄p = ǫ − 2∆ + (38b) σ ν̄ 0 4πr2 (me c2 )2 hEν̄ i with Lν and Lν̄ the neutrino and antineutrino luminosities, σ0 = 2.569 × 10−44 cm2 and ǫν = hEν2 i/hEν i the ratio between second moment of the neutrino spectrum and the average neutrino energy (similarly for antineutrinos) and ∆ = 1.2933 MeV the proton-neutron mass difference. Inserting this expressions for the neutrino and antineutrino absorption rates and assuming that Lν ∼ Lν̄ we get that proton-rich ejecta (Ye > 0.5) occur whenever 4∆ > ǫν̄ − ǫν . During the first seconds after bounce neutrino and antineutrinos luminosities are dominated by matter being accreted in the proto-neutron star that results in spectra for neutrinos and antineutrinos that satisfy the previous condition and the composition is proton-rich. Later as the luminosities are dominated from the cooling of the proto-neutron star the antineutrino average energy increases and matter is expected to become neutron rich [72] Once weak interactions freeze-out and the value of Ye is set the evolution of matter follows different paths depending if we are in proton rich ejecta or neutron rich ejecta (see figure 8). As the matter expands and cools α particles form and at lower temperatures some of them can assemble 12 C either by the triple alpha reaction (proton-rich ejecta) or the sequence α(αn, γ)9 Be(α, n)12 C (neutron-rich ejecta). The carbon nuclei will capture additional α particles until iron group or even heavier nuclei are formed [73,74]. The amount of nuclei formed depends both in the entropy of the ejecta and the expansion timescale. Large entropies mean a larger amount of photons present and larger photodissociation rates reducing the efficiency with which 12 C is produced. In fast expansions the three body reactions responsible of the buildup of 12 C freeze-out relatively soon due to the quadratic dependence in density. Both effects reduce the amount of nuclei synthesized leaving large amounts of free protons or neutrons. If enough free protons or neutrons are left, the nuclei act as “seed” for the formation of heavier elements via proton captures (νp-process in proton rich ejecta) or neutron captures (r-process in neutron rich ejecta). 4.1 Nucleosynthesis in proton-rich ejecta: the νp-process The early, proton-rich ejecta consist of two components. On the one hand there is material that comes from the convecting postshock region and is expelled when the explosion is launched and the shock accelerates (‘hot bubble ejecta’). Much of this material starts a rather slow expansion from large distances from the neutrino-radiating neutron star, is quite dense, has modest entropies (s ∼ 15 − −30 kB per nucleon), and is slightly neutron-rich (Ye & 0.47) or moderately proton-rich with Ye . 0.52 [75]. This material experiences little effect from neutrino-interactions during nucleosynthesis. This is in strong contrast to the matter ejected in the second component, which is the early neutrino-driven wind. The wind comes from the surface of the hot neutron star, is strongly heated by neutrinos, and has to make its way out of the deep gravitational well of the compact remnant. Therefore the wind has rather high entropies, short expansion timescales and can become quite proton-rich (Ye ∼ 0.57, [66,67]). Eur. Phys. J. Special Topics 156, 123 (2008) 17 Moving into cooler regions, protons and neutrons in this wind matter assemble first into C and then, by a sequence of (p, γ), (α, γ) and (α, p) reactions into even-even N = Z nuclei like 56 Ni, 60 Zn and 64 Ge, with some free protons left, and with enhanced abundances of 45 Sc, 49 Ti and 64 Zn solving a longstanding nucleosynthesis puzzle [66,67]. In the absence of a sizable neutrino fluence, this nucleosynthesis sequence resembles explosive hydrogen burning on the surface of an accreting neutron star in a binary (the rp-process, [76]) and matter flow would basically end at 64 Ge as this nucleus has a β halflive (≈ 64 s) which is much longer than the expansion timescale and proton captures are prohibited by the small reaction Q value. However, the wind material is ejected in the presence of an extreme flux of neutrinos and antineutrinos. While νe -induced reactions have no effect as all neutrons are bound in nuclei with rather large Q-values for neutrino capture, antineutrino absorption on the free protons yield a continuous supply of free neutrons with a density of free neutrons of 1014 –1015 cm−3 for several seconds, when the temperatures are in the range 1–3 GK [77]. These neutrons, not hindered by Coulomb repulsion, are readily captured by the heavy nuclei in a sequence of (n, p) and (p, γ) reactions in this way effectively by-passing the nuclei with long beta-halflives like 64 Ge and allowing the matter flow to proceed to heavier nuclei. Fröhlich et al. argue that all core-collapse supernovae will eject hot, explosively processed matter subject to neutrino irradiation and that this novel nucleosynthesis process (called νpprocess) will operate in the innermost ejected layers producing neutron-deficient nuclei above A > 64 [77]. However, how far the mass flow within the νp-process can proceed, strongly depends on the environment conditions, most noteably on the Ye value of the matter [75,77,78]. Obviously the larger Ye , the larger the abundance of free protons which can be transformed into neutrons by antineutrino absorption. The reservoir of free neutrons produced by antineutrino absorptions is also larger if the luminosities and average energies of antineutrinos are large or the wind material expands slowly. Ref [75] provides simple estimates of this parameters. 12 Mi/(M ejXi,⊙) 10 2 10 1 10 0 10 −1 40 50 60 70 80 90 100 110 120 A Fig. 9. Production factors (ejected mass of a given isotope divided by the total ejected mass and normalized to the solar mass fraction of the element) from six hydrodynamical trajectories corresponding to the early proton rich wind obtained in the explosion of a 15 M⊙ star [79]. Figure 9 shows the production factors resulting from several trajectories corresponding to the early proton-rich wind from the protoneutron star resulting of the explosion of a 15 M⊙ star [79]. (These trajectories have also been studied in reference [75].) No production of nuclei above A = 64 is obtained if antineutrino absorption reactions are neglected. The production of light p-process nuclei like 84 Sr, 94 Mo and 96,98 Ru is also clearly seen in the figure. Thus 18 Eur. Phys. J. Special Topics 156, 123 (2008) the νp process offers the explanation for the production of these light p-nuclei, which was yet unknown [80]. However, simulations fail to reproduce the observed abundance of 92 Mo, the most abundant p-nucleus in nature. It is, however, observed that 92 Mo is significantly produced in slightly neutron-rich winds with Ye values between 0.47 and 0.49 as they might be found in a later phase of the explosion (a few seconds after bounce) [78]. Here the α-rich freeze-out overabundantly produces 90 Zr from which some matter flow is carried to 92 Mo by successive proton captures [78]. Ref. [81] has recently analyzed the production of 92 Mo in proton rich ejecta concluding that in order to achieve a 92 Mo/ 94 Mo ratio consistent with the solar abundances the proton separation energy of 93 Rh must be Sp = 1.64 ± 0.1 MeV. This value is not consistent with the value of Sp = 2.0 ± 0.008 recently measured at JYFLTRAP [82]. Before concluding that proton rich ejecta cannot produce 92 Mo it will be necessary to reevaluate all the (p, γ) and (n, p) reactions in the region using the new available experimental masses and explore the influence that the presence of isomeric states can have in the final nucleosynthesis. 25 0 4.2 Nucleosynthesis in neutron-rich ejecta: the r-process 100 20 ra 0 bu nd an ce s Known mass Known half−life r−process waiting point (ETFSI−Q) 98 lar 96 94 92 r−path So 90 88 86 188190 186 84 82 80 184 180182 178 176 78 0 76 164 168 172 162 166 170 174 160 158 156 154 152 150 140 144 148 138 142 146 134136 130132 128 15 74 72 70 68 66 64 62 0 −1 54 40 126 124 122 120 116118 112114 110 108 106 104 100102 98 N=126 10 −3 46 44 52 10 10 42 48 0 50 −2 10 10 10 1 60 58 56 N=184 96 92 94 38 86 88 90 84 36 82 80 34 78 7476 32 72 30 70 6668 28 64 62 26 60 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 N=82 N=50 Fig. 10. The figure shows the range of r-process paths, defined by their waiting point nuclei. After decay to stability the abundance of the r-process progenitors produce the observed solar r-process abundance distribution. The r-process paths run generally through neutron-rich nuclei with experimentally unknown masses and half lives. In this calculation a mass formula based on the ETFSI model and special treatment of shell quenching [83] has been adopted. (courtesy of K.-L. Kratz and H. Schatz). The rapid neutron-capture process (r-process) is responsible for the synthesis of approximately half of the nuclei in nature beyond Fe [84,85]. It requires neutron densities which are high enough to make neutron capture faster than β decay even for neutron excess nuclei 15–30 units from the stability line. These conditions enable the production of neutron-rich nuclei close to the dripline via neutron capture and (γ, n) photodisintegration during the r-process. Once the neutron source ceases, the progenitor nuclei decay either via β − or α emission or by fission towards stability and form the stable isotopes of elements up to the heaviest species Th, U and Pu. Due to the relatively small neutron separation energies in nuclei with Nmag + 1, where Nmag = 50, 82, 126, 184, the r-process flow at magic neutron numbers comes to a halt requiring Eur. Phys. J. Special Topics 156, 123 (2008) 19 several β decays to proceed. As the half lives of these magic nuclei are large compared to “regular” r-process nuclides, they determine the dynamical timescale of the r-process. Furthermore, much matter is accumulated at these ‘waiting points’ resulting in the observed peak structure in the r-process abundance distribution (see figure 10) Despite many promising attempts the actual site of the r-process has not been identified yet. However, parametric studies have given clear evidence that the observed r-process abundances cannot be reproduced at one site with constant temperature and neutron density [86]. Thus the abundances require a superposition of several (at least three) r-process components. This likely implies a dynamical r-process in an environment in which the conditions change during the duration of the process. Recent observations of metal-poor stars show that the relative abundance of elements heavier than Z ≃ 56, except for the radioactive actinides, exhibits a striking consistency with the observed solar abundances of these elements, while elements lighter than Z = 56 are underabundant relative to a scaled solar r-process pattern that matches the heavy element abundances [87, 88]. These observations indicate that the astrophysical sites for the synthesis of light and heavy r-process nuclides are different [89,90]. The exact site and operation for both types of r-process is not known. There are clear indications that the process responsible for the production of heavy elements is universal [91] while the production of lighter elements (in particular Sr, Y and Zr) has a more complex Galactic history [92]. The astrophysical sites for the r-process are still heavily debated. Depending on the thermodynamical conditions and in particular the entropy they can be classified in low-entropy and high-entropy sites. Low-entropy sites include the decompression of cold neutron star material [93,94], prompt explosions of ONeMg cores [95,96] and jets from accretion disks [97,98]. High entropy sites include the neutrino-driven wind from the nascent neutron star in corecollapse supernova [68,69,99,72] and shocked surface layers of exploding ONeMg cores [100, 101]. Currently, the neutrino-driven wind from the nascent neutron star in a core-collapse supernova is the favored scenario for r-process nucleosynthesis. In this environment, neutrino emission from the cooling of the just formed neutron star produces an outflow of baryonic matter. This matter expands rapidly and cools, once charged-particle reactions freeze out (alpha-rich freezeout), elements heavier than iron are produced. As discussed in section 4 the nucleosynthesis in this ejecta is very sensitive to the entropies and expansion times scales. Parametric studies have shown that the neutron-to-seed ratio available for making the r-process goes like s3 /τ , with s the entropy per nucleon and τ the expansion time scale. That is, increasing the entropy or decreasing the dynamic timescale results in smaller production of seed nuclei and hence a larger number of free neutron captures per heavy nucleus. In order to form the heavier nuclei r-process nuclei, the platinum peak and the actinides, a short dynamical timescale (few milliseconds), high entropies (above 150kB ) and low electron fractions (Ye < 0.5) are required [65]. These wind parameters are determined by neutron star properties like the mass and radius, and by the spectra and luminosities of neutrinos and antineutrinos emitted from the neutron star [64]. The major challenge for current hydrodynamical models is how to achieve the large entropies required. A recent study [72] has shown that the interaction of the wind with previous ejecta of the supernova produces a reverse shock that substantially increases the entropy of the ejected matter. However, this increase takes place after the alpha-rich freeze-out when the neutron-to-seed ratio has already been determined and the r-process is ongoing. In this case it is not clear if the increased entropy can have nucleosynthesis consequences [102]. Before discussing the basic features of network calculations of r-process it is convenient to describe the nuclear physics input required. 4.2.1 Nuclear r-process input Arguably the most important nuclear ingredient in r-process simulations are the nuclear masses as they determine the flow-path. Unfortunately nearly all of them are experimentally unknown and have to be theoretically estimated. Traditionally this is done on the basis of parametrizations to the known masses. Although these empirical mass formulae achieve rather remarkable 20 Eur. Phys. J. Special Topics 156, 123 (2008) fits to the data (the standard deviation is of order 700 keV [103,104]), extrapolation to unknown masses appears less certain and different mass formulae can predict quite different trends for the very neutron-rich nuclei of relevance to the r-process. The most commonly used parametrizations are based on the finite range droplet model (FRDM), developed by Móller and collaborators [105,106], and on the ETFSI (Extended Thomas-Fermi with Strutinski integral) model of Pearson [107,108]. A new era has been opened very recently, as for the first time, nuclear mass tables have been derived on the basis of nuclear many-body theory (Hartree-Fock-Bogoliobov model) [109–111] rather than by parameter fit to data. The nuclear halflives strongly influence the relative r-process abundances. In a simple β-flow equilibrium picture (discussed below) the elemental abundance is proportional to the halflife, with some corrections for β-delayed neutron emission [70]. As r-process halflives are longest for the magic nuclei, these waiting point nuclei determine the minimal r-process duration time; i.e. the time needed to build up the r-process peak around A ∼ 200. We note, however, that this time depends also crucially on the r-process path and can be as short as a few 100 milliseconds if the r-process path runs close to the neutron dripline. There are a few milestone halflife measurements including the N = 50 waiting point nuclei 78 Ni [112] and the N = 82 waiting point nuclei 130 Cd [113] and 129 Ag [114]. Although no halflives for N = 126 waiting points have yet been determined, there has been decisive progress towards this goal recently [115]. 1000 FRDM DF3+QRPA SM HFB Exp. T 1/2 (ms) 100 10 41 42 43 44 45 46 47 48 49 50 Charge Number Z Fig. 11. Comparison of various theoretical halflife predictions with data for the N = 82 r-process waiting points. (from [116]) These data play crucial roles in constraining and testing nuclear models which are still necessary to predict the bulk of halflives required in r-process simulations. It is generally assumed that the halflives are dominated by allowed Gamow-Teller (GT) transitions, with forbidden transitions contributing noriceably for the heavier r-process nuclei [117]. The β decays only probe the weak low-energy tail of the GT distributions and provide quite a challenge to nuclear modeling as they are not constrained by sumrules. Traditionally the estimate of the halflives are based on the quasiparticle random phase approximation on top of the global FRDM or ETFSI models. Recently halflives for selected (spherical) nuclei have been presented using the QRPA approach based on the microscopic Hartree-Fock-Bogoliubov method [118] or a global density functional [119]; in particular the later approach achieved quite good agreement with data for spherical nuclei in different ranges of the nuclear chart. Applications of the interacting shell model [120,121,116] have yet been restricted to waiting point nuclei with magic neutron Eur. Phys. J. Special Topics 156, 123 (2008) 21 numbers. Here, however, this model which accounts for correlations beyond the QRPA approach, obtains quite good results (for an example see Fig. 11). We remark that, besides a good description of the allowed (and forbidden) transition matrix elements, the models should also provide an accurate reproduction of the Qn values. Further, r-process simulations require rates for neutron capture (whenever the (n, γ) ⇄ (γ, n) equilibrium approximation is not valid) and for the various fission processes (neutroninduced, beta-delayed, spontaneous, perhaps neutrino induced) together with the corresponding yield distributions. If the r-process occurs in strong neutrino fluences, different neutrino-induced charged-current (e.g. (νe , e− )) and neutral-current (e.g. (ν, ν ′ )) reactions, which are often accompanied by the emission of one or several neutrons [122–126], have to be modelled and included as well. We mention that the occurence of low-lying dipole strength in neutron rich nuclei, as predicted by nuclear models [127], can have noticeable effects on neutron capture cross sections [21,85]. 4.2.2 r-process network calculations Assuming that the astrophysical conditions allow for the occurrence of an r-process and that enough neutrons per seed are available the evolution of nuclear abundances has to be computed solving a system of differential equations like equation (27) that can include more than 6000 nuclear species [102], assuming some dependence of the temperature and density with time that can either be obtained from a full hydrodynamical simulation [72], a steady state approximation of the neutrino-wind [99,128] or simple analytical parametrizations [65,129]. A simplified set of equations is obtained assuming that during the r-process the dominating reactions are neutron capture, beta decay and photodissociations. (In addition fission needs to be considered if the neutron-to-seed ratio is large enough to produce the fissioning nuclei.) This approximation neglects the possibility that charge particle reactions involving light nuclei can occur even after the alpha-rich freeze-out [130,131]. In this case equation (27) simplifies: Ẏ (Z, A) = nn hσv(Z, A − 1)iY (Z, A − 1) + λγ (Z, A + 1)Y (Z, A + 1) + J X j=0 λβjn (Z − 1, A + j)Y (Z − 1, A + j) − nn hσv(Z, A)i + λγ (Z, A) + J X j=0 λβjn (Z, A) Y (Z, A) (39) where nn is the neutron density, nn hσv(Z, A)i is the thermal averaged neutron-capture rate and λγ (Z, A) the photodissociation rate for a nucleus A Z, while λβjn (Z, A) is the β − decay rate of A Z with emission of j delayed neutrons (up to a maximum of J). If the assumption is made that the neutron abundance (Yn = nn mu /ρ) varies slowly enough to be evolved explicitly, it can be assumed that nn is constant over a timestep. In this case the network can be divided into separate pieces for each isotopic chain and solve then sequentially, beginning with the lowest Z [132]. Figure 12 shows the evolution of temperature and neutron density during an r-process calculation. In addition, the right panel show the evolution of the average lifetime of a nucleus for neutron capture, photodissociation and beta-decay defined as: P 1 Z,A nn hσv(Z, A)iY (Z, A) P = (40a) τ(n,γ) Z,A Y (Z, A) P 1 Z,A λγ (Z, A)Y (Z, A) P = (40b) τ(γ,n) Z,A Y (Z, A) Eur. Phys. J. Special Topics 156, 123 (2008) 3.5 10 27 3 Nn 2 10 21 1.5 1 10 19 T 10 0.5 0 0.01 17 10 15 0.1 10 13 10 1 Time Scales (s) T (GK) 10 23 Nn (cm −3) 10 25 2.5 10 3 10 2 10 1 10 0 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 10 −7 10 −8 10 −9 0.01 500 n/seed (γ,n) 400 β − 300 200 n/seed 22 100 (n,γ) 0.1 1 0 10 Time (s) Time (s) Fig. 12. (right panel) Evolution of temperature and neutron density during an r-process calculations based in an adiabatic expansion with constant entropy s = 500 kB and velocity of 4500 km (corresponding to an expansion timescale τ = 50ms). This value of entropy and expansion time scale was chosen to produce a large enough neutron-to-seed ratio to allow to reach the region where fission takes place. The left panel shows the evolution of the neutron-to-seed ratio and average lifetime of a nucleus for neutron capture, photodissociation and beta-decay. See text for their definition. 1 = τβ P λβ (Z, A)Y (Z, A) P . Z,A Y (Z, A) Z,A (40c) the origin of time corresponds to the moment when the alpha-rich freeze-out takes place. At this time we have large temperatures and neutron-densities making the lifetime for neutron capture and photodissociation identical and much shorter than the beta-decay rate. During this phase the r-process takes place in (n, γ) ⇄ (γ, n) equilibrium. In this case using equations (17) and (19) the abundances in an isotopic chain are given by the simple relation: nn hσv(Z, A)i Y (Z, A + 1) = = Y (Z, A) λγ (Z, A + 1) 3/2 3/2 A+1 2π~2 G(Z, A + 1) Sn (Z, A + 1) exp nn mu kT A 2G(Z, A) kT (41) where Sn the neutron separation energy. For each isotopic chain, the above equation defines a nucleus that has the maximum abundance and which is normally known as waiting point nucleus as the flow of neutron captures “waits” for this nucleus to beta-decay. The set of waiting point nuclei constitutes the r-process path. The maximum of the abundance distribution can be determined setting the left-hand side of eq. (41) to 1, which results in a value of Sn that is the same for all isotopic chains for a given neutron density and temperature: T9 3 0 Sn (M eV ) = (42) 34.075 − log nn + log T9 5.04 2 where T9 is the temperature in units of 109 K and nn is the neutron density in cm−3 . Equation (42) implies that the r-process proceeds along lines of constant neutron separation energies towards heavy nuclei that for typical conditions during the r-process corresponds to Sn0 ∼ 2– 3 MeV. Due to pairing, the most abundance isotopes have always an even neutron number. For this reason, it may be more appropriate to characterize the most abundance isotope in an isotopic chain as having a two-neutron separation energy S2n = 2Sn0 [133]. If (n, γ) ⇄ (γ, n) equilibrium is valid P it is sufficient to consider the time evolution of the total abundance of an isotopic chain Y (Z) = A Y (Z, A) as the abundances of different isotopes are fully determined by equation (41). From equation (39) we can determine the time evolution of Y (Z) obtaining: Ẏ (Z) = λβ (Z − 1)Y (Z − 1) − λβ (Z)Y (Z) (43) Eur. Phys. J. Special Topics 156, 123 (2008) 10 0 10 0 β− (γ,n) (n,γ) 10 −1 10 −1 10 −2 lifetime (s) lifetime (s) 10 −2 10 −3 10 −4 10 −5 10 −6 10 −7 20 23 10 −3 10 −4 β− (γ,n) (n,γ) 10 −5 T = 1.2 GK Nn = 2.3 × 10 25 cm −3 30 T = 0.4 GK Nn = 5.0 × 10 23 cm −3 10 −6 40 50 60 70 10 −7 20 30 40 Z 50 60 70 Z Fig. 13. Average lifetimes for neutron capture, photodissociation and beta-decay for different isotopic chains during the r-process calculations. The left panel corresponds to a time when (n, γ) ⇄ (γ, n) equilibrium is valid. In the second panel due to the low temperatures the photodissociation rates are very low and the r-process proceeds under conditions where the lifetime for beta-decay and neutron capture are similar. P P where λβ (Z) = A λβ (Z, A)Y (Z, A)/ A λY (Z, A). In this case the r-process evolution is independent of the neutron-capture rates, only beta-decays are necessary for equation (43) and masses via Sn in equation (41). If the r-process proceeds in (n, γ) ⇄ (γ, n) equilibrium and its duration is larger than the beta decay lifetimes of the nuclei present, equation 43 tries to reach an equilibrium denoted as steady β-flow that satisfies: λβ (Z − 1)Y (Z − 1) = λβ (Z)Y (Z) (44) In this case the peaks at A = 130 and 195 in the solar r-process distribution can be attributed to the large β-decay rates of the waiting point nuclei with N = 82 and 126. The left panel of figure 13 shows average lifetimes for neutron capture, photodissociation and beta-decay for different isotopic chains defined similarly than equation (40) but restricting the sums to an isotopic chain. The lifetimes reach a maximum for Z = 48 and Z = 70 corresponding to N = 82 and N = 126. 400 Total (n,fission) beta fission spontaneous fission neutrino induced fission 0.1 300 200 n/seed n/seed Fission rate (s −1) 1 0.01 100 0.001 0 1 2 3 4 5 0 Time (s) Fig. 14. Evolution of the fission rates for all different fission channels for the same r-process calculation shown in the previous figures. The posterior evolution during the r-process depends on the available neutrons and expansion timescale. If the expansion proceeds very fast as the neutron density decreases neutron 24 Eur. Phys. J. Special Topics 156, 123 (2008) captures become inefficient even if neutrons are still available and the r-process freezes out. If the initial neutron-to-seed ratio and expansion timescale is low enough the freeze-out occurs when neutrons are exhausted. In any case during the freeze-out the progenitor nuclei β-decay back to the stability. The final r-process patter may differ from the freeze-out pattern due the production of neutrons via beta-delayed neutron emission and subsequent captures [134,135]. If the initial neutron-to-seed ratio is large (& 100) the r-process can reach the region where fission takes place beyond N = 184 [136,137]. In this case all fission channels (neutron induced, beta-delayed, neutrino induced and spontaneous fission) need to be considered. In addition to the appropriate fission rates, fission yield distributions are also necessary. Figure 14 shows the rate at which fission takes place due to the different channels. As every fission produces at least two fragments fission results in an increase of the abundance of heavy nuclei present. If we denote as Yh the total abundance of heavy nuclei, the fission rate can be defined as Ẏh /Yh where one includes the appropriate fission channel in the calculation of the time derivative. Fission rates for individual nuclei have been computed as described in [138] supplemented by fission yields computed using the ABLA code [139,140] using the Myers & Świaţecki fission barriers [141]. The figure shows that neutron induced fission is the dominating channel during the whole duration of the r-process. If neutron-to-seed ratio is large enough to induce fission as matter expands it will cool and for sufficiently low temperatures the photodissociation rates become negligible. However, the neutron densities are still large enough to keep the r-process going. Under this circumstances the r-process proceeds by a competition of (n, γ) and beta-decay rates. This possibility was already suggested by Blake and Schramm in 1976 [142] and has recently been “reinvented” in ref. [143]. Figure 13 (right panel) shows the average lifetimes for neutron capture, photodissociation and beta-decay for different isotopic chains once these conditions have been achieved. The figure shows clearly that the photodissociation rates are much smaller except near magic neutron numbers where both neutron capture rates and beta decays became small and the neutron separation energies are small enough to allow for photodissociations. A clear proportionality between beta-decay and neutron capture rates can also be seen in the figure. In order to accurately determine the r-process evolution during this phase it is necessary to know both beta-decays and neutron capture rates. Moreover, during this phase the r-process moves to nuclei located farther from the stability than during the (n, γ) ⇄ (γ, n) equilibrium phase. Once neutrons become to be exhausted the path tends to move back to the stability and finally the freeze-out takes place. 10 −1 10 10 −1 s = 350, n/seed = 116 s = 400, n/seed = 186 s = 450, n/seed = 245 s = 500, n/seed = 417 FRDM −2 10 −3 10 −4 10 −5 10 −5 Y Y 10 −3 −4 10 −6 10 −6 10 −7 10 −7 −8 10 −8 10 10 10 −9 60 90 120 150 A 180 210 240 s = 350, n/seed = 116 s = 400, n/seed = 186 s = 450, n/seed = 245 s = 500, n/seed = 417 ETFSI−Q 10 −2 10 −9 60 90 120 150 180 210 240 A Fig. 15. R-process abundances obtained in simulations which explore the potential impact of fission. The calculations are performed for different neutron-to-seed ratios and two different mass tabulations. The solid circles show the solar system r-process abundance distribution. (from [144]) Figure 15 show the impact of different masses in the final r-process abundances. While the studies, which use the FRDM masses [105], give quite similar abundance distributions for different neutron-to-seed ratios once a threshold value is overcomed, the calculations based on the ETFSI-Q [83] masses yield abundance results which vary strongly with the assumed value. This strikingly different behavior can be traced back to differences in the predicted masses for Eur. Phys. J. Special Topics 156, 123 (2008) 25 very neutron rich nuclei. The FRDM mass tabulation predicts that certain nuclei just above the magic neutron number N = 82 act as obstacles for the r-process matter flow holding back material and ensuring that more free neutrons are available when part of the matter reaches heavy nuclei with N = 184 to guarantee production of nuclides even beyond these waiting points. The ETFSI-Q mass tabulation does not present such obstacles on the r-process path and, as no matter is kept back at relatively small mass numbers, less neutrons are available once the matter flow reaches N = 184 suppressing matter flow beyond this waiting point [144]. We will know which of the two mass models is more realistic once masses of very neutron rich nuclei will be measured; this is one of the important aims at radioactive ion-beam facilities like FAIR, RIBF and SPIRAL 2. It should be stressed that observations of r-process abundances in old halo stars in our galaxy show always the same pattern between mass numbers A = 130 and 195 [87,88] while the patterns differ for A < 130. It is intriguing that these old stars in completely different locations of the Milky Way, which have witnessed only a few, and importantly not the same, supernova outbursts, show identical r-process patterns between the second and third peaks as obtained when averaged over the chemical history of the galaxy (solar r-process abundance). The fact that r-process simulations including fission and based on the FRDM mass model yield just such a behavior is certainly interesting. However, the nuclear and astrophysical input in these simulations is yet too uncertain to draw any conclusions. The work presented here has benefitted from a close and intensive collaboration with Karlheinz Langanke during the last decade. 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