**Meet the editor**

Professor İbrahim Küçük received both his MSc. degree in 1987 and the PhD. degree in 1993 from the Middle East Technical University of Ankara, Turkey. He has joined as an instructor at the Erciyes University, Kayseri, Turkey. Currently he is a professor at the same university, head of department of Astronomy and Space Sciences. He has published several papers, primarily in

the fields of stellar evolution of low mass stars and radio astronomy. He is studying for the foundation of a national radio astronomy observatory, which will become the first in Turkey.

Contents

**Preface IX** 

**Part 1 Nucleosynthesis 1** 

Boris Pritychenko

Chapter 4 **The r-Process of Nucleosynthesis:** 

**The Puzzle Is Still with Us 61**  Marcel Arnould and Stephane Goriely

Chapter 5 **Diffuse Emission of 26Al and 60Fe in the Galaxy 89** 

Chapter 6 **Energy Generation Mechanisms in Stellar Interiors 123** 

S. Capozziello, L. Consiglio, M. De Laurentis,

**and Its Generalizations – Lie Symmetry Analysis 131** 

**From the Dark Matter Search to Alternative Hypotheses 151** 

M. L. Pumo

Wei Wang

İbrahim Küçük

Chapter 7 **The Lane-Emden-Fowler Equation** 

Chaudry Masood Khalique

**Part 2 High Energy Astrophysics 149** 

G. De Rosa and C. Di Donato

Chapter 8 **The Missing Matter Problem:** 

Chapter 1 **Nuclear Excitation Processes in Astrophysical Plasmas 3**  G. Gosselin, P. Mohr, V. Méot and P. Morel

Chapter 2 **Stellar Nucleosynthesis Nuclear Data Mining 21** 

Chapter 3 **The s-Process Nucleosynthesis in Massive Stars: Current Status and Uncertainties due to Convective Overshooting 41** 

## Contents

## **Preface XI**


	- **Part 2 High Energy Astrophysics 149**

X Contents



## **Part 3 Cosmology 341**


## Preface

The object of this book is to present a broad range of well worked out, recent theoretical and observational studies in Astrophysics. The contributions presented here include stellar nucleosynthesis and nuclear excitation processes, high energy astrophysics and cosmology. I am greatly indebted to authors, who presented valuable papers. Finally, I thank Romina Skomersic for her patience and constant support.

> **İbrahim Küçük** Erciyes University, Science Faculty, Astronomy and Space Sciences Department, Kayseri, Turkey

**Part 1** 

**Nucleosynthesis** 

## **Part 1**

**Nucleosynthesis** 

**1** 

**Nuclear Excitation Processes** 

G. Gosselin1, P. Mohr2,3, V. Méot1 and P. Morel1

In general, nuclear transitions are almost independent of the atomic environment of the nucleus. This feature is a basic prerequisite for the widely used nuclear chronometers (with the most famous example of 14C, the so-called radiocarbon dating). However, a closer look at the details of nuclear transitions shows that under special circumstances the atomic environment may affect nuclear transitions. This is most obvious for electron capture decays where the nucleus captures an electron (typically from the lowest K-shell). A nice example for the experimental verification of this effect is the dependence of the electron capture halflife of 7Be on the chemical form of the beryllium sample (Ohtsuki et al., 2004). Also the half-

electron may remain in the (otherwise completely occupied) K-shell, thus enhancing the decay Q-value and decay rate. An experimental verification was found for 187Re (Bosch et al., 1996). As electron densities in solids may also vary with temperature (e.g. in the Debye-Hückel model), -decay half-lives may also depend on temperature. However, the latest

not confirm earlier claims in this direction (Farkas et al., 2009). The relevance of temperature

Contrary to the above mentioned -decays where the role of electrons in the environment of the nucleus is obvious, the present study investigates electromagnetic transitions in nuclei. We also do not analyze electron screening where stellar reaction rates between charged particles at extremely low energies are enhanced because the repulsive Coulomb force between the positively charged nuclei is screened by the electrons in the stellar plasma. Details on electron screening can also be found in this book (Kücük, 2012) and in the latest

The electromagnetic transitions under study in this chapter are extremely important in almost any astrophysical scenario. Capture reactions like (p,), (n,), and (,) play key roles


**1. Introduction** 

lives of -

Yokoi, 1987).

study of the decay branching between -

review of solar fusion reactions (Adelberger et al., 2011).

and density dependence of -

**in Astrophysical Plasmas** 



*1CEA,DAM,DIF, Arpajon* 

*3ATOMKI, Debrecen* 

*1France 2Germany 3Hungary* 

*2Diakonie-Klinikum, Schwäbisch Hall* 

## **Nuclear Excitation Processes in Astrophysical Plasmas**

G. Gosselin1, P. Mohr2,3, V. Méot1 and P. Morel1 *1CEA,DAM,DIF, Arpajon 2Diakonie-Klinikum, Schwäbisch Hall 3ATOMKI, Debrecen 1France 2Germany 3Hungary* 

## **1. Introduction**

In general, nuclear transitions are almost independent of the atomic environment of the nucleus. This feature is a basic prerequisite for the widely used nuclear chronometers (with the most famous example of 14C, the so-called radiocarbon dating). However, a closer look at the details of nuclear transitions shows that under special circumstances the atomic environment may affect nuclear transitions. This is most obvious for electron capture decays where the nucleus captures an electron (typically from the lowest K-shell). A nice example for the experimental verification of this effect is the dependence of the electron capture halflife of 7Be on the chemical form of the beryllium sample (Ohtsuki et al., 2004). Also the halflives of - -decays may be affected by the environment: for fully ionized nuclei the emitted electron may remain in the (otherwise completely occupied) K-shell, thus enhancing the decay Q-value and decay rate. An experimental verification was found for 187Re (Bosch et al., 1996). As electron densities in solids may also vary with temperature (e.g. in the Debye-Hückel model), -decay half-lives may also depend on temperature. However, the latest study of the decay branching between - -decay and +-decay/electron capture in 74As could not confirm earlier claims in this direction (Farkas et al., 2009). The relevance of temperature and density dependence of - -decay has been studied in detail in the review (Takahashi and Yokoi, 1987).

Contrary to the above mentioned -decays where the role of electrons in the environment of the nucleus is obvious, the present study investigates electromagnetic transitions in nuclei. We also do not analyze electron screening where stellar reaction rates between charged particles at extremely low energies are enhanced because the repulsive Coulomb force between the positively charged nuclei is screened by the electrons in the stellar plasma. Details on electron screening can also be found in this book (Kücük, 2012) and in the latest review of solar fusion reactions (Adelberger et al., 2011).

The electromagnetic transitions under study in this chapter are extremely important in almost any astrophysical scenario. Capture reactions like (p,), (n,), and (,) play key roles

Nuclear Excitation Processes in Astrophysical Plasmas 5

important because such transitions may be able to produce and/or destroy isomers and thus affect s-process nucleosynthesis. The s-process path is located along the valley of stability. The level schemes of the nuclei under study are well-known; this holds in particular for the excitation energies and spins. This information allows for a careful study of the influence of

Details of the s-process are given in this book (Pumo, 2012) and in a recent review (Käppeler et al., 2011). Here we repeat very briefly the most important properties of s-process nucleosynthesis. The main component of the s-process operates in low-mass AGB stars. Two alternating neutron sources are active. The 13C(,n)16O reaction operates for about 104 to 105 years at low temperatures below 10 keV; in most cases this temperature is too low to affect isomer production or destruction. The 22Ne(,n)25Mg reaction is activated for a few years during so-called helium shell flashes at temperatures around 25 keV and densities of about 103 g/cm3. (Gallino et al., 1998). Under these conditions dramatic variations of isomer production and destruction rates can be expected (Ward & Fowler, 1980; Mohr et al. 2007; Gintautas et al., 2009; Mohr et al., 2009; Hayakawa et al., 2010). It has to be noted that the stellar transition rates may exceed the experimentally accessible ground state contribution

(Belic et al., 2002; Mohr et al., 2007b; Rauscher et al., 2011) by orders of magnitude.

Two different temperature dependencies can be found for such low-energy γ-transitions which should not be mixed up. First, the total transition rate between states is given by the sum over all contributing branchings; all these different contributions vary strongly with temperature because of the exponential temperature dependence of the surrounding blackbody radiation. Second, each individual transition may additionally be modified by the plasma environment; this may lead to an additional temperature and density dependence of individual transitions with low -ray energy. The latter effect is the main subject of this

As example we have chosen the nucleus 171Tm. It has a ground state with *Jπ* = 1/2+ and a low-lying first excited state with *Jπ* = 3/2+ at the excitation energy of 5.04 keV. This state decays by a M1 transition with small E2 admixture (mixing δ = 0.021±0.001) with a halflife of *T*1/2 = 4.77±0.08 ns. Because of its low energy, this transition is highly converted (internal conversion coefficient α = 1408±55). All data have been taken from the latest data evaluation (Baglin, 2002). 171Tm is located on a neutron-rich branch of the s-process and may be reached either via the branching at 169Er in the 168Er(n,)169Er(n,) 170Er(n,) 171Er(β- )171Tm reaction chain or via the branching at 170Tm in the 169Tm(n,)170Tm(n,)171Tm chain. It is interesting to note that the destruction of 171Tm in the 171Tm(n,)172Tm capture reaction proceeds mainly via neutron capture in the thermally excited 3/2+ state

As mentioned above, nuclear excitation in astrophysical plasmas may be significantly modified by the electronic environment -transitions of relatively low energy. Particles from the plasma other than photons interact with the nucleus and may excite it to an upper level. Thermodynamic conditions in the plasma may substantially alter the electronic environment of the nucleus and also perturb the de-excitation process of internal conversion. As a rule of thumb, every transition with an energy lower than 100 keV must be looked at, as internal

the plasma.

chapter.

(Rauscher et al., 2011).

**2. Modification of a particular transition** 

in hydrostatic and explosive burning of stars, in the neutrino production of our sun, and in the synthesis of heavy elements in the so-called s-process and r-process. Photodisintegration reactions like (,p), (,n) and (,) define the reaction path in the so-called γ-process which produces a significant amount of the rare p-nuclei. In addition, half-lives of isomeric states may be affected under stellar conditions via photon-induced excitation of so-called intermediate states.

As we will show in this chapter, -transitions which are most affected by the electronic environment are found in heavy nuclei and are characterized by relatively low -transition energies below approximately 100 keV. First of all, astrophysical processes have to be identified where such -transitions play an important role.

In early burning stages of stars from hydrogen burning up to silicon burning heavier nuclei are synthesized mainly by capture reactions along the valley of stability. Typical Q-values of these capture reactions between light nuclei are of the order of several MeV. In these scenarios -transitions are practically not affected by the surrounding plasma. A possible exception in the 7Be(p,)8B reaction will be discussed separately as a special example later in this chapter.

The synthesis of heavy nuclei proceeds mainly via neutron capture reactions in the slow and rapid neutron capture processes (s-process, r-process). The s-process path is located close to stability, and typical Q-values for neutron capture reactions are again of the order of several MeV. The corresponding capture -rays are also not significantly affected by the environment. As the r-process operates close to the neutron dripline, typical Q-values decrease down to about 2-3 MeV or even below. However, under typical r-process conditions an equilibrium between the (n,) capture and (,n) photodisintegration reaction is found, and the r-process path becomes mainly sensitive to the neutron separation energies, but almost independent of the corresponding (n,) and (,n) cross sections. Although there may be some influence of the plasma environment on the low-energy -transitions in the rprocess, there is no significant influence on the outcome of the r-process.

Contrary to the s-process and the r-process, the so-called rp-process proceeds via proton captures on the neutron-deficient side of the chart of nuclides close to the proton dripline. The Q-values of these (p,) reactions may become small. However, in general not much is known on nuclei on the path of the rp-process, and thus any discussion of the influence of the surrounding plasma on low-energy -transitions in the rp-process must remain quite speculative and is omitted in this chapter.

In the so-called p-process or -process existing heavy seed nuclei are destroyed in the thermal photon bath of a hot environment by (,p), (,n) and (,) reactions leading to the production of the low-abundance p-nuclei. Again, the required -ray energy is of the order of several MeV, by far too high for a significant influence of the plasma environment.

Further details on the various nucleosynthesis processes can be found in the latest textbooks (Iliadis, 2007; Rolfs and Rodney, 1988) and in several contributions to this book (Matteucci, 2012; Pumo, 2012; Arnould and Goriely, 2012).

At first view, it seems that the plasma environment is not able to play a significant role in any of the above processes. However, a closer look at the s-process nucleosynthesis shows that there are a number of cases where low-energy γ-transitions turn out to be extremely 4 Astrophysics

in hydrostatic and explosive burning of stars, in the neutrino production of our sun, and in the synthesis of heavy elements in the so-called s-process and r-process. Photodisintegration reactions like (,p), (,n) and (,) define the reaction path in the so-called γ-process which produces a significant amount of the rare p-nuclei. In addition, half-lives of isomeric states may be affected under stellar conditions via photon-induced excitation of so-called

As we will show in this chapter, -transitions which are most affected by the electronic environment are found in heavy nuclei and are characterized by relatively low -transition energies below approximately 100 keV. First of all, astrophysical processes have to be

In early burning stages of stars from hydrogen burning up to silicon burning heavier nuclei are synthesized mainly by capture reactions along the valley of stability. Typical Q-values of these capture reactions between light nuclei are of the order of several MeV. In these scenarios -transitions are practically not affected by the surrounding plasma. A possible exception in the 7Be(p,)8B reaction will be discussed separately as a special example later in

The synthesis of heavy nuclei proceeds mainly via neutron capture reactions in the slow and rapid neutron capture processes (s-process, r-process). The s-process path is located close to stability, and typical Q-values for neutron capture reactions are again of the order of several MeV. The corresponding capture -rays are also not significantly affected by the environment. As the r-process operates close to the neutron dripline, typical Q-values decrease down to about 2-3 MeV or even below. However, under typical r-process conditions an equilibrium between the (n,) capture and (,n) photodisintegration reaction is found, and the r-process path becomes mainly sensitive to the neutron separation energies, but almost independent of the corresponding (n,) and (,n) cross sections. Although there may be some influence of the plasma environment on the low-energy -transitions in the r-

Contrary to the s-process and the r-process, the so-called rp-process proceeds via proton captures on the neutron-deficient side of the chart of nuclides close to the proton dripline. The Q-values of these (p,) reactions may become small. However, in general not much is known on nuclei on the path of the rp-process, and thus any discussion of the influence of the surrounding plasma on low-energy -transitions in the rp-process must remain quite

In the so-called p-process or -process existing heavy seed nuclei are destroyed in the thermal photon bath of a hot environment by (,p), (,n) and (,) reactions leading to the production of the low-abundance p-nuclei. Again, the required -ray energy is of the order

Further details on the various nucleosynthesis processes can be found in the latest textbooks (Iliadis, 2007; Rolfs and Rodney, 1988) and in several contributions to this book (Matteucci,

At first view, it seems that the plasma environment is not able to play a significant role in any of the above processes. However, a closer look at the s-process nucleosynthesis shows that there are a number of cases where low-energy γ-transitions turn out to be extremely

of several MeV, by far too high for a significant influence of the plasma environment.

process, there is no significant influence on the outcome of the r-process.

speculative and is omitted in this chapter.

2012; Pumo, 2012; Arnould and Goriely, 2012).

identified where such -transitions play an important role.

intermediate states.

this chapter.

important because such transitions may be able to produce and/or destroy isomers and thus affect s-process nucleosynthesis. The s-process path is located along the valley of stability. The level schemes of the nuclei under study are well-known; this holds in particular for the excitation energies and spins. This information allows for a careful study of the influence of the plasma.

Details of the s-process are given in this book (Pumo, 2012) and in a recent review (Käppeler et al., 2011). Here we repeat very briefly the most important properties of s-process nucleosynthesis. The main component of the s-process operates in low-mass AGB stars. Two alternating neutron sources are active. The 13C(,n)16O reaction operates for about 104 to 105 years at low temperatures below 10 keV; in most cases this temperature is too low to affect isomer production or destruction. The 22Ne(,n)25Mg reaction is activated for a few years during so-called helium shell flashes at temperatures around 25 keV and densities of about 103 g/cm3. (Gallino et al., 1998). Under these conditions dramatic variations of isomer production and destruction rates can be expected (Ward & Fowler, 1980; Mohr et al. 2007; Gintautas et al., 2009; Mohr et al., 2009; Hayakawa et al., 2010). It has to be noted that the stellar transition rates may exceed the experimentally accessible ground state contribution (Belic et al., 2002; Mohr et al., 2007b; Rauscher et al., 2011) by orders of magnitude.

Two different temperature dependencies can be found for such low-energy γ-transitions which should not be mixed up. First, the total transition rate between states is given by the sum over all contributing branchings; all these different contributions vary strongly with temperature because of the exponential temperature dependence of the surrounding blackbody radiation. Second, each individual transition may additionally be modified by the plasma environment; this may lead to an additional temperature and density dependence of individual transitions with low -ray energy. The latter effect is the main subject of this chapter.

As example we have chosen the nucleus 171Tm. It has a ground state with *Jπ* = 1/2+ and a low-lying first excited state with *Jπ* = 3/2+ at the excitation energy of 5.04 keV. This state decays by a M1 transition with small E2 admixture (mixing δ = 0.021±0.001) with a halflife of *T*1/2 = 4.77±0.08 ns. Because of its low energy, this transition is highly converted (internal conversion coefficient α = 1408±55). All data have been taken from the latest data evaluation (Baglin, 2002). 171Tm is located on a neutron-rich branch of the s-process and may be reached either via the branching at 169Er in the 168Er(n,)169Er(n,) 170Er(n,) 171Er(β- )171Tm reaction chain or via the branching at 170Tm in the 169Tm(n,)170Tm(n,)171Tm chain. It is interesting to note that the destruction of 171Tm in the 171Tm(n,)172Tm capture reaction proceeds mainly via neutron capture in the thermally excited 3/2+ state (Rauscher et al., 2011).

## **2. Modification of a particular transition**

As mentioned above, nuclear excitation in astrophysical plasmas may be significantly modified by the electronic environment -transitions of relatively low energy. Particles from the plasma other than photons interact with the nucleus and may excite it to an upper level. Thermodynamic conditions in the plasma may substantially alter the electronic environment of the nucleus and also perturb the de-excitation process of internal conversion. As a rule of thumb, every transition with an energy lower than 100 keV must be looked at, as internal

not.

**2.1 Radiative excitation** 

induced emission.

(Hamilton, 1975):

excitation rate:

the number of modes:

balance.

Nuclear Excitation Processes in Astrophysical Plasmas 7

processes. Here, we consider that electrons and photons are at LTE, but obviously nuclei are

Radiative excitation occurs through the resonant absorption of a photon. Most astrophysical plasmas have a large blackbody component in their photon spectrum. A blackbody photon can easily be absorbed by a nucleus if its energy is very close to a nuclear transition energy. However, the huge number of blackbody photons ensures that the resulting rate is significant. Two de-excitation processes compete with photon absorption: spontaneous and

The microscopic cross section is a Breit and Wigner resonant capture cross section

*<sup>J</sup> h hE*

Here is the photon frequency, the gamma width of the excited nuclear level and its total width. By folding this cross section with the blackbody distribution, we deduce an

2 1 ln 2 1

The induced de-excitation rate can be deduced from the spontaneous rate by multiplying by

2 3

*c I*

*h T* 

2

2

(2)

4

(3)

(5)

2 2

 

1 *f i <sup>r</sup>*

ln 2

*f i*

*J J*

1

*r*

*e*

*E kT*

the transition radiative half-life.

(4)

*e*

 2 2

 

2 1

*J T*

*f e E <sup>i</sup> J J kT*

*J*

*ind sp d d*

*n*

where I is the blackbody radiative intensity. This gives a total de-excitation rate:

reaches a value in the same order of magnitude as the nuclear transition energy.

ln 2

*d E J J kT*

*T*

*f i r*

This total de-excitation rate and the excitation rate satisfy the principle of detailed

As mentioned above, all excitation processes were calculated for 171Tm. Fig. 1 shows excitation and induced emission rates becoming significant when the plasma temperature

*e*

 

2 1 22 1

*f i <sup>J</sup> h c*

where Tr is the radiative temperature and *<sup>f</sup> <sup>i</sup> <sup>J</sup> <sup>J</sup> T*

conversion can significantly contribute to the transition rate. Some examples for several transition energies have already been shown earlier (Gosselin et al., 2010). The two levels involved need not include the ground state, but can also be built on an isomeric state.

At least four different electromagnetic excitation processes may be able to excite a nucleus under typical astrophysical plasma conditions (Gosselin et al., 2010):


Another non-electromagnetic excitation process is inelastic neutron scattering. It will not be dealt with in here, as it is strongly dependent on the specific astrophysical plasma in which it occurs. As the neutron spectrum is not directly related to the thermodynamic conditions of the plasma, but rather to the astrophysical site under study, it is impossible to plot an excitation rate as a function of the temperature as for the other processes.

All these electromagnetic processes must be dealt with along with their inverse processes, respectively photon emission, internal conversion, bound internal conversion (BIC), superelastic scattering of electrons (a scattering process where the scattered electron gains some energy from the nucleus) and neutrons.

Describing each process is a two-step undertaking. The first step is a microscopic description of the excitation process which uses quantum mechanics formalism and aims at calculating a cross section (when there is an incident particle) or a transition rate (when there is none, such as with NEET). The electronic environment of the nucleus is described with a relativistic average atom model (RAAM) (Rozsnyai, 1972) from which an atomic potential can be extracted which depends on the density and the temperature of the plasma.

The second step is to derive a macroscopic plasma transition rate for all processes. Thermodynamics and plasma physics in the RAAM model are used to get distribution functions of photons or electrons and build the corresponding transition rate. Except for NEET, such a model is able to provide reliable values of electronic shells binding energies and occupancies, as well as a distribution function of free electrons.

If the plasma can be considered to be at local thermodynamic equilibrium (LTE), the excitation and de-excitation rate of each process are related to each other by:

$$\frac{\lambda\_e}{\lambda\_d} = \frac{2\left\|\mathbf{J}\_f + \mathbf{1}\right\|}{2\left\|\mathbf{J}\_i + \mathbf{1}\right\|} e^{-\frac{\Delta E}{kT}} \tag{1}$$

In this expression, e is the excitation rate, d the de-excitation rate, Ji and Jf the spins of the initial and final states in the nucleus and E the nuclear transition energy. This relation is known as the principle of detailed balance and expresses the micro-reversibility of excitation processes. Here, we consider that electrons and photons are at LTE, but obviously nuclei are not.

#### **2.1 Radiative excitation**

6 Astrophysics

conversion can significantly contribute to the transition rate. Some examples for several transition energies have already been shown earlier (Gosselin et al., 2010). The two levels involved need not include the ground state, but can also be built on an isomeric state.

At least four different electromagnetic excitation processes may be able to excite a nucleus

Radiative excitation. A photon from the blackbody spectrum in the plasma is absorbed

 Nuclear Excitation by Electron Capture (NEEC). A free electron from the plasma is captured onto an empty atomic shell, giving its energy to the nucleus. This is also

 Nuclear Excitation by Electron Transition (NEET). A loosely bound electron makes a transition to a deeper atomic shell and gives its energy to the nucleus (Morel et al.,

Another non-electromagnetic excitation process is inelastic neutron scattering. It will not be dealt with in here, as it is strongly dependent on the specific astrophysical plasma in which it occurs. As the neutron spectrum is not directly related to the thermodynamic conditions of the plasma, but rather to the astrophysical site under study, it is impossible to plot an

All these electromagnetic processes must be dealt with along with their inverse processes, respectively photon emission, internal conversion, bound internal conversion (BIC), superelastic scattering of electrons (a scattering process where the scattered electron gains some

Describing each process is a two-step undertaking. The first step is a microscopic description of the excitation process which uses quantum mechanics formalism and aims at calculating a cross section (when there is an incident particle) or a transition rate (when there is none, such as with NEET). The electronic environment of the nucleus is described with a relativistic average atom model (RAAM) (Rozsnyai, 1972) from which an atomic potential can be extracted which depends on the density and the temperature of the plasma. The second step is to derive a macroscopic plasma transition rate for all processes. Thermodynamics and plasma physics in the RAAM model are used to get distribution functions of photons or electrons and build the corresponding transition rate. Except for NEET, such a model is able to provide reliable values of electronic shells binding energies

If the plasma can be considered to be at local thermodynamic equilibrium (LTE), the

2 1 2 1

*J*

*J*

*d i*

*f e kT*

In this expression, e is the excitation rate, d the de-excitation rate, Ji and Jf the spins of the initial and final states in the nucleus and E the nuclear transition energy. This relation is known as the principle of detailed balance and expresses the micro-reversibility of excitation

*E*

(1)

*e*

under typical astrophysical plasma conditions (Gosselin et al., 2010):

known as Inverse Internal Conversion (Gosselin & Morel, 2004).

excitation rate as a function of the temperature as for the other processes.

and occupancies, as well as a distribution function of free electrons.

excitation and de-excitation rate of each process are related to each other by:

by the nucleus (Ward & Fowler, 1980).

energy from the nucleus) and neutrons.

Inelastic scattering of electrons (Gosselin et al., 2009).

2004).

Radiative excitation occurs through the resonant absorption of a photon. Most astrophysical plasmas have a large blackbody component in their photon spectrum. A blackbody photon can easily be absorbed by a nucleus if its energy is very close to a nuclear transition energy. However, the huge number of blackbody photons ensures that the resulting rate is significant. Two de-excitation processes compete with photon absorption: spontaneous and induced emission.

The microscopic cross section is a Breit and Wigner resonant capture cross section (Hamilton, 1975):

$$\sigma(h\nu) = \frac{2J\_f + 1}{2\left(2J\_i + 1\right)} \pi \hbar^2 c^2 \frac{\Gamma\_r \Gamma}{\left(h\nu\right)^2 \left[\left(h\nu - \Delta E\right)^2 + \frac{\Gamma^2}{4}\right]}\tag{2}$$

Here is the photon frequency, the gamma width of the excited nuclear level and its total width. By folding this cross section with the blackbody distribution, we deduce an excitation rate:

$$\mathcal{A}\_{\epsilon} = \frac{2J\_f + 1}{2J\_i + 1} \frac{\ln 2}{T\_{\dot{l}\_f \to l\_i}^{\gamma}} \frac{1}{e^{kT\_r} - 1} \tag{3}$$

where Tr is the radiative temperature and *<sup>f</sup> <sup>i</sup> <sup>J</sup> <sup>J</sup> T* the transition radiative half-life.

The induced de-excitation rate can be deduced from the spontaneous rate by multiplying by the number of modes:

$$\mathcal{A}\_d^{ind} = n\_\nu \mathcal{A}\_d^{sp} = \frac{c^2 I\_\nu}{2 \ln \nu^3} \frac{\ln 2}{T\_{I\_f \to I\_i}^{\gamma}} \tag{4}$$

where I is the blackbody radiative intensity. This gives a total de-excitation rate:

$$\mathcal{A}\_d = \frac{\ln 2}{T\_{I\_f \to I\_i}^{\gamma}} \frac{e^{\frac{\Delta E}{kT\_r}}}{e^{\frac{\Delta E}{kT\_r}} - 1} \tag{5}$$

This total de-excitation rate and the excitation rate satisfy the principle of detailed balance.

As mentioned above, all excitation processes were calculated for 171Tm. Fig. 1 shows excitation and induced emission rates becoming significant when the plasma temperature reaches a value in the same order of magnitude as the nuclear transition energy.

Nuclear Excitation Processes in Astrophysical Plasmas 9

2 1 ln 2 <sup>1</sup>

*<sup>J</sup> Tf E f E <sup>J</sup> <sup>T</sup>*

(8)

*e e FD r FD b*

with Eb the binding energy of the bound electron and fFD is the Fermi-Dirac distribution.

2 1

*f*

*f i*

 

*i J J*

Fig. 2. NEEC excitation and Internal Conversion de-excitation of 171Tm.

nuclear transition threshold becoming less and less important.

**2.3 Nuclear excitation by electron transition** 

condition (0) can more often be achieved.

At low temperatures, Fig. 2 shows a near constant internal conversion rate which is very close to the laboratory value. For higher temperatures, the number of bound electrons decreases as the atom is ionized, and the number of allowed conversions must decrease. At low temperatures, the NEEC rate is very small as there are few free electrons to be captured and few vacant atomic states on which they could be captured. The NEEC rate then rises as the temperature increases to reach a maximum. The decrease at the higher temperatures can be attributed to the rising kinetic energy of the free electrons, the fraction of which below the

Nuclear Excitation by Electron Transition (NEET) occurs when a loosely bound electron makes a transition to a deeper atomic shell and gives its energy to the nucleus. This may happen when the electronic and nuclear transition energies are very close to each other (separated by less than the atomic widths). NEET requires at least one electron on the outer atomic shell and at least a vacancy on the inner atomic shell. The energy difference between the atomic and the nuclear transition energy is called the mismatch and is denoted by . In the laboratory, such restrictive conditions can only be achieved for a very small number of nuclei. In astrophysical plasma, various conditions of density and temperature can be encountered, with a huge number of different electronic configurations at various charge states. The electronic shell binding energies are modified and the energy resonance

Fig. 1. Radiative excitation and de-excitation of 171Tm.

#### **2.2 Nuclear excitation by electron capture**

NEEC is the inverse process of internal conversion. A free electron from the plasma is captured onto an atomic shell and the excess energy is used to excite the nucleus. It still has not been observed in the laboratory despite some attempts in channeling experiments (Kimball et al., 1991) and some projects with EBIT or EBIS (Marss, 2010). Considering NEEC in plasmas has first been proposed by Doolen (Doolen, 1978) in plasma at LTE.

If an electron has a kinetic energy lower than the nuclear excitation energy, the NEEC cross section can be expressed by the Fermi golden rule (Messiah, 1961):

$$\sigma\_{\rm NEEC}\left(E\right) = \frac{2\,\pi}{\hbar \,\upsilon\_e} \left| \left\langle \psi\_f \phi\_b \right| H \left| \psi\_i \phi\_r \right\rangle \right|^2 \rho\_b\left(E\right) \tag{6}$$

where ve is the incident electron speed, i and f the nuclear initial and final states wave functions, b and r the bound and free electron wave functions and b(E) the total final state density. The matrix element is directly related to the internal conversion coefficient (Hamilton, 1975), which gives a resonant electron capture cross section:

$$\sigma\_{\rm NECC}\left(E\right) = \frac{\pi\hbar^2}{2m\_eE} \frac{2J\_f + 1}{2J\_i + 1} \frac{1\,\text{m}\,\text{2}}{T\_{I\_f \to I\_i}^{\gamma}} \alpha \frac{\hbar\Gamma}{\left(E - E\_r\right)^2 + \left(\frac{\Gamma}{2}\right)^2} \tag{7}$$

where *f i <sup>J</sup> <sup>J</sup> T* is the radiative half-life of the transition, the internal conversion coefficient, Er the resonance energy and the nuclear level width.

A NEEC rate in plasma can then be derived by folding this cross section with the free electron distribution:

8 Astrophysics

NEEC is the inverse process of internal conversion. A free electron from the plasma is captured onto an atomic shell and the excess energy is used to excite the nucleus. It still has not been observed in the laboratory despite some attempts in channeling experiments (Kimball et al., 1991) and some projects with EBIT or EBIS (Marss, 2010). Considering NEEC

If an electron has a kinetic energy lower than the nuclear excitation energy, the NEEC cross

 

where ve is the incident electron speed, i and f the nuclear initial and final states wave functions, b and r the bound and free electron wave functions and b(E) the total final state density. The matrix element is directly related to the internal conversion coefficient

2 1 ln 2

*f i*

is the radiative half-life of the transition, the internal conversion coefficient,

 

 

*e i J J <sup>r</sup>*

A NEEC rate in plasma can then be derived by folding this cross section with the free

 <sup>2</sup> <sup>2</sup> *NEEC f b ir b e*

*E HE*

   

*E E*

2

 

(7)

(6)

2

2

in plasmas has first been proposed by Doolen (Doolen, 1978) in plasma at LTE.

*v* 

section can be expressed by the Fermi golden rule (Messiah, 1961):

(Hamilton, 1975), which gives a resonant electron capture cross section:

*<sup>J</sup> <sup>E</sup>*

2

2 21

*f*

*mE J T*

*NEEC*

Er the resonance energy and the nuclear level width.

where

*f i <sup>J</sup> <sup>J</sup> T*

electron distribution:

Fig. 1. Radiative excitation and de-excitation of 171Tm.

**2.2 Nuclear excitation by electron capture** 

$$\lambda\_e = \frac{2}{2} \frac{l\_f + 1}{l\_i + 1} \frac{\ln 2}{T\_{l\_f \to l\_i}^{\gamma}} \alpha \left( T\_e \right) f\_{FD} \left( E\_r \right) \left[ 1 - f\_{FD} \left( E\_b \right) \right] \tag{8}$$

with Eb the binding energy of the bound electron and fFD is the Fermi-Dirac distribution.

Fig. 2. NEEC excitation and Internal Conversion de-excitation of 171Tm.

At low temperatures, Fig. 2 shows a near constant internal conversion rate which is very close to the laboratory value. For higher temperatures, the number of bound electrons decreases as the atom is ionized, and the number of allowed conversions must decrease. At low temperatures, the NEEC rate is very small as there are few free electrons to be captured and few vacant atomic states on which they could be captured. The NEEC rate then rises as the temperature increases to reach a maximum. The decrease at the higher temperatures can be attributed to the rising kinetic energy of the free electrons, the fraction of which below the nuclear transition threshold becoming less and less important.

#### **2.3 Nuclear excitation by electron transition**

Nuclear Excitation by Electron Transition (NEET) occurs when a loosely bound electron makes a transition to a deeper atomic shell and gives its energy to the nucleus. This may happen when the electronic and nuclear transition energies are very close to each other (separated by less than the atomic widths). NEET requires at least one electron on the outer atomic shell and at least a vacancy on the inner atomic shell. The energy difference between the atomic and the nuclear transition energy is called the mismatch and is denoted by . In the laboratory, such restrictive conditions can only be achieved for a very small number of nuclei. In astrophysical plasma, various conditions of density and temperature can be encountered, with a huge number of different electronic configurations at various charge states. The electronic shell binding energies are modified and the energy resonance condition (0) can more often be achieved.

Nuclear Excitation Processes in Astrophysical Plasmas 11

Fig. 4. Atom-Nucleus coupling matrix element of (3s1/2 –6s1/2) atomic transition for 171Tm.

are involved. The NEET rate becomes:

 

*e*

*NEET*

NEET rate higher than 103 s-1 are shown).

not be avoided.

An accurate calculation requires a good knowledge of nuclear and atomic wave functions for every configuration. However, the huge number of electronic configurations in this approach, which mirrors the DCA (Detailed Configuration Accounting) approach used to determine atomic spectra (Abdallah et al., 2008), makes such a calculation a prohibitive task. Therefore, we replace the detailed spectrum of atomic transitions by a Gaussian envelope, whose mean energy and statistical standard deviation are extracted from the RAAM (Faussurier et al., 1997). This quicker approach works the best when only outer atomic shells

2 1 , 1

The NEET rate as a function of electronic temperature is presented in Fig. 5. Many atomic transitions contribute to the total NEET rate, the 3s1/2–6s1/2 and 3p1/2-6p3/2 transitions being the two dominant ones around 4 keV (only the M1 and E2 transitions with a maximum

The average atom model cannot be steadily applied for atomic transitions involving deep shells because their average occupation numbers differ highly from the NEET requirements of at least a vacancy in the inner shell and an electron in the outer shell. In such cases, the mismatch of the RAAM mean configuration can be very different from the mismatches of the real configurations on which NEET is possible. When this discrepancy gets higher than the statistical standard deviation, a detailed configuration approach such as DCA can then

Fully comprehensive DCA calculations are still out of reach of the fastest available supercomputers. However, a careful selection of the atomic transitions may significantly

reduce the number of electronic configurations and may soon be an accessible goal.

*<sup>N</sup> TN R e*

2

<sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup>

2

(11)

<sup>1</sup> 1,2 <sup>2</sup> <sup>2</sup>

The probability of NEET occurring on an isolated and excited atom is given by:

$$P\_{NEET}\left(\mathcal{S}\right) = \frac{\left|R\_{1,2}\right|^2}{\delta^2 + \left(\frac{\Gamma\_1 + \Gamma\_2}{2}\right)^2} \left(1 + \frac{\Gamma\_2}{\Gamma\_1}\right) \tag{9}$$

where 1 and 2 represent the total width (atomic and nuclear) of the initial and final configurations, respectively, and |R1,2| is the atom-nucleus matrix coupling element (Morel et al., 2004). The NEET probability reaches a maximum when the mismatch is zero. Mismatch variations for a 3s-6s atomic transition on Fig. 3 exhibit matching conditions around a temperature of 4 keV for a plasma density of 100 g/cm3.

Fig. 3. Mismatch of (3s1/ –6s1/2) atomic transition for 171Tm.

The atom-nucleus coupling matrix element is little sensitive to temperature as illustrated on Fig. 4. As the temperature increases, there are less bound electrons. This reduces screening of the nucleus by the remaining bound electrons, whose orbitals are closer to the nucleus. The overlap between the electron and the nucleus wave functions is larger, which increases the coupling.

In plasma, the NEET rate can be estimated as a summation over all initial configurations of the rate of creation of such a configuration multiplied by the NEET probability:

$$\mathcal{A}^{\text{NEET}}\left(\rho, T\_{\epsilon}\right) = \sum\_{\alpha} P\_{\alpha}\left(\rho, T\_{\epsilon}\right) \frac{\Gamma\_{\alpha}}{\hbar} N\_{1} \left(1 - \frac{N\_{2}}{\Omega\_{2}}\right) \frac{\left|R\_{\alpha, \beta}\right|^{2}}{\delta\_{\alpha, \beta}^{2} + \left(\frac{\Gamma\_{\alpha} + \Gamma\_{\beta}}{2}\right)^{2}} \tag{10}$$

Here, N1 and N2 are the initial electronic occupations of the two atomic shells involved in the transition, of degeneracy 1 and 2, corresponding to outer and inner shell, respectively.

10 Astrophysics

*R*

2

 

2

 

where 1 and 2 represent the total width (atomic and nuclear) of the initial and final configurations, respectively, and |R1,2| is the atom-nucleus matrix coupling element (Morel et al., 2004). The NEET probability reaches a maximum when the mismatch is zero. Mismatch variations for a 3s-6s atomic transition on Fig. 3 exhibit matching conditions

The atom-nucleus coupling matrix element is little sensitive to temperature as illustrated on Fig. 4. As the temperature increases, there are less bound electrons. This reduces screening of the nucleus by the remaining bound electrons, whose orbitals are closer to the nucleus. The overlap between the electron and the nucleus wave functions is larger, which increases

In plasma, the NEET rate can be estimated as a summation over all initial configurations of

Here, N1 and N2 are the initial electronic occupations of the two atomic shells involved in the transition, of degeneracy 1 and 2, corresponding to outer and inner shell, respectively.

2

 

> 

> > 2

 

, 2 1 2 2 2 ,

(10)

*N R*

 

the rate of creation of such a configuration multiplied by the NEET probability:

, ,1

*T PT N*

 

*e e*

*NEET*

1,2 2 2 2 1 1 2

1

(9)

The probability of NEET occurring on an isolated and excited atom is given by:

*NEET*

around a temperature of 4 keV for a plasma density of 100 g/cm3.

Fig. 3. Mismatch of (3s1/ –6s1/2) atomic transition for 171Tm.

the coupling.

*P*

Fig. 4. Atom-Nucleus coupling matrix element of (3s1/2 –6s1/2) atomic transition for 171Tm.

An accurate calculation requires a good knowledge of nuclear and atomic wave functions for every configuration. However, the huge number of electronic configurations in this approach, which mirrors the DCA (Detailed Configuration Accounting) approach used to determine atomic spectra (Abdallah et al., 2008), makes such a calculation a prohibitive task. Therefore, we replace the detailed spectrum of atomic transitions by a Gaussian envelope, whose mean energy and statistical standard deviation are extracted from the RAAM (Faussurier et al., 1997). This quicker approach works the best when only outer atomic shells are involved. The NEET rate becomes:

$$\mathcal{X}^{NET}\left(\rho, T\_e\right) = \frac{2\pi}{\hbar} N\_1 \left(1 - \frac{N\_2}{\Omega\_2}\right) \left|\mathcal{R}\_{1,2}\right|^2 \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{\delta^2}{2\sigma^2}}\tag{11}$$

The NEET rate as a function of electronic temperature is presented in Fig. 5. Many atomic transitions contribute to the total NEET rate, the 3s1/2–6s1/2 and 3p1/2-6p3/2 transitions being the two dominant ones around 4 keV (only the M1 and E2 transitions with a maximum NEET rate higher than 103 s-1 are shown).

The average atom model cannot be steadily applied for atomic transitions involving deep shells because their average occupation numbers differ highly from the NEET requirements of at least a vacancy in the inner shell and an electron in the outer shell. In such cases, the mismatch of the RAAM mean configuration can be very different from the mismatches of the real configurations on which NEET is possible. When this discrepancy gets higher than the statistical standard deviation, a detailed configuration approach such as DCA can then not be avoided.

Fully comprehensive DCA calculations are still out of reach of the fastest available supercomputers. However, a careful selection of the atomic transitions may significantly reduce the number of electronic configurations and may soon be an accessible goal.

Nuclear Excitation Processes in Astrophysical Plasmas 13

Fig. 6. WKB Inelastic electron scattering cross section of 171Tm.

Fig. 7. Inelastic electron scattering excitation rate of 171Tm.

Fig. 8 summarizes excitation rates of the four electromagnetic processes present in plasma. NEEC dominates at low temperatures, radiative excitation at the hottest temperatures and NEET in between. The electron inelastic scattering never dominates although this situation

**2.5 Lifetime evolution** 

could change at higher plasma density.

Fig. 5. NEET excitation of 171Tm.

## **2.4 Inelastic electron scattering**

In astrophysical plasmas, there usually exists a huge number of free electrons. If the nuclear transition energy is low enough, a significant part of these electrons might be able to excite the nucleus through Coulomb excitation. A semi-classical theory works well for high -above 1 MeV- energy ions (Alder et al., 1956), whereas a more sophisticated quantum model is required to deal with electrons in the keV range. With an unscreened atomic potential, a WKB approximation can be successfully implemented (Gosselin et al., 2009). The WKB cross section is very close to the more exact usual DWBA quantum approach (Alder et al., 1956) but much computationally heavier.

Cross sections exhibit usually low values in the 10-30 cm2 range which can be at least partially compensated by the huge number of free electrons in high temperature plasmas (above the nuclear transition energy). However, these cross sections also exhibit a nonphysical behavior close to the energy threshold where it does not drop to zero as it should do, as can be seen on Fig. 6. This can be explained by an "acceleration" of the incident electron by the unscreened potential as the global neutrality of the atom is not verified. Using a screened potential in the future will allow to get rid of this artifact.

By folding this cross section with the free electron distribution, we easily deduce an electron inelastic scattering excitation rate in plasma as shown on Fig. 7. It is negligible at low temperatures when there are very few free electrons, and these electrons do not have a high enough energy to be above the threshold. This changes when the temperature reaches values around the nuclear transition energy. At high temperatures, the excitation rate does not vary much as the lowering cross section is compensated by the increasing velocity of the electrons.

12 Astrophysics

In astrophysical plasmas, there usually exists a huge number of free electrons. If the nuclear transition energy is low enough, a significant part of these electrons might be able to excite the nucleus through Coulomb excitation. A semi-classical theory works well for high -above 1 MeV- energy ions (Alder et al., 1956), whereas a more sophisticated quantum model is required to deal with electrons in the keV range. With an unscreened atomic potential, a WKB approximation can be successfully implemented (Gosselin et al., 2009). The WKB cross section is very close to the more exact usual DWBA quantum approach (Alder et al., 1956)

Cross sections exhibit usually low values in the 10-30 cm2 range which can be at least partially compensated by the huge number of free electrons in high temperature plasmas (above the nuclear transition energy). However, these cross sections also exhibit a nonphysical behavior close to the energy threshold where it does not drop to zero as it should do, as can be seen on Fig. 6. This can be explained by an "acceleration" of the incident electron by the unscreened potential as the global neutrality of the atom is not verified.

By folding this cross section with the free electron distribution, we easily deduce an electron inelastic scattering excitation rate in plasma as shown on Fig. 7. It is negligible at low temperatures when there are very few free electrons, and these electrons do not have a high enough energy to be above the threshold. This changes when the temperature reaches values around the nuclear transition energy. At high temperatures, the excitation rate does not vary much as the lowering cross section is compensated by the increasing velocity of the

Using a screened potential in the future will allow to get rid of this artifact.

Fig. 5. NEET excitation of 171Tm.

**2.4 Inelastic electron scattering** 

but much computationally heavier.

electrons.

Fig. 6. WKB Inelastic electron scattering cross section of 171Tm.

Fig. 7. Inelastic electron scattering excitation rate of 171Tm.

#### **2.5 Lifetime evolution**

Fig. 8 summarizes excitation rates of the four electromagnetic processes present in plasma. NEEC dominates at low temperatures, radiative excitation at the hottest temperatures and NEET in between. The electron inelastic scattering never dominates although this situation could change at higher plasma density.

Nuclear Excitation Processes in Astrophysical Plasmas 15

both the ground and excited states to the LTE (for the nuclear levels) relative populations, that is populations whose ratio is given by the Boltzmann factor. For high temperatures and a higher spin for the excited state, the population of the excited state reaches a higher value

As shown before in this chapter, thermalization between the ground state and the first excited state of 171Tm is achieved on a timescale far below 1 second, i.e. much faster than the timescales of the astrophysical s-process. This result remains valid as long as the levels under study are connected by a direct γ-transition or a γ-cascade. A similar result has been derived explicitly for the ground state band of 176Lu (Gintautas et al., 2009). Thus, because of the prompt thermalization within γ-bands it is obvious that the stellar transition rate of any photon-induced reaction has to be calculated including all contributions of thermally excited states. This holds in particular for (γ,n) and (γ,α) reactions for the astrophysical γ-process, but also for the photodestruction of isomers in the s-process. Because the required photon energies for (γ,n) and (γ,α) reactions is of the order of several MeV, the influence of the surrounding plasma remains small, and we focus on the photodestruction of isomers in the

The *K*-isomers are found in heavy deformed nuclei. Transitions between states with large differences in the *K* quantum number are strongly suppressed by selection rules. Thus, there are no direct transitions between low-*K* states and high-*K* states. As a consequence, thermalization has to proceed via higher-lying so-called intermediate states (IMS) with intermediate *K* quantum number which have a decay branching to the low-*K* and to the high-*K* part of the excitation spectrum of the respective nucleus. The transition rates from the low-*K* side and the high-*K* side of the spectrum to the IMS define the timescale for thermalization. Typically, these IMS are located at excitation energies above 500 keV. They decay down to the lowest states with low and high *K* by γ-cascades where energies of the individual γ-transitions are obviously much smaller than the excitation energy, i.e. the energies may be as low as 100 keV or even below. This is the energy region where the plasma effects become important. The full formalism for the calculation of stellar reaction rates is given in an earlier work (Gosselin et al., 2010). The essential result is that the stellar reaction rate λ\* for transitions from the low-*K* to the high-*K* states (and reverse) can be

> \* 1 2 1 2

where Γ1 and Γ2 are the total decay widths to the low-*K* and the high-*K* states respectively in the deformed nucleus under stellar conditions (i.e., summed over all levels, each particular transition width Γi→f being modified by the plasma environment according to the discussion in this work). It should be noted that the smaller of the two widths Γ1 and Γ2 essentially defines the stellar reaction rate λ\* for transitions from the low-*K* to the high-*K* states because

(12)

derived by a formula similar to Eq. (2) which leads to

the larger width cancels out in the above equation.

than that of the ground state.

**3. Stellar transition rates** 

following.

Fig. 8. Nuclear excitation rate in plasma of 171Tm.

Highly variable rates of excitation also means highly variable rates of de-excitation which means that the nuclear lifetime of the excited level now depends on the plasma temperature. Fig. 9 shows these variations which cover nearly two orders of magnitude in the particular example of 171Tm at 100 g/cm3. Some other nuclei even exhibit larger variations, such as 201Hg (Gosselin et al., 2007) where the lifetime can be increased by a factor of more than 104.

Fig. 9. Lifetime in plasma of 171Tm.

However, this lifetime is an extension of the usual notion of lifetime in the laboratory. This plasma lifetime is the characteristic time required to get from any relative populations of both the ground and excited states to the LTE (for the nuclear levels) relative populations, that is populations whose ratio is given by the Boltzmann factor. For high temperatures and a higher spin for the excited state, the population of the excited state reaches a higher value than that of the ground state.

## **3. Stellar transition rates**

14 Astrophysics

Highly variable rates of excitation also means highly variable rates of de-excitation which means that the nuclear lifetime of the excited level now depends on the plasma temperature. Fig. 9 shows these variations which cover nearly two orders of magnitude in the particular example of 171Tm at 100 g/cm3. Some other nuclei even exhibit larger variations, such as 201Hg (Gosselin et al., 2007) where the lifetime can be increased by a

However, this lifetime is an extension of the usual notion of lifetime in the laboratory. This plasma lifetime is the characteristic time required to get from any relative populations of

Fig. 8. Nuclear excitation rate in plasma of 171Tm.

factor of more than 104.

Fig. 9. Lifetime in plasma of 171Tm.

As shown before in this chapter, thermalization between the ground state and the first excited state of 171Tm is achieved on a timescale far below 1 second, i.e. much faster than the timescales of the astrophysical s-process. This result remains valid as long as the levels under study are connected by a direct γ-transition or a γ-cascade. A similar result has been derived explicitly for the ground state band of 176Lu (Gintautas et al., 2009). Thus, because of the prompt thermalization within γ-bands it is obvious that the stellar transition rate of any photon-induced reaction has to be calculated including all contributions of thermally excited states. This holds in particular for (γ,n) and (γ,α) reactions for the astrophysical γ-process, but also for the photodestruction of isomers in the s-process. Because the required photon energies for (γ,n) and (γ,α) reactions is of the order of several MeV, the influence of the surrounding plasma remains small, and we focus on the photodestruction of isomers in the following.

The *K*-isomers are found in heavy deformed nuclei. Transitions between states with large differences in the *K* quantum number are strongly suppressed by selection rules. Thus, there are no direct transitions between low-*K* states and high-*K* states. As a consequence, thermalization has to proceed via higher-lying so-called intermediate states (IMS) with intermediate *K* quantum number which have a decay branching to the low-*K* and to the high-*K* part of the excitation spectrum of the respective nucleus. The transition rates from the low-*K* side and the high-*K* side of the spectrum to the IMS define the timescale for thermalization. Typically, these IMS are located at excitation energies above 500 keV. They decay down to the lowest states with low and high *K* by γ-cascades where energies of the individual γ-transitions are obviously much smaller than the excitation energy, i.e. the energies may be as low as 100 keV or even below. This is the energy region where the plasma effects become important. The full formalism for the calculation of stellar reaction rates is given in an earlier work (Gosselin et al., 2010). The essential result is that the stellar reaction rate λ\* for transitions from the low-*K* to the high-*K* states (and reverse) can be derived by a formula similar to Eq. (2) which leads to

$$
\lambda^\* \propto \frac{\Gamma\_1 \Gamma\_2}{\Gamma\_1 + \Gamma\_2} \tag{12}
$$

where Γ1 and Γ2 are the total decay widths to the low-*K* and the high-*K* states respectively in the deformed nucleus under stellar conditions (i.e., summed over all levels, each particular transition width Γi→f being modified by the plasma environment according to the discussion in this work). It should be noted that the smaller of the two widths Γ1 and Γ2 essentially defines the stellar reaction rate λ\* for transitions from the low-*K* to the high-*K* states because the larger width cancels out in the above equation.

Nuclear Excitation Processes in Astrophysical Plasmas 17

179Hf(n,γ)180mHf( β-)180mTa reaction chain (Schumann & Käppeler, 1999; Beer & Ward, 1981;

A common problem in all above production scenarios is the survival of 180Ta in its isomeric state. If the production occurs in a high-temperature environment like any supernova explosion, then 180Ta is produced in thermal equilibrium between the low-*K* ground state band and the high-*K* isomeric band. Following the evolution of the isomer-to-ground state ratio, freeze-out is found around 40 keV with a survival probability of 180Ta in its isomeric state of 0.38±0.01 (Hayakawa et al., 2010). Under s-process conditions with its lower temperature, it is not clear how much 180Ta can survive in the isomeric state because of mixing between hotter and cooler areas of the thermally pulsing AGB star (Mohr et al., 2007). The resulting yield of 180Ta depends sensitively on the properties of the lowest

There is indirect confirmation for the existence of IMS which couple the low-*K* ground state and the high-*K* isomeric state from photoactivation experiments (Belic et al., 2002; Collins et al., 1990). However, no direct γ-transition has been observed up to now. Based on reasonable estimates for transition strengths it has been suggested (Mohr et al., 2007) that the lowest IMS is located at an excitation energy of 594 keV with *J*=(5). This state decays with a transition energy of 72 keV and a half-life of about 16 ns to the low-*K* side of 180Ta. A weak branch to the high-K side can be expected via a transition to the 7+ state at 357 keV. This may lead to thermalization of 180Ta within days at the s-process temperatures around

Because of the low transition energy of only 72 keV, a significant modification of the corresponding transition strength can be expected. Enhancements of the radiative strength of up to a factor of 10 for this transition have been calculated (Gosselin et al., 2010) which are mainly based on NEEC. However, unfortunately the influence on the stellar transition rate between low-*K* states and high-*K* states remains very small. The stellar transition rate as given in Eq. (12) is essentially defined by the weak branch of the 594 keV state which has a transition energy of more than 200 keV. As transitions with such high energies are practically not affected by the plasma environment, the stellar transition rate remains almost

**B reaction, and solar neutrino production** 

The 7Be(p,γ)8B reaction is the key reaction for the production of high-energy neutrinos in our sun. It is one of the very few examples for a capture reaction between light nuclei where low-energy γ-rays play a significant role. The Q-value of this reaction is extremely low (137 keV), and together with the most effective energy of about 18 keV (at temperatures around 15 million Kelvin, typical for the center of our sun) we find a transition energy of about 155 keV in the low-mass (*Z*=5) nucleus 8B. The γ-energy of 155 keV is still too high to be significantly influenced by the surrounding plasma. Although experimental conditions in the laboratory (either a proton beam and neutral 7Be target or a 7Be beam in arbitrary charge state on a neutral hydrogen target) are quite different from the stellar environment of 7Be, the cross sections from laboratory experiments do not require a plasma correction for the electromagnetic transition strength to calculate the stellar reaction rates, as e.g. summarized in the NACRE compilation (Angulo et al., 1999)

intermediate state (IMS) which couples the low-*K* and high-*K* bands.

Yokoi & Takahashi, 1983; Mohr et al., 2007).

25 keV (Mohr et al., 2007).

unchanged.

**B, the <sup>7</sup>**

**Be(p,γ) 8**

**4.3 <sup>8</sup>**

## **4. Some selected examples**

In the following section we discuss three examples in greater detail. The first two examples are so-called *K*-isomers which are relevant in the astrophysical s-process (176Lu and 180Ta). The last example is the 7Be(p,)8B reaction with its low reaction Q-value of 137 keV.

## **4.1 176Lu in the astrophysical s-process**

176Lu is a so-called s-only nucleus because it is synthesized only in the astrophysical sprocess. It is produced either in its high-*K J*π;*K*=7- ;7 ground state with a half-life of 38 gigayears (quasi-stable for the s-process) or in its low-lying low-*K J*π;*K*=1- ;0 isomer at 123 keV with a half-life of less than 4 hours. The 175Lu(n,)176Lu reaction produces most of 176Lu in the low-*K* isomeric state which decays by β emission to 176Hf. 176Lu can only survive if the isomer is coupled to the high-*K* ground state (Heil et al., 2008). The most relevant IMS for this coupling is located at 839 keV with *J*π;*K*=5- ;4, although other IMS have been suggested very recently (Gintautas et al., 2009; Dracoulis et al., 2010).

The IMS at 839 keV decays predominantly to the high-*K* part of 176Lu; thus, the decay branch to the low-*K* part defines the stellar reaction rate. The decay properties of the IMS at 839 keV are well known from various γ-spectroscopic studies (Doll et al., 1999; Klay et al., 1991; Lesko et al., 1991). The lowest γ-ray energy is 123 keV, and three further γ-rays are observed at higher energies. Because of the relatively high energies the stellar reaction rate is only weakly affected. The dominating effect is NEEC in this case. However, the modification of the stellar transition rate remains far below a factor of two in the astrophysically relevant energy region (Gosselin et al., 2010).

## **4.2 180Ta and its uncertain nucleosynthetic origin**

180Ta is the rarest nucleus in our solar system (Lodders, 2003), and it is the only nucleus which does not exist in its ground state, but in an isomeric state. The ground state is a low-*K* state with *J*π;*K*=1+;1 and a short half-life of 8.154 hours. The isomer is located at an excitation energy of 77 keV; it is a high-*K* state with *J*π;*K*=9-;9. Because of its huge *K* quantum number, its decay to the ground state is highly suppressed, and also the energetically possible βdecays to 180Hf and 180W are largely hindered. The half-life of the isomer is unknown with a lower limit of 7.1 x 1015 years (Hult et al, 2006; Wu & Niu, 2003). Despite significant effort, the nucleosynthetic origin of 180Ta is still uncertain.

Various astrophysical sites and corresponding processes have been suggested for the nucleosynthesis of 180Ta. Very recently it has been concluded that a large contribution to the solar abundance can be produced in the neutrino burst during type II supernovae in the socalled ν-process by the 180Hf(ν,e-)180Ta reaction (Hayakawa et al., 2010). Alternatively, in the same astrophysical site the classical p- or γ-process may produce some 180Ta by photodestruction of 181Ta in the 181Ta(γ,n)180Ta reaction (Arnould & Goriely 2003; Utsunomiya et al., 2006). Similar conditions for the temperature of several billions Kelvin (or *kT* ≈ 200 – 300 keV) occur in type Ia supernovae, and it has been found that some 180Ta can also be made in that site (Travaglio et al., 2011). In addition, some 180Ta may also be produced in the s-process via β-decay of thermally excited 179Hf to 179Ta and subsequent neutron capture in the 179Ta(n,γ)180Ta reaction or via isomeric β-decay of 180mHf in the 16 Astrophysics

In the following section we discuss three examples in greater detail. The first two examples are so-called *K*-isomers which are relevant in the astrophysical s-process (176Lu and 180Ta).

176Lu is a so-called s-only nucleus because it is synthesized only in the astrophysical s-

with a half-life of less than 4 hours. The 175Lu(n,)176Lu reaction produces most of 176Lu in

isomer is coupled to the high-*K* ground state (Heil et al., 2008). The most relevant IMS for this coupling is located at 839 keV with *J*π;*K*=5-;4, although other IMS have been suggested

The IMS at 839 keV decays predominantly to the high-*K* part of 176Lu; thus, the decay branch to the low-*K* part defines the stellar reaction rate. The decay properties of the IMS at 839 keV are well known from various γ-spectroscopic studies (Doll et al., 1999; Klay et al., 1991; Lesko et al., 1991). The lowest γ-ray energy is 123 keV, and three further γ-rays are observed at higher energies. Because of the relatively high energies the stellar reaction rate is only weakly affected. The dominating effect is NEEC in this case. However, the modification of the stellar transition rate remains far below a factor of two in the astrophysically relevant

180Ta is the rarest nucleus in our solar system (Lodders, 2003), and it is the only nucleus which does not exist in its ground state, but in an isomeric state. The ground state is a low-*K* state with *J*π;*K*=1+;1 and a short half-life of 8.154 hours. The isomer is located at an excitation energy of 77 keV; it is a high-*K* state with *J*π;*K*=9-;9. Because of its huge *K* quantum number, its decay to the ground state is highly suppressed, and also the energetically possible βdecays to 180Hf and 180W are largely hindered. The half-life of the isomer is unknown with a lower limit of 7.1 x 1015 years (Hult et al, 2006; Wu & Niu, 2003). Despite significant effort,

Various astrophysical sites and corresponding processes have been suggested for the nucleosynthesis of 180Ta. Very recently it has been concluded that a large contribution to the solar abundance can be produced in the neutrino burst during type II supernovae in the socalled ν-process by the 180Hf(ν,e-)180Ta reaction (Hayakawa et al., 2010). Alternatively, in the same astrophysical site the classical p- or γ-process may produce some 180Ta by photodestruction of 181Ta in the 181Ta(γ,n)180Ta reaction (Arnould & Goriely 2003; Utsunomiya et al., 2006). Similar conditions for the temperature of several billions Kelvin (or *kT* ≈ 200 – 300 keV) occur in type Ia supernovae, and it has been found that some 180Ta can also be made in that site (Travaglio et al., 2011). In addition, some 180Ta may also be produced in the s-process via β-decay of thermally excited 179Hf to 179Ta and subsequent neutron capture in the 179Ta(n,γ)180Ta reaction or via isomeric β-decay of 180mHf in the

;7 ground state with a half-life of 38 giga-

emission to 176Hf. 176Lu can only survive if the

;0 isomer at 123 keV

The last example is the 7Be(p,)8B reaction with its low reaction Q-value of 137 keV.

years (quasi-stable for the s-process) or in its low-lying low-*K J*π;*K*=1-

**4. Some selected examples** 

**4.1 176Lu in the astrophysical s-process** 

the low-*K* isomeric state which decays by β-

energy region (Gosselin et al., 2010).

process. It is produced either in its high-*K J*π;*K*=7-

very recently (Gintautas et al., 2009; Dracoulis et al., 2010).

**4.2 180Ta and its uncertain nucleosynthetic origin** 

the nucleosynthetic origin of 180Ta is still uncertain.

179Hf(n,γ)180mHf( β-)180mTa reaction chain (Schumann & Käppeler, 1999; Beer & Ward, 1981; Yokoi & Takahashi, 1983; Mohr et al., 2007).

A common problem in all above production scenarios is the survival of 180Ta in its isomeric state. If the production occurs in a high-temperature environment like any supernova explosion, then 180Ta is produced in thermal equilibrium between the low-*K* ground state band and the high-*K* isomeric band. Following the evolution of the isomer-to-ground state ratio, freeze-out is found around 40 keV with a survival probability of 180Ta in its isomeric state of 0.38±0.01 (Hayakawa et al., 2010). Under s-process conditions with its lower temperature, it is not clear how much 180Ta can survive in the isomeric state because of mixing between hotter and cooler areas of the thermally pulsing AGB star (Mohr et al., 2007). The resulting yield of 180Ta depends sensitively on the properties of the lowest intermediate state (IMS) which couples the low-*K* and high-*K* bands.

There is indirect confirmation for the existence of IMS which couple the low-*K* ground state and the high-*K* isomeric state from photoactivation experiments (Belic et al., 2002; Collins et al., 1990). However, no direct γ-transition has been observed up to now. Based on reasonable estimates for transition strengths it has been suggested (Mohr et al., 2007) that the lowest IMS is located at an excitation energy of 594 keV with *J*=(5). This state decays with a transition energy of 72 keV and a half-life of about 16 ns to the low-*K* side of 180Ta. A weak branch to the high-K side can be expected via a transition to the 7+ state at 357 keV. This may lead to thermalization of 180Ta within days at the s-process temperatures around 25 keV (Mohr et al., 2007).

Because of the low transition energy of only 72 keV, a significant modification of the corresponding transition strength can be expected. Enhancements of the radiative strength of up to a factor of 10 for this transition have been calculated (Gosselin et al., 2010) which are mainly based on NEEC. However, unfortunately the influence on the stellar transition rate between low-*K* states and high-*K* states remains very small. The stellar transition rate as given in Eq. (12) is essentially defined by the weak branch of the 594 keV state which has a transition energy of more than 200 keV. As transitions with such high energies are practically not affected by the plasma environment, the stellar transition rate remains almost unchanged.

#### **4.3 <sup>8</sup> B, the <sup>7</sup> Be(p,γ) 8 B reaction, and solar neutrino production**

The 7Be(p,γ)8B reaction is the key reaction for the production of high-energy neutrinos in our sun. It is one of the very few examples for a capture reaction between light nuclei where low-energy γ-rays play a significant role. The Q-value of this reaction is extremely low (137 keV), and together with the most effective energy of about 18 keV (at temperatures around 15 million Kelvin, typical for the center of our sun) we find a transition energy of about 155 keV in the low-mass (*Z*=5) nucleus 8B. The γ-energy of 155 keV is still too high to be significantly influenced by the surrounding plasma. Although experimental conditions in the laboratory (either a proton beam and neutral 7Be target or a 7Be beam in arbitrary charge state on a neutral hydrogen target) are quite different from the stellar environment of 7Be, the cross sections from laboratory experiments do not require a plasma correction for the electromagnetic transition strength to calculate the stellar reaction rates, as e.g. summarized in the NACRE compilation (Angulo et al., 1999)

Nuclear Excitation Processes in Astrophysical Plasmas 19

Alder, K.; Bohr, A.; Huus, T.; Mottelson, B.; & Winther, A. (1956). *Review of Modern Physics*,

Arnould, M. & Goriely, S. (2012). The r-process of nucleosynthesis: The puzzle is still with

Baglin, C. M. (2002). *Nuclear Data Sheets,* Vol 96, pp 399-610; available online at

Basunia, M. S. (2006). *Nuclear Data Sheets,* Vol 107, pp 791-1026; available online at

Abdallah, J. & Sherrill, M.E. (2008). *High Energy Density Physics,* Vol 83, pp 195-245

us, In: *Nucleosynthesis*, InTech, 978-953-308-32-9, Rijeka, Croatia.

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Gosselin, G.; Méot, V. & Morel, P. (2007). *Physical Review C,* Vol 76, pp 044611

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Kimball, J. C.; Bittel, D. & Cue, N. (1991). *Physics Letters A,* Vol 152, pp 367-370

Gintautas, V.; Champagne, A. E.; Kondev, F. G. & Longland, R. (2009). *Physical Review C,* Vol

Gosselin, G.; Pillet, N.; Méot, V.; Morel, P. & Dzyublik, A. Ya. (2009). *Physical Review C,* Vol

Hamilton, W. D. (1975). *The Electromagnetic Interaction in Nuclear Spectroscopy,* Elsevier, ISBN

Hayakawa, T.; Mohr, P.; Kajino, T.; Chiba, S. & Mathews, G. J. (2010). *Physical Review C,* Vol

Hult, M.; Gasparro, J.; Marissens, G.; Lindahl, P.; Wäthjen, U.; Johnston, P. & Wagemans, C.

Iliadis, C. (2007). *Nuclear Physics of Stars*, Wiley-VCH, 978-3-527-40602-9, Weinheim,

Käppeler, F.; Gallino, R.; Bisterzo, S. & Aoki, W. (2011). *Review of Modern Physics,* Vol 83, pp

Adelberger, E. G.; et al. (2011). *Review of Modern Physics,* Vol 83, pp 195-245

Angulo, C.; et al. (1999). *Nuclear Physics A,* Vol 656, pp 3-187 Arnould, M. & Goriely, S. (2003). *Physics Reports,* Vol 384, pp 1-84

Beer, H. & Ward, R. A. (1981). *Nature,* Vol 291, pp 308-310 Belic, D.; et al. (2002). *Physical Review C,* Vol 65, pp 035801

Doll, C.; et al. (1999). *Physical Review C,* Vol 59, pp 492-499 Doolen, G. D. (1978). *Physical Review C,* Vol 18, pp 2547-2559

Farkas, J.; et al. (2009). *Journal of Physics G,* Vol 36, pp 105101

Gallino, R.; et al. (1998). *Astrophysical Journal,* Vol 497, pp 388-403

Gosselin, G. & Morel, P. (2004). *Physical Review C,* Vol 70, pp 064603

Heil, M.; et al. (2008). *Astrophysical Journal,* Vol 673, pp 434-444

(2006). *Physical Review C,* Vol 74, pp 054311

Bosch, F.; et al. (1996). *Physical Review Letters,* Vol 77, pp 5190-5193 Collins, C. B.; et al. (1990). *Physical Review C,* Vol 42, pp R1813-R1816

Dracoulis, G. D.; et al. (2010). *Physical Review C,* Vol 81, pp 011301(R)

**6. Acknowledgments** 

Vol 28 pp 432-542

80, pp 015806

79, pp 014604

82, pp 055801

Germany.

157-193

0 444 10519 O, New York, USA

**7. References** 

This work was supported by OTKA (NN83261).

*http://www.nndc.bnl.gov/ensdf*.

*http://www.nndc.bnl.gov/ensdf*

or in a compilation of solar fusion cross sections (Adelberger et al., 2011). However, two further effects of the surrounding plasma have to be kept in mind here: (i) The electron capture decay of 7Be is significantly affected under solar conditions because electrons have to be captured from the surrounding plasma instead of the bound *K*-shell electrons for 7Be in a neutral beryllium atom. (ii) Electron screening affects all capture cross sections at very low energies. Because the electronic environment is different in laboratory experiments and under stellar conditions, different corrections have to be applied here.

## **5. Conclusion**

The description of nuclear excitation in hot astrophysical plasma environment requires an accurate knowledge of each individual excitation process. The dominant processes are photo-excitation, where a photon close to the resonant nuclear transition energy is absorbed by the nucleus, NEEC and NEET, where an electron from the continuum (NEEC) or an outer electronic shell (NEET) is captured in a vacancy of the electronic cloud, and inelastic electron scattering, where an incident electron gives a part of its energy to excite the nucleus.

Results for the excitation of the first isomeric state of 171Tm clearly exhibit a strong dependence upon the plasma temperature. The nuclear lifetime under plasma conditions is more than one order of magnitude higher than the laboratory value. All these calculations are made at Local Thermodynamic Equilibrium (LTE), a condition encountered in many astrophysical plasmas.

However, nuclear excitation models in plasma need to be elaborated further. NEET rates require to take into account detailed electronic configuration, as the mean RAAM configuration is more often than not far from a real configuration on which NEET can occur. For high density plasmas, electron inelastic scattering is a major process and screening effects will have to be added to the description.

In most astrophysical scenarios the influence of the surrounding plasma on astrophysical reaction rates will remain relatively small because the γ-transition energies are too high to be significantly affected by the electronic environment. Note that typical γ-transition energies exceed by far 1 MeV whereas plasma effects become most important below 100 keV. Nevertheless, it should be always kept in mind that γ-transitions with energies below about 100 keV may be modified dramatically. As soon as such a transition defines the stellar transition rate, the calculated stellar reaction rate without consideration of plasma effects may be wrong; this error may reach one order of magnitude in special cases, i.e. for very low γ-transition energies.

Such cases with small γ-transition energies (and thus huge influence of the plasma environment) can be found in particular in the astrophysical s-process where the production and destruction of so-called *K*-isomers proceeds by low-energy γ-transitions which connect the low-*K* and high-*K* parts of the excitation spectrum of heavy nuclei via intermediate states. However, for the two most prominent examples (176Lu and 180Ta) it is found that plasma effects remain relatively small for the resulting stellar reaction rates although one particular transition in 180Ta is enhanced by about a factor of 10.

## **6. Acknowledgments**

This work was supported by OTKA (NN83261).

## **7. References**

18 Astrophysics

or in a compilation of solar fusion cross sections (Adelberger et al., 2011). However, two further effects of the surrounding plasma have to be kept in mind here: (i) The electron capture decay of 7Be is significantly affected under solar conditions because electrons have to be captured from the surrounding plasma instead of the bound *K*-shell electrons for 7Be in a neutral beryllium atom. (ii) Electron screening affects all capture cross sections at very low energies. Because the electronic environment is different in laboratory experiments and under stellar conditions, different corrections have to be

The description of nuclear excitation in hot astrophysical plasma environment requires an accurate knowledge of each individual excitation process. The dominant processes are photo-excitation, where a photon close to the resonant nuclear transition energy is absorbed by the nucleus, NEEC and NEET, where an electron from the continuum (NEEC) or an outer electronic shell (NEET) is captured in a vacancy of the electronic cloud, and inelastic electron scattering, where an incident electron gives a part of its

Results for the excitation of the first isomeric state of 171Tm clearly exhibit a strong dependence upon the plasma temperature. The nuclear lifetime under plasma conditions is more than one order of magnitude higher than the laboratory value. All these calculations are made at Local Thermodynamic Equilibrium (LTE), a condition encountered in many

However, nuclear excitation models in plasma need to be elaborated further. NEET rates require to take into account detailed electronic configuration, as the mean RAAM configuration is more often than not far from a real configuration on which NEET can occur. For high density plasmas, electron inelastic scattering is a major process and screening

In most astrophysical scenarios the influence of the surrounding plasma on astrophysical reaction rates will remain relatively small because the γ-transition energies are too high to be significantly affected by the electronic environment. Note that typical γ-transition energies exceed by far 1 MeV whereas plasma effects become most important below 100 keV. Nevertheless, it should be always kept in mind that γ-transitions with energies below about 100 keV may be modified dramatically. As soon as such a transition defines the stellar transition rate, the calculated stellar reaction rate without consideration of plasma effects may be wrong; this error may reach one order of magnitude in special cases, i.e. for very low

Such cases with small γ-transition energies (and thus huge influence of the plasma environment) can be found in particular in the astrophysical s-process where the production and destruction of so-called *K*-isomers proceeds by low-energy γ-transitions which connect the low-*K* and high-*K* parts of the excitation spectrum of heavy nuclei via intermediate states. However, for the two most prominent examples (176Lu and 180Ta) it is found that plasma effects remain relatively small for the resulting stellar reaction rates although one

particular transition in 180Ta is enhanced by about a factor of 10.

applied here.

**5. Conclusion** 

energy to excite the nucleus.

astrophysical plasmas.

γ-transition energies.

effects will have to be added to the description.


**1. Introduction**

with observed values.

nucleosynthesis research.

**2. Nucleosynthesis and its data needs**

In the past 100 years, astronomy, astrophysics and cosmology have evolved from the observational and theoretical fields into more experimental science, when many stellar and planetary processes are recreated in physics laboratories and extensively studied (Boyd et al., 2009). Many astrophysical phenomena have been explained using our understanding of nuclear physics processes, and the whole concept of stellar nucleosynthesis has been introduced. The importance of nuclear reactions as a source of stellar energy was recognized by Arthur Stanley Eddington as early as 1920 (Eddington, 1920). Later, nuclear mechanisms by which hydrogen is fused into helium were proposed by Hans Bethe (Bethe, 1939). However,

**Stellar Nucleosynthesis Nuclear Data Mining** 

*National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY* 

**2**

*USA* 

Boris Pritychenko

Further developments helped to identify the Big Bang, stellar and explosive nucleosynthesis processes that are responsible for the currently-observed variety of elements and isotopes (Burbidge et al., 1957; Cameron, 1957; Hoyle, 1946; Merrill, 1952). Today, nuclear physics is successfully applied to explain the variety of elements and isotope abundances observed in stellar surfaces, the solar system and cosmic rays via network calculations and comparison

A comprehensive analysis of stellar energy production, metallicity and isotope abundances indicates the crucial role of proton-, neutron- and light ion-induced nuclear reactions and *α*-, *β*-decay rates. These subatomic processes govern the observables and predict the star life cycle. Calculations of the transition rates between isotopes in a network strongly rely on theoretical and experimental cross section and decay rate values at stellar temperatures. Consequently, the general availability of nuclear data is of paramount importance in stellar

This chapter will provide a review of theoretical and experimental nuclear reaction and structure data for stellar and explosive nucleosynthesis and modern computation tools and methods. Examples of evaluated and compiled nuclear physics data will be given. Major

Nucleosynthesis is an important nuclear astrophysics phenomenon that is responsible for presently observed chemical elements and isotope abundances. It started in the early Universe and presently proceeds in the stars. The Big Bang nucleosynthesis is responsible for a relatively high abundance of the lightest primordial elements in the Universe from 1H to 7Li,

nuclear databases and their input for nucleosynthesis calculations will be discussed.

neither of these contributions explained the origin of elements heavier than helium.

Klay, N.; et al. (1991). *Physical Review C,* Vol 44, pp 2801-2838


## **2**

## **Stellar Nucleosynthesis Nuclear Data Mining**

## Boris Pritychenko

*National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY USA* 

## **1. Introduction**

20 Astrophysics

Kücük, I. (2012). Screening Factors and Thermonuclear Reaction Rates for Low Mass Stars,

Marss, R. (2010), *XI'th International Symposium on EBIS and EBIT*, 7-*10 April 2010*, UCRL-

Matteucci, F. (2012). Stellar Nucleosynthesis, In: *Nucleosynthesis*, InTech, 978-953-308-32-9,

Messiah, A. (1961). *Quantum Mechanics,* North Holland, ISBN 0 471 59766 X, Amsterdam,

Mohr, P.; Fülöp, Zs. & Utsunomiya, H. (2007b). *European Physical Journal A,* Vol 32, pp 357-

Mohr, P.; Bisterzo, S.; Gallino, R.; Käppeler, F.; Kneissl, U. & Winckler, N. (2009). *Physical* 

Morel, P.; Méot, V.; Gosselin, G.; Gogny, D. & Yunnes, W. (2004). *Physical Review A,* Vol 69,

Ohtsuki, T.; Yuki, H.; Muto, M.; Kasagi, J. & Ohno, K. (2004). *Physical Review Letters,* Vol 93,

Pumo, M. L. (2012). The s-process nucleosynthesis in massive stars: current status and

Rauscher, T.; Mohr, P.; Dillmann, I. & Plag, R. (2011). *Astrophysical Journal,* Vol 738, pp 143 Rolfs, C. E. & Rodney, W. S. (1988). *Cauldrons in the Cosmos*, The University of Chicago Press,

Takahashi, K. & Yokoi, K. (1987). *Atomic Data Nuclear Data Tables,* Vol 36, pp 375-409

Travaglio, C.; Roepke, F.; Gallino, R. & Hillebrandt, W. (2011). *Astrophysical Journal,* Vol 739,

Utsunomiya, H.; Mohr, P.; Zilges, A. & Rayet, M. (2006). *Nuclear Physics A,* Vol 777, pp 459-

Wu, S. C. & Niu, H. (2003). *Nuclear Data Sheets,* Vol 100, pp 483-705; available online at

uncertainties due to convective overshooting, In: *Nucleosynthesis*, InTech, 978-953-

Mohr, P.; Käppeler, F. & Gallino, R. (2007). *Physical Review C,* Vol 75, pp 012802(R)

In: *Nucleosynthesis*, InTech, 978-953-308-32-9, Rijeka, Croatia.

Klay, N.; et al. (1991). *Physical Review C,* Vol 44, pp 2801-2838

PRES-427008, Livermore, California, USA

Rijeka, Croatia.

*Review C,* Vol 79, pp 045804

308-32-9, Rijeka, Croatia.

0-226-72456-5, Chicago, IL, USA.

*http://www.nndc.bnl.gov/ensdf*

Rozsnyai, B. F. (1972). *Physical Review A,* Vol 5, pp 1137-1149

Yokoi, K. & Takahashi, K. (1983). *Nature,* Vol 305, pp 198-200

Schumann, M. & Käppeler, F. (1999). *Physical Review C,* Vol 60, pp 025802

Ward, R. A. & Fowler, W. A. (1980). *Astrophysical Journal,* Vol 238, pp 266-286

Holland

pp 063414

pp 112501

pp 93

478

369

Lesko, K. T.; et al. (1991). *Physical Review C,* Vol 44, pp 2850-2864 Lodders, K. (2003). *Astrophysical Journal,* Vol 591, pp 1220-1247

> In the past 100 years, astronomy, astrophysics and cosmology have evolved from the observational and theoretical fields into more experimental science, when many stellar and planetary processes are recreated in physics laboratories and extensively studied (Boyd et al., 2009). Many astrophysical phenomena have been explained using our understanding of nuclear physics processes, and the whole concept of stellar nucleosynthesis has been introduced. The importance of nuclear reactions as a source of stellar energy was recognized by Arthur Stanley Eddington as early as 1920 (Eddington, 1920). Later, nuclear mechanisms by which hydrogen is fused into helium were proposed by Hans Bethe (Bethe, 1939). However, neither of these contributions explained the origin of elements heavier than helium.

> Further developments helped to identify the Big Bang, stellar and explosive nucleosynthesis processes that are responsible for the currently-observed variety of elements and isotopes (Burbidge et al., 1957; Cameron, 1957; Hoyle, 1946; Merrill, 1952). Today, nuclear physics is successfully applied to explain the variety of elements and isotope abundances observed in stellar surfaces, the solar system and cosmic rays via network calculations and comparison with observed values.

> A comprehensive analysis of stellar energy production, metallicity and isotope abundances indicates the crucial role of proton-, neutron- and light ion-induced nuclear reactions and *α*-, *β*-decay rates. These subatomic processes govern the observables and predict the star life cycle. Calculations of the transition rates between isotopes in a network strongly rely on theoretical and experimental cross section and decay rate values at stellar temperatures. Consequently, the general availability of nuclear data is of paramount importance in stellar nucleosynthesis research.

> This chapter will provide a review of theoretical and experimental nuclear reaction and structure data for stellar and explosive nucleosynthesis and modern computation tools and methods. Examples of evaluated and compiled nuclear physics data will be given. Major nuclear databases and their input for nucleosynthesis calculations will be discussed.

## **2. Nucleosynthesis and its data needs**

Nucleosynthesis is an important nuclear astrophysics phenomenon that is responsible for presently observed chemical elements and isotope abundances. It started in the early Universe and presently proceeds in the stars. The Big Bang nucleosynthesis is responsible for a relatively high abundance of the lightest primordial elements in the Universe from 1H to 7Li,

0 10 20 30 40 50 60 70 80 90 100

Z

Fig. 2. Solar system elemental abundances; data are taken from (Grevesse & Sauval, 1998).

nucleosynthesis processes are strongly affected by the astrophysical site conditions.

Dillmann et al., 2006) and general nuclear science and industry databases. Dedicated libraries are optimized for nuclear astrophysics applications, and contain pre-selected data that are often limited to the original scope. In many cases, these sources reflect the present state of nuclear physics, when experimental data are not always available or limited to a single measurement. Such limitation highlights the importance of theoretical calculations that strongly dependent on nuclear models. Another problem arises from the fact that

Stellar Nucleosynthesis Nuclear Data Mining 23

To broaden the scope of the traditional nuclear astrophysics calculations, we will investigate applicability of nuclear physics databases for stellar nucleosynthesis data mining. These databases were developed for nuclear science, energy production and national security applications and will provide complementary astrophysics model-independent results.

This section will briefly consider the light elements and concentrate on the active research subject of production of medium and heavy elements beyond iron via slow and rapid neutron capture and associated data needs. Finally, photo disintegration and proton capture

2

**3. Stellar nucleosynthesis**

nucleosynthesis processes will be reviewed.

4

log(Nel/N

H) + 12

6

8

10

12

and it precedes stars formation and stellar nucleosynthesis. The general consistency between theoretically predicted and observed lightest elements abundances serves as a strong evidence for the Big Bang theory (Kolb & Turner, 1988).

The currently-known variety of nuclei and element abundances is shown in Fig. 1 and Fig. 2. These Figures indicate a large variety of isotopes in Nature (Anders & Grevesse, 1989) and the strong need for additional nucleosynthesis mechanisms beyond the Big Bang theory.

Fig. 1. The chart of nuclides. Stable and long-lived (>1015 s) nuclides are shown in black. Courtesy of NuDat Web application (*http://www.nndc.bnl.gov/nudat*).

These additional mechanisms have been pioneered by Eddington (Eddington, 1920) via introduction of a revolutionary concept of element production in the stars. Present nucleosynthesis models explain medium and heavy element abundances using the stellar nucleosynthesis that consists of burning (explosive) stages of stellar evolution, photo disintegration, and neutron and proton capture processes. The model predictions can be verified through the star metallicity studies and comparison of calculated isotopic/elemental abundances with the observed values, as shown in Fig. 2 .

Nowadays, there are many well-established theoretical models of stellar nucleosynthesis (Boyd et al., 2009; Burbidge et al., 1957); however, they still cannot reproduce the observed abundances due to many parameter uncertainties. The flow of the nuclear physics processes in the network calculations is defined by the nuclear masses, reaction, and decay rates, and strongly correlated with the stellar temperature and density. These calculations depend heavily on our understanding of nuclear physics processes in stars, and the availability of high quality nuclear data.

The common sources of stellar nucleosynthesis data include KADONIS, NACRE and REACLIB dedicated nuclear astrophysics libraries (Angulo et al., 1999; Cyburt et al., 2010;

2 Will-be-set-by-IN-TECH

and it precedes stars formation and stellar nucleosynthesis. The general consistency between theoretically predicted and observed lightest elements abundances serves as a strong evidence

The currently-known variety of nuclei and element abundances is shown in Fig. 1 and Fig. 2. These Figures indicate a large variety of isotopes in Nature (Anders & Grevesse, 1989) and the

strong need for additional nucleosynthesis mechanisms beyond the Big Bang theory.

Fig. 1. The chart of nuclides. Stable and long-lived (>1015 s) nuclides are shown in black.

These additional mechanisms have been pioneered by Eddington (Eddington, 1920) via introduction of a revolutionary concept of element production in the stars. Present nucleosynthesis models explain medium and heavy element abundances using the stellar nucleosynthesis that consists of burning (explosive) stages of stellar evolution, photo disintegration, and neutron and proton capture processes. The model predictions can be verified through the star metallicity studies and comparison of calculated isotopic/elemental

Nowadays, there are many well-established theoretical models of stellar nucleosynthesis (Boyd et al., 2009; Burbidge et al., 1957); however, they still cannot reproduce the observed abundances due to many parameter uncertainties. The flow of the nuclear physics processes in the network calculations is defined by the nuclear masses, reaction, and decay rates, and strongly correlated with the stellar temperature and density. These calculations depend heavily on our understanding of nuclear physics processes in stars, and the availability of

The common sources of stellar nucleosynthesis data include KADONIS, NACRE and REACLIB dedicated nuclear astrophysics libraries (Angulo et al., 1999; Cyburt et al., 2010;

Courtesy of NuDat Web application (*http://www.nndc.bnl.gov/nudat*).

abundances with the observed values, as shown in Fig. 2 .

high quality nuclear data.

for the Big Bang theory (Kolb & Turner, 1988).

Fig. 2. Solar system elemental abundances; data are taken from (Grevesse & Sauval, 1998).

Dillmann et al., 2006) and general nuclear science and industry databases. Dedicated libraries are optimized for nuclear astrophysics applications, and contain pre-selected data that are often limited to the original scope. In many cases, these sources reflect the present state of nuclear physics, when experimental data are not always available or limited to a single measurement. Such limitation highlights the importance of theoretical calculations that strongly dependent on nuclear models. Another problem arises from the fact that nucleosynthesis processes are strongly affected by the astrophysical site conditions.

To broaden the scope of the traditional nuclear astrophysics calculations, we will investigate applicability of nuclear physics databases for stellar nucleosynthesis data mining. These databases were developed for nuclear science, energy production and national security applications and will provide complementary astrophysics model-independent results.

## **3. Stellar nucleosynthesis**

This section will briefly consider the light elements and concentrate on the active research subject of production of medium and heavy elements beyond iron via slow and rapid neutron capture and associated data needs. Finally, photo disintegration and proton capture nucleosynthesis processes will be reviewed.

for sustainable nuclear fusion reaction

**3.1.2 Triple alpha process**

**3.1.3 CNO cycle**

**3.2 s-process**

Next, the deuterium produced in the first stage can fuse with another hydrogen

The hydrogen burning in stars leads to the production of the helium core at a star's center.

Stellar Nucleosynthesis Nuclear Data Mining 25

*<sup>α</sup>* <sup>+</sup> *<sup>α</sup>* <sup>⇔</sup> <sup>8</sup>*Be* <sup>−</sup> 93.7*keV*

Carbon-Nitrogen-Oxygen cycle leads to the production of elements heavier than carbon and

Further burning includes carbon, neon, oxygen and silicon burning. Here, we have high-temperature and density burning stages when photonuclear processes produce additional *α*-particles and create a complex network of reactions with light particles. These reactions produce fusion nuclei up to the Fe-Ni peak. The Fe-Ni region nuclei are the most tightly bound in Nature and fusion reactions stop. Further element production proceeds via

The slow-neutron capture (s-process) is responsible for creation of ∼50 % of the elements beyond iron. In this region, neutron capture becomes dominant because of the increasing Coulomb barrier and decreasing binding energies. This s-process takes place in the Red Giants and AGB stars, where neutron temperature (*kT*) varies from 8 to 90 keV. A steady supply of

<sup>13</sup>*C*(*α*, *n*)

Fig. 2 indicates a high abundance of 56Fe nuclei due to termination of the explosive nucleosynthesis. It is natural to assume that iron acts as a seed for the neutron capture

<sup>16</sup>*O*

<sup>22</sup>*Ne*(*α*, *n*)25*Mg* (7)

Finally, 4He will be produced in the pp I and pp II branches.

consists of major CNO-I (regeneration of carbon and alpha particle)

and minor CNO-II branches (production of 16O and proton):

neutron and proton captures and photo disintegration.

neutrons is available due to H-burning and He-flash reactions

**3.1.4 Advanced stages of stellar burning**

Further helium burning goes through the 3*α* process

*<sup>σ</sup>* <sup>∼</sup> <sup>3</sup>*E*4.5 <sup>×</sup> <sup>10</sup>−24*<sup>b</sup>* (2)

*<sup>d</sup>* <sup>+</sup> *<sup>p</sup>* <sup>→</sup><sup>3</sup> *He* <sup>+</sup> *<sup>γ</sup>* <sup>+</sup> 5.49*MeV* (3)

<sup>8</sup>*Be* <sup>+</sup> *<sup>α</sup>* <sup>⇔</sup> <sup>12</sup>*C*<sup>∗</sup> <sup>+</sup> 7.367*MeV* (4)

<sup>12</sup>*<sup>C</sup>* <sup>→</sup><sup>13</sup> *<sup>N</sup>* <sup>→</sup><sup>13</sup> *<sup>C</sup>* <sup>→</sup><sup>14</sup> *<sup>N</sup>* <sup>→</sup><sup>15</sup> *<sup>O</sup>* <sup>→</sup><sup>15</sup> *<sup>N</sup>* <sup>→</sup><sup>12</sup> *<sup>C</sup>* (5)

<sup>15</sup>*<sup>N</sup>* <sup>→</sup><sup>16</sup> *<sup>O</sup>* <sup>→</sup><sup>17</sup> *<sup>F</sup>* <sup>→</sup><sup>17</sup> *<sup>O</sup>* <sup>→</sup><sup>14</sup> *<sup>N</sup>* <sup>→</sup><sup>15</sup> *<sup>O</sup>* <sup>→</sup><sup>15</sup> *<sup>N</sup>* (6)

#### **3.1 Burning phases of stellar evolution**

Fusion reactions are responsible for burning phases of stellar evolution. These reactions produce light, tightly-bound nuclei and release energy. The process of new element creation proceeds before nuclear binding energy reaches maximum value in the Fe-Ni region. Four important cases will be reviewed: pure hydrogen burning, triple alpha process, CNO cycle, and stellar burning. These processes take place in stars with a mass similar to our Sun, as shown in Fig. 3. The data needs for these processes are addressed in the IAEA FENDL (Aldama & Trkov, 2004) and EXFOR (Experimental Nuclear Reaction Data) (NRDC, 2011) databases.

Fig. 3. Cross section of a Red Giant showing nucleosynthesis and elements formed. Courtesy of Wikipedia (*http://en.wikipedia.org/wiki/Stellar\_nucleosynthesis*).

#### **3.1.1 Pure hydrogen burning**

Hydrogen is the most abundant element in the Universe. The proton-proton chain dominates stellar nucleosynthesis in stars comparable to our Sun

$$p + p \to d + \mathcal{J}^+ + \nu + 0.42 MeV \tag{1}$$

Further analysis of the pp-process (Burbidge et al., 1957) indicates extremely low cross sections for E<1 MeV nuclear projectiles and explains the necessity of large target mass and density for sustainable nuclear fusion reaction

$$
\sigma \sim 3E^{4.5} \times 10^{-24}b \tag{2}
$$

Next, the deuterium produced in the first stage can fuse with another hydrogen

$$d + p \rightarrow^3 He + \gamma + 5.49 MeV \tag{3}$$

Finally, 4He will be produced in the pp I and pp II branches.

#### **3.1.2 Triple alpha process**

The hydrogen burning in stars leads to the production of the helium core at a star's center. Further helium burning goes through the 3*α* process

$$\begin{aligned} \alpha + \alpha &\Leftrightarrow \, ^8Be - 93.7 \,\text{keV} \\ ^8Be + \alpha &\Leftrightarrow \, ^{12}C^\* + 7.36 \,\text{MeV} \end{aligned} \tag{4}$$

#### **3.1.3 CNO cycle**

4 Will-be-set-by-IN-TECH

Fusion reactions are responsible for burning phases of stellar evolution. These reactions produce light, tightly-bound nuclei and release energy. The process of new element creation proceeds before nuclear binding energy reaches maximum value in the Fe-Ni region. Four important cases will be reviewed: pure hydrogen burning, triple alpha process, CNO cycle, and stellar burning. These processes take place in stars with a mass similar to our Sun, as shown in Fig. 3. The data needs for these processes are addressed in the IAEA FENDL (Aldama & Trkov, 2004) and EXFOR (Experimental Nuclear Reaction Data) (NRDC, 2011)

Fig. 3. Cross section of a Red Giant showing nucleosynthesis and elements formed. Courtesy

Hydrogen is the most abundant element in the Universe. The proton-proton chain dominates

Further analysis of the pp-process (Burbidge et al., 1957) indicates extremely low cross sections for E<1 MeV nuclear projectiles and explains the necessity of large target mass and density

*<sup>p</sup>* <sup>+</sup> *<sup>p</sup>* <sup>→</sup> *<sup>d</sup>* <sup>+</sup> *<sup>β</sup>*<sup>+</sup> <sup>+</sup> *<sup>ν</sup>* <sup>+</sup> 0.42*MeV* (1)

of Wikipedia (*http://en.wikipedia.org/wiki/Stellar\_nucleosynthesis*).

stellar nucleosynthesis in stars comparable to our Sun

**3.1.1 Pure hydrogen burning**

**3.1 Burning phases of stellar evolution**

databases.

Carbon-Nitrogen-Oxygen cycle leads to the production of elements heavier than carbon and consists of major CNO-I (regeneration of carbon and alpha particle)

$$^{12}\text{C} \rightarrow ^{13}N \rightarrow ^{13}\text{C} \rightarrow ^{14}N \rightarrow ^{15}O \rightarrow ^{15}N \rightarrow ^{12}\text{C} \tag{5}$$

and minor CNO-II branches (production of 16O and proton):

$$^{15}N \to ^{16}O \to ^{17}F \to ^{17}O \to ^{14}N \to ^{15}O \to ^{15}N \tag{6}$$

#### **3.1.4 Advanced stages of stellar burning**

Further burning includes carbon, neon, oxygen and silicon burning. Here, we have high-temperature and density burning stages when photonuclear processes produce additional *α*-particles and create a complex network of reactions with light particles. These reactions produce fusion nuclei up to the Fe-Ni peak. The Fe-Ni region nuclei are the most tightly bound in Nature and fusion reactions stop. Further element production proceeds via neutron and proton captures and photo disintegration.

#### **3.2 s-process**

The slow-neutron capture (s-process) is responsible for creation of ∼50 % of the elements beyond iron. In this region, neutron capture becomes dominant because of the increasing Coulomb barrier and decreasing binding energies. This s-process takes place in the Red Giants and AGB stars, where neutron temperature (*kT*) varies from 8 to 90 keV. A steady supply of neutrons is available due to H-burning and He-flash reactions

$$\begin{aligned} \, ^{13}\text{C}(\mathfrak{a},\mathfrak{n})^{16}\text{O} \\ \, ^{22}\text{Ne}(\mathfrak{a},\mathfrak{n})^{25}\text{Mg} \end{aligned} \tag{7}$$

Fig. 2 indicates a high abundance of 56Fe nuclei due to termination of the explosive nucleosynthesis. It is natural to assume that iron acts as a seed for the neutron capture

0.1 1 10 100

Stellar Nucleosynthesis Nuclear Data Mining 27

Fig. 4. ENDF/B-VII.1 and EXFOR libraries 56Fe(n,*γ*) cross sections (Chadwick et al., 2006;

It is commonly known that for the equilibrium *<sup>s</sup>*-process-only nuclei product of �*σMaxw*

adjusted for nuclear astrophysics models and are essentially model-independent.

and solar-system abundances (*N*(*A*)) is preserved (Rolfs & Rodney, 1988)

where *T*<sup>9</sup> is temperature expressed in billions of Kelvin, *NA* is an Avogadro number. *T*<sup>9</sup> is

The stellar equilibrium conditions provide an important test for the s-process nucleosynthesis in mass regions between neutron magic numbers N=50,82,126 (Arlandini et al., 1999). To investigate this phenomenon, we will consider ENDF libraries. These data were never

ENDF library is a core nuclear reaction database containing evaluated (recommended) cross sections, spectra, angular distributions, fission product yields, thermal neutron scattering, photo-atomic and other data, with emphasis on neutron-induced reactions. ENDF library

Incident Neutron Energy (keV)

11.6045 × *T*<sup>9</sup> = *kT* (10)

*<sup>σ</sup>AN*(*A*) <sup>=</sup> *<sup>σ</sup>A*−1*N*(*A*−1) <sup>=</sup> *constant* (11)

*<sup>γ</sup>* (*kT*)�

ENDF/B-VII.1

Allen 1982

Macklin 1964

Shcherbakov 1977

10-5

10-2

101

Cross Section (b)

104

56Fe(n,γ)

NRDC, 2011) for astrophysical range of energies.

related to the *kT* in MeV units as follows

reactions that eventually produce medium and heavy elements. The neutron capture time of s-process takes approximately one year. Consequently, the process path lies along the nuclear valley of stability up to the last long-lived nucleus of 209Bi.

In the giant stars, neutron reaction rates define the elemental abundances, and in some cases branching points (created by the competition between neutron capture and *β*-decay) strongly affect the heavy isotope production rates. Therefore, special attention has to be paid to the branching points at 79Se,134Cs,147Pm, 151Sm, 154Eu, 170Yb and 185W, and the neutron poison (absorption) 16,18O,22Ne(n,*γ*) reactions. The s-process astrophysical site conditions imply the following data needs

	- **–** neutron-induced
	- **–** charged particle

Convincing proof of s-process existence and its role in Nature could come from the calculation of isotopic abundances and comparison with observed values (Anders & Grevesse, 1989; Grevesse & Sauval, 1998). Present-day s-process nucleosynthesis calculations often are based on the dedicated nuclear astrophysics data tables, such as works of (Bao et al., 2000), and (Rauscher & Thielemann, 2000). These data tables contain quality information on Maxwellian-averaged cross sections (�*σMaxw <sup>γ</sup>* (*kT*)�) and astrophysical reaction rates (*R*(*T*9)). However, it is essential to produce complementary neutron-induced reaction data sets for an independent verification and to expand the boundaries of the existing data tables.

Recent releases of ENDF/B-VII evaluated nuclear reaction libraries (Chadwick et al., 2006) and publication of the Atlas of Neutron Resonances reference book (Mughabghab, 2006) created a unique opportunity of applying these data for non-traditional applications, such as s-process nucleosynthesis (Pritychenko et al., 2010). Many neutron cross sections for astrophysical range of energies, including 56Fe(n,*γ*) reaction as shown in Fig. 4, are available in the ENDF (Evaluated Nuclear Data File) and EXFOR libraries. The feasibility study of the evaluated nuclear data for s-process nucleosynthesis will be presented below.

#### **3.2.1 Calculation of Maxwellian-averaged cross sections and uncertainties**

The Maxwellian-averaged cross section can be expressed as

$$
\langle \sigma^{\text{Max}w}(kT) \rangle = \frac{2}{\sqrt{\pi}} \frac{(m\_2/(m\_1+m\_2))^2}{(kT)^2} \int\_0^\infty \sigma(E\_n^L) E\_n^L e^{-\frac{E\_n^L w\_2}{kT(m\_1+m\_2)}} dE\_{n\prime}^L \tag{8}
$$

where *k* and *T* are the Boltzmann constant and temperature of the system, respectively, and *E* is an energy of relative motion of the neutron with respect to the target. Here, *E<sup>L</sup> <sup>n</sup>* is a neutron energy in the laboratory system and *m*<sup>1</sup> and *m*<sup>2</sup> are masses of a neutron and target nucleus, respectively.

The astrophysical reaction rate for network calculations is defined as

$$R(T\_9) = N\_A \langle \sigma v \rangle = 10^{-24} \sqrt{(2kT/\mu)N\_A \sigma^{Maxwell}} (kT),\tag{9}$$

6 Will-be-set-by-IN-TECH

reactions that eventually produce medium and heavy elements. The neutron capture time of s-process takes approximately one year. Consequently, the process path lies along the nuclear

In the giant stars, neutron reaction rates define the elemental abundances, and in some cases branching points (created by the competition between neutron capture and *β*-decay) strongly affect the heavy isotope production rates. Therefore, special attention has to be paid to the branching points at 79Se,134Cs,147Pm, 151Sm, 154Eu, 170Yb and 185W, and the neutron poison (absorption) 16,18O,22Ne(n,*γ*) reactions. The s-process astrophysical site conditions imply the

Convincing proof of s-process existence and its role in Nature could come from the calculation of isotopic abundances and comparison with observed values (Anders & Grevesse, 1989; Grevesse & Sauval, 1998). Present-day s-process nucleosynthesis calculations often are based on the dedicated nuclear astrophysics data tables, such as works of (Bao et al., 2000), and (Rauscher & Thielemann, 2000). These data tables contain quality information on

However, it is essential to produce complementary neutron-induced reaction data sets for an

Recent releases of ENDF/B-VII evaluated nuclear reaction libraries (Chadwick et al., 2006) and publication of the Atlas of Neutron Resonances reference book (Mughabghab, 2006) created a unique opportunity of applying these data for non-traditional applications, such as s-process nucleosynthesis (Pritychenko et al., 2010). Many neutron cross sections for astrophysical range of energies, including 56Fe(n,*γ*) reaction as shown in Fig. 4, are available in the ENDF (Evaluated Nuclear Data File) and EXFOR libraries. The feasibility study of the

independent verification and to expand the boundaries of the existing data tables.

evaluated nuclear data for s-process nucleosynthesis will be presented below.

**3.2.1 Calculation of Maxwellian-averaged cross sections and uncertainties**

(*m*2/(*m*<sup>1</sup> + *m*2))<sup>2</sup> (*kT*)<sup>2</sup>

is an energy of relative motion of the neutron with respect to the target. Here, *E<sup>L</sup>*

where *k* and *T* are the Boltzmann constant and temperature of the system, respectively, and *E*

energy in the laboratory system and *m*<sup>1</sup> and *m*<sup>2</sup> are masses of a neutron and target nucleus,

 ∞ 0

*σ*(*E<sup>L</sup> <sup>n</sup>* )*E<sup>L</sup> n e* <sup>−</sup> *<sup>E</sup><sup>L</sup>*

*<sup>n</sup> <sup>m</sup>*<sup>2</sup> *kT*(*m*1+*m*2) *dE<sup>L</sup>*

(2*kT*/*μ*)*NAσMaxw*(*kT*), (9)

*<sup>n</sup>*, (8)

*<sup>n</sup>* is a neutron

The Maxwellian-averaged cross section can be expressed as

<sup>√</sup>*<sup>π</sup>*

The astrophysical reaction rate for network calculations is defined as

*<sup>R</sup>*(*T*9) = *NA*�*σv*� <sup>=</sup> <sup>10</sup>−<sup>24</sup>

�*σMaxw*(*kT*)� <sup>=</sup> <sup>2</sup>

*<sup>γ</sup>* (*kT*)�) and astrophysical reaction rates (*R*(*T*9)).

valley of stability up to the last long-lived nucleus of 209Bi.

following data needs • Reaction rates

• Half-lives

respectively.

**–** neutron-induced **–** charged particle

Maxwellian-averaged cross sections (�*σMaxw*

Fig. 4. ENDF/B-VII.1 and EXFOR libraries 56Fe(n,*γ*) cross sections (Chadwick et al., 2006; NRDC, 2011) for astrophysical range of energies.

where *T*<sup>9</sup> is temperature expressed in billions of Kelvin, *NA* is an Avogadro number. *T*<sup>9</sup> is related to the *kT* in MeV units as follows

$$11.6045 \times T\_{\theta} = kT \tag{10}$$

It is commonly known that for the equilibrium *<sup>s</sup>*-process-only nuclei product of �*σMaxw <sup>γ</sup>* (*kT*)� and solar-system abundances (*N*(*A*)) is preserved (Rolfs & Rodney, 1988)

$$
\sigma\_A \mathbf{N}\_{(A)} = \sigma\_{A-1} \mathbf{N}\_{(A-1)} = constant \tag{11}
$$

The stellar equilibrium conditions provide an important test for the s-process nucleosynthesis in mass regions between neutron magic numbers N=50,82,126 (Arlandini et al., 1999). To investigate this phenomenon, we will consider ENDF libraries. These data were never adjusted for nuclear astrophysics models and are essentially model-independent.

ENDF library is a core nuclear reaction database containing evaluated (recommended) cross sections, spectra, angular distributions, fission product yields, thermal neutron scattering, photo-atomic and other data, with emphasis on neutron-induced reactions. ENDF library

",'0 -

**Proceedings of Summer Nuclear Data Week 2011**

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Stellar Nucleosynthesis Nuclear Data Mining 29

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Fig. 6. The NNDC website (*http://www.nndc.bnl.gov*) provides access to nuclear databases and Maxwellian-averaged cross sections and astrophysical reaction rates online calculations.

where *σ*(*E*1), *E*<sup>1</sup> and *σ*(*E*2), *E*<sup>2</sup> are cross section and energy values for the corresponding energy bin. The last equation is a good approximation of neutron cross section values for

separate energy bins using Doppler-broadened cross sections and the Wolfram Mathematica online integrator (Wolram, 2011). Further, summing integrals for all energy bins will produce

Grevesse, 1989) are plotted in Fig. 7. They reveal that the ENDF/B-VII.1 library data closely replicate a two-plateau plot (Rolfs & Rodney, 1988). The current result provides a powerful

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a sufficiently dense energy grid. This allowed deduction of �*σMaxw*

a precise ENDF value for the Maxwellian-averaged cross section.

The product values of the ENDF/B-VII.1 �*σMaxw*

testimony for stellar nucleosynthesis.

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evaluations cover all neutron reaction channels within 10−<sup>5</sup> eV - 20 MeV energy range. In many cases, evaluations contain information on neutron cross section covariances. An example of ENDF/B-VII.1library 56Fe(n,*γ*) neutron cross section covariances (uncertainties) is shown in Fig. 5.

Fig. 5. ENDF/B-VII.1 library 56Fe(n,*γ*) cross section covariances (Zerkin et al., 2005).

The evaluated neutron libraries are based on theoretical calculations using EMPIRE, TALYS and Atlas collection of nuclear reaction model codes (Herman et al., 2007; Koning et al., 2008; Mughabghab, 2006) that are often adjusted to fit experimental data (NRDC, 2011). The model codes are essential for neutron cross section calculations of short-lived radioactive nuclei where experimental data are not available. The ENDF data files are publicly available from the NNDC (National Nuclear Data Center) Sigma Web Interface: *http://www.nndc.bnl.gov/sigma* (Pritychenko & Sonzogni, 2008).

Previously, Maxwellian-averaged cross sections and astrophysical reaction rates were produced using the Simpson method for the linearized ENDF cross sections (Pritychenko et al., 2010) . A similar effort was completed at the Japanese Atomic Energy Agency (Nakagawa et al., 2005). Fig. 6 shows the NNDC website (Pritychenko et al., 2006) that provides access to nuclear databases and the NucRates Web application (*http://www.nndc.bnl.gov/astro*). The NucRates application was designed for online calculations of �*σMaxw*(*kT*)� and *<sup>R</sup>*(*T*9) using all major evaluated nuclear reaction libraries, astrophysical neutron-induced reactions and *kT* values ranging from 10−<sup>5</sup> eV to 20 MeV.

The Simpson method allowed quick calculation of integral values; however, the degree of precision was within ∼1%. Precision can be improved with the linearized ENDF files because the cross section value is linearly-dependent on energy within a particular bin (Pritychenko, 8 Will-be-set-by-IN-TECH

evaluations cover all neutron reaction channels within 10−<sup>5</sup> eV - 20 MeV energy range. In many cases, evaluations contain information on neutron cross section covariances. An example of ENDF/B-VII.1library 56Fe(n,*γ*) neutron cross section covariances (uncertainties)

Fig. 5. ENDF/B-VII.1 library 56Fe(n,*γ*) cross section covariances (Zerkin et al., 2005).

The evaluated neutron libraries are based on theoretical calculations using EMPIRE, TALYS and Atlas collection of nuclear reaction model codes (Herman et al., 2007; Koning et al., 2008; Mughabghab, 2006) that are often adjusted to fit experimental data (NRDC, 2011). The model codes are essential for neutron cross section calculations of short-lived radioactive nuclei where experimental data are not available. The ENDF data files are publicly available from the NNDC (National Nuclear Data Center) Sigma Web Interface: *http://www.nndc.bnl.gov/sigma*

Previously, Maxwellian-averaged cross sections and astrophysical reaction rates were produced using the Simpson method for the linearized ENDF cross sections (Pritychenko et al., 2010) . A similar effort was completed at the Japanese Atomic Energy Agency (Nakagawa et al., 2005). Fig. 6 shows the NNDC website (Pritychenko et al., 2006) that provides access to nuclear databases and the NucRates Web application (*http://www.nndc.bnl.gov/astro*). The NucRates application was designed for online calculations of �*σMaxw*(*kT*)� and *<sup>R</sup>*(*T*9) using all major evaluated nuclear reaction libraries, astrophysical neutron-induced reactions and *kT*

The Simpson method allowed quick calculation of integral values; however, the degree of precision was within ∼1%. Precision can be improved with the linearized ENDF files because the cross section value is linearly-dependent on energy within a particular bin (Pritychenko,

is shown in Fig. 5.

(Pritychenko & Sonzogni, 2008).

values ranging from 10−<sup>5</sup> eV to 20 MeV.

)('+(\*1,! "(-%\*!1+"+ "("' )\*,&',('\* 1 \$'(/% &',+ **About Us Comments/Questions** "+%"&\*

Fig. 6. The NNDC website (*http://www.nndc.bnl.gov*) provides access to nuclear databases and Maxwellian-averaged cross sections and astrophysical reaction rates online calculations.

2010)

$$
\sigma(E) = \sigma(E\_1) + (E - E\_1) \frac{\sigma(E\_2) - \sigma(E\_1)}{E\_2 - E\_1} \,\tag{12}
$$

where *σ*(*E*1), *E*<sup>1</sup> and *σ*(*E*2), *E*<sup>2</sup> are cross section and energy values for the corresponding energy bin. The last equation is a good approximation of neutron cross section values for a sufficiently dense energy grid. This allowed deduction of �*σMaxw <sup>γ</sup>* (*kT*)� definite integrals for separate energy bins using Doppler-broadened cross sections and the Wolfram Mathematica online integrator (Wolram, 2011). Further, summing integrals for all energy bins will produce a precise ENDF value for the Maxwellian-averaged cross section.

The product values of the ENDF/B-VII.1 �*σMaxw <sup>γ</sup>* (30*keV*)� times solar abundances (Anders & Grevesse, 1989) are plotted in Fig. 7. They reveal that the ENDF/B-VII.1 library data closely replicate a two-plateau plot (Rolfs & Rodney, 1988). The current result provides a powerful testimony for stellar nucleosynthesis.

0 100 200 300

A

Fig. 8. Maxwellian-averaged cross section uncertainties for ENDF/B-VII.1, Low-Fidelity and

KADONIS libraries (Chadwick et al., 2006; Dillmann et al., 2006; Little et al., 2008).

0

100

200

0

KADONIS

Low-Fidelity

ENDF/B-VII.1

Stellar Nucleosynthesis Nuclear Data Mining 31

100

Maxwellian-averaged Cross Section Uncertainties, %

200

0

100

200

Fig. 7. ENDF/B-VII.1 library product of neutron-capture cross section (at 30 keV in mb) times solar system abundances (relative to Si = 106) as a function of atomic mass for nuclei produced only in the *s*-process.

The predictive power of stellar nucleosynthesis calculations depends heavily on the neutron cross section values and their covariances. To understand the unique isotopic signatures from the presolar grains, ∼1% cross section uncertainties are necessary (Käppeler, 2011). Unfortunately, present uncertainties are often much higher, as shown in Fig. 8.

The situation gets even more complex after considering s-process branches where *β*-decay and neutron capture rates are

$$
\lambda\_{\mathfrak{n}} \sim \lambda\_{\mathfrak{B}} \tag{13}
$$

Here, the stellar thermal environment may affect *β*-decay rates and change the process path. Branching is particularly important for unstable isotopes such as 134Cs. This isotope can either decay to 134Ba, if neutron flux is low or capture neutron and produce 135Cs. The 134Cs *β*-decay lifetime may vary from <sup>∼</sup>1 y to 30 day over the temperature range of (100-300)×106 K, which further complicates the calculations.

Finally, the calculated values of Maxwellian-averaged cross sections at *kT*=30 keV, for selected s-process nuclei, derived from the JENDL-4.0, ROSFOND 2010 and ENDF/B-VII.1 libraries (Chadwick et al., 2006; Shibata et al., 2011; Zabrodskaya et al., 2007) have been produced and shown in Table 1. The tabulated results are compared with the KADONIS values (Dillmann et al., 2006). Due to a limited number of ENDF covariance files, Low-Fidelity cross section covariances (Little et al., 2008) were used to calculate uncertainties for ENDF/B-VII.1 data.

The complete sets of ENDF s-process nucleosynthesis data sets are available for download from the NucRates Web application *http://www.nndc.bnl.gov/astro*. These complimentary data 10 Will-be-set-by-IN-TECH

ENDF/B-VII.1

*λ<sup>n</sup>* ∼ *λβ* (13)

10

σ(mb)\*

produced only in the *s*-process.

and neutron capture rates are

further complicates the calculations.

N**(**/10

6 Si**)**

100

100 120 140 160 180 200

Fig. 7. ENDF/B-VII.1 library product of neutron-capture cross section (at 30 keV in mb) times

The predictive power of stellar nucleosynthesis calculations depends heavily on the neutron cross section values and their covariances. To understand the unique isotopic signatures from the presolar grains, ∼1% cross section uncertainties are necessary (Käppeler, 2011).

The situation gets even more complex after considering s-process branches where *β*-decay

Here, the stellar thermal environment may affect *β*-decay rates and change the process path. Branching is particularly important for unstable isotopes such as 134Cs. This isotope can either decay to 134Ba, if neutron flux is low or capture neutron and produce 135Cs. The 134Cs *β*-decay lifetime may vary from <sup>∼</sup>1 y to 30 day over the temperature range of (100-300)×106 K, which

Finally, the calculated values of Maxwellian-averaged cross sections at *kT*=30 keV, for selected s-process nuclei, derived from the JENDL-4.0, ROSFOND 2010 and ENDF/B-VII.1 libraries (Chadwick et al., 2006; Shibata et al., 2011; Zabrodskaya et al., 2007) have been produced and shown in Table 1. The tabulated results are compared with the KADONIS values (Dillmann et al., 2006). Due to a limited number of ENDF covariance files, Low-Fidelity cross section covariances (Little et al., 2008) were used to calculate uncertainties for ENDF/B-VII.1 data. The complete sets of ENDF s-process nucleosynthesis data sets are available for download from the NucRates Web application *http://www.nndc.bnl.gov/astro*. These complimentary data

solar system abundances (relative to Si = 106) as a function of atomic mass for nuclei

Unfortunately, present uncertainties are often much higher, as shown in Fig. 8.

Atomic Mass

Fig. 8. Maxwellian-averaged cross section uncertainties for ENDF/B-VII.1, Low-Fidelity and KADONIS libraries (Chadwick et al., 2006; Dillmann et al., 2006; Little et al., 2008).

demonstrated strong connections between nuclear astrophysics and nuclear structure. Further progress will require mass and half-life measurements of unstable nuclei. The r-process data

Stellar Nucleosynthesis Nuclear Data Mining 33

Recent r-process estimates (Cowan & Thielemann, 2004) demonstrate sharp abundance peaks for the A=130,195 (N=82,126) nuclei with large N/Z ratios and another broad peak at A=160.

For neutron closed shells, nuclei most likely will experience *β*-decay rather than absorb

Thus near neutron closed shells, the relationship between abundances and *β*-decay lifetimes

From the last formula, one can conclude that closed shell nuclei with the largest half-lives will have the largest abundances. In order to obtain the complete picture, all half-lives and decay modes have to be determined. A list of properties relevant to r-process A∼130 nuclei is shown in Table 2. It includes *β*-decay half-lives and emission probabilities for delayed neutrons. The delayed neutrons provide an additional neutron source for the r-process and may shift the location of the abundance peak. The tabulated data were taken from the Evaluated Nuclear Structure Data File (ENSDF) database (Burrows, 1990) and the relation

The regularly updated ENDF and ENSDF database evaluations could provide valuable for r-process data in the actinide region. The extremely neutron-rich superheavy fission nuclei play an important role in element production. Presently, these nuclei can be studied only with theoretical model calculations. These calculations could be calibrated using the existing actinide data for Maxwellian-averaged neutron cross sections, half-lives, spontaneous fission

An example of nuclear reaction and structure data sets for the Z=90-110 region is shown in Table 3. The tabulated values demonstrate an increasingly complex nature of nuclear decay for Z>95 nuclei where spontaneous fission and *β*-decay play an important role. Spontaneous fission fragments are of interest to the r-process studies. Complimentary information on fission fragments distribution can be obtained from the Sigma Web interface

Finally, for equilibrium conditions one can deduce r-process analog of the equation 11

*dNZ*,*A*/*dt* = *<sup>λ</sup>Z*−<sup>1</sup>*NZ*−<sup>1</sup> − *<sup>λ</sup>ZNZ* (14)

*dNZ*,*A*/*dt* = *NZ*−1,*A*−1/*τZ*−<sup>1</sup> − *NZ*,*A*/*τ<sup>Z</sup>* (16)

*λZNZ* = *NZ*/*τ<sup>Z</sup>* (15)

*NZ*−1/*τZ*−<sup>1</sup> = *NZ*/*τ<sup>Z</sup>* (17)

*τ* = *T*1/2/0.693 (18)

needs can be summarized as follows

• neutrino interaction rates • fission probabilities

• fission products distribution

between nuclear lifetime and half-life is

and delayed neutrons probabilities.

*http://www.nndc.bnl.gov/sigma* (Pritychenko & Sonzogni, 2008).

Further analysis of r-process abundances shows (Boyd, 2008)

• nuclear masses • *β*-decay half-lives • n-capture rates

another neutron

is


Table 1. Evaluated nuclear reaction and KADONIS libraries Maxwellian-averaged neutron capture cross sections in mb at *kT*=30 keV for *s*-process nuclei.

sets demonstrate a strong correlation between nuclear astrophysics and nuclear industry data needs, the large nuclear astrophysics potential of ENDF libraries, and a perspective beneficial relationship between both fields.

#### **3.3 r-process**

The detailed analysis of stable and long-lived nuclei indicates the large number of isotopes that lie outside of the s-process path peaks near A=138 and 208. In addition, the large gap between s-process nucleus 209Bi and 232Th,235,238U effectively terminates the *s*-process at 210Po. Production of the actinide neutron-rich nuclei cannot be explained by the *s*-process nucleosynthesis and requires introduction of rapid neutron capture or r-process. In this case, neutron capture timescale has to be less than typical *β*-decay lifetimes of ∼ms for neutron-rich nuclides. It implies neutron fluxes 1010-1011 higher than those of the s-process. Such conditions can be found in *ν*-driven core-collapse supernova and neutron stars. From here, one may conclude that r-process temperature depends on the site and may lie within a (0.5-10) <sup>×</sup>109 K range. However, it is still not clear where r-process takes place and how it proceeds.

Among many unknowns of the r-process is process path. The path is defined by the nuclear masses and *β*-decay half-lives. The 2003 & 2011 experimental Atomic Mass Evaluations (Audi et al., 2003; Audi & Meng, 2011) do not cover nuclei far from stability near the r-process expected path. To resolve this problem, theoretical mass calculations based on the FRDM and other models have been performed with 25% uncertainties (Aprahamian, 2011). This calculation helped to identify the list of critical nuclei along the r-process path and demonstrated strong connections between nuclear astrophysics and nuclear structure. Further progress will require mass and half-life measurements of unstable nuclei. The r-process data needs can be summarized as follows

• nuclear masses

12 Will-be-set-by-IN-TECH

sets demonstrate a strong correlation between nuclear astrophysics and nuclear industry data needs, the large nuclear astrophysics potential of ENDF libraries, and a perspective beneficial

The detailed analysis of stable and long-lived nuclei indicates the large number of isotopes that lie outside of the s-process path peaks near A=138 and 208. In addition, the large gap between s-process nucleus 209Bi and 232Th,235,238U effectively terminates the *s*-process at 210Po. Production of the actinide neutron-rich nuclei cannot be explained by the *s*-process nucleosynthesis and requires introduction of rapid neutron capture or r-process. In this case, neutron capture timescale has to be less than typical *β*-decay lifetimes of ∼ms for neutron-rich nuclides. It implies neutron fluxes 1010-1011 higher than those of the s-process. Such conditions can be found in *ν*-driven core-collapse supernova and neutron stars. From here, one may conclude that r-process temperature depends on the site and may lie within a (0.5-10) <sup>×</sup>109 K range. However, it is still not clear where r-process takes place and how it

Among many unknowns of the r-process is process path. The path is defined by the nuclear masses and *β*-decay half-lives. The 2003 & 2011 experimental Atomic Mass Evaluations (Audi et al., 2003; Audi & Meng, 2011) do not cover nuclei far from stability near the r-process expected path. To resolve this problem, theoretical mass calculations based on the FRDM and other models have been performed with 25% uncertainties (Aprahamian, 2011). This calculation helped to identify the list of critical nuclei along the r-process path and

Isotope JENDL-4.0 ROSFOND 2010 ENDF/B-VII.1 KADONIS 42-Mo- 96 1.052E+2 1.035E+2 1.035E+2±1.700E+1 1.120E+2±8.000E+0 44-Ru-100 2.065E+2 2.062E+2 2.035E+2±3.949E+1 2.060E+2±1.300E+1 46-Pd-104 2.700E+2 2.809E+2 2.809E+2±4.923E+1 2.890E+2±2.900E+1 48-Cd-110 2.260E+2 2.346E+2 2.349E+2±4.263E+1 2.370E+2±2.000E+0 50-Sn-116 9.115E+1 1.002E+2 1.003E+2±1.875E+1 9.160E+1±6.000E-1 52-Te-122 2.644E+2 2.639E+2 2.349E+2±4.882E+1 2.950E+2±3.000E+0 52-Te-123 8.138E+2 8.128E+2 8.063E+2±1.063E+2 8.320E+2±8.000E+0 52-Te-124 1.474E+2 1.473E+2 1.351E+2±2.697E+1 1.550E+2±2.000E+0 54-Xe-128 2.582E+2 2.826E+2 2.826E+2±6.823E+1 2.625E+2±3.700E+0 54-Xe-130 1.333E+2 1.518E+2 1.518E+2±2.835E+1 1.320E+2±2.100E+0 56-Ba-134 2.301E+2 2.270E+2 2.270E+2±4.038E+1 1.760E+2±5.600E+0 56-Ba-136 7.071E+1 7.001E+1 7.001E+1±1.087E+1 6.120E+1±2.000E+0 60-Nd-142 3.557E+1 3.701E+1 3.343E+1±4.251E+1 3.500E+1±7.000E-1 62-Sm-148 2.361E+2 2.444E+2 2.449E+2±4.507E+1 2.410E+2±2.000E+0 62-Sm-150 4.217E+2 4.079E+2 4.227E+2±3.607E+2 4.220E+2±4.000E+0 64-Gd-154 9.926E+2 1.010E+3 9.511E+2±1.096E+2 1.028E+3±1.200E+1 66-Dy-160 8.702E+2 8.293E+2 8.328E+2±6.769E+1 8.900E+2±1.200E+1 72-Hf-176 5.930E+2 4.529E+2 4.531E+2±4.896E+1 6.260E+2±1.100E+1 80-Hg-198 1.612E+2 1.612E+2 1.613E+2±1.635E+1 1.730E+2±1.500E+1 82-Pb-204 8.355E+1 7.242E+1 7.242E+1±7.624E+0 8.100E+1±2.300E+0 Table 1. Evaluated nuclear reaction and KADONIS libraries Maxwellian-averaged neutron

capture cross sections in mb at *kT*=30 keV for *s*-process nuclei.

relationship between both fields.

**3.3 r-process**

proceeds.


Recent r-process estimates (Cowan & Thielemann, 2004) demonstrate sharp abundance peaks for the A=130,195 (N=82,126) nuclei with large N/Z ratios and another broad peak at A=160. Further analysis of r-process abundances shows (Boyd, 2008)

$$d\mathbf{N}\_{\mathbf{Z},\mathbf{A}}/dt = \lambda\_{\mathbf{Z}-1}\mathbf{N}\_{\mathbf{Z}-1} - \lambda\_{\mathbf{Z}}\mathbf{N}\_{\mathbf{Z}} \tag{14}$$

For neutron closed shells, nuclei most likely will experience *β*-decay rather than absorb another neutron

$$
\lambda\_{\mathbf{Z}} \mathbf{N}\_{\mathbf{Z}} = \mathbf{N}\_{\mathbf{Z}} / \mathbf{\tau}\_{\mathbf{Z}} \tag{15}
$$

Thus near neutron closed shells, the relationship between abundances and *β*-decay lifetimes is

$$dN\_{\rm Z,A}/dt = N\_{\rm Z-1,A-1}/\tau\_{\rm Z-1} - N\_{\rm Z,A}/\tau\_{\rm Z} \tag{16}$$

Finally, for equilibrium conditions one can deduce r-process analog of the equation 11

$$N\_{\mathbf{Z}-1}/\tau\_{\mathbf{Z}-1} = N\_{\mathbf{Z}}/\tau\_{\mathbf{Z}}\tag{17}$$

From the last formula, one can conclude that closed shell nuclei with the largest half-lives will have the largest abundances. In order to obtain the complete picture, all half-lives and decay modes have to be determined. A list of properties relevant to r-process A∼130 nuclei is shown in Table 2. It includes *β*-decay half-lives and emission probabilities for delayed neutrons. The delayed neutrons provide an additional neutron source for the r-process and may shift the location of the abundance peak. The tabulated data were taken from the Evaluated Nuclear Structure Data File (ENSDF) database (Burrows, 1990) and the relation between nuclear lifetime and half-life is

$$
\pi = T\_{1/2} / 0.693 \tag{18}
$$

The regularly updated ENDF and ENSDF database evaluations could provide valuable for r-process data in the actinide region. The extremely neutron-rich superheavy fission nuclei play an important role in element production. Presently, these nuclei can be studied only with theoretical model calculations. These calculations could be calibrated using the existing actinide data for Maxwellian-averaged neutron cross sections, half-lives, spontaneous fission and delayed neutrons probabilities.

An example of nuclear reaction and structure data sets for the Z=90-110 region is shown in Table 3. The tabulated values demonstrate an increasingly complex nature of nuclear decay for Z>95 nuclei where spontaneous fission and *β*-decay play an important role. Spontaneous fission fragments are of interest to the r-process studies. Complimentary information on fission fragments distribution can be obtained from the Sigma Web interface *http://www.nndc.bnl.gov/sigma* (Pritychenko & Sonzogni, 2008).

Isotope *σ*(*n*, *F*), mb T1/2, y SF, % *α*-decay, % *β*-decay, % 232Th 1.672E1 1.40E10 1.1E-9 100 ? 235U 1.376E3 7.04E8 7E-9 100 ? 238U 7.839E1 4.468E9 5.45E-5 100 ? 237Np 9.953E2 2.144E6 <sup>≤</sup>2E-10 100 ? 239Pu 1.885E3 2.411E4 3.1E-10 100 ? 241Am 7.767E2 432.6 3.6E-10 100 ? 250Cm 3.192E2 8.3E3 74 18 8 250Bk 1.209E3 3.212 h ? ? 100 252Cf 2.133E3 2.645 3.092 96.908 ? 255Es 3.108E2 39.8 d 0.0041 8 92 255Fm 2.776E3 20.07 h 2.4E-5 100 ?

Stellar Nucleosynthesis Nuclear Data Mining 35

Table 3. Nuclear reaction and structure properties of several actinides. Reaction cross sections were calculated from ENDF/B-VII.1 library at *kT*=400 keV and decay data were taken from the ENSDF database (*http://www.nndc.bnl.gov/ensdf*). The ? symbol was used

> Reaction EXFOR NSR p,*γ* 396 2162 p,n 666 2571 p,*α* 337 1031 *α*,*γ* 144 522 *α*,n 343 1321 *α*,p 166 848

Table 4. Total number of p-process reaction entries in EXFOR (*http://www-nds.iaea.org/exfor*)

These reactions have been compiled in the EXFOR database since 70ies, and the database content is shown in Fig. 9. For historic reasons, it is relatively complete for neutron-, proton- and alpha-induced reaction compilations, and has a limited number of compilations for heavy-ion and photonuclear reactions. The IAEA EXFOR Web interface: *http://www-nds.iaea.org/exfor* (Zerkin et al., 2005) allows user-friendly nuclear astrophysics data search using multiple parameters, such as target, nuclear reaction, cross section, and energy range. The interface is currently used for a *p*-process nuclei data mining operation at ATOMKI (Szücs et al., 2010). EXFOR is the best source of experimental nuclear reaction data; however,

To overcome this problem the Nuclear Science References (NSR) database (*http://www.nndc.bnl.gov/nsr*) (Pritychenko et al., 2011) is recommended. Table 4 shows EXFOR and NSR database content for the ATOMKI project scope of (p,*γ*), (p,n), (p,*α*), (*α*,*γ*), (*α*,n) and (*α*,p) reactions. The tabulated data indicate a factor of 3-5 difference between two databases. This is mostly due to the fact that multiple article can be combined into a single

Another important tool for p-process nucleosynthesis studies is the nuclear reaction reciprocity theorem. It allows extracting a reaction cross section if an inverse reaction is

and NSR (*http://www.nndc.bnl.gov/nsr*) databases as of August 2011.

it is not complete, and it takes ∼1-2 y before article compilation is completed.

EXFOR entry and gaps in the EXFOR coverage.

where data were not available.


Table 2. Properties of neutron-rich nuclides relevant to the A=130 r-process peak. All data are taken from the ENSDF database (*http://www.nndc.bnl.gov/ensdf*) (Burrows, 1990). The ? symbol was used where data were not available.

#### **3.4 p-process**

A detailed analysis of the Fig. 1 data indicates between 29 and 35 proton-rich nuclei that cannot be produced in the s- or r-processes. A significant fraction of these nuclei originate from the *γ*-process (Boyd, 2008; Woosley & Howard, 1978). This process could take place in Type-II supernovae at (2-3)×T9. It begins with (*γ*,n) reactions that synthesize proton-rich heavy nuclei that are followed by charged-particle emitting reactions. Such process includes an extensive reaction network consisting of approximately 20,000 reactions and 2,000 nuclei. Due to lack of experimental data, p-process network calculations are often based on theoretical model predictions. This situation can be improved via addition of the known experimental reaction cross sections.

14 Will-be-set-by-IN-TECH

Isotope T1/2, msec *β*−-decay, % *β*-n Emission, %

Table 2. Properties of neutron-rich nuclides relevant to the A=130 r-process peak. All data are taken from the ENSDF database (*http://www.nndc.bnl.gov/ensdf*) (Burrows, 1990). The ?

A detailed analysis of the Fig. 1 data indicates between 29 and 35 proton-rich nuclei that cannot be produced in the s- or r-processes. A significant fraction of these nuclei originate from the *γ*-process (Boyd, 2008; Woosley & Howard, 1978). This process could take place in Type-II supernovae at (2-3)×T9. It begins with (*γ*,n) reactions that synthesize proton-rich heavy nuclei that are followed by charged-particle emitting reactions. Such process includes an extensive reaction network consisting of approximately 20,000 reactions and 2,000 nuclei. Due to lack of experimental data, p-process network calculations are often based on theoretical model predictions. This situation can be improved via addition of the known experimental

symbol was used where data were not available.

**3.4 p-process**

reaction cross sections.

127Ag <sup>109</sup>±25 100 ? 128Ag <sup>58</sup>±5 100 ? 129Ag <sup>46</sup>±7 100 ? 130Ag <sup>≈</sup>50 ? ? 130Cd <sup>162</sup>±7 100 3.5±1.0 131Cd <sup>68</sup>±3 100 3.5±1.0 132Cd <sup>97</sup>±10 100 60±<sup>15</sup> 131In <sup>280</sup>±30 100 <sup>≤</sup>2.0±0.3 132In <sup>207</sup>±6 100 6.3±0.9 133In <sup>165</sup>±3 100 85±<sup>10</sup> 134In <sup>140</sup>±4 100 <sup>65</sup> 135In <sup>92</sup>±10 100 <sup>&</sup>gt;<sup>0</sup> 130Sn <sup>223200</sup>±4200 100 ? 131Sn <sup>56000</sup>±5000 100 ? 132Sn <sup>39700</sup>±800 100 ? 133Sn <sup>1460</sup>±30 100 0.0294±<sup>24</sup> 134Sn <sup>1050</sup>±11 100 17±<sup>13</sup> 135Sn <sup>530</sup>±20 100 21±<sup>3</sup> 136Sn <sup>250</sup>±30 100 30±<sup>5</sup> 137Sn <sup>190</sup>±60 100 58±<sup>15</sup> 138Sn >0.000408 ? ? 139Sn ?? ? 140Sn ?? ? 136Sb <sup>923</sup>±14 100 16.3±3.2 137Sb <sup>450</sup>±50 100 49±<sup>10</sup> 138Sb <sup>≥</sup>0.0003 ? ? 139Sb >0.00015 ? ? 137Te <sup>2490</sup>±50 100 2.99±<sup>16</sup> 138Te <sup>1400</sup>±400 100 6.3±2.1 139Te >0.00015 ? ?


Table 3. Nuclear reaction and structure properties of several actinides. Reaction cross sections were calculated from ENDF/B-VII.1 library at *kT*=400 keV and decay data were taken from the ENSDF database (*http://www.nndc.bnl.gov/ensdf*). The ? symbol was used where data were not available.


Table 4. Total number of p-process reaction entries in EXFOR (*http://www-nds.iaea.org/exfor*) and NSR (*http://www.nndc.bnl.gov/nsr*) databases as of August 2011.

These reactions have been compiled in the EXFOR database since 70ies, and the database content is shown in Fig. 9. For historic reasons, it is relatively complete for neutron-, proton- and alpha-induced reaction compilations, and has a limited number of compilations for heavy-ion and photonuclear reactions. The IAEA EXFOR Web interface: *http://www-nds.iaea.org/exfor* (Zerkin et al., 2005) allows user-friendly nuclear astrophysics data search using multiple parameters, such as target, nuclear reaction, cross section, and energy range. The interface is currently used for a *p*-process nuclei data mining operation at ATOMKI (Szücs et al., 2010). EXFOR is the best source of experimental nuclear reaction data; however, it is not complete, and it takes ∼1-2 y before article compilation is completed.

To overcome this problem the Nuclear Science References (NSR) database (*http://www.nndc.bnl.gov/nsr*) (Pritychenko et al., 2011) is recommended. Table 4 shows EXFOR and NSR database content for the ATOMKI project scope of (p,*γ*), (p,n), (p,*α*), (*α*,*γ*), (*α*,n) and (*α*,p) reactions. The tabulated data indicate a factor of 3-5 difference between two databases. This is mostly due to the fact that multiple article can be combined into a single EXFOR entry and gaps in the EXFOR coverage.

Another important tool for p-process nucleosynthesis studies is the nuclear reaction reciprocity theorem. It allows extracting a reaction cross section if an inverse reaction is

**4. Conclusion**

**5. References**

3-183.

886-900.

pp. 337-676.

probabilities have been considered.

Distribution FE, December 2004.

Guelph, Ontario, Canada.

Finally, a review of stellar nucleosynthesis and its data needs has been presented. Several nuclear astrophysics opportunities and the corresponding computation tools and methods have been identified. Complimentary sets of nuclear data for s-process nucleosynthesis have been produced. These, nuclear astrophysics model-independent data sets are based on the latest evaluated nuclear libraries and low-fidelity covariances data. This analysis indicates that the nucleosynthesis processes and respective abundances are strongly affected by both nuclear reaction cross sections and nuclear structure effects. Several nuclear structure data sets that include *β*-decay half-lives, spontaneous fission, and delayed neutron emission

Stellar Nucleosynthesis Nuclear Data Mining 37

Present results demonstrate a wide range of uses for nuclear reaction cross sections and structure data in stellar nucleosynthesis. They provide additional benchmarks and build a bridge between nuclear astrophysics and nuclear industry applications. Further work will

We are grateful to M. Herman (BNL) for the constant support of this project, to S. Goriely (Universite Libre de Bruxells) and R. Reifarth (Goethe University) for productive discussions, and to V. Unferth (Viterbo University) for a careful reading of the manuscript. This work was sponsored in part by the Office of Nuclear Physics, Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-98CH10886 with Brookhaven Science Associates, LLC.

Aldama, D.L. & Trkov, A. (2004). FENDL-2.1 Update of an evaluated nuclear data

Anders, E. & Grevesse, N. (1989). Abundances of the elements - Meteoritic and solar.

Angulo, C.; Arnould, M.; Rayet, M.; Descouvemont, P.; Baye, D.; Leclercq-Willain, C.; Coc, A.;

Aprahamian, A. (2011). R-process mass sensitivities. *Proc. XIV International Symposium on*

Arlandini, C.; Käppeler, F. ; Wisshak, K.; Gallino, R.; Lugaro, M.; Busso, M. & Straniero, O.

Audi, G.; Wapstra, A.H. & Thibault, C. (2003). The AME2003 atomic mass Evaluation (II).

Audi, G. & Meng, W. (2011). Atomic Mass Evaluation 2011. *Private Communication*.

*Geochimica et Cosmochimica Acta*, Vol. 53, January 1989, pp. 197-214.

library for fusion applications. *International Atomic Energy Agency*, INDC(NDS)-467,

Barhoumi, S.; Aguer, P.; Rolfs, C.; Kunz, R.; Hammer, J.W.; Mayer, A.; Paradellis, T.; Kossionides, S.; Chronidou, C.; Spyrou, K.; Degl'Innocenti, S.; Fiorentini, G.; Ricci, B.; Zavatarelli, S.; Providencia, C.; Wolters, H.; Soares, J.; Grama, C.; Rahighi, J.; Shotter, A. & Lamehi Rachti, M. (1999). A compilation of charged-particle induced thermonuclear reaction rates. *Nuclear Physics A*, Vol. 656, No.1, August 1999, pp.

*Capture γ-ray Spectroscopy*, August 28-September 2, 2011, University of Guelph,

(1999). Neutron Capture in Low-Mass Asymptotic Giant Branch Stars: Cross Sections and Abundance Signatures. *The Astrophysical Journal*, Vol. 525, November 1999, pp.

Tables, graphs, and references. *Nuclear Physics A*, Vol. 729, Issue 1, December 2003,

include extensive data analysis, neutron physics, and network calculations.

Fig. 9. Time and content evolution of the EXFOR database. Initially database scope was limited to neutron-induced reaction cross sections, later scope expansion included charge particle and photo-nuclear reactions.

known. For 1 + 2 → 3 + 4 and 3 + 4 → 1 + 2 processes the cross section ratio is

$$\frac{\sigma\_{34}}{\sigma\_{12}} = \frac{m\_3 m\_4 E\_{34} (2J\_3 + 1)(2J\_4 + 1)(1 + \delta\_{12})}{m\_1 m\_2 E\_{12} (2J\_1 + 1)(2J\_1 + 1)(1 + \delta\_{34})},\tag{19}$$

where *E*12, *E*<sup>34</sup> are kinetic energies in the c.m. system, J is angular momentum, and *δ*12=*δ*34=0.

#### **3.5 rp-process**

The rp-process (rapid proton capture process) consists of consecutive proton captures onto seed nuclei to produce heavier elements. It occurs in a number of astrophysical sites including X-ray bursts, novae, and supernovae. The rp-process requires a high-temperature environment (∼109 K) to overcome Coulomb barrier for charged particles. This process contributes to observed abundances of light and medium proton-rich nuclei and compliments the p-process. The rp-process data needs include


#### **4. Conclusion**

16 Will-be-set-by-IN-TECH

Alpha

 Neutron Proton

Photo-nuclear

1960 1970 1980 1990 2000 2010

<sup>=</sup> *<sup>m</sup>*3*m*4*E*34(2*J*<sup>3</sup> <sup>+</sup> <sup>1</sup>)(2*J*<sup>4</sup> <sup>+</sup> <sup>1</sup>)(<sup>1</sup> <sup>+</sup> *<sup>δ</sup>*12) *m*1*m*2*E*12(2*J*<sup>1</sup> + 1)(2*J*<sup>1</sup> + 1)(1 + *δ*34)

where *E*12, *E*<sup>34</sup> are kinetic energies in the c.m. system, J is angular momentum, and *δ*12=*δ*34=0.

The rp-process (rapid proton capture process) consists of consecutive proton captures onto seed nuclei to produce heavier elements. It occurs in a number of astrophysical sites including X-ray bursts, novae, and supernovae. The rp-process requires a high-temperature environment (∼109 K) to overcome Coulomb barrier for charged particles. This process contributes to observed abundances of light and medium proton-rich nuclei and compliments

, (19)

Fig. 9. Time and content evolution of the EXFOR database. Initially database scope was limited to neutron-induced reaction cross sections, later scope expansion included charge

known. For 1 + 2 → 3 + 4 and 3 + 4 → 1 + 2 processes the cross section ratio is

Year

5k

particle and photo-nuclear reactions.

**3.5 rp-process**

• nuclear masses • proton capture rates *σ*<sup>34</sup> *σ*<sup>12</sup>

the p-process. The rp-process data needs include
