Gabriel Martínez Pinedo - Research Topics · Nuclearphysicsaspectsofstellarevolutionandheavy...
Transcript of Gabriel Martínez Pinedo - Research Topics · Nuclearphysicsaspectsofstellarevolutionandheavy...
Nuclear physics aspects of stellar evolution and heavyelement nucleosynthesis
Gabriel Martínez Pinedo
The origin of cosmic elements: Past and Present Achievements, Future Challenges,Barcelona, June 10–11, 2013
Nuclear Astrophysics Virtual Institute
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Outline
1 Introduction
2 Astrophysical reaction ratesDirect capture ratesResonant reactionsNuclear Statistical Equilibrium
3 Nucleosynthesis heavy elements
4 Astrophysical SitesNeutrino-driven windsNeutron star mergers
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What is Nuclear Astrophysics?
Nuclear astrophysics aims at understanding the nuclear processes thattake place in the universe.
These nuclear processes generate energy in stars and contribute to thenucleosynthesis of the elements and the evolution of the galaxy.
Hydrogen mass fraction X = 0.71
Helium mass fraction Y = 0.28
Metallicity (mass fraction of everything else) Z = 0.019
Heavy Elements (beyond Nickel) mass fraction 4E-6
0 50 100 150 200 250
mass number
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
num
be
r fra
ctio
n
D-nuclei12C,16O,20Ne,24Mg, …. 40Ca
Fe peak
(width !)
s-process peaks (nuclear shell closures)
r-process peaks (nuclear shell closures)
AuFe Pb
U,Th
GapB,Be,Li
general trend; less heavy elements
3. The solar abundance distribution
+ +
Elemental
(and isotopic)
composition
of Galaxy at
location of solar
system at the time
of it’s formation
solar abundances:
Bulge
Halo
Disk
Sun
K. Lodders, Astrophys. J. 591, 1220-1247 (2003)
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Cosmic Cycle
interstellarmedium
densemolecular
cloudscondensation
star formation (~3%)
mixing
SN explosion
106-1010 y
102-106y
108 y
compactremnant
(WD,NS,BH)
M~104...6 Mo
SNIa
~90%
infalldustdust
SNR's &hot
bubbles
winds
starsM > 0.08 Mo
stars
Galact ic
halo
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Composition of the Universe after Big BangMatter Composit ion
76
24
8.00E-08
70.683
27.431
1.886
0
20
40
60
80
H He Metals
Mas
s F
ract
ion
(%
)
after Big Bang Nucleosynthesis
in Solar System
Stars are responsible of destroing Hydrogen and producing “metals”.
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Nuclear Alchemy: How to make Gold in Nature?
Pieter Bruegel (The Elder): The Alchemist
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Nuclear Alchemy: How to make Gold in Nature?
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Nucleosynthesis processes
In 1957 Burbidge, Burbidge, Fowler and Hoyle and independentlyCameron, suggested several nucleosynthesis processes to explain theorigin of the elements.
Ni (Z=28)
Sn (Z=50)
Pb (Z=82)
2 8
2028
50
82
126
162
184
νp-process“neutrino-proton process”rp-process“rapid proton process” via unstable proton-rich nuclei through proton capture
r-process“rapid process” viaunstable neutron-rich nuclei
Proton dripline(edge nuclear stability)
Neutron dripline(edge nuclear stability)
s-process“slow process” via chain of stable nuclei throughneutron capture
Fusion up to iron
Number of neutrons
Nu
mb
er o
f p
roto
ns
Big Bang Nucleosynthesis
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Where does the energy come from?Energy comes from nuclear reactions in the core.
41H→ 4He + 2e+ + 2νe + 26.7 MeV
E = mc2
The Sun converts 600 million tons ofhydrogen into 596 million tons of heliumevery second. The difference in mass isconverted into energy. The Sun willcontinue burning hydrogen during 5 billionsyears.Energy released by H-burning:6.45 × 1018 erg g−1 = 6.7 MeV/nucSolar Luminosity: 3.85 × 1033 erg s−1
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Nuclear Binding Energy
Liberated energy is due to the gain in nuclear binding energy.
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Type of processes
Transfer (strong interaction)
15N(p, α)12C, σ ' 0.5 b at Ep = 2.0 MeV
Capture (electromagnetic interaction)
3He(α, γ)7Be, σ ' 10−6 b at Ep = 2.0 MeV
Weak (weak interaction)
p(p, e+ν)d, σ ' 10−20 b at Ep = 2.0 MeV
b = 100 fm2 = 10−24 cm2
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Types of reactions
Nuclei in the astrophysical environment can suffer different reactions:
Decay56Ni→ 56Co + e+ + νe15O + γ → 14N + p
dna
dt= −λana
In order to dissentangle changes in the density (hydrodynamics) fromchanges in the composition (nuclear dynamics), the abundance isintroduced:
Ya =na
n, n ≈
ρ
mu= Number density of nucleons (constant)
dYa
dt= −λaYa
Rate can depend on temperature and density
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Types of reactions
Nuclei in the astrophysical environment can suffer different reactions:
Capture processesa + b→ c + d
dna
dt= −nanb〈σv〉
dYa
dt= −
ρ
muYaYb〈σv〉
decay rate: λa = ρYb〈σv〉/mu
3-body reactions:3 4He→ 12C + γ
dYαdt
= −ρ2
2m2u
Y3α〈ααα〉
decay rate: λα = Y2αρ
2〈ααα〉/(2m2u)
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Reaction ratesConsider na and nb particles per cubic centimeter of species a and b. Thenumber of nuclear reactions per unit of time and volume
a + A→ B + b
is given by:
raA =na(va)nA(vA)
(1 + δaA)σ(v)v, v = |ua − uA| (relative velocity)
In stellar environment the velocity (energy) of particles follows a thermaldistribution that depends of the type of particles.
Nuclei (Maxwell-Boltzmann)
n(v)dv = n4πv2( m2πkT
)3/2exp
(−
mv2
2kT
)dv = nφ(v)dv
Electrons, Neutrinos (if thermal) (Fermi-Dirac)
n(p)dp =g
(2π~)3
4πp2
e(E(p)−µ)/kT + 1dp
photons (Bose-Einstein)
n(p)dp =2
(2π~)3
4πp2
epc/kT − 1dp
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Stellar reaction rate
The product σv has to be averaged over the velocity distribution φ(v)(Maxwell-Boltzmann)
〈σv〉 =
∫ ∞
0φ(v)σ(v)vdv
that gives:
〈σv〉 = 4π( m2πkT
)3/2 ∫ ∞
0v3σ(v) exp
(−
mv2
2kT
)dv, m =
mamb
ma + mb
or using E = mv2/2
〈σv〉 =
(8πm
)1/2 1(kT )3/2
∫ ∞
0σ(E)E exp
(−
EkT
)dE (1)
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Cross section determination
The calculation of the cross section requires the determination of the wave function forthe system projectile (a) and target (A) for a particular value of energy E. This requiressolutions of the Schrodinger equation for a potential
V(r) = Vnuclear(r) + Vcoulomb(r) + Vcentrifugal(r)
Nuclear potential: complicated form with strong dependence on energy, E,angular momentum, J and parity, π (due to the internal structure of the target andprojectile). It is of very short range: R = 1.2(A1/3
a + A1/3A ) fm.
Coulomb potential (only for charged particles):
V(r) =ZaZAe2
r
Centrifugal barrier:
V(r) =~2l(l + 1)
2mr2
cross section suppressed for high l values. Normally s-wave (l = 0) and p-wave(l = 1) dominate.
Cross section is mainly determined by long range behaviour of the potential
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Cross section
The general form of the total cross section for the formation of a nucleuswith AC = Aa + AA and ZC = Za + ZA
a + A→ C → B + b
σ(E) = πo2∑
l
(2l + 1)Tl, o =~
mv=
~√
2mE
Tl transmission coefficient through the potential barrier.The problem reduces to a calculation of the tunneling probabilitythrough a barrier.
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Neutron capture
Fig. 2.7 Three-dimensional square-well potential of radius R0 and po-
tential depth V0. The horizontal line indicates the total particle energy
E. For the calculation of the transmission coefficient, it is necessary
to consider a one-dimensional potential step that extends from −∞ to
+∞. See the text.
A + n→ B + γ
σn(E) = πo2∑
(2l + 1)Tl,n(E)Pγ(E + Q)Tn transmision coefficient, Pγ probability of gamma emission, E neutronenergy (∼ keV), Q = mA + mn − mB = S n, Q � En.
σn(E) ∝ o2Tn(E), Tn(E) = vPl(E)
Pl(En), propability tunneling through the centrifugal barrier of momentum l.Normally s-wave dominates and P0(E) = 1.
σn(E) ∝1v2 v =
1v, 〈σnv〉 = constant
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Charged-particle reactionsStars’ interior is a neutral plasma made of charged particles (nuclei andelectrons). Nuclear reactions proceed by tunnel effect. For the p + p reactionthe Coulomb barrier is 550 keV, but the typical proton energy in the Sun is only1.35 keV.
Assuming s-wave dominates:
σ(E) = πo2T0(E), T0 = exp{−
2~
∫ Rc
Rn
√2m[V(r) − E]dr
}
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S-factor
For the coulomb potential and assuming that Rn ≈ 0 � Rc the integralgives:
T0 = e−2πη = eb/E1/2, η =
ZaZAe2
~
√m2E
η is the Sommerfeld parameter that accounts for tunneling through acoulomb barrier.We can rewrite the cross section as:
σ(E) =1E
S (E)e−2πη
S is the so-called S -factor and accounts for the short distancedependence of the cross section on the nuclear potential. It is expectedto be only mildly dependent on Energy.
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S-factor
S factor makes possible accurate extrapolations to low energy.
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Gamow window
Using definition S factor:
〈σv〉 =
(8πm
)1/2 1(kT )3/2
∫ ∞
0S (E) exp
[−
EkT−
bE1/2
]dE
exp[−
EkT−
bE1/2
]≈ exp
[−
3E0
kT
]exp
− (E − E0
∆/2
)2
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Gamow windowAssuming the S factor is constant over the gamow window and approximatingthe integrand by a Gaussian one gets:
〈σv〉 =
(2m
)1/2∆
(kT )3/2 S (E0) exp(−
3E0
kT
)with
E0 =
(bkT
2
)2/3
= 1.22(Z2aZ2
AAT 26 )1/3 keV
∆ =4√
3
√E0kT = 0.749 (Z2
aZ2AAT 5
6 )1/6 keV
(A = m/mu and T6 = T/106 K)Examples for solar conditions (T = 15 × 106 K):
reaction E0 (keV) ∆/2 (keV) exp(−3E0/kT ) T dependencep + p 5.9 3.2 1.1 × 10−6 T 3.6
14N + p 26.5 6.8 1.8 × 10−27 T 20
12C + α 56.0 9.8 3.0 × 10−57 T 42
16O + 16O 237.0 20.2 6.2 × 10−239 T 182
Reaction rate depends very sensitively on temperature
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Direct reactions
So far we have discussed the so-called “direct reactions” in which thereaction proceeds directly to a bound nuclear state:
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S-factor 3He(3He, 2p)4He
FIG. 4 (color online). The data, the best quadratic þ screening
result for S33ðEÞ, and the deduced best quadratic fit (line) and
allowed range (band) for Sbare33
. See text for references. Ade
lber
geretal,
Rev.
Mod
.Phy
s.83
,195
(201
1)
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3He(α, γ)7Be
T. Neff, Phys. Rev. Lett. 106, 042502 (2011)
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Resonant reactions
The cross section can also have contributions from resonances that canbe seen like quasi-bound states. During the reaction a quasi-bound,compound, state forms that decays by particle or gamma emission.
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Simple example of resonances
Square-well potential
Fig. 2.7 Three-dimensional square-well potential of radius R0 and po-
tential depth V0. The horizontal line indicates the total particle energy
E. For the calculation of the transmission coefficient, it is necessary
to consider a one-dimensional potential step that extends from −∞ to
+∞. See the text.
Radial wave function R(r) = u(r)/r:
uin = A sin(Kr) K = 1~
√2m(E + V0)
uout = C sin(kr + δ0) k = 1~
√2mE
Fig. 2.8 (a) Ratio of wave function intensities in the interior (r < R0)
and exterior (r > R0) region, |A|2/|C|2, and (b) phase shift δ0
versus total energy E for the scattering of neutrons (2m/ℏ2 =0.0484 MeV−1fm
−2) by a square-well potential (Fig. 2.7). For the po-
tential depth and the radius, values of V0 = 100 MeV and R0 = 3 fm,
respectively, are assumed. The curves show the resonance phenome-
non.
Enhanced transmission, resonance,appears at some particular energies.
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Cross section example
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Resonance cross sectionThe cross section for capture through an isolated resonance is given bythe Breit-Wigner formula:
a + A→ C → B + b
σ(E) = πo2 (2JC + 1)(1 + δaA)(2Ja + 1)(2JA + 1)
ΓaAΓbB
(E − Er)2 + (Γ/2)2 , o =1k
=~
pwith Γ = ΓaA + ΓbB + . . . (sum over all partial widths for all possibledecay channels). They depend on energy.Particle width
Γ(l)(E) =2~RvPl(E)θ2
l
Photon width
Γ(l)γ (E) =
8πl[(2l + 1)!!]2 B(ωl)E(2l+1)
γ , ω = Electric or Magnetic
Incoming particle, E = EaA
Outgoing particle, E = EaA + Q ≈ Q (Q � EaA, independent of energy)
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Example
Partial widths: For example theoretical calculations (Herndl et al. PRC52(95)1078)
Direct
Sp=3.34 MeV
Res.
Weak changes in
gamma widthStrong energy dependence
of proton width 62
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Reaction rate for a narrow resonance
If we assume a narrow resonance (Γ � Er and Γ � kT ) the astrophysicalreaction rate [see eq. (1)] is given by:
〈σv〉 =
(8πm
)1/2 1(kT )3/2 Er exp
(−
Er
kT
)πo2
rωΓaAΓbB
Γ∫ ∞
0
Γ
(E − Er)2 + (Γ/2)2 dE
to give:
〈σv〉 =
(2π
mkT
)3/2
~2(ωγ)r exp(−
Er
kT
)ωγ is denoted the resonant strength
ω =2JC + 1
(2Ja + 1)(2JA + 1), γ =
ΓaAΓbB
Γ
Typically Γ = ΓaA + ΓbB
if ΓaA � ΓbB then γ = ΓaA
if ΓaA � ΓbB then γ = ΓbB
reaction rate determined by smaller width
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Example
71
The Gamow window moves to higher energies with increasing temperature – therefore
different resonances play a role at different temperatures.
Gamov Window:
0.1 GK: 130-220 keV
1 GK: 500-1100 keV
0.5 GK: 330-670 keV
But note: Gamov window has
been defined for direct reaction
energy dependence !
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Helium Burning
Once hydrogen is exhausted the stellar core is made mainly ofhelium. Hydrogen burning continues in a shell surrounding the core.4He + p produces 5Li that decays in 10−22 s.Helium survives in the core till the temperature become largeenough (T ≈ 108 K) to overcome the coulomb barrier for4He + 4He. The produced 8Be decays in 10−16 s. However, thelifetime is large enough to allow the capture of another 4He:
3 4He→ 12C + γ
Hoyle suggested that in order to account for the large abundance ofCarbon and Oxygen, there should be a resonance in 12C that speedsup the production.12C can react with another 4He producing 16O
12C + α→ 16O + γ
These are the two main reactions during helium burning.
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Level scheme
Fig. 5.30 Energy level diagrams for the most important nuclides par-
ticipating in helium burning. The numbers in square brackets represent
the energy of the ground state of the system AZXN + 4
2He2 with respect
to the ground state of the nucleus A+4Z+2YN+2 (that is, the Q-value of the
(α,γ) reaction on AZXN or the α-particle separation energy of A+4
Z+2YN+2).
All information is adopted from Ajzenberg-Selove (1990) and Audi,
Wapstra and Thibault (2003).
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Tripple alpha rate
The tripple alpha rate can be written as:
〈ααα〉 = 6 · 33/2(
2π~mαkT
)3Γ7.6
~e−Er/kT
with Er = 379 keV
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alpha capture on 12C
16
resonance
(high lying)
resonance
(sub threshold)
E1E1E2 DC
resonance
(sub threshold)
E2
some tails of resonances
just make the reaction
strong enough …
complications: • very low cross section makes direct measurement impossible
• subthreshold resonances cannot be measured at resonance energy
• Interference between the E1 and the E2 components
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Alpha capture stops on 16O
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Inverse reactions
Let’s have the reaction
a + A→ B + γ Q = ma + mA − mB
We are interested in the inverse reaction. One can use detailed-balanceto determine the inverse rate. Simpler using the concept of chemicalequilibrium.
dna
dt= −nanA〈σv〉aA + (1 + δaA)nBλγ = 0(
nanA
nB
)eq
= (1 + δaA)λγ
〈σv〉aA
Using equilibrium condition for chemical potentials: µa + µA = µB
µ(Z, A) = m(Z, A)c2+kT ln
n(Z, A)G(Z,A)(T )
(2π~2
m(Z, A)kT
)3/2 , G(Z,A)(T ) =∑
i
(2Ji+1)e−Ei/(kT )
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Inverse reactions
One obtains:(nanA
nB
)eq
=GaGA
GB
(mamA
mB
)3/2 (kT
2π~2
)3/2
e−Q/kT
Finally, we obtain:
λγ =GaGA
(1 + δaA)GB
(mamA
mB
)3/2 (kT
2π~2
)3/2
e−Q/kT 〈σv〉
For a reaction a + A→ B + b (Q = ma + mA − mB − mb):
〈σv〉bB =(1 + δbB)(1 + δaA)
GaGA
GbGB
(mamA
mbmB
)3/2
e−Q/kT 〈σv〉aA
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Rate Examples: 4He(αα, γ)
0.1 1 10
Temperature (GK)
10−3
10−2
10−1
100
101
102
103
Rat
e (s
−1)
4He(αα,γ) nα = 1029 cm−3
12C(γ,αα)4He
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Rate Examples: (p, γ)
0.1 1 10
Temperature (GK)
100
101
102
103
104
105
106
107
108
109
1010
Rat
e (s
−1)
64Ga(p,γ) np = 1027 cm−3
65Ge(γ,p)
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Rate Examples: (α, γ)
0.1 1 10
Temperature (GK)
10−5
10−4
10−3
10−2
10−1
100
101
102
103
Rat
e (s
−1)
94Kr(α,γ) nα = 1027 cm−3
98Sr(γ,α)
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Rate examples: (n, γ)
0.1 1 10
Temperature (GK)
100
101
102
103
104
105
106
107
108
109
1010
Rat
e (s
−1)
130Cd(n,γ) nn = 1022 cm−3
131Cd(γ,n)130Cd beta decay
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Summary
General features reaction rates:
Reactions involving neutral particles, neutrons, are almostindependent of the temperature of the environment.
Charged particles reactions depend very strongly on temperature(tunneling coulomb barrier).
At high temperatures inverse reactions (γ, n), (γ, p), (γ, α) becomeimportant, nγ ∝ T 3
We can distinguish two different regimes:
Nuclear reactions are slower than dynamical time scales (expansion,contraction,. . . ) of the system: Hydrodynamical burning phases.
Nuclear reactions are faster than dynamical time scales: ExplosiveNucleosynthesis.
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Stellar Evolution
0 30 60 90 120 150 180 210 240
Mass Number A
−10
−8
−6
−4
−2
0
Bin
din
g E
ner
gy
per
nu
cleo
n (
MeV
)
1H
4He
16O
12C
56Fe 62Ni
208Pb
232Th
238U
Stellar Evolution
Composition stellar core is given by Nuclear Statistical Equilibrium (NSE).
Nuclear reactions operate in a time scale much shorter than any other timescale inthe system (dynamical timescale).
Composition is given by a minimum of the Free Energy: F = U − TS . Under theconstrains of conservation of number of nucleons and charge neutrality.
Nuclear Statistical Equilibrium favors free nucleons at high temperatures and iron group
nuclei at low temperatures for constant entropy (adiabatic evolution).
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Nuclear Statistical Equilibrium
The minimum of the free energy is obtained when:
µ(Z, A) = (A − Z)µn + Zµp
implies that there is an equilibrium between the processes responsiblefor the creation and destruction of nuclei:
A(Z,N)� Zp + Nn + γ
Processes mediated by the strong and electromagnetic interactionsproceed in a time scale much sorter than any other evolutionary timescale of the system.
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Nuclear abundances in NSE
Nuclei follow Boltzmann statistics:
µ(Z, A) = m(Z, A)c2 + kT ln
n(Z, A)GZ,A(T )
(2π~2
m(Z, A)kT
)3/2with GZ,A(T ) the partition function:
GZ,A(T ) =∑
i
(2Ji + 1)e−Ei(Z,A)/kT ≈π
6akTexp(akT ) (a ∼ A/9 MeV)
Results in Saha equation (B(Z, A) = Zmpc2 + (A − Z)mnc2 − M(Z, A)c2):
Y(Z, A) =GZ,A(T )A3/2
2A
(ρ
mu
)A−1
YZp YA−Z
n
(2π~2
mukT
)3(A−1)/2
eB(Z,A)/kT
Composition depends on two parameters: Yp,Yn. Determined fromconservation laws:∑
i
AiYi (Nucleon number conservation),∑
i
ZiYi = Ye (Charge conservation)
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Composition in NSE
Photon entropy per nucleon sγ ≈ 7 photon-to-baryon ratio
For a given temperature the composition is determined by entropy andYe, electron-to-nucleon ratio, that is determined by weak interactionprocesses.
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Dependence on Ye
With reduced Ye the peak of the nuclear abundance distribution movesfrom 56Ni to heavier neutron rich nuclei.
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Signatures and nucleosynthesis processes
Solar system abudances contain signatures of nuclear structure andnuclear stability.
They are the result of different nucleosynthesis processes operating indifferent astrophysical environments and the chemical evolution of thegalaxy.
0 20 40 60 80 100 120 140 160 180 200 220
Mass Number
10−2
100
102
104
106
108
1010
Abundance r
ela
tive t
o S
ilic
on =
10
6
He
D
H
Li
Be
B
CONeSiS
Ca
Fe
Ni
GeSr
XeBa
Pt
Pb
r sr s
interstellarmedium
densemolecular
cloudscondensation
star formation (~3%)
mixing
SN explosion
106-1010 y
102-106y
108 y
compactremnant
(WD,NS,BH)
M~104...6 Mo
SNIa
~90%
infalldustdust
SNR's &hot
bubbles
winds
starsM > 0.08 Mo
stars
Galact ic
halo
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Nucleosynthesis beyond iron (Traditional view)
Neutron Number
Cha
rge
Num
ber
The stable nuclei beyond ironcan be classified in threecategories depending of theirorigin:
s-process
r-process
p-process (γ-process)
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Nucleosynthesis beyond iron (Traditional picture)
Three processes contribute to the nucleosynthesis beyond iron:s-process, r-process and p-process (γ-process).
30
40
50
60
70
80
40 60 80 100 120
Z
N
p-drip
n-drip
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
80 100 120 140 160 180 200
Ab
undan
ces
[Si=
10
6]
A
180Ta
m
138La
ssr
r
p
152Gd
164Er
115Sn 180
W
s-process: relatively low neutron densities, nn = 1010−12 cm−3, τn > τβ
r-process: large neutron densities, nn > 1020 cm−3, τn < τβ.
p-process: photodissociation of s-process material.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Heavy elements and metal-poor starsCowan & Sneden, Nature 440, 1151 (2006)
30 40 50 60 70 80 90Atomic Number
−8
−6
−4
−2
0
Rel
ativ
e lo
g ε
30 40 50 60 70 80 90−8
−6
−4
−2
0
Stars rich in heavy r-process elements (Z > 52)and poor in iron (r-II stars, [Eu/Fe] > 1.0).
Robust abundance patter for Z > 52,consistent with solar r-process abundance.
These abundances seem the result of eventsthat do not produce iron. [Qian & Wasserburg,Phys. Rept. 442, 237 (2007)]
Possible Astrophysical Scenario: Neutron starmergers.
Stars poor in heavy r-process elements butwith large abundances of light r-processelements (Sr, Y, Zr)
Production of light and heavy r-processelements is decoupled.
Astrophysical scenario: neutrino-drivenwinds from core-collapse supernova
40 50 60 70 80
Atomic Number (Z)
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
log ε
(Z
)
EuHD 122563 (Honda et al. 2006)
translated pattern of CS 22892-052 (Sneden et al. 2003)
Ag
Y
PdMo
Ru
Nb
SrZr
(b)
Honda et al, ApJ 643, 1180 (2006)
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution metalicitySneden, Cowan & Gallino 2008
Core-collapse Supernovae
[Mg/Fe]
[Eu/Fe]
[Fe/H]
1.5
1.0
0.5
0
–0.5
1.5
1.0
0.5
0
–0.5
–3 –2 –1 0
a
b
Type Ia
r-process occurs already at early galactic historyr-process is related to rare events not correlated with Iron. Largescatter at large low metalicities.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
The r-process
The r-process is responsible for the synthesis of half the nuclei withA > 60 including U, Th and maybe the super-heavies.
100
150
200
250
10−310
−210
−110
0101
r−process waiting point (ETFSI−Q)
Known mass
Known half−life
N=126
N=82
Sol
ar r a
bund
ance
s
r−process path
28
30 32 34 36 38 40 42 44 46 48 50 52 54 56 58
60
62
64
66 68
70
72
74 76
78
80
82
84
86
88 90
92 94
96
98
100 102
104
106
108
110
112 114
116 118
120
122
124
126
128
130 132
134 136
138
140 142 144 146 148
150
152
154
156
158
160
162
164 166 168 170 172 174
176
178
180 182
184
186
188 190
26
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
N=184
30
32
28
Main parameter determining the nucleosynthesis is the neutron-to-seed ratio ns .
A f = Ai + ns
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Adiabatic expansions
In astrophysical environments, it is normally a good aproximation toassume that there is no exchange of heat with the environment. In thiscase the expansion is adiabatic, entropy is constant.
Normally the entropy is dominated by radiation, photons,
sγ ∼T 3
ρ
In this case the abundances of different nuclear species assuming NuclearStatistical Equilibrium are:
Y(Z,N) ∼YZ
p YNn
sA−1γ
(kT
muc2
)3(A−1)/2
eB(Z,N)/kT
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution composition: assuming NSE
Adiabatic expansion from high temperatures:
s = 50, T = 10 GK, ρ = 8.47 × 106 g cm−3, Ye = 0.48
0 10 20 30 40 50 60
Neutron number
0
10
20
30
40
50
Pro
ton n
um
ber
20
18
16
14
12
10
8
6
4
2
0
log Y
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution composition: assuming NSE
Adiabatic expansion from high temperatures:
s = 50, T = 8 GK, ρ = 3.78 × 106 g cm−3, Ye = 0.48
0 10 20 30 40 50 60
Neutron number
0
10
20
30
40
50
Pro
ton n
um
ber
20
18
16
14
12
10
8
6
4
2
0
log Y
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution composition: assuming NSE
Adiabatic expansion from high temperatures:
s = 50, T = 6 GK, ρ = 1.44 × 106 g cm−3, Ye = 0.48
0 10 20 30 40 50 60
Neutron number
0
10
20
30
40
50
Pro
ton n
um
ber
20
18
16
14
12
10
8
6
4
2
0
log Y
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution composition: assuming NSE
Adiabatic expansion from high temperatures:
s = 50, T = 4 GK, ρ = 3.76 × 105 g cm−3, Ye = 0.48
0 10 20 30 40 50 60
Neutron number
0
10
20
30
40
50
Pro
ton n
um
ber
20
18
16
14
12
10
8
6
4
2
0
log Y
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution composition: assuming NSE
Adiabatic expansion from high temperatures:
s = 50, T = 2 GK, ρ = 3.47 × 104 g cm−3, Ye = 0.48
0 10 20 30 40 50 60
Neutron number
0
10
20
30
40
50
Pro
ton n
um
ber
20
18
16
14
12
10
8
6
4
2
0
log Y
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
A recipe to produce elements heavier than Iron
The above discussion explains how Iron is produced in supernova. How can weproduce heavier elements?
0 30 60 90 120 150 180 210 240Mass Number A
−10
−8
−6
−4
−2
0B
indi
ng E
nerg
y pe
r nu
cleo
n (M
eV) n, p
4He
16O
12C
56Fe 62Ni
208Pb
232Th
238U
This can be achieved
Having many photons per nucleon. High entropy environments.
Having very fast expansions.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Why does it work?
Nuclei with A = 5 and A = 8 are not stable.
Nuclei heavier than A = 7 can only be produced by 3-body reactions:
3 4He→ 12C + γ
2 4He + n→ 9Be + γ
suppressed due to low densities (creation ∼ ρ3) and high temperatures(destruction ∼ ρnγ = ρT 3).
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
α-rich freeze out
Timescale for destruction alpha particles (creation heavy nuclei):
1 2 3 4 5 6 7 8 9 10
Temperature (GK)
10−4
10−3
10−2
10−1
100
101
102
103
Tim
esca
le (
s)
entropy = 10
entropy = 100
entropy = 1000
by reaction2 4He + n→ 9Be + γ
At the end of the alpha-rich freeze out the matter is composed ofneutrons (protons), alphas, and few heavy nuclei.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Freeze-outs as a function of entropy and Ye
(B. S. Meyer, Phys. Rept. 227, 257 (1993))
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Neutron to seed in adiabatic expansions
100 200 300
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0 1 2
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
Entropy Entropy(linear scale) (log scale)
Ele
ctr
on A
bundan
ce
2501501001 10 20 50
in a contour plot as a function of initial entropy S and for an expansion timescale of 0.05 s as expected from SNe II conditionsFIG. 9.ÈYn/Y
seedY
e
From Freiburghaus et al., Astrophys. J. 516, 381 (1999)
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Neutron to seed in adiabatic expansions
neutron-to-seed ns ∼ (1 − 2Ye)s3/τdyn, T (t) = T0e−t/τdyn
0 100 200 300 400 500 600S
0.2
0.3
0.4
0.5
Ye
τdyn = 0.0039 s 0.039 s 0.195 s
Combinations Ye, s, and τdyn necessary for producing the A = 195 peak(195Pt), ns = 100.From Y.-Z. Qian, Prog. Part. Nucl. Phys. 50, 153 (2003)
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Evolution Abundances
Calculation assuming: s = 250 k, Ye = 0.4, τdyn = 8 ms, T (t) = T0e−t/τdyn
012345678910
Temperature (GK)
10−6
10−5
10−4
10−3
10−2
10−1
100
Abundan
ce
neutronsprotons
alpha
heavy
0.001 0.01 0.1 1 10
Time (s)
−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
log
(Yn/Y
seed
)
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Which are the heaviest seed produced?
Involves an equilibrium between (α, γ) and (γ, α):
(Z,N) + α� (Z + 2,N + 2) + γ
µ(A) + µα = µ(A + 4)
Y(Z + 2, A + 4)Y(Z, A)
= nα
(2π~2
mukT
)3/2 (A + 4
4A
)3/2 G(Z + 2, A + 4)G(Z, A)
exp[
Qα(Z + 2, A + 4)kT
]The maximum for a line of constant Ye or η = (N − Z)/A occurs when:
Qα(MeV) =T9
11.605ln
7.9 × 105T 3/29
ρ5Yα
For typical values T9 = 3, Yα = 0.2, ρ5 = 0.3 we have Qα = 4.7 MeV
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Contours constant Qα
α separation energies determine the heaviest nuclei that are build beforeα-rich freeze out.
From Woosley & Hoffman, Astrophys. J. 395, 202 (1992).
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Nuclear physics needs
Masses (Sn)(location of the path)
E-decay half-lives(abundance andprocess speed)
Fission rates and distributions:• n-induced• spontaneous• E-delayed E-delayed n-emission
branchings(final abundances)
n-capture rates• for A>130
in slow freezeout• for A<130
maybe in a “weak” r-process ?
Seed productionrates (DDD,DDn, D2n, ..)
Q-physics ?
figure from H. Schatz
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Classical r-process, (n, γ)� (γ, n) equilibrium
(γ,n) photodisintegration
Equilibrium favors“waiting point”
β-decay
Temperature: ~1-2 GKDensity: 300 g/cm3 (~60% neutrons !)
Prot
on n
umbe
r
Seed
Rapid neutroncapture
neutron capture timescale: ~ 0.2 µs
Need: • mix of suitable heavy seed nuclei (A=56-90) and neutrons• sufficient large number density of neutrons (max at least ~1e24 cm-3)• sufficient large neutron/seed ratio (at least ~100)
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
(n, γ)� (γ, n) equilibrium
If the r-process occurs in (n, γ)� (γ, n) equilibrium:
µ(Z, A + 1) = µ(Z, A) + µn
Y(Z, A + 1)Y(Z, A)
= nn
(2π~2
mukT
)3/2 (A + 1
A
)3/2 G(Z, A + 1)2G(Z, A)
exp[S n(Z, A + 1)
kT
]The maximum of the abundance defines the r-process path:
S 0n(MeV) =
T9
5.04
(34.075 − log nn +
32
log T9
)For nn = 5 × 1021 cm−3 and T = 1.3 GK corresponds at S n = 3.23 MeV,
S 2n = 6.46 MeV
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Two-neutron separation energies
100 120 140 160 180 200A
0
1
2
3
4
5
6S2n/2
(MeV
)
FRDM
100 120 140 160 180 200A
0
1
2
3
4
5
6
S2n/2
(MeV
)
ETFSI-Q
100 120 140 160 180 200A
0
1
2
3
4
5
6
S2n/2
(MeV
)
HFB-17
100 120 140 160 180 200A
0
1
2
3
4
5
6
S2n/2
(MeV
)
DuZu
A. Arcones, GMP, Phys. Rev. C 83, 045809 (2011)
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Impact of nuclear masses on r-process abundances
Currently available mass models show big differences in the predictedmasses before and after the neutron shell closures, where one expectstransitions from deformed to spherical nuclei.
125 130 135 140 145 150 155 160
Mass Number
0
4
8
12
S2n (
MeV
)
rpro
cess
reg
ion
FRDMETFSI−QData
Cd (Z=48) Isotopes
FAIR reach
160 165 170 175 180 185 190 195 200 205
Mass Number
0
4
8
12
16
S2
n (
MeV
)
FRDMETFSI−QData
Er (Z=68) Isotopes
FAIR reach
rpro
cess
reg
ion
110 120 130 140 150 160 170 180 190 200 210
Mass Number
10−9
10−8
10−7
10−6
10−5
10−4
Ab
un
dan
ce
Solar AbundancesFRDMETFSI−Q
Impact on abundances
(A. Arcones & GMP, Phys. Rev. C 83, 045809 (2011))
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
FAIR: a new era in our understanding of the r-process
the FAIR reach for nuclear masses.
58
65 r-processpath
rp-processpath
20
28
50
82
8
8
20
28
50
82
126
masses measured at the FRS-ESR
stable nuclei
nuclides with known masses
Will be measured with SUPER-FRS-CR-NESR
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
r-process Astrophysical sites
Core-collapse supernova
Neutrino-winds from protoneutronstars.
Aspherical explosions, Jets,Magnetorotational Supernova, . . .[Winteler et al, ApJ 750, L22 (2012)]
Neutrino-induced r-process in Helayers [Banerjee et al., PRL 106, 201104(2011)]
Neutron star mergers
Dynamically ejected matter frommerger (Possible observationalconsequences)
Winds from accretion disks aroundblack holes [Wanajo & Janka, ApJ 746,180 (2012)]
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Neutrino emission from the proto-neutron star
− −
T ~ 1/R���������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������
���������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������
νe
−eν
µ µ τ τν , ν , ν , ν
Neutrino detection from SN1987A
0.0 5.0 10.0 15.0Time (seconds)
0.0
10.0
20.0
30.0
40.0
50.0
Ene
rgy
(MeV
)
Kamiokande IIIMB
Gravitational binding energy: Egrav ≈ GM2/R ∼ 1053 erg.Neutrino emission lasts around 10 s with energies Eν ∼ 10 MeV.Enormous neutrino fluxes around the neutron star surface:Φν = 1043 cm−2 s−1 at 20 km. Gravitational binding energynucleon ∼ 100 MeV.With Eν ∼ 10 MeV the typical neutrino-nucleon cross section is10−41 cm2. This results in interaction times of 10 ms.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Neutrinos and Explosive Nucleosynthesis
Main processes:
νe + n� p + e−
ν̄e + p� n + e+
Neutrino interactions determine theproton to neutron ratio.Proton rich ejecta〈Eν̄e 〉 − 〈Eνe 〉 < 4(mn − mp) ≈ 5.2 MeV
Early neutron-rich ejecta:r-process
Late proton-rich ejecta:νp-process
α, n
α, p α, p, nuclei
α, n, nuclei
R in
km
102
103
104
105
Rν
31.4
He
Ni
Si
PNS
ORns ~10
Neutrino cooling and
Neutrino-driven wind
n, p
νp-process
r-process
M(r) in M
νe,µ,τ, νe,µ,τ
νe,µ,τ, νe,µ,τ –
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Neutrino-driven winds and r-process
Woosley et al, ApJ 433, 229 (1994),suggested neutrino-driven windsas the r-process site.
High entropy conditions notconfirmed by any other group,Takahashi, Witti, Janka, A&A 286,857 (1994) . . .
Mass Number
Ab
un
da
nce
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Impact neutrino mean energies and YeSimulations for a 15 M� star [GMP, Fischer, Lohs, Huther, PRL 109, 251104 (2012)]
1 1.5 2 2.5 3
2
3
4
5
6
7
Lu
min
osity [
10
51
erg
s−
1]
(Un,Up) = 0
RMF (Un,Up)
0.5 1 1.5 2 2.5 30
10
20
30
40
50
60
70
80
90
100
En
tro
py p
er
Ba
ryo
n,
s [
kB]
0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
Time After Bounce [s]
Ele
ctr
on
Fra
ctio
n,
Ye
0.5 1 1.5 2 2.5 37
8
9
10
11
12
13
14
15
Time After Bounce [s]
Me
an
En
erg
y [
Me
V]
νeν̄eνµ/τν̄µ/τ
40 50 60 70 80 90 100 110
Mass Number A
10−4
10−3
10−2
10−1
100
101
102
103
104
105
Over
pro
duct
ion F
acto
r
Neutrino winds produce neutron-rich ejecta at early times
Neutron-richness sensitive to nuclear symmetry energy[see also Roberts, Reddy, & Shen, PRC 86, 065803 (2012)]
Entropy is not large enough to produce elements heavier than A ∼ 120 (Z ∼ 50).
Late ejecta may become proton-rich due to suppression of charge-currentprocesses at high densities [Fischer, GMP, Hempel, Liebendörfer, PRD 85, 083003(2012)].
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Neutron star mergers: Short gamma-ray bursts and r-process
Mergers are expected to eject around 0.01 M� of veryneutron rich material.
They are also promissing sources of gravitational waves.
Electromagnetic emission is expected from the radioactivedecay of heavy nuclei produced by the r-process.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
r-process abundances frommergers
60 90 120 150 180 210 240A
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
YFRDMDuflo ZukerETFSI−Q
Only elements above A > 120 are produced.
Robust abundance pattern due to fission.
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Transients from r-process ejecta
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Radioactive heating and light curve
The r-process heating at late timesgoes like t−1.3.
Similar to nuclear waste fromterrestrial reactors.
Independent of the ejectacomposition.
Independent of the nuclear massmodel.
Light curve reaches peakbrightness at times of 1 day,reaching luminosities 1000 timesthose of a typical Nova.
Results are sensitive to photonopacities.
Metzger, GMP, Darbha, Quataert, Arcones etal, MNRAS 406, 2650 (2010)
10−6
10−5
10−4
10−3
10−2
10−1
100
101
102
103
Time (days)
108
1010
1012
1014
1016
1018
1020
Ener
gy g
ener
atio
n (
erg s
−1 g
−1)
56NiFissionBeta decay
Total heating
0.1 1 10
Time (days)
1040
1041
1042
Lum
inosi
ty (
erg s
−1)
Efficiency = 1.0
Efficiency = 0.5
Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites
Bibliography
W. N. Cottingham and D. A. Greenwood, An Introduction to NuclearPhysics, Cambridge University Press (Cambridge, 2001)
C. Iliadis, Nuclear Physics of Stars, Wiley-VCH (Weinheim, 2007)
G. Martínez-Pinedo, Selected topics in Nuclear Astrophysics, Eur. Phys. J.Special Topics 156, 123 (2008)
S. E. Woosley, A. Heger, and T. A. Weaver, The evolution and explosion ofmassive stars, Rev. Mod. Phys. 74, 1015 (2002)
H.-Th. Janka, K. Langanke, A. Marek, G. Martínez-Pinedo and B. Müller,Theory of core-collapse supernovae, Phys. Repts. 442, 38 (2007)
B. S. Meyer, The r-, s-, and p-Processes in Nucleosynthesis, Ann. Rev. Astron.& Astrop. 32, 153 (1994)
Y.-Z. Qian, The origin of the heavy elements: Recent progress in theunderstanding of the r-process, Prog. Part. Nucl. Phys. 50, 153 (2003)
M. Arnould, S. Goriely and K. Takahashi, The r-process of stellarnucleosynthesis: Astrophysics and nuclear physics achievements andmysteries, Phys. Repts. 450, 97 (2007)