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

Introduction Astrophysical reaction rates Nucleosynthesis heavy elements Astrophysical Sites

Outline

1 Introduction

2 Astrophysical reaction ratesDirect capture ratesResonant reactionsNuclear Statistical Equilibrium

3 Nucleosynthesis heavy elements

4 Astrophysical SitesNeutrino-driven windsNeutron star mergers


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)


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


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”.


Nuclear Alchemy: How to make Gold in Nature?

Pieter Bruegel (The Elder): The Alchemist


Nuclear Alchemy: How to make Gold in Nature?


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


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


Nuclear Binding Energy

Liberated energy is due to the gain in nuclear binding energy.


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


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


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)


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


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)


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


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.


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


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

}


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.


S-factor

S factor makes possible accurate extrapolations to low energy.


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


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


Direct reactions

So far we have discussed the so-called “direct reactions” in which thereaction proceeds directly to a bound nuclear state:


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)


3He(α, γ)7Be

T. Neff, Phys. Rev. Lett. 106, 042502 (2011)


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.


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.


Cross section example


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)


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


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


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 !


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.


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).


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


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


Alpha capture stops on 16O


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 )


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


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


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)


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)


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


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.


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).


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.


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)


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.


Dependence on Ye

With reduced Ye the peak of the nuclear abundance distribution movesfrom 56Ni to heavier neutron rich nuclei.


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


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)


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.


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)


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.


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


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


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




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




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




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




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


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.


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).


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.


Freeze-outs as a function of entropy and Ye

(B. S. Meyer, Phys. Rept. 227, 257 (1993))


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)


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)


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

)


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


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).


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


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)


(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


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)


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))


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


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)]


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.


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 –


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


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)].


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.


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.


Transients from r-process ejecta


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


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)

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