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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
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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
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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. I also would like to thank Almudena Arcones, Ivan Borzov, David Dean,
Carla Fröhlich, Alexander Heger, Raphael Hix, Thomas Janka, Andrius Juodagalvis, Aleksandra Kelic,
Hans Peter Loens, Matthias Liebendörfer, Tony Mezzacappa, Bronson Messer, Darko Mocelj, Peter
von Neumann-Cosel, Kaori Otsuki, Achim Richter, Jorge Sampaio, Karl-Heinz Schmidt, Friedrich-Karl
Thielemann and Stan Woosley for many discussions and fruitful collaborations.
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