asignatura de física nuclear universidad de santiago de ... · física nuclear, tema 15. 2. josé...

1

Universidad de Santiago de Compostela

Asignatura de Física NuclearCurso académico 2011/2012

Tema 15

Nucleosíntesis primordial y estelar

Física Nuclear, Tema 15 José Benlliure

2Física Nuclear, Tema 15 José Benlliure

Decays and reactions in the stellar plasma

1.1 Gamma-ray transitionsIn a hot plasma, excited states in a given nucleus are thermally populated through photon absorption, Coulombexcitations by surrounding ions, inelastic particle scattering or other mechanisms. The time scale for excitationand de-excitation is much shorter than stellar hydrodynamics time scales. Contrary to lab investigations wheredecaying or reacting nuclei are in their ground state, these excited states will play an important role in stellardecays or reactions. At thermal equilibrium the probability for populating a given state

can be obtained as:

kTE

kTE

eg

egNN

P /

/

The decay of 26Al represents a clear example of the role of excited states in the nuclear media. This nucleus is though to be produced In type II supernovae during the explosive carbon and neon burning phases. This nucleus decays tothe first excited state (1809 keV) in 26Mg. The observation of this gamma ray inseveral -ray telescopes as COMPTEL aboard the Gamma Ray Observatory isa major prove for nucleosynthesis processes in the Universe.

Since the first excited state in 26Al is an isomer decaying to the ground state in26Mg, the observed intensities of 1809 keV -rays can only be transformed in nucleosynthesis rate of 26Al if one takes into consideration the populations of the different excited states in 26Al, and in particular the isomeric state.

g=2j+1 being j the spin of the state

3 José Benlliure

1.2 Weak interactions-decay:In a hot plasma, excited states in a parent nucleus are thermallypopulated and may also undergo -decay transitions to thedaughter nucleus. Even stable nuclei may undergo a -decay in the stellar medium. The total -decay rate will be given by the weighted sum of the individual transition rates ij according to:

i j

ijiP * The sum over i and j runs over the parent and daughter

states and Pi can be obtained from the previous equation.

Under these conditions, -decay becomes temperature dependent by also density dependent at sufficiently large values of density when the electron gas is degenerate, limiting the number of final states available forthe electron emission.

Temperature dependenceof the decay rate of 26Al


Física Nuclear, Tema 15

4 José Benlliure

1.2 Weak interactionsElectron capture:At the temperature typical of the stellar interior most nuclei posses few, if any, bound electrons. Being theirdecay constant for bound electron capture very small. However, and due to the density of free electrons inthe stellar medium, nuclei can decay through the capture of free electrons. The probability for this process is proportional to the electron density and inversely proportional to the average electron velocity.

At low densities, the kinetic energies of the free electrons are usually small. At very high densities, however, the (Fermi) energy of the degenerate electrons may become sufficiently large to cause nuclei to undergocontinuum capture of energetic electrons, even if they are stable in the laboratory.

Pair production:At high temperatures pair production can become and effective process. Then positron capture should be considered in addition to the continuum electron capture.

Neutrino energy loss mechanism:This mechanism known as Urca process becomes important at high temperatures and densities and consistsof alternate electron captures and --decays involving the same pair of parent and daughter nuclei being the netresult of two subsequent decays of a neutrino anti-neutrino pair.

eXeXXXeX NAZN

AZN

AZN

AZN

AZ ),(),( -

11 L



5 José Benlliure

1.3 Particle induced reactionsIn the stellar medium reactions are produced by collisions of two moving particles therefore the reaction rate (reactions per time and volume unit) will be determined by the number of colliding particles per volume unit, their relative velocity and their interaction cross section.

)(1001 vvNNr

In a stellar plasma at thermodynamic equilibrium the velocity of the constituent particles follows a given distribution. Then we can generalize the expression for the reaction rate taking into account the possibilityof having identical colliding particles as:

0 01

011001101001 )1(

)()(

vNN

vNNdvvvvPNNr

Since nuclei in the plasma move non relativistically their relative velocities can be described by a Maxwell-Boltzmann distribution, then:

Em

mdE

mEe

kTmdEEPdvve

kTmdvvP kTEkTvm

224

2)( 4

2)( 01

0101

/2/3

012)2/(2/3

01 201

)( )(107318.3)()(18 113

0 0

/605.11

10

102/3

9

10/

2/3

2/1

0101

9

smolcmdEeEE

MMMM

TdEeEE

kTmvN TkTE

A

The reaction rate per mol is defined as::

With E given in MeV, T9 =T/109 in kelvin, , M in units of u and the cross section in barn.



6 José Benlliure

1.4 Photon induced reactionsIf one of the colliding particles is a photon the reaction is called photodisintegration (+30+1). Considering that photons move with the speed of light we can write the reaction rate and the corresponding decay constant as:

0

3

3

033 )()()3( )()(

dEEEcNNr

dEEEcNNr

The energy distribution of photons can be obtained from the Planck’s radiation law u(E).

dEe

Ehc

dEEEu

dEEN kTE 1)(8)(

)( /

2

3

Then, the final expression for the decay rate will be:

0 /

2

23 )(1

8)3(

dEEe

Ech kTE

In general photodisintegration are endothermic reactions, then thelower integration limit in the previous equation will be Q3 .



(decay probability per nucleus per second)(reactions per second)

7 José Benlliure

1.5 Abundance evolutionThe rate of change of the abundance of nucleus 0 due to reactions with nucleus 1 can be expressed as:

011001011

001

1

0 )1()0(

)0( vNNrNNdtdN

From these equations we obtain the following relations:

011

1011

11

1

011

1

0110101

01

1

0

0101

0101

)0(1)0(

1)1(

)0(

)0()1(1

)1()0(

vNMXvN

vNMX

vNrN

NNr

A

A

The decay constant of a nucleus for destruction via particle-induced reactions depends explicitly on the stellardensity and implicitly on temperature. If a nucleus 0 can be destroyed by different reactions its total lifetime is:

i i )0(

1)0(

1



Xi =mass fraction


Cross sections and reaction rates

2.3 Abundance evolutionThe final abundance of a nucleus can be obtained takinginto account all creation and destruction mechanisms (reactions, -decays, photodisintegrations,…)

n o pipiioiniin

kj l mmimlilijkkj

i

NNvNN

NNvNNdtdN

,,

,,,

-

In a realistic situation we should consider the evolution of not just one nuclide, but of several species simultaneously. Such a system of coupled, nonlinear ordinary differential equations is called a nuclearreaction network. Very often, equilibrium conditions together with the reciprocity theorem helps in solvingthese nuclear reaction networks.


Cross sections and reaction rates

2.4 Forward and reverse reactionsThe cross sections of a forward and reverse reaction are related by the reciprocity theorem. If we considerreactions involving particles with rest mass 0+12+3 we obtain the following relation:

)1()1(

)12)(12()12)(12(

01

23

2323

0101

32

10

2301

0123

EmEm

jjjj

kTQ

kTE

kTE

A

A emm

jjjj

dEeE

dEeE

mm

vNvN /

3/2

23

01

0132

2310

001

/230101

023

/012323

2/1

23

01

2301

0123 2301

01

23

)1)(12)(12()1)(12)(12(

For reactions involving photons, 0+1+3, we obtain these other expressions:

)1(1

)12(2)12)(12(

012

012

01

3

10

301

013

EEcm

jjj

001

/301012/3

2/1

01

03012

012

01

013

10/

2

23

301 01

)(8

)1)(12()12)(12(

18

)3(

dEeEkTN

m

dEEEcm

jjj

e

Ech

vNkTEA

kTE

A


1.6 Reaction rate equilibriaThe net reaction rate considering forward and reverse reactions is obtained as:

)1()1( 23

012332

01

23011001232301

vNNvNNrrr

being the equilibrium condition (r=0):

kTQnormnorm

normnorme

mm

GGGG

jjjj

vv

NNNN /

3/2

01

23

10

32

10

32

012301

230123

10

32 2301

)12)(12()12)(12(

)1()1(

while for reactions involving photons:

301

30110013301 )3(

)1(N

vNNrrr

kTQnormnorm

norme

GGG

jjj

kTmhv

NNN /

10

3

10

32/3

01

3/22301

0110

3 301

)12)(12()12(

)(1

2)3()1(1

This last expression is known as the Saha statistical equation.


i

kTEinorm

i g

egG

i

/


1.7 Nuclear energy generationThe nuclear energy generation in stars is given by the Q value of the thermonuclear reactions taking place in thestellar medium. The energy production per unit time and unit mass is given by::

1

0

01

2301

01

2301102301230123012301 )1()1(

dtdNQvNNQrQ

At higher temperatures also the reverse process must be considered being the net energy production:01232301

The time integrated released energy is obtained from:

finalinitial

N

N

NNN

NQdNQdtfinal

init

0010

1001

230110

01

23012301

)(

)()1(

)()1(

0

0



1.8 Nonresonant reactions induced by charged particles

max

02 12

l

lll T

kabs

21

2121

22210

1989534.0 2

2expMMMM

EZZeeZZ

EmT

hl

Absorption cross section of charged particles are dominated bythe 1/k2=1/E dependence term and the transmission coefficient of the Coulomb barrier Tl

Astrophysical S-factorIn order to performed reliable extrapolations of the measuredcross sections at energies of astrophysical interest nuclear Astrophysics introduces the astrophysics S-factor removing the 1/E dependence of the absorption cross sectionsand the s-wave Coulomb barrier probability.

)(1)( 2 ESeE

E



1.9 Nonresonant reactions induced by charged particlesWith the definition of the S-factor we can write the nonresonant reaction rate as:

0

/202/3

2/1

010

/22/3

2/1

01 )(8)(

)(8 dEeeS

kTN

mdEeESe

kTN

mvN kTEAkTEA

A

The Gamow peak:The major contribution to the reaction rate will come from energies where the product of the Maxwell-Boltzmann distribution and the Gamow factor is near its maximum. This product is known as the Gamow peak and represents the relatively narrow energy range over which most nuclear reactions occur in a stellar plasma.The location of the maximum of the Gamow peak (E=E0 ) can be obtained from the derivative of the product of these two terms:

01122

22/3

0

01210

210

01

0

kTE

meZZkTEeZZ

Em

dEd

EEhh

3/1

29

10

1021

20

3/120122

10

2

0 122.0)(2

T

MMMMZZkTmeZZE

h



1.10 Nonresonant reactions induced by neutronsNeutrons that are produced in a star quickly thermalise and their velocities are given by Maxwell-Boltzmann distributions. Although neutron induced reactions can lead to the emission of charged particles (e.g. n,p) in general these are exothermic reaction being the corresponding barrier transmission coefficients constants, then for s waves:

Ev

Tkabs

1112max

02

l

lll

Considering higher order partial waves the general expression for the reaction rates of reactions induced byneutrons assuming low neutron energies compared to the neutron binding energy is:

dEeEdEeEEkTN

mvN kTEkTEA

A

0

/2/1

0

/2/3

2/1

01)(

)(8 l

The integrand El+1/2e-E/kT represents the stellar energy window in which mostof the nonresonant neutron-induced reactions take place. The maximum of theIntegrand occurs at Emax =(l+1/2)kT



1.11 Nonresonant reactions induced by photonsFor nonresonant charged-particle emission reactions induced by photons the decay constant in terms of theastrophysical S-factor is given by the expression:

001

/2/0

001

01

2/

30101301)()()3( dEeeeESdEES

EeeQE kTEkTQkTE

Again, the integrand in this equation represents the Gamow peak centered around. 3010 QEEeff

If we consider now reactions emitting neutrons with an energy smaller than the neutron binding energy weobtain the following equation:

0

2/1301

/

0

2/101

/301)3(

dEQEedEEeQE kTEkTE ll

In this case the energy window for (,n) reactions will be located at

neff QkTE 2/1l



1.12 Narrow-resonance reaction ratesPossible resonances is the reaction cross sections could have an important impact on the reaction rates. If we consider narrow resonances (

less than few keV). Isolated resonances can be described by the Breit-Wigner formula:

4/)12)(12()1)(12(

4)( 22

10

012

r

baBW EEjj

JE

where ji are the spins of the target and projectile, J and E are the spin and energy of the resonance, i are the resonance partial widths of entrance andexit channel and is the total resonances width and. Emk 012/2/2 h

The reaction rate for a single narrow resonance can then be obtained as:

dEe

EEw

kTmNdEeEE

kTN

mvN kTE

r

baA

kTEBW

AA

0

/222/3

01

2

0

/2/3

2/1

01 4/)(2)(

)(8 h

where )11)(12(/)1)(12( 1001 jjJw



1.13 Total reaction ratesFor the calculation of the total reaction rates, all processes contributing significantly to the reaction mechanism in the effective stellar energy range have to be taken into account.

tnonresonanAk

k

resonancesbroadA

i

i

resonancesnarrowAtotalA vNvNvNvN


18

- t=10-6 s, T=1013 K (E~1000 MeV)Nucleons and anti-nucleons annihilate but are not created any moreleptons and neutrinos made possible the conversion between protons and neutrons via weakinteractions.

A mechanism should account for the present imbalance between matter and radiation (~10-9)and between matter and anti-matter CP violating decays.

First moments:- t=10-12 s, T=1016 K (E~1000 GeV)

At this temperature matter and radiation is in equilibrium, all species of particles and anti-particles are created but also annihilated.


Primordial nucleosynthesis

2.1 The early Universe: particle physics era

?

,

qq

nnppee

epn

enp

e

e

19

Proton and neutron concentrations are in equilibrium being more abundant protons becausethey are lighter than neutrons.

- t=10-2 s, T=1011 K (E~10 MeV)Electrons are the only remaining leptons, (mc2=105 MeV) and

(mc2=1784 MeV) are no longerproduced and they decay to electrons or annihilate but neutrinos are produced in neutral weak interactions.

Ffísica Nuclear, Tema 15 José Benlliure

2.1 The early Universe: particle physics era

o

oee

o

Zee

Zee

Zee

kTcmm

p

n pneNN /2

- t=1 s, T=1010 K (E~1 MeV)Neutrino interactions are no longer important and they expand freely.



2.2 The early Universe: nuclear physics era- t=190 s, T=109 K (E<0.5 MeV)

Positrons were no longer produced and they annihilated, then the proportion of protons and neutrons was fixed.

7/1~87/13/ pn NN

At this moment, protons and neutrons underwent a series of nuclear reactions leading to the transformation of protons and neutrons in 4He with no free neutrons surviving. After this processthe Universe was composed of protons, helium, electrons, photons and neutrinos.

- t~500000 y, T=5 103 K (E~0.43 eV)Under these conditions, ions and electrons combined forming a neutral gas. From this moment onthe Universe changed from being radiation dominated to be matter dominated. The consequence is that the previously opaque Universe became transparent since radiation could travel unscatteredthrough space. Indeed, the background radiation we observe now was produced at that time.


21

dnp


The first nuclear reaction that took place in the Universe was the production of deuterium in the proton andneutron fusion reaction.

2.3 First nuclear reactions

At high temperatures, the reverse reaction occurs as quickly as deuterium production, and there is no effective accumulation of deuterium nuclei. Taking into account that the photon energy necessary for photo-dissociation is 2.225 MeV one can obtain that deuterium production is effective at temperatures below T=9 108 K and thus, for a time t=250 s.

The final abundance of deuterium will depend on the neutron half life but also on the ratio between photonsand baryons densities at the early Universe. The comparison between calculated and measured abundances allow to determine one of the main cosmological parameters, the baryon density of the Universe.

E

>2.225 MeV

1 2 3 4 5MeV




2.3 First nuclear reactionsd+n 3H+d+d 3H+pd+p 3He+d+d 3He+nn+3He 3H+pd+3H 4He+nd+3He 4He+p

3H+4He 7Li+3He+4He 7Be+n+7Be 7Li+p7Be 7Li+e-+p+7Li 8Be 4He+4He

The main reactions in the Big Bang nucleosynthesis process leads mostly to the transformation of the existing neutrons in the early Universe into 4He nuclei. Additionally, some deuterium 3He, 3H andheavier species as 7Be, 7Li and 6Li are also produced.



2.4 Big Bang nucleosynthesis and cosmologyThe final abundances of different nuclear species produced duringthe Big Bang nucleosynthesys process are very much dependenton some fundamental cosmological parameters such as the baryondensity of the Universe.

Nuclear models allow to determine the expected abundances of lightnuclei produced after the Big Bang as a function of the Universebaryon density with high accuracy.

Primordial abundances obtained from astronomic abundances ordetailed characterizations of the temperature fluctuations in theCosmic background radiation (CBR) together with Big Bangnucleosynthesis models help in putting some light in fundamental questions such as:- the baryon density of the Universe- galactic chemical evolution- particle physics standard model


Nucleosíntesis estelar

HHeCONeSiFe

HHeCNeOSiFe

25

Ciclo p-pMeV 7.24224 4

eeHep


BeHeHe 743

Estrellas de la secuencia principalT ~ 8-55 MK

3.1 Combustión hidróstatica de hidrógeno

Nucleosíntesis estelar: A<60

26

Ciclo CNO

HeCpN

eNOOpNNpC

eCNNpC

e

e

41215

1515

1614

1413

1313

1312

MeV 7.24224 4 eeHep


Estrellas de la secuencia principal con una composición inicial que incluye C, N y OT ~ 8-55 MK

La captura de protones es mucho más lenta que las desintegraciones radiactivas



27

Los ciclos en los que participan núcleos con A>20 solo son posibles si existen núcleos semillas de esa masa ya que la tasa de la reacción 19F(p,)20Ne es al menos tres órdenes de magnitud inferior al de la reacción19F(p,)16O


Ciclos superioresNe-NaMg-AlSi-pS-Cl



28

MgNeHeNeOHeOCHe

MeVQCBeHeMeVQBeHeHe

24204

20164

16124

1284

844

) 37.7( ) 091.0(


Escenarios: estrellas AGB, T ~ 0.1-0.4 GK3.2 Combustión hidrostática de helio


29

SOOnSOOpPOOHeSiOOHeMgOO

MgCCpNaCCHeNeCC

321616

311616

311616

4281616

4241616

241212

231212

4201212

2


3.3 Combustión hidrostática de elementos pesados


30

Escenarios: estrellas AGB masivas o sistemasbinarios (Novas), T ~ 0.1-0.4 GKLa captura de protones compite con la desintegración .


Ciclo CNO caliente

3 4 5 6 7 8

9 10

111213

14

C (6)N (7)

O (8) F (9)

Ne (10)Na (11)

Mg (12)

3 4 5 6 7 8

9 10

111213

14

C (6)N (7)

O (8) F (9)

Ne (10)Na (11)

Mg (12)

3 4 5 6 7 8

9 10

111213

C (6)N (7)

O (8) F (9)

Ne (10)Na (11)

Mg (12)

T1/2 =1.7s

3

flow

T9 < 0.08 GK

T9 ~ 0.08-0.1 GK

T9 ~ 0.3 GK

4.1 Combustión explosiva de hidrógeno


31

Escenarios: sistemas binarios (X-ray burst), T > 0.5 GKLa captura de protones compite con la desintegración .


Ruptura del ciclo CNO

3 4 5 6 7 8

9 10

111213

14

C (6)N (7)

O (8) F (9)

Ne (10)Na (11)

Mg (12)

3

flow

T9 >~ 0.3 15O()19Ne18Ne(,p)21NaT9 >~ 0.6

4.2 Combustión explosiva de hidrógeno y helio

Nucleosíntesis estelar: A>60

32

- novas- X-ray bursts (sistemas binarios)

Escenarios astrofísicos

Reacciones- (p,)


4.3 Captura rápida de protones: proceso rp



4.2 Captura rápida de protones: proceso rp



4.3 Procesos de captura de neutrones



s-only

FeCoNi

Rb

GaGe

ZnCu

SeBr

As

ZrY

Sr

Kr(n,)

()

()

r-pro

cess

p-pro

cess

r-only

p-only

neutron number

prot

on n

umbe

r



37

r process

s process

p process




38

Número de neutrones

Núme

rode

proto

nes

semilla(n,

(,n

n ~ 107-8 cm-3

n

n ~ 1024 cm-3

n

proceso condiciones escala escenarioproceso scaptura de n

T ~ 0.1 GKn ~ 1-103 a, n ~107-8cm-3

~ 1-105-6 a estrellas AGB

proceso rcaptura de n

T ~ 1-2 GKn ~ s, n ~1024-26cm-3

< 1 s supernovas IIestre. Neutrones

proceso pcaptura de p

T ~ 2-3 GK ~ 1 s supernovas II




39Física Nuclear, Tema 15 José BenlliureFeCoNi

Rb

GaGe

ZnCu

SeBr

As

ZrY

Sr

Kr(n,)

()

()

63Ni, t1/2 =100 y64Cu, t1/2 =12 h, 40% (), 60% ()

79Se, t1/2 =65 ky

80Br, t1/2 =17 min, 92% (), 8% ()

85Kr, t1/2 =11 y

4.4 El proceso s de captura lenta de neutrones



free neutrons are unstable

they must be produced in situ

most likely candidates as neutron source are:

13C(,n)16O 22Ne(,n)25Mg

22Ne

18O(,)

+)

14N

18F(,)

25Mg

(,n)

13N

16O

13C

(,n)+)

12C(p,)

astrophysical site:

core He burning (and shell C-burning) in massive stars (e.g. 25 solar masses)T8 ~ 2.2 – 3.5

astrophysical site:

He-flashes followed by H mixing into 12C enriched zoneslow-mass (1.5 - 3 Msun ) TP-AGB starsT8 ~ 0.9 – 2.7

contribution to weak s-process contribution to main s-process

4.4 El proceso s de captura lenta de neutrones


41

N

Z

capacerrada

Línea deestabilidad

punto deespera

Camino del proceso r: equilibrio(n,)

(,n),masas

kT)(SkTmπ

A+A

A)G(Z,)+AG(Z,=

A)Y(Z,)+AY(Z,

nu

n /exp212

112/32

h


4.5 El proceso r de captura rápida de neutrones


42

N

Z

capacerrada


punto deespera

Camino del proceso r: equilibrio(n,)

(,n),masas

Velocidad del proceso r: desintegración ,vidas medias

Fin y ciclo del proceso r: fisión,barreras y modos de fisiónFísica Nuclear, Tema 15 José Benlliure



43

n >1023 cm-3

T~109 K Tiempo de irradiación Semilla inicial

Modelos independientes del escenario: estáticos o dinámicos

n = 1020

n = 1026

Entropía Tiempo de expansión Física nuclear




44

Colapso de E.N: Supernovas IIFrecuen./año y galaxia 10-5 – 10-4 2.2 10-2

Masa ejectada (Mo ) 4 10-3 – 4 10-2 10-6 – 10-5

Colapso de un sistema binariode estrellas de neutronesSupernova tipo II

Modelos hidrodinámicos de evolución estelar




45

-e epn

enpe

e- npe

,e,,e,-ee

He2p2n

C3He 12

Fe,...Si,Mg,,OXHe

T>1010 K

T~6 109 K

T~4 109 K

T~2 109 Kproceso r

proceso

n,12 C

n,pn,

n,O,Ne,Mg,Si,Fe,…

102 10103104105

R(km)

M(Mo )3

1,4

2

flujo

de ne

utrino

s(~

10 s)

Expansión adiabática Alta entropía Equilibrio nuclear Abundancia de neutrones


4.5 El proceso r de captura rápida de neutrones: explosiones supernova


46

Colapso de un sistema binario deestrellas de neutrones

Baja entropía Abundancia de neutrones




47

Producción de núcleos muy lejos de la estabilidad

Captura de neutrones: (camino del proceso r)- Secciones eficaces de captura- Masas de los núcleos participantes (n,)

(,n)

Desintegración beta: (velocidad del proceso y curva de abundancias)- vidas medias- energías de la transición Q

(masas)- tipos de transición- emisión retardada de neutrones

Fisión: (fin y ciclo del proceso r)- Probabilidades de fisión (barreras y densidades)- Distribuciones de productos de fisión- fisión inducida por neutrones- fisión inducida por neutrinos- fisión retardada por la desintegración

N

Z

capacerrada


punto deespera


4.5 El proceso r de captura rápida de neutrones: inputs de física nuclear


asignatura de física nuclear universidad de santiago de ... · física nuclear, tema 15. 2. josé...

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