Fisica -specialistica.pdf · timento di Fisica di Perugia and of INFN, in particular of the...

Universita degli studi di Perugia

Facolta di Scienze Matematiche, Fisiche eNaturali

Tesi di Laurea Magistrale in

Fisica

The 13C(α,n)16O reaction rate. Recent

estimates, new measurements through

the Trojan Horse Method and their

astrophysical consequences.

Relatore Candidato

Prof. Busso Maurizio Maria Trippella Oscar

Prof. Spitaleri Claudio

Anno Accademico 2010/2011

Universita degli studi di Perugia

Facolta di Scienze Matematiche, Fisiche eNaturali

Tesi di Laurea Magistrale in

Fisica

The 13C(α,n)16O reaction rate. Recent

estimates, new measurements through

the Trojan Horse Method and their

astrophysical consequences.

Relatore Candidato

Prof. Busso Maurizio Maria Oscar Trippella

Prof. Spitaleri Claudio

Anno Accademico 2010/2011

...We are stardust, we are golden

We are billion year old carbon...

Woodstock - Crosby, Stills, Nash and Young

3

CHAPTER

ONE

INTRODUCTION.

Historically, stars have been part of religious practices and used for celestialnavigation and orientation: there are examples of astronomical studies allaround the world from Egypt to Greece, from the Maya population to theChinese one. However, it was only due to European researchers during andafter the XVIth century that astronomy assumed its modern role as a sci-ence. A special role was obviously played by the introduction of the telescopeby Galileo Galilei in the XVIIth century; the subsequent search for physicalexplanations for the motion and appearance of stars founded astrophysics.This is the branch of astronomy that today studies the structure, evolution,chemical composition and physical properties of stars and galaxies. Impor-tant conceptual progresses on the physical behaviour of stars occurred duringthe twentieth century because of new theoretical approaches, the applica-tion of modern physics and the advent of more accurate photometric andspectroscopic measurements. During the first decades of the XXth century,results from nuclear physics research, in particular the discovery of the enor-mous energy stored in the nuclei, led astrophysicists to guess that reactionsamong nuclear species were the source of the stellar power (Rolf & Rodney,1988; Eddington et al., 1920). Since then, nuclear astrophysics has playeda key role in providing the interpretation of astrophysical observations. Inthis sense, using the observational evidence coming from stellar atmospheresand the experimental evidence coming from nuclear experiments aimed atstudying specific nuclear reactions, nuclear astrophysics can determine howthe processes of nuclear fusion drive the structural changes and promotesstellar evolution.

This thesis is a particular example of the role played by nuclear astro-physics, as it covers the steps from the nuclear measurement of a reactionrate of astrophysical interest (the 13C(α,n)16O reaction) up to the study ofthe stellar consequences implied by a reaction rate change. These conse-quences concern the release of neutrons and the ensuing n-capture nucle-

5

osynthesis in low mass stars. The above mentioned reaction is importantbecause it is considered as the dominant neutron source active in stars witha mass included in the range 0.8 - 3 M⊙, which actively contribute to thenucleosynthesis of heavy nuclei through neutron capture processes.

Roughly a half of all elements heavier than iron in the universe were pro-duced in this way, in the so-called s (slow) process (Burbidge et al., 1957),which basically includes neutron-induced capture reactions and beta de-cays. The term slow, used to distinguish this mechanism from a rapid one(r-process, occurring in supernovae), refers to the fact that the neutron-capture timescale is in general longer than for the decay of unstable nuclei,which fact requires typical neutron densities of about 106 − 1010n/cm3.

In order to set the stages for the nuclear astrophysics processes of interest,I shall first discuss the typical evolutionary phases for a star of one solar mass(assumed to represent a low mass star in general). A particular emphasiswill be dedicated to the Asymptotic Giant Branch (AGB) stage when, afterthe exhaustion of helium at the center, the representative point in the H-Rdiagram ascends for a second time towards the red giant branch (RGB),asymptotically approaching it.

During this phase, and more specifically in the Thermally Pulsing-AGB,the C-O core is surrounded by two shells of helium and hydrogen burningalternatively. There is a helium rich intershell region between the two shellsthat becomes almost completely convective at intervals, while the tempera-ture suddenly increases: it is the so-called thermal pulse (TP). The thermalpulse is repeated many times (from ∼ 5 to 50 cycles) before the envelope iscompletely eroded by mass loss, so nucleosynthesis products manufacturedby He burning and the s-process at its bottom are carried to the surface. Inthe intershell region 12C is abundant. The existence, now proven, of mixingepisodes carrying protons downward from the envelope yields the formationof a p- and 12C-rich layer after each thermal pulse. There, after the ignitionof the H shell, p-captures generate the so-called 13C pocket. In this contextI shall discuss how neutrons are released thanks to the 13C(α,n)16O reactionand s-processing occurs in AGB stars, in the radiative inter-pulse phases.The typical stellar environment in which our reaction takes place correspondsto T ∼ 0.09−0.1×109 K. In such conditions, the other main neutron source,the 22Ne(α,n)25Mg reaction, is switched off, as it needs higher temperaturesto be activated.

In the above conditions big problems affecting our knowledge of reac-tion rates are related to the effects of the Coulomb barrier for the charged-particle-induced reactions and to electron screening. The presence of thebarrier implies an exponential suppression for the cross section and does notallow a direct measurement at the energies of astrophysical interest. Crosssection measurements at such low energies must also cope with a low signal-to-noise ratio, which can be improved only in underground experimentalfacilities, such as LUNA at the Gran Sasso National Laboratories.

6

At present, existing direct measurements for the reaction 13C(α,n)16O,collected in the NACRE compilation by Angulo et al. (1999), stop at theminimum value of 280 keV (Drotleff et al., 1993), whereas the region ofastrophysical interest, the so-called ”Gamow window”, corresponds to 190± 90 keV at a temperature of 0.1 × 109 K. Below the limit reached bemeasurements only a theoretical extrapolation is possible. Various types ofapproaches have been tried over the years to extend the measurement ofthe cross section into the region of astrophysical interest. The main aim ofthese efforts is to improve the accuracy of the measurement, reducing theuncertainty, which sometimes exceeds 300%. The major source of error isthe presence of a subthreshold resonance corresponding to the excited stateof 17O (Eres = 6.356 Mev or Ec.m. = −3 keV). The most recent works in theliterature are oriented towards a substantial lowering of the reaction rate,because it is believed that the role of the resonance mentioned above wasoverestimated in the past.

In this context I participated to a new experiment at the Florida StateUniversity, made by the ASFIN2 collaboration (centered at LaboratorioNazionale del Sud) applying an indirect technique called “Trojan HorseMethod”. The THM is based on a quasi-free break-up process and allows toextract the cross section of the two-body reaction (of astrophysical interest):

x+ a→ c+ C (a.1)

from a suitable three-body one:

A+ a→ c+ C + s (a.2)

Here A acts as the Trojan Horse nucleus, being a cluster x ⊕ s structure.In the hypothesis of the TH-nucleus quasi-free break-up, s represents thespectator of the virtual 2-body reaction of interest for astrophysics.

Our experiment was performed by measuring the sub-Coulomb 13C(α,n)16O scat-tering within the interaction region via the THM, applied to the 13C(6Li,n16O)d re-action in the quasi-free kinematics regime. However, the final result derivingby the Trojan Horse method is not complete yet, because data analysis isstill under development and will be finalized in the next months.

Since the result derived from the THM is not yet applicable, it was de-cided to check what would be the consequences for n-capture nucleosynthe-sis if the presently-accepted rate were to change by some substantial factor.Presently, the rate most commonly used is that suggested by Drotleff et al.(1993). A decrease of its values by roughly a factor of 3 would correspondapproximately to the alternative indications by Kubono et al. (2003). I shallshow that a result in this direction would imply substantial changes in the op-eration of the crucial s-process branching at 85Kr with respect to what is as-sumed today. Elements far from this region would be essentially unchanged.I also analyzed the effects of an increase in the rate by Drotleff et al. (1993)

7

by the same factor of 3, noting that the changes would be more widespreadover the s-process path and would introduce remarkable changes in our ideason the solar abundance distribution. These results encourage a deeper studyof the 13C(α,n)16O reaction.

This thesis would not have been possible without the help of the Dipar-timento di Fisica di Perugia and of INFN, in particular of the LaboratoriNazionali del Sud and of the Perugia and Catania Sections. Thanks are dueto INFN for providing me with a fellowship covering the expenses of thestages in Catania and in Tallahassee (Florida).

8

CONTENTS

1 Introduction. 5

2 Final evolutionary stages for low mass stars. 112.1 pre-AGB phases. . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Asymptotic Giant Branch (AGB) stars and Thermal Pulse. . 152.3 The third dredge-up. . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Nucleosynthesis and observations for AGB stars. . . . . . . . 21

3 s-Process nucleosynthesis in AGB stars. 253.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 The classical analysis of the s process. . . . . . . . . . . . . . 27

3.3 Evolution and nucleosynthesis in the AGB stages. . . . . . . . 313.4 The neutron source 13C(α,n)16O. . . . . . . . . . . . . . . . . 343.5 Possible future scenarios. . . . . . . . . . . . . . . . . . . . . 38

4 Cross sections of nuclear reactions at low energies. 39

4.1 Coulomb barrier and penetration factor. . . . . . . . . . . . . 404.2 Cross section, astrophysical factor and reaction rate. . . . . . 414.3 Gamow peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Direct measurements and experimental problems. . . . . . . 45

4.5 Indirect methods for nuclear astrophysics . . . . . . . . . . . 49

5 Measure of the 13C(α,n)16O reaction through the THM. 515.1 Theory of the Trojan Horse method. . . . . . . . . . . . . . . 525.2 Plane Wave Impulse Approximation. . . . . . . . . . . . . . . 54

5.3 Current measurement status . . . . . . . . . . . . . . . . . . . 585.4 The Trojan Horse Method applied to the 13C(α,n)16O reaction. 625.5 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . 645.6 Position Sensitive Detectors (PSDs). . . . . . . . . . . . . . . 68

5.7 The position calibration. . . . . . . . . . . . . . . . . . . . . . 705.8 Energy calibration. . . . . . . . . . . . . . . . . . . . . . . . . 725.9 Data Analysis and future work. . . . . . . . . . . . . . . . . . 73

9

CONTENTS

6 On the astrophysical consequences of changes in the 13C(α,n)16O rate. 796.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Effects of reducing the rate by a factor of three. . . . . . . . . 806.3 Effects of increasing the rate by a factor of three. . . . . . . . 84

7 Conclusions 89

8 Ringraziamenti. 101

A Main thermonuclear reactions in pre-AGB phases. 103A.1 Hydrogen (H) burning. . . . . . . . . . . . . . . . . . . . . . . 103

A.1.1 pp-Chain. . . . . . . . . . . . . . . . . . . . . . . . . . 103A.1.2 CNO-cycle. . . . . . . . . . . . . . . . . . . . . . . . . 105

A.2 Helium (He) burning: triple-α process. . . . . . . . . . . . . . 106

10

CHAPTER

TWO

FINAL EVOLUTIONARY STAGES FOR LOW MASS

STARS.

Stars, like for example the Sun, are gaseous objects that shine of proper lightbecause of thermonuclear fusion reactions occurring in their interior produc-ing electromagnetic energy and neutrinos. They are considered as the forgesof universe because the whole set of elements (excluding initial abundancesof nuclei lighter than 12C, which are created during the first minutes afterthe Big Bang) are produced in stars. The main cause of heating, contrac-tion and density increase in stars is the total gravitational energy of thestellar mass. Generally speaking, the larger is the mass, the higher is thecentral temperature allowing reactions among heavier elements. Theoreti-cal and experimental studies on the reaction rates showed that fusion can,in sequence, occur among: hydrogen (H), helium (He), carbon (C), neon(Ne), oxygen (O), magnesium (Mg) and silicon (Si). If the initial mass ofa star is less than about Mmin ∼ 0.08 M⊙ (M⊙ being the so-called solarmass, corresponding to about 1.9891×1030 kg), the temperature is not highenough to start hydrogen burning. In this work I shall limit my discussion tostars belonging to the mass range 0.8− 3 M⊙, the so-called Low Mass Stars(hereafter LMS). They experience only hydrogen and helium burning beforeelectron degeneracy in a C-O core stops the proceeding of stellar evolution.

Concerning this concept of electron degeneracy, it is the state in whichmatter has such high values of density ρ and pressure P that electronsbecome a Fermi condensate, whose pressure effectively stops the slow gravi-tational contraction of the star, thus preventing the appropriate conditionsto start thermonuclear reactions. In practice, particles of mass mp have avery small mean free path l, to the point that they are almost in contact toeach other. This means that:

l ∼(

1

n

)1/3

=

(

µmH

ρ

)1/3

=

(

mp

ρ

)1/3

(2.1)

11

2.1. pre-AGB phases.

has a numerical value close to the particle dimension, defined by the DeBroglie’s wavelength:

λ =h

mpv(2.2)

where v indicates the thermal velocity v =

√

3kBT

mp. Then:

(

mp

ρ

)1/3

=h

mp

√

mp

3kBT(2.3)

from which I get ρ:

ρ1/3 =m

5/6p

√3kBT

h(2.4)

ρ =

(√3kBT

h

)3

T 3/2m5/2p ∝ T 3/2m5/2

p (2.5)

This is the critical density at which particles begin to degenerate and cannotbe described any more by a Maxwell-Boltzmann distribution. Such a criticaldensity is lower when the particle mass is lower: hence, electrons degeneratebefore atomic nuclei. The occurrence of electron degeneracy depends onthe stellar temperature and initial mass, in the sense that lower massesdegenerate more easily having a lower internal temperature.

Let’s briefly discuss the main evolutionary stages of a typical low-massstar making use of a schematic view of the track followed by the stellarrepresentative point in the Hertzsprung-Russell diagram (hereafter H-R di-agram). This is a plot reporting the absolute magnitudes or luminosities ofstars versus their spectral types or effective temperatures and is a very usefultool, providing important information about stellar structure and evolution.In particular, I shall concentrate on the structure of the so-called asymp-totic giant branch (AGB) stars. These stars are climbing for the secondtime along the red giant branch; here they experience thermal instabilities,or pulses, from the He shell activating on the border of the degenerate C-Ocore. Following a pulse, AGB stars provide to mix to the surface fresh car-bon (which is the main product of incomplete helium burning) and s-processisotopes.

2.1 pre-AGB phases.

At first, I discuss the pre-AGB evolution adopting a typical model of a 1M⊙ star, introducing the required terminology and physics when necessary.For clarity, I present in Figure 2.1 the track followed by the stellar repre-sentative point in the H-R diagram. Stars are born from gas clouds in theinterstellar medium (ISM) thanks to the gravitational collapse of a massive

12


Figure 2.1: Schematic evolution in the H-R diagram of a 1 M⊙ stellarmodel and solar metallicity. All the major evolutionary phases discussed inthe text are indicated. The plot reports bolometric magnitude Mbol versuseffective temperature Teff .

fragment of a cloud. The ISM, in the physical conditions just described, ismainly composed of atoms and molecules of hydrogen and heavy elements.Sir James Jeans, in the twenties, laid down the quantitative circumstancesallowing a cold gas cloud in the ISM to become gravitationally unstableand to condense into a proto-star. Starting from the Virial theorem andassuming a spherical mass, he deduced the so-called Jeans’ mass (MJ ):

MJ =

(

2 · 1035T3/2

n1/2

)

(2.6)

In equation (2.6) I indicate the cloud temperature with T , while n corre-sponds to the particle number density in the same zone. The numerical valueof the Jeans’ mass, expressed in grams, depends on temperature and densityand in typical conditions of interstellar clouds corresponds to about 1000M⊙.Hence, if a cloud is more massive than this critical value the collapse can

13


occur. After the gravitational collapse, the representative point of a star inthe H-R diagram moves along a line called Hayashi track, from the nameof the Japanese physicist who derived it, characterized by heat transportoccurring through convention. The luminosity decreases while the surfacetemperature Teff is almost constant because of the decreasing radius. Then,the representative point moves to a track of increasing temperatures (Henyeytrack), until it stops on the Main Sequence (hereafter MS) that correspondsto reaching central temperatures and densities (T = 107 K, ρ = 100 g/cm3)sufficient to start hydrogen fusion. Core hydrogen burning starts on the so-called zero age main sequence (ZAMS) and the star remains near this zonefor 80 - 90% of its life. The main effect is the transformation of four protonsinto a nucleus of 4He, with a release of energy of about Q = 26MeV (thisroughly corresponds to the Q-value resulting from the chain of reactions, seeAppendix A). For initial temperatures lower than about 18 × 106 K, cor-responding to an initial mass of about 1.3 M⊙, reactions proceed throughdirect fusions of protons (the so-called pp-chain); for higher temperaturesthe CNO cycle prevails. This last process needs non-zero initial abundancesof carbon, nitrogen and oxygen (CNO), which act as catalysts for the con-version of hydrogen into helium. Figure 2.2 shows the relative efficiency ofthe two processes as a function of temperature. For the mass range of our

Figure 2.2: Produced energy per unit time and stellar mass versus temper-ature, for the pp-chain and the CNO cycle. For stars with M > 1.3 M⊙ theCNO cycle prevails in the energy production. The vertical line shows thetemperature T0 at which the energy production is the same for the twomechanisms.

interest, during the whole main sequence the stellar structure consists in a

14

2.2. Asymptotic Giant Branch (AGB) stars and Thermal Pulse.

H-burning core, a large He-rich inert buffer and a relatively thin convectiveenvelope. When, because of hydrogen exhaustion, the nuclear processes failto contrast the gravitational pressure, the hydrostatic equilibrium is brokenand the core starts to contract. At this stage stars leave the main sequencewhile the central He core becomes electron degenerate and nuclear burningis established in a shell surrounding this core. Simultaneously, the star ex-pands and the outer layers become convective. Convection extends quitedeeply inward (in mass), and the star ascends the (first) red giant branch(hereafter RGB). Helium is the most abundant element in the stellar core,while the remaining hydrogen buffer has at its base a thin burning shell.The envelope inward extension enriches the surface with materials recentlyaffected by p-captures and this determines a modification of the chemicalabundances; in particular, a significant depletion of 12C and 15N and an in-crease of 4He, 13C and 14N occur. Oxygen isotopes experience changes too,with an increase in 17O and a depletion in 18O (Boothroyd & Sackmann,1999; Charbonnel, 1994).

The activation of the H-burning shell increases the stellar luminosity andthe star leaves the MS toward the RGB on the H-R diagram. Here, the He-core continues to contract and heat. Neutrino energy losses from the centercause the temperature maximum to move outward, as shown in Figure 2.1.Eventually, triple alpha reactions (4He(2α,γ)12C), which rapidly increase thecore luminosity, are ignited at the point of maximum temperature, but witha degenerate equation of state. The temperature and density (∼ 108 K and∼ 107 g/cm3 ) are decoupled, as the equilibrium of a degenerate gas doesnot depend on T . In such a case He-burning ignition can occur only in anexplosive way (the He-flash). Following this, the star quickly moves to theHorizontal Branch, where it burns 4He gently in a convective core, and H ina shell (which provides most of the luminosity). Helium burning increasesthe mass fraction of 12C and 16O (the latter through the further reaction12C(α,γ)16O) and the outer regions of the convective core become stable tothe Schwarzschild’s criterion for convection. It is however unstable to theLedoux’s stability rule. This situation is referred to as semi-convection. Atcore He exhaustion, the star shrinks again and has to carry out the excessenergy, generated by gravitational contraction of the C-O core and by Heburning in a shell. The representative point in the H-R diagram, for low-mass stars, asymptotically approaches the RGB track and is therefore knownas the AGB stage.

2.2 Asymptotic Giant Branch (AGB) stars and

Thermal Pulse.

Every star less massive than about 8 M⊙ evolves into an asymptotic giantbranch star with an electron-degenerate core composed of carbon and oxy-

15


gen. The ascent of the AGB begins following the exhaustion of helium at thecenter. The phenomenon was discovered by Schwarzschild & Harm (1965)in LMS and then confirmed by Weigart et al. (1966) in more massive stars.Model AGB stars are confined to a very small region of the theoretical H-Rdiagram, all with surface temperatures in the range 2500 − 6000 K, in aregion near the RGB track. At core He exhaustion, the star, whose masshas been reduced by stellar winds by up to 10%, starts to be powered byHe burning in a shell and partly by the release of potential energy from thegravitationally contracting C-O core. The central density rapidly increases(above 105 g/cm3) and the C-O core degenerates and cools down with a hugeenergy loss by plasma neutrinos. In LMS core burning is completely pre-vented by degeneracy and one can note that there exists a relation betweenthe luminosity and the mass of the degenerate core: L ∼ 104(MCO − 0.5)where L and MCO are measured in solar unities.

During the early phases (E-AGB), for all stars less massive than about3 M⊙, the energy output from the He shell forces the star to expand adcool so that the H shell remains substantially inactive. When the E-AGBphase is terminated, the H shell is reignited, and from then on it dominatesthe energy production, whereas the He shell is almost inactive (LHe/LH ∼10−3). Late on the AGB, the stellar structure, schematically represented inFigure 2.3, is characterized by a C-O core, two shells (an inner of helium andan outer of hydrogen) burning alternatively, separated by a thin He-rich layerin radiative equilibrium, (∼ 10−2 M⊙, the so-called intershell region) and anextended convective envelope. A thermal pulse occurs when the amount of

Figure 2.3: Stellar structure of a star in the thermally-pulsing AGB phase.

16


He synthesized by the H shell is high enough to be compressed and heatedas requested for its re-ignition. The first thermal pulse determines the endof the early AGB stage and the beginning of the second part of the AGB,defined as thermally-pulsing (TP-AGB). When LMS begin this phase, withC-O cores of mass 0.5 < MCO/M⊙ < 0.6, they are brighter than the tip ofthe red giant branch (logL ∼ 3.3). During the quiescent hydrogen-burningphase, the temperatures and densities in the helium-rich layers below theburning shell increase together with the mass of these same layers. Oncethe mass of the helium-rich region exceeds a critical value, the rate at whichenergy is emitted by helium burning becomes larger than the rate at whichit can escape via radiative losses, and a thermonuclear runaway ensues.

Although the degree of electron degeneracy of the He-rich material isweak, this thermonuclear runaway occurs because the thermodynamic timescale needed to locally expand the gas is much longer than the nuclear burn-ing time scale of the 3α-reactions. The power generated blows up to 108L⊙

(most of which being spent to expand the structure); radiative mechanismscannot transmit all this energy and the intershell region from radiative be-comes convective. Then, the freshly synthesized products of He burning(such as 12C, whose resulting mass fraction in the top layers of the intershellregion is X(12C) ∼ 0.25) are mixed over the whole intershell. Afterwards, thestar readjusts its structure and the thermal instability pushes outward thelayers of material located above the He-burning shell. The temperature andthe density at the base of the H-rich envelope decrease and the H-burningshell is quenched. As a consequence, the intershell region becomes radia-tive again. The above process is repeated many times (from about 5 to 50)until the envelope is completely eroded by mass loss, which strongly affectsthe AGB phase. The illustration (see Figure 2.4) shows the structure ofa TP-AGB star over time, showing with thick black lines the base of theconvective envelope, the H-burning shell, and the He-burning shell. Theregion between the H and He shells is the helium intershell. Horizontal graybars represent zones where protons can partially penetrate the He layers,because the convective eddies do not stop abruptly at the convective bor-der, but have a decreasing profile of temperatures. When H burning in theshell starts, these protons build fresh 13C through the 12C(p,γ)13N(β+ν)13Creaction. This subsequently undergoes alpha captures through (α,n)16O,releasing neutrons. In current models, 13C is naturally burned under radia-tive conditions before being ingested in the convective zone of the followingthermal pulse. Note that proton penetration into the He-rich layers cannotoccur in other ways. In particular, the convective thermal pulse does notreach the H-burning shell, despite it can extend very close to it. An entropybarrier is present, during the thermal instability, between the intershell re-gion and the base of the stellar envelope, preventing the direct penetrationof convection from the He-rich layers into the H shell.

After the expansion and cooling of the envelope, the stellar structure

17


Figure 2.4: Illustration of the structure of a thermally pulsing-asymptoticgiant branch star over time.

shrinks. Because of the low density, the ratio of the gas pressure to the radi-ation pressure decreases and the local temperature gradient increases. Theadiabatic temperature gradient approaches its minimum allowed value for afully ionized gas plus radiation and convection from the envelope penetratesbelow the H-He discontinuity, beyond the former position of the now inac-tive H shell. He-shell burning continues radiatively for another few thousandyears, and then H-shell burning starts again. After a limited number of TPs,when the mass of the H-exhausted core reaches 0.6 M⊙ and the H shell isinactive, the mentioned penetration of the convective envelope reaches downto regions of the He intershell previously affected by the TP so that newlysynthesized materials can be mixed to the surface (third dredge-up, TDU).

TDU is so-called because it is very similar to a previous mixing episode,named second dredge-up (experienced only by intermediate mass stars dur-ing the E-AGB phase). However, the occurrence of TDU is much faster andit is expected to repeat many times. The star undergoes recurrent TDUepisodes, whose efficiency depends on the physics of the convective borders.The TDU is influenced by the parameters affecting the H-burning rate, suchas the metallicity, the mass of the H-exhausted core, and the mass of theenvelope, which in turn depends on the effectiveness of mass loss by stellarwinds [see the discussion in Straniero et al. (2006)].

During the TP-AGB phase, the envelope becomes progressively enrichedin primary 12C and in s-process elements (the s process will be discussedin the third chapter). As mentioned, a few protons penetrate into the toplayers of the He intershell at TDU. At hydrogen re-ignition, these protons

18

2.3. The third dredge-up.

are captured by the abundant 12C forming 13C in a thin region of the Heintershell (13C pocket). Hence, neutrons are released in the pocket underradiative conditions by the 13C(α,n)16O reaction at about T ∼ 0.9× 108 K.This neutron exposure lasts for about 10 - 20 thousand years with a verylow neutron density (106 to 107 n/cm−3). The pocket, strongly enriched ins-process elements, is then engulfed by the subsequent convective TP. At themaximum extension of the convective TP, when the temperature at the baseof the convective zone exceeds 3× 108 K, a second neutron burst is poweredfor a few years by the marginal activation of the 22Ne(α,n)25Mg reaction.This neutron burst is characterized by a low neutron exposure and a highneutron density up to 1010 n/cm−3, depending on the maximum temperaturereached at the bottom of the thermal pulse.

Summing up, the main characteristics of the He-burning shell in AGBstars, from the point of view of the nuclear processes occurring, are related tothe development of thermal instabilities called shell flashes or thermal pulses.The four phases of such a thermal pulse can be summarized essentially asfollows.

1. During the first stage almost all of the surface luminosity is providedby the H-shell. This phase lasts for 104 to 105 years, depending on thecore-mass.

2. The He-shell suddenly starts burning very strongly, producing lumi-nosities up to ∼ 108L⊙. The energy deposited by these He-burning reactionsis too large to be transported by radiative processes and a convective shelldevelops, which extends from the He-shell almost to the H-shell. This con-vective zone includes mostly He (about 72-75%) and 12C (about 22 - 25%),and lasts for about 200 years.

3. During the so-called power-down phase, were the He shell begins todie out and the convection is shut-off, the previously released energy drivesa substantial expansion, pushing the H-shell to such low temperatures anddensities that it is extinguished.

4. The dredge-up phase follows, where the convective envelope, in re-sponse to the cooling of the outer layers, extends inward and, in later pulses,beyond the H-He discontinuity (where the H-shell was previously sited) andcan even penetrate the flash-driven convective zone which was produced bythe He-shell. This phenomenon allows ashes from both He and H burningto be mixed to the surface. This is the so-called TDU, accounting for theexistence of carbon stars enriched in s-process elements in the late stages ofthe AGB.

2.3 The third dredge-up.

A crucial problem for the production of new nuclei in the intershell region,and for their mixing into the envelope where they can be observed was found

19

2.3. The third dredge-up.

since the first numerical models for TP-AGB stages. In 1977, Iben drewattention to the fact that the direct penetration of convention, associated toa thermal pulse, into the H-shell is inhibited by an entropy barrier placedbetween the He-intershell and the envelope. For this reason, hydrogen can’tapproach zones where He is burning until the entropy excess is carried out,causing expansion and cooling of the envelope. The stellar structure shrinksand the base of the convective envelope sinks below the interface between thetwo shells. This event, as already mentioned, is know as the third dredge-upor TDU. The depth and efficiency of the dredge-up phenomenon typicallygrows from pulse to pulse; it is measured through the so-called dredge-upparameter λ, defined as:

λ ≡ ∆MTDU

∆MH(2.7)

This is the ratio between the mass carried to the surface at each thermalpulse, ∆MTDU , and the mass processed by the H-burning shell during theinterpulse phase, ∆MH . Generally speaking, the whole TP-AGB evolutiondepends on stellar mass, and this is particularly true for the third dredge-up.TDU is influenced by the parameters affecting the H-burning rate, such asthe metallicity, the core and the envelope mass. In particular, there is astrong dependence of the evolutionary properties of AGB stars on the initialmetallicity (Z) and the value of λ increases when Z decreases. The amountof material dredged-up in a single episode (∆MTDU ) initially increases whenthe core mass increases, then decreases, when the mass loss erodes a sub-stantial fraction of the envelope. Mass loss also determines the number ofthermal pulses: the higher the stellar mass is, the larger is the number ofthermal pulses.

A lot of problems still affect the determination of the TDU efficiency.They include in particular the opacity tables (that give the kν , couplingradiation to matter) and the value of the free parameter αP characterizingthe so-called mixing length lM treatment of convection. This last quantitydetermines the mean free path of a convective eddy in units of the pressurescale HP . One can use it to describe the transport of heat in convective con-ditions. In all evolutionary calculations for AGB stages, αP is maintainedconstant to a value calibrated on the solar model. At first, TDU was easilydiscovered in models of stars belonging to Population II (low metallicity) andin intermediate mass stars (IMS) with massive envelopes. Then Lattanzio(1989) and subsequently Straniero et al. (1995), using the Schwarzschild cri-terion for convention and values of the αP parameter in excess of ∼ 1.5 (thevalue accepted today is ∼ 2.1), succeeded in finding TDU also in LMS ofPopulation I, thus explaining the existence of carbon stars of low luminosityin the solar neighborhoods.

Subsequently, new opacity tables stimulated a number of calculations ofAGB models by various groups (Vassiliadis & Wood, 1993; Straniero et al.,1995; Forestini & Charbonnel, 1997; Frost et al., 1998). Since these im-

20

2.4. Nucleosynthesis and observations for AGB stars.

provements, an agreement on the method to describe TDU was achieved.Some of the new models found third dredge-up, and this was establishedas a self-consistent process agreed upon by researchers. However, the com-plexities of the AGB structure, involving extreme contrasts in local matterproperties, the use of the mixing-length theory for describing convectivetransport, and the short duration of the interpulse phases available for mix-ing, continue to make it difficult to address the problem from first principles.

In summary, concerning TDU events, only most recent stellar modelsconfirmed numerically its existence for initial masses as low as about 1.5M⊙, in typical solar conditions. In fact, AGB stars belonging to GalacticGlobular Clusters, whose initial mass are of the order of 0.8 − 0.9 M⊙, donot show the enhancement of carbon and s elements, which is the signatureof the TDU. Moreover, depending on stellar physical parameters, there is aminimum envelope mass for which TDU takes place. The efficiency of TDUis connected with the chemical composition; for given values of the core andenvelope masses, it is deeper in low metallicity stars, where H burning isless efficient. Actually, the propagation of the convective instability is self-sustained due to the increase of the local opacity that occurs because freshhydrogen (high opacity) is brought by convection into the He-rich layers(low opacity). In general, TDU occurs only after some initial, less intensethermal pulses and ends when the envelope mass becomes smaller than about0.4 M⊙, while thermal instabilities of the He shell are still active.

2.4 Nucleosynthesis and observations for AGB stars.

The evolutionary phases briefly outlined above are important because of thenucleosynthesis of heavy elements that was demonstrated observationally tooccur there. Several years before stellar model could address the problem,Merril (1952) discovered that the chemically peculiar S stars (characterizedby C/O ∼ 0.7 - 0.9), enriched in elements heavier than iron, contain theunstable isotope 99Tc (τ = 2 × 105 years) in their spectra. It was clearthat ongoing nucleosynthesis occurred in situ in their interior and that theproducts were mixed to the surface. The fact that Tc is widespread in S starsand also in the more evolved C stars (C/O > 1) was subsequently confirmedby many workers on a quantitative basis. It is therefore not surprisingthat red giants in the TP-AGB phase were suggested as the site for the sprocesses as early as in the 1960s (Sanders, 1967). AGB stars are well knownas the main site where the s-process occurs, i.e. where the slow addition ofneutrons proceeding along the valley of β-stability generates about 50% ofnuclei beyond the Fe-peak (for a recent review see Busso et al., 2004).

The main neutron source for s processing is now recognized to be the13C(α,n)16O reaction, whose activation however depends on still uncer-tain mixing mechanisms for protons. In this case they must inject hy-

21


drogen from the envelope into the He-rich region, during the TDU phe-nomenon. Here protons react on the abundant 12C, producing 13C throughthe 12C(p,γ)13N(β+ν)13C chain. Stellar model calculations (see e.g. Gallino et al.,1998; Straniero et al., 1997) showed that any 13C produced in the radiativeHe-rich layers at dredge-up burns locally before a convective pulse devel-ops. The temperature is rather low for He-burning conditions (0.9× 108 K,or 8 keV), and the average neutron density never exceeds 1 × 107 n/cm3.As a consequence of neutron captures, a pocket of s-enhanced material isformed and subsequently engulfed into the next pulse. Here s-elements aremixed over the whole He intershell by convection and are slightly modifiedby the marginal activation of the 22Ne source. They are then brought tothe surface during the following episode of TDU. The 22Ne source providesonly a small contribution in low mass stars, which is nevertheless significant,because it occurs at higher temperature and neutron densities, which cantherefore explain several details of s-process branching reactions dependingon the environment conditions.

AGB stars are important manufacturing sites also for other elementsand isotopes. I can broadly divide them into two groups: the H-burningproducts (mainly coming from regions across and above the H-burning lay-ers) and He-burning products (mainly coming from He-rich zones, above thedegenerate C-O core). Several such nuclei of both groups are suitable fordirect observational tests in either evolved stars or in their descendants andthe diffuse Planetary Nebulae generated by their mass loss.

Over the years several studies provided the observational basis for neutron-capture nucleosynthesis models in AGB stars, in particular for discriminat-ing between the competing neutron sources. Coupling of high-resolutionspectroscopic observations with sophisticated stellar atmosphere models al-lowed the determination of heavy-element abundances in AGB stars (seeGustaffson, 1989, for a discussion). In particular, (Smith & Lambert, 1985,1986, 1990) and Plez et al. (1992) revealed that MS and S stars show anincreased concentration of s-process elements. Despite large observationaluncertainties, this was recognized to apply also to C stars, characterized bya photospheric C/O ratio above unity (Utsumi, 1970, 1985; Kilston et al.,1985; Olofsson et al., 1993; Busso et al., 1995). More recent studies are nowaviable (Abia et al., 2001, 2002), based on high-resolution spectra. This haslead to strong revisions in the quantitative s-element abundances. N starswere confirmed to be of near solar metallicity, but they show on average<[ls/Fe]>= +0.67±0.10 and <[hs/Fe]>= +0.52±0.29, which is significantlylower than estimated by Utsumi and is more similar to S star abundances(Smith & Lambert, 1990; Busso et al., 2001). This revision allowed the ex-tension to C(N) stars of the generally good agreement between observeds-process abundances and theoretical predictions of s-process nucleosynthe-sis in AGB stars (Gallino et al., 1998; Busso et al., 1999). Such comparisonsconfirm also for C(N) stars the existence of an intrinsic spread in the abun-

22


dance of 13C burnt, and allow us to place observed AGBs with differents-process and carbon enrichment along simple evolutionary sequences (seeFigure 2.5).

Figure 2.5: Observations of the logarithmic ratios [ls/Fe] of light s ele-ments (Y, Zr) with respect to the logarithmic ratios between heavy (Ba,La, Nd, Sm) and light (Y, Zr) slow neutron capture (s) elements. Symbolsrefer to different types of s-enriched stars. Stars with the higher s-elementenrichments are C-rich (adapted from Busso et al., 1995).

Direct information on AGB nucleosynthesis can also be derived spec-troscopically from stars belonging to the post-AGB phase and evolving tothe blue (see Figure 2.1) after envelope ejection (Gonzalez & Wallerstein,1992; Waelkens et al., 1991; Decin et al., 1998). Since the pioneering workof McClure et al. (1980) and McClure (1984), another source of informationhas come from the observation of surface abundances for the binary rela-tives of AGB stars, that is, for the various classes of binary sources whoseenhanced concentrations of n-rich elements are caused by mass transfer ina binary system (Pilachowski et al., 1998; Wallerstein et al., 1997). In sum-mary, direct observations contain compelling evidence that AGB stars arethe main astrophysical site for the s process and provide abundant con-straints on its occurrence: its neutron exposure, correlation with 12C pro-duction, inferred masses of parent stars, etc...

23


24

CHAPTER

THREE

S-PROCESS NUCLEOSYNTHESIS IN AGB STARS.

In this section of my thesis I present a discussion of nucleosynthesis processesoccurring in the final evolutionary stages of stars with moderate mass, whenthey climb for the second time along the red giant branch (the so-calledAsymptotic Giant Branch, or AGB, phase), with particular attention forslow neutron captures.

I dedicate most of the space to low mass stars (0.8 − 3 M⊙) where thedominant neutron source is the reaction 13C(α,n)16O as they are now rec-ognized as the most important contributors to the s process. I also presenta short review of the researches on s-process nucleosynthesis, starting fromthe first hypotheses of a release of neutron in convective layers, and summa-rizing the improvements that subsequently led to a crisis in the traditionalideas and to a new scenario in which slow-neutron capture in AGB starsoccurs in radiative interpulse phases.

In particular, I underline the fact that, in order to understand quantita-tively the complexity of s-process nucleosynthesis in the galaxy, we still needa more accurate knowledge of the 13C(α,n)16O reaction rate. In this context,a new measurement of this cross section, performed with the Trojan HorseMethod, will be presented and discussed in the second part of this thesis.

3.1 Introduction.

All elements not created in the Big Bang are produced through thermonu-clear reactions in stellar environments. A fundamental paper on stellar nu-cleosynthesis, now recognized as the basis of any subsequent study, waswritten by E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoylein 1957 (Burbidge et al., 1957, often referred as B2FH). These authors de-scribed the processes of hydrogen and helium fusion, the burning of elementswith an intermediate mass (from carbon to silicon) and the production ofheavier elements above iron through neutron captures. The Coulomb bar-

25

3.1. Introduction.

rier of iron is too high to be overcome by charged particle interactions so tocreate elements heavier than Fe only reactions involving neutrons can be atplay. Following B2FH, neutron addiction reactions can be divided accord-ing to their time scale, as compared with those for competing β-decays ofunstable nuclei encountered along the neutron capture path.

Following the usual definition, I call r(rapid) process the set of neutron-addition reactions that occur on time scales so short to prevail over thedecay times (τ) of unstable nuclei τ ≫ (σφ)−1 even when they are ratherfar from the valley of beta stability (see Figure 3.1). This sets the typicaltime scales to be smaller than a few seconds. In the previous expression Iused σ for the cross section and φ to indicate the neutron flux in the burningregion of a star. The r process can occur in supernovae, where huge neutronfluxes (about 1023 n/cm3) allow the creation of very heavy (A209) and veryneutron-rich elements. In such conditions a stable nucleus can capture manyneutrons before it decays. On the other hand, in hydrostatic evolutionary

Figure 3.1: The valley of β-stability. Illustration of the neutron-capturepath, followed by processes responsible for the formation of 50% of the nucleibetween iron and the actinides.

stages one meets less extreme conditions of temperature and neutron den-sity, so that the neutron flow proceeds along the valley of beta stability,where the lifetime of unstable nuclei is generally shorter than the neutroncapture time scale. Typical neutron densities in this case range from about106 to 1010 n/cm3. The corresponding neutron-capture nucleosynthesis isthen called s(slow) process; in it, elements are produced through a series ofsubsequent neutron captures on stable nuclei followed by a β-decay whenan unstable nucleus is encountered. In the s process only rarely neutron

26

3.2. The classical analysis of the s process.

captures can compete in time scale with weak interactions. However, thesefew cases are important, as the flow encounters a branching point where theabundances of the nearby nuclei inform us on the physical conditions (neu-tron density, temperature, etc...). About half of all elements heavier thaniron are produced in a stellar environment through s processes.

Many improvements on the first ideas by Burbidge et al. (1957) weresoon presented, thanks to increased precision in the measurements of isotopeabundances from meteorites and of neutron capture cross sections. Variousreviews dealing with the s process, and with connected stellar and nuclearissues have been published over the years, especially for the asymptotic giantbranch (AGB) stars where neutron-rich elements are produced in the innerregions and then carried to the surface by a series of mixing phenomenaknown under the name of third dredge-up (referred in the following as TDU).

3.2 The classical analysis of the s process.

Here I briefly present the general features of s-process nucleosynthesis start-ing from the B2FH article that opened the road for the modern theories ofheavy element production in stars. Clayton et al. (1961) and Seeger et al.(1965) provided the mathematical tools that outlined the so-called ”phe-nomenological approach” or ”classical analysis” of the process, i.e. an ana-lytical formulation based only on nuclear properties and abundance system-atics.

The starting point of this analysis was the study of the distribution ofthe σ Ns products between neutron-capture cross sections and s-processabundances. The mentioned authors built the experimental distribution ofσNs values, using data on the neutron-capture cross sections then availableand on the solar system isotopic composition. This was then compared witha model σNs curve, by computing analytically the s-process contributionsNs to each isotope. As a consequence, the ratio (N(A) − Ns(A))/N(A)yielded a prediction on the fractional abundances due to the more complexr process. In a slow neutron-capture process, the abundance of an isotopeAth varies in time through destruction and creation mechanisms:

dN(A)

dt= N(A− 1)nn 〈σ((A − 1), v)v〉 −N(A)nn 〈σ(A, v)v〉 (3.1)

where 〈σ(A, v)v〉 indicates the Maxwellian-averaged product of cross sectionand relative velocity, and nn is the neutron density. In the simple expressionof equation (3.1) only stable nuclei of atomic mass number A − 1 and A,affected only by neutron captures are considered, without branchings. It isconvenient to replaces time with the time-integrated neutron flux, or neutronexposure τ , through the substitution:

τn =

∫

nnvTdt (3.2)

27


This differential equation then becomes:

dN(A)

dτ= N(A− 1)nn 〈σ((A− 1), v)v〉 −N(A)nn 〈σ(A, v)v〉 (3.3)

In steady state conditions, production equals destruction, the time derivativevanishes and 〈σ(A)N(A)〉 = const.

This simplified relation is rather well satisfied in the experimental solar-system distribution of σNs values for heavy nuclei, over large intervals ofthe atomic mass number A. A modern version of this curve is presentedin Figure 3.2, (taken from Kappeler et al., 2011). The curve appears oftensmooth, but is interrupted by steep drops at nuclei where a neutron shellclosure occurs, (their number of neutrons are then called magic neutronnumbers, N = 50, 82 and 126. For s-process elements N = 50 occurs for A= 88 - 90, in the region of Sr - Y - Zr, which are often called ls (or light-s)elements. N = 82 occurs at Ba - La - Ce, called hs (or heavy-s) elements.Finally, N = 126 occurs at A = 208 - 209, at the end of the stable nucleidistribution, and involves 208Pb and 209Bi. The solar abundances show s-

Figure 3.2: The characteristic product of cross section times s-processabundance 〈σ(A)N(A)〉, plotted as a function of mass number. The thicksolid line represents the main component obtained by means of the classicalmodel, and the thin line corresponds to the weak component in massivestars (see text). Symbols denote the empirical products for the s-only nuclei.Some important branchings of the neutron-capture chain are indicated aswell.

process peaks at the atomic mass numbers of the above elements, because(n,γ) cross sections for neutron magic nuclei are very small. Clayton and coworkers derived two main conclusions:

28


1. the whole distribution of s-element abundances above Fe in the solarsystem requires more than one s-process mechanism (or component) occur-ring in separated astrophysical environments in order to bypass the bottle-necks introduced by neutron magic nuclei. One of the components of theprocess had to account for the s nuclei of A ≤ 88 (the weak s-component),and a second one was necessary for nuclei with 88 ≤ A ≤ 208 (the maincomponent). A third (strong) component was also initially assumed forproducing roughly 50% of 208Pb that was missing. This was subsequentlyproven to be simply due to low metallicity AGB stars with high neutronexposures (Busso et al., 1995; Gallino et al., 1998). In this paper I concen-trate my attention on the behaviour of elements from Sr to Pb, i.e. the maincomponent.

2. In order to allow the neutron flux to pass through the bottlenecks,Clayton et al. (1961) approximated what is, in nature, a limited number ofrepeated neutron irradiations with a continuous distribution of decreasingneutron fluxes, in which many nuclei capture a relatively small number ofneutrons and few nuclei capture a large number of them. The reason for thisapproximation is that it can expressed by a continuous function (a power-law or an exponentially-decreasing function) yielding simplified solutions. Inparticular, they adopted a distribution of neutron exposures

ρ(τ) = Kexp(−τ/τ0) (3.4)

where ρ(τ)dτ represents the number of seed nuclei (mainly 56Fe) exposed toan integrated flux between τ and τ + dτ . Their choice soon became verypopular, because it allows an exact analytic solution for the set of equations:

σ(A)Ns(A) = GN56τ0

A∏

Ai=56

[

1 + (σ(Ai)τ0)−1]−1

(3.5)

where the only degrees of freedom are: 1. the fraction G of solar Fe nucleiirradiated, and 2. the mean neutron exposure τ0. Ns(A) represents the partof the abundance NA due to the slow neutron capture.

Concerning the main component, the mean exposure τ0 was originallyestimated to be around 0.2 mbarn−1, but was updated over the years withthe improvements in the nuclear data, up to around 0.3 mbarn−1 (at 30keV).

The success of the exponential distribution of neutron exposure was aresult of its mathematical convenience and also of the fact that Ulrich (1973)showed how the AGB phases of intermediate mass stars can indeed mimic anexponential form, under the assumption that neutrons are released duringthe convective instabilities of He-shell. He showed that the exponentialdistribution derives from the overlap factor r between subsequent convectivepulses, if a constant exposure ∆τ is produced in every pulse.

29


In fact, after N pulses the fraction of material experiencing an exposureτ = N∆τ is rN = rr/∆r. This is an exact solution if the neutron densityand the temperature don’t change during the s-process. The classical analy-sis rapidly became a technique sophisticated enough to account for reactionbranchings along the s-path, contrary to the simple assumptions implied byequation (3.1). Even at the low neutron densities characterizing the s pro-cess, the competition between captures and decays has still to be consideredfor a number of crucial unstable isotopes, like 79Se, 85Kr, 148Pm and 151Sm.For them, the probability of a neutron capture is high enough to competewith the beta decay.

Application of the branching analysis to specific ramifications of the pro-cess was since then used for inferring the stellar parameters (average neutrondensity, temperature, electron density). It was also shown byWard & Newman(1978) that the branchings held information on the pulsed nature of the neu-tron flux. For each branching, a branching ratio fβ can be defined by com-paring the rates for β-decay and neutron capture, so that fβ = λβ/(λβ+λn),where λn = Nn 〈σ〉b vT . Here 〈σ〉b is the Maxwellian averaged (n,γ) crosssection of the nucleus at the branching point.

In the case of a branching, the curve describing the product σNs isdivides in two ramifications and each branch is studied separately. Alsothe existence of metastable isomeric states of nuclei, for example of 85Kr,pointed to that result. The method briefly described so far was continuously

Figure 3.3: The complex branching of 85Kr.

updated over the past three decades, to take into account progresses in neu-tron capture cross-sections measured along the s path. The level of accu-

30

3.3. Evolution and nucleosynthesis in the AGB stages.

racy reached today in cross-section measurements has finally demonstratedthat the phenomenological approach, based on an exponential distributionof exposures, can no longer be seen as an acceptable approximation of the sprocess. Hence, we now recognize that the classical analysis of the s process,after its many important contributions in the past, in now superseded.

3.3 Evolution and nucleosynthesis in the AGB stages.

Stars of the Asymptotic Giant Branch are the final evolutionary stage (forthermonuclear reaction) of low and intermediate mass stars. Even below 8M⊙ the AGB evolutionary scenario and related nucleosynthesis significantlychange with the mass of the star. In the following I review the properties ofAGB for stars of low mass. The quantitative results have been derived fromrecently published AGB models computed by several authors, in particular:Straniero et al. (1997) For clarity, I first discuss the previous phases of stel-lar evolution before the representative point of a star in H-R diagram goesto AGB zone, confining to stars between 0.8 and 3 M⊙: the so-called LMS(Low Mass Star). The upper mass limit for AGB stars marks the inferiormass limit for massive stars, those that, after He exhaustion in the core,burn C, Ne, O and Si, form a degenerate iron core and, eventually, collapse.The precise value of this limit is not well defined because it depends by themetallicity. The lower limit, instead, corresponds to the mass value to reachthe inner temperature of about 10 million of degree (measured in Kelvin)necessary to start hydrogen combustion. Hydrogen burning follows the reac-tions of pp-chain but, if temperature in star is bigger than about 18×106 K,the CNO-cycle is the main energy source. This stage was the longest in stel-lar life, it was the so-called main sequence (MS). Core hydrogen goes on untilH is exhausted in the core over a mass fraction is close to 10%. A schematicview of track followed by the stellar representative point is given by the H-Rdiagram (see Figure 2.1). Then the He core shrinks, while the stellar radiusincrease to carry out the energy produced by the H-burning shell. As con-sequence of envelope expansion, the stellar representative point in the H-Rdiagram moves to the red and to increase luminosity, and then climbs a trackcalled the red giant branch (RGB). While the envelope expands outward,convection penetrates into region that had already experienced partial C-Nprocessing or proton captures and it carried to surface part of them. Athelium core exhaustion, star become powered by He burning in a shell, sothe large energy output pushes the representative point in a track that, forlow mass star, asymptotically approaches the former RGB and is thereforeknown as the AGB. The AGB stage is characterized by a degenerate coremade of C-O whose pressure is mainly provided by degenerate electrons, bytwo shells (of H and He), and by an extended convective envelope and it canbe divided in two stages: E-AGB and TP-AGB. During the early phases (E-

31


AGB) C-O core can increase and warm because of helium burning in shell. Instar with M > 2 msb a second dredge-up can occur delivering some elementsfrom hydrogen shell to surface. After E-AGB the two shells are separatedby a thin layer in radiative equilibrium: the so-called He-intershell. As shellH burning proceeds while the He shell is inactive (LHe/LH < 10−3), themass of the He intershell MH −MHe increases (owing to sinking of newlyformed He) and attains higher densities and temperatures. This results ina dramatic increase of the He-burning rate for short period of time: the so-called Thermal Pulse (hereafter TP). Thermal pulses are real thermonuclearflashes repeating at regular time lapse (the so-called interpulse during whichHe-shell remains inactive) and during which He burns in semi-explosive con-ditions, as in the case of degenerated core. In fact, these events are caused bycombination of two main factors: intrinsic instability of thin shells and thepartial degeneration. Since the unstable thermal configuration the emissionof energy due to He-shell begin to oscillate with increasing amplitude until athermal pulse is created with a typical power of about 105L⊙. The radiativestate of the He intershell is thereby interrupted, and the shell then becomesalmost completely convective. This results in a mixing process called thirddredge-up (hereafter TDU), which carries processing material to surface. Inthis way it is possible to study internal process, so the discovery of 99Tc byMerril in 1952 was a proof to affirm that also heavy elements are createdin stars. From the structural point of view, the TDU is very similar to thesecond dredge-up however, its occurrence is much faster and is expected torepeat many times. Modelling TDU was always very difficult; it was relatedto the choice of the opacity tables and, in the framework of the mixing-length theory, to the value of αP (the ratio of the mixing length l and thepressure scale height HP ). Now the main energy source is helium and starhas to readjust its structure expanding too radiate the energy surplus. Theprocess is repeated many times (about 10-50 cycles) before the envelopeis completely eroded by mass loss. This evolutionary phase is usually re-ferred to as the TP-AGB (Thermally-Pulsing AGB). The Figure 3.4 showsthe internal structure of a thermal-pulse-asymptotic giant branch star as afunction of time. One can easily looks at the alternate motion (in mass) ofthe two shells following the position in mass of the H-burning shell (MH),of the He-burning shell (MHe) and of the bottom border of the convectiveenvelope (MCE). During the whole AGB stage a star loses a big part of itsconvective envelupe. Then one of the most severe uncertainties still affectingAGB models concerns mass loss. The duration of the AGB and the numberof TPs, the amount of mass dredged up, the impact of stellar winds on in-terstellar abundances and many other important predictions depend on theassumed mass loss rate. The available data indicate that this rate rangesbetween 10−8 and 10−4M⊙/ yr (Loup et al., 1993). Studies of Mira andsemi-regular variables show that mass loss is not a monotonically increasingfunction of time, and the star certainly encounters variations in its mass

32


Figure 3.4: Plot of the internal structure of a TP-AGB star as a function oftime, for a 3M⊙ model with Z = 0.02 (Straniero et al., 1997). The positionsin mass of the H-burning shell (MH), of the He-burning shell (MHe), and ofthe bottom border of the convective envelope (MCE) are shown. Convectivepulses (shown in Figure 2.4) occupy almost the whole intershell region duringthe sudden advancement in mass of the He shell. The periodic penetrationof the envelope into the He intershell (third dredge-up) is clearly visible.This model reaches the C star phase (C/O > 1) at the 26th pulse. Pulsesfrom 17 to 32 are shown.

loss efficiency, until a final violent (perhaps dynamical) envelope ejectionoccurs. The pressure radiation in envelope, increasing after helium burning,is the responsible of solar wind injection. In this phase AGB star pumps ininterstellar medium about or most than 70% of their whole mass in the formof dust and gas until it is completely expelled leaving the naked core. Thisis the post-AGB stage. The representative point of core nebula describes abig excursion in temperature. It goes toward the blue zone because it showsthe internal and hotter zones and the warm coming from stellar surface isenough to ionize the material. A star now is surrounded by a brilliant zone,the so-called planetary nebula. In the main time luminosity decrease veryquickly because mass loss extinguishes the thermonuclear reactions in twoshell H and He then star came under the track of main sequence. This isthe white dwarfs stage, the final phase of life of a low mass star, where itradiates its residual energy travelling along a diagonal line, the so-calledcooling sequence.

33

3.4. The neutron source 13C(α,n)16O.

3.4 The neutron source 13C(α,n)16O.

There are two important neutron sources in typical AGB conditions: the13C(α,n)16O reaction, originally introduced by Cameron et al. (1954) andthe 22Ne(α,n)25Mg reaction; also this one was suggested by Cameron et al.(1960). 22Ne is naturally produced in the He intershell starting from theoriginal CNO nuclei present in the star at its birth and transformed mainlyinto 14N by the operation of the H-burning shell. In He-rich layers 14N isconsumed through the chain:

14N(α,γ)18F(β+ν)18O(α,γ)22Ne

Due to its natural occurrence, this neutron source was the first to be exploredin stellar models to describe s-process, mostly for stars in mass range 4-8M⊙, known as Intermediate Mass Stars (IMS). This source produces a highneutron density of about 1010 − 1012 n/cm3 and needs a temperature largerthan 3− 3.2× 108 K to be activated. The maximum temperature achievedin LMS at the bottom of TPs barely reaches T = 3× 108 K, hence the 22Nesource is only marginally at play. At the beginning of the eighties, this factpushed some authors to reanalyze the conditions for the activation of thealternative 13C(α,n)16O source that had been previously largely ignored.

This second reaction is activated at relatively low temperatures (T =0.8 − 1.0 × 108 K) and can therefore easily explain why the abundancesof s-elements are highly enhanced in low mass AGB stars, where the tem-perature is low. The idea was confirmed by further observations, includingthe abundance trends of heavy s-elements in not evolved stars of both thegalactic halo and the disk.

In order to allow the 13C(α,n)16O reaction to be the main neutron sourcefor s-processing at low temperatures, two conditions must be met.

1. A mechanism for injecting protons into the He-rich region must befound, so that interacting with the abundant 12C they can produce 13C inHe intershell.

2. The amount of 13C thus obtained must burn through the 13C(α,n)16O re-action in layers where the temperature is low (T ≤ 0.8 − 1.0 × 108 k) tomaintain the neutron density low. The reaction 13C(α,n)16O is consideredto be the main source of neutrons for the s-process in low mass stars duringthe asymptotic giant branch phase. However, producing neutrons through13C-burning is more difficult than through 22Ne burning, mainly becauseone needs some mixing process suitable to bring protons into the He inter-shell: indeed, the amount of 13C naturally left behind by H burning is byfar insufficient to drive significant neutron captures.

In the He-rich layers of AGB stars one has then to start from a 13Cabundance built locally at H-reignition, through small amounts of protonsdiffused down from the envelope into the intershell region. The direct en-

34


gulfment of protons from the H shell when convective instabilities developis instead inhibited by an entropy barrier at the H shell.

Since the occurrence of the third dredge-up forces the hydrogen-rich andthe carbon-rich layers to establish a contact, this will naturally producesome mixing at the H/He interface: by chemical diffusion during the inter-pulse phase (for which it is difficult to define a quantitative approach) orby hydrodynamical effects induced by convective overshooting, or even frombuoyancy in magnetic fields.

The assumption that proton mixing occurs during the third dredge-up,forming a 13C-pocket whose mass was left as a free parameter proved to bea fruitful approach (Gallino et al., 1998). Subsequently, observations andchemical evolution models for the galaxy guided the research, indicatingthat the average efficiency of the mixing processes at TDU must be suchthat the reservoir of 13C reaches a mass of a few 10−4 M⊙ (Travaglio et al.,1999; Busso et al., 2001). Afterwards, possible physical mechanisms for pro-ducing a 13C pocket of the suitable mass and with the suitable abundancedistribution have been extensively investigated by different authors, in orderto find a more secure basis for s-process nucleosynthesis in stars.

In order to provide a suitable site for s-processing the 13C reservoir mustbe formed through a limited number of protons captures by the chain ofreactions:

12C(p,γ)13N(β+ν)13C

Too efficient proton captures, indeed, activate a full CN cycling, leading to14N production through the 13C(p,γ)14N reaction, and 14N is a very efficientabsorber for neutrons, which would inhibit the captures on heavier nuclei.

In general, one expects a zone close to H-He interface, where more pro-tons are expected and where the subsequent burning produces mainly 14N:this region is not useful for s-processing, but will manufacture a lot of 15Nfrom neutron captures on 14N. Here the subsequent convective instability ofthe He-shell produces abundant 19F, from 15N(α,γ) reactions. Below thisregion the decaying abundance of protons creates the conditions suitable forforming almost pure 13C and hence to activate efficiently the 13C(α,n)16O re-action and the neutron capture nucleosynthesis processes. Later, when theconvective instability of the He-shell develops and attains its maximumstrength, the temperature reaches value of typically 3 × 108 K, the 22Nesource is marginally activated, providing a small neutron burst of higherpeak neutron density. This second neutron burst was recognized as beingable to explain several details of the solar s-process abundance distribu-tion, for nuclei after reaction branchings requiring a relatively high neutrondensity (1010 n/cm3). An important point concerns the time scale of 13Cburning. Actually, the first models (Kappeler et al., 1990) assumed thatthe locally-produced 13C could remain essentially inactive until the nextconvective instability, when it would be ingested and burned at the typical

35


Figure 3.5: Two successive thermal pulses (in particular, the 29th and

30th) for the 3 M⊙ model with Z = Z⊙ are shown in their relative positionsas calculated from the stellar model. The shaded zone is the 13C pocket,in which protons are captured by 12C. In the figure on the left, ingestionand burning of 13C in a pulse is based on the older models. 13C(α,n)16O isfirst burned convective, producing the major neutron exposure, followed bya small exposure from the 22Ne(α,n)25Mg neutron source in the pulse. Thenewer model, as shown in the second illustration, states that 13C burnsin the thin radiative layer where it is produced, releasing neutrons locally.After ingestion into the convective intershell region, this is then followed bya second small neutron exposure from the marginal activation of the 22Nesource.

temperature of 1.5 × 108 K, characteristic of the first phases of a thermalpulse. Subsequently, it was understood Straniero et al. (1995, 1997) thatthe neutron release by 13C burning starts very early, before the convectiveinstability develops. It therefore occurs in radiative and not in convectiveconditions and at very low temperatures, as mentioned. All 13C nuclei avail-able below the H shell were found by Straniero et al. (1997) to be consumedby the 13C(α,n)16O reaction before the growth of the next instability. Theneutron density in each layer scales with the local 13C abundance, reachingat most 107 n/cm3. The thermal velocity is close to 8 keV. The convec-tive pulse driven by each thermal instability simply dilutes the s-processproducts over the whole intershell zone and exposes it to the new neutronflux from 22Ne burning. The seed material in the next 13C-pocket is there-fore a combination of nuclei present in the H burning ashes from the upperintershell, and of the s-processed material left behind in the lower part of

36


the intershell zone at the quanching of the previous convective instability.The thermal pulse history is represented schematically in Figure 3.5. Thethin zone q indicates the position of the 13C-pocket where neutrons arereleased. The fraction r of the mass of the convective He shell contains s-processed material from the previous pulses; the fraction 1− r contains theH-shell burning ashes (with fresh Fe-seeds) swept by the convective pulse.Using the reaction rate by Drotleff et al. (1993), the duration of the 13Cconsumption, including the effects of some delayed neutron recycling by the12C(n,γ)13C(α,n)16O chain, is about 20000 years, leaving several thousandyears before the growth of the next convective instability (at least 30000 yrin 2 M⊙ stars). However, the reaction rate for (α,n) captures on 13C is veryuncertain at the very low energies at play. I shall discuss extensively theimplications of this in the rest of this thesis. Based on the above analysis,

Figure 3.6: Schematic representation of the thermal pulse history and ofs-processing in the interpulse periods.

s-process nucleosynthesis in AGB stages can be summarized as occurring indifferent phases:

1. penetration of a small amount of protons into the top layers of thecool He intershell (to form a proton pocket);

2. formation of a 13C pocket at H reignition;3. release of neutrons by the 13C(α,n)16O reaction when the region

is subsequently compressed and heated to T = 0.8 − 1.0 × 108 K. Here sprocessing takes place locally under radiative conditions generating an s-enhanced pocket;

4. ingestion into the convective thermal pulse, where the s-enhancedpocket is mixed with H-burning ashes from below the H shell (Fe seeds,

37

3.5. Possible future scenarios.

14N) and with material s-processed in the previous pulses;5. exposure to a small neutron irradiation at high nn by the 22Ne source

over the mixed material in the pulse;6. occurrence of the TDU episode after the quenching of the thermal

instability, so that part of the s-processed and 12C-rich material is mixedinto the envelope;

7. repetition of the above cycle until the TP phase is over.

3.5 Possible future scenarios.

On the basis of the scenario described above, it was shown by Travaglio et al.(1999) that the chemical evolution of s-elements up to the solar formationage could be well reproduced. Very recently, however, observations of openclusters by our group D’Orazi et al. (2009); ? revealed that the above pic-ture is insufficient to account for the s-element enrichment in the more recentgalactic disk, where an s-process enhancement larger than in the Sun exists.This indicates that AGB stars of very small mass (M < 1.5M⊙), contribut-ing in the Galaxy only after the solar formation, must produce s-elementsmore efficiently than more massive stars. They should therefore have moreextended 13C pockets. These enlarged 13C reservoirs would cover regions ofthe star where a higher temperature (10 keV) is present and would inducehigher n-densities.

Due to this new scenario and to the warnings already presented on theuncertainty in in the present rate for the 13C(α,n)16O reaction, there is nowa strong need to clarify this rate. This can be illustrated as follows.

1. For stellar masses above 1.5 M⊙. The neutron density at 8keV is solow that a possible increase of the rate would have minimal effects, unless itis larger than a factor of 3-5. More relevant would be a possible reductionof the rate with respect to the values indicated by NACRE. This is a realpossibility, if the rate is less affected than so far assumed by the contributionof a sub-threshold resonance. In such a case, 13C might have insufficienttime to burn in the interpulse phase, and would end up burning, at leastpartially, in the convective thermal pulse. Here the extra energy generatedwould be crucial and might induce phenomena like a shell- splitting, withstrong changes in the neutron density and large modifications in our presentpicture of the s-process.

2. For masses below 1.5 M⊙, both an increase and a decrease of the ratemight be critical, as the slightly higher temperature spanned by the 13C-pocket would emphasize the effects on the otherwise low n-density. Again,some 13C in the cooler layers might remain unburned, with the same desta-bilizing effects described at point 1.

38

CHAPTER

FOUR

CROSS SECTIONS OF NUCLEAR REACTIONS AT

LOW ENERGIES.

Nuclear reactions have a fundamental role in many astrophysical environ-ments because they provide the energy to sustain their luminosity over theirlifetimes and also because they are responsible of nucleosynthesis of elementsin stars. Usually one can refer to these as thermonuclear reactions becausethe star contracts converting gravitational energy into thermal energy, un-til the temperature and density become high enough to ignite them. In astellar environment at thermodynamic equilibrium, velocities and energiesof interacting nuclei follow the Maxwell-Boltzmann distribution with typicaltemperatures depending on stellar mass and evolution stage: from 106 to109 K. So, nuclear reactions take places at very low energies, of the order ofa few keV, because of the equation E = kBT , where kB is the Boltzmannconstant. An accurate knowledge at typical astrophysical energies of thereaction rates and therefore of the cross sections is highly desirable becausethey affect the different stellar evolutionary phases as well as the estimates ofthe chemical element abundances. Uncertainties of reaction rates are usuallyhigh in stellar conditions because of difficulties to implement experiments atsuch low energies.

As I have already said in previous chapters, in the low mass star AGBphases the region between the H shell and the He shell (He-intershell)is affected by brief convective instabilities, (thermal pulses), due to thesudden ignition of He burning in the He shell. In these conditions the13C(α,n)16O reaction is the main neutron source for the s process working inradiative conditions in a thin layer at the top of the intershell (13C-pocket)during the interpulse periods. The rate for α captures on 13C is measuredat high energy only, while for stellar energies its values are deduced by ex-trapolation. It is therefore necessary, and this is the main goal of this thesis,to determine the 13C(α,n)16O reaction rate in the unexplored energy zone.In order to set the stage for this task, in this chapter I will preliminary in-

39

4.1. Coulomb barrier and penetration factor.

troduce some concepts and a general discussion of the theoretical problemsinvolved in the study of the nuclear processes in astrophysics as reported indetail in Rolf & Rodney (1988).

4.1 Coulomb barrier and penetration factor.

In stars, nuclear reactions take place between charged particles because theatoms are in most cases completely stripped of their atomic electrons. It isassumed that they are almost completely ionized because of the typical hightemperature conditions (around a few keV at least). This high temperatureis on the other hand needed to permit the reactions, because nuclei arepositively charged and repel each other with a Coulomb force proportionalto their nuclear charge. Nuclear reactions are therefore inhibited by theCoulomb barrier whose height (in MeV) is given, in CGS units, by:

EC =Z1Z2e

2

Rn(4.1)

where Rn = R1 +R2 is the nuclear radius, Z1 and Z2 represent the integralcharges of the interacting nuclei. Classically, a reaction can occur only be-tween particles with energies higher than Ec. Incident projectiles at lowerenergies would reach the closest distance to the nucleus at the turning pointRC and would not penetrate the Coulomb barrier. Figure 4.1 representsthe schematic view of the effective potential resulting when one combinesthe very strong and attractive nuclear potential with the electromagneticpotential. Consequently, if this is the case, the fractions of particles whoseenergies exceeds the Coulomb barrier is negligible and it seems necessary anhigher stellar temperature. This obstacle was removed when Gamow (1928)showed that, in according to the quantum mechanics, there is a small butfinite probability for the particles with energies E < EC to penetrate thebarrier: this is the so-called tunnel effect. One might define the penetrationfactor, which is the basis of the tunnel effect, through the following ratio(Clayton et al., 1983):

T =|χ(Rn)|2|χ(Rc)|2

(4.2)

where the upper quantity represents the probability of finding the particlesat the interaction radius, and the other one at the classical turning point ofthe Coulomb barrier. It can be calculated by solving the radial part of theSchrodinger equation:

d2χl

dr2+

2µ

h2[E − Vl(r)] = 0 (4.3)

40

4.2. Cross section, astrophysical factor and reaction rate.

Figure 4.1: Schematic representation of the combined nuclear and Coulombpotentials. The plot reports the total potential V (r) versus the relativedistance r between the two interacting particle.

where Vl(r)is the potential for the lth partial wave resulting when the cen-trifugal potential term is also present (Clayton et al., 1983)

Vl(r) =l(l + 1)h2

2µr2+Z1Z2e

2

r(4.4)

At low energies or, equivalently, where the classical turning point is muchlarger than the nuclear radius, equation (4.2) can be approximated by thesimpler expression giving the so-called Gamow factor:

T = e−2πη (4.5)

with the Sommerfeld parameter, η = Z1Z2e2/hv, depending only on the

relative velocity of the two interacting particles and their charges. At lowenergy, below the Coulomb barrier, tunneling probability has an approxi-mate expression that drops exponentially with (4.5).

4.2 Cross section, astrophysical factor and reac-tion rate.

In stellar objects, the production of nuclear energy and the synthesis ofelements proceeds through fusion reactions until all light nuclei are con-verted to iron (A ∼ 60), corresponding to the maximum binding energy

41


for nucleon. More complex reactions lead to the production of the heavierelements. These processes can take place through a slow (s) or a rapid (r) se-quence of neutron captures with respect to the rate of β-decays of the nucleijust formed. The rate of each fusion reactions depends on the astrophysicalconditions and it can vary by several orders of magnitude for different tem-perature and density. I now briefly present the formalism adopted in orderto derive the astrophysical rates of charged-particle-induced reactions. Ingeneral, a nuclear reaction can be written symbolically as:

x+X −→ y + Y (4.6)

where x represent the projectile and X the target in the entrance channel,while Y is the residual nucleus and y the ejectile, which together constitutethe exit channel. In order to have a description for the nuclear process, inastrophysical environments we require the introduction of a ”cross section”.The cross section is defined as the probability that a given nuclear reactionwill take place. It is used to determine how many reactions occur per unittime and unit volume providing important information on energy productionin stars. Classically, this cross section σ depends only on the combinedgeometrical area of the projectile and the target nucleus. Since all nuclearcross sections are of the order of 10−24 cm2 (or lower), for convection andconvenience, a new unit of area, the barn (b), equal to 10−28 m2 has beendefined for cross sections. In reality, since nuclear reactions are governed bythe laws of quantum mechanics, the cross section must be described by theenergy-depend quantity

σ = πλ2DB

1

E(4.7)

where λDB represents the De Broglie wavelength:

λDB =mx +mX

mX

h

(2mxEx)2

(4.8)

For charged-particle nuclear reaction the cross section is strongly suppressedby Coulomb and centrifugal barriers and it drops rapidly for E < Ec. Re-calling equation (4.5) and (4.7), it is possible to factorize the cross sectionas:

σ =1

ES(E)e−2πη (4.9)

where S(E) is the so-called nuclear or astrophysical factor and contains allnuclear effects. The astrophysical factor is a much more useful quantity be-cause for non-resonant reactions it is a smoothly varying function of energy.Figure 4.2 shows that S(E) varies much less rapidly with beam energy thanthe cross section and it allows an easier procedure for extrapolating the en-ergy behaviour at astrophysical energies. As just discussed, nuclear cross sec-tions are in general energy-dependent or, equivalently, velocity-dependent,

42


Figure 4.2: In the upper panel, the cross section σ(E) of a charged-particle-induced nuclear reaction in shown. There is a rapid exponential decreasedown to EL, which is the lower limit for the beam energy at which experimen-tal measurements can be made. So, as the lower panel shows, extrapolationto lower energies is more reliable if one uses the S(E) factor.

σ = σ(v), where v represents the relative velocity between projectile and thetarget nucleus. Starting from cross section, I can introduce another impor-tant quantity, the so-called reaction rate, to describe the nuclear process inastrophysical scenarios. The reaction rate is defined as the number of givenreactions per unit volume per unit times (this gives an idea of the velocityof the reaction).

rxX =1

1 + δxXNxNX 〈σ(v)v〉 (4.10)

where the product NxNX represents the total number of pairs of non-identical nuclei X and x. For identical particles the Kronecker symbol δxXis introduced, otherwise each pair would be counted twice. The bracketedquantity 〈σ(v)v〉 is referred to as the reaction rate per particles pair and it isthe mean of the product σ(v)v over all the possible energies, weighted overthe Maxwell-Boltzmann distribution:

φ(v) = 4πv2(

m

2πkBT

)3/2

exp

(−mv22kT

)

(4.11)

43

4.3. Gamow peak.

by introducing the center of mass energy E =1

2µv2 with µ representing the

reduced mass of interacting particles, the reaction rate is then expressed as:

rxX =1

1 + δxXNxNX

(

8

πµ

)1/2 1

(kbT )3/2

∫

∞

0

σ(E)Eexp

(

− E

kBT

)

dE

(4.12)Here r is expressed in units of reactions per cubic centimeter per second. Itsequation characterizes the reaction rate at a given stellar temperature T and,during stellar evolution, its temperature changes. Then it’s important tohave information of the value of this rate for each temperature or, equivantly,to have information for each energy. It is also desirable to obtain r in thesame, analytic expression for 〈σ(v)v〉 in terms of temperature T .

4.3 Gamow peak.

Starting from the mathematical expression for the nuclear reaction ratefound in the previous section (4.12), one can easily calculate the theoret-ical best condition for reactions taking place in stellar environments. Thenif equation (4.9) is inserted into equation (4.12), one obtains:

〈σv〉 =(

8

µπ

)1/2 1

(kBT )3/2

∫

∞

0

S(E)exp

(

− E

kBT−(

EG

E

)1/2)

dE (4.13)

where the symbol EG is the so-called Gamow energy. The integrand ofthe equation (4.13), because of the limited dependence of S(E) from E, isgoverned by the combination of two exponential terms: the first representsthe Maxwell-Boltzmann distribution and the second one is the probabilityof tunneling through the Coulomb barrier. The maximum of the integrandis reached at an energy E0:

E0 =

(

kBT

2

)3/2

E1/2G (4.14)

The convolution of the two functions results into a peak, the so-calledGamow peak, centered near the energy E0 and generally much larger thankBT . The maximum value of the integrand will be:

Imax = exp

(

2E0

kBT

)

(4.15)

which depends strongly on the Coulomb barrier. As it can see from Figure4.3 the Gamow peak has an effective width ∆, which is referred as Gamowwindow, wherein most of reactions take place:

∆ =4√3(E0kBT )

1/2 (4.16)

44

4.4. Direct measurements and experimental problems.

Figure 4.3: The Gamow peak is the result of the convolution of two func-tions: the Maxwell-Boltzmann distribution and the quantum mechanicaltunneling function through the Coulomb barrier. The energetic region rele-vant for the astrophysical investigation (the zone with gray lines) is aroundthe value E0.

Usually the effective energy for thermonuclear reactions ranges from fewkeV to about a hundred keV depending on both the reaction and the astro-physical site in which the reaction occurs. However, the nuclear processesof astrophysical interest occur at energies that in general are too low fordirect measurement in laboratory, as discussed in the next chapters. Thesedifficulties are related to different problems and usually the standard solu-tion is to measure the cross section or, equivalently, the S-factor over a widerange of energies and to the lowest energies possible and then to extrapolatethe data downward to E0 with the help of theoretical arguments and othermethods.

4.4 Direct measurements and experimental prob-lems.

The direct measurement of the cross sections in the low-energy conditionsunder which thermonuclear reactions take place between charged particles instars is a hard task. First of all, the Coulomb barrier between the interactingparticles is usually of the order of 1 MeV while reactions, mainly inducedaround Gamow peak, are often centered in the range from 1 keV to a fewhundred keV. The cross section then is strongly suppressed by an exponential

45


factor (4.)

σ(E) ∝ exp(−2πη) (4.17)

At the energy corresponding to the Gamow peak, the cross section is ofthe order of nano or pico barns. A low cross section means a low number ofparticles collected (Nd), the so-called signal events, on the detector accordingto the equation

Nd ∝ σNiτ∆Ω (4.18)

where Ni is the number of incident particles τ is the thickness of the targetand ∆Ω the solid angle covered by detector. So, different ways are possiblein order to increase Nd:

1. the use of detectors with large solid angles;

2. the use of a more intense beam current (with cautions, not to damagethe target);

3. the use of thicker target that however implies a worse energy resolu-tion.

Even if Nd might increase by improving the experimental setup, thisnumber is affected by the background noise events Nb, coming from thecosmic rays, from natural radiation or from the electronic noise introducedby the experimental setup. For a successful experiment it is important toreach the condition:

Nd

Nb≫ 1 (4.19)

This ratio can be adjusted by increasing the detected particles or by reducingthe background noise, with the following techniques:

1. using of very low-noise electronics.

2. performing nuclear astrophysics experiments in underground labora-tories, as the Laboratori Nazionali del Gran Sasso.

3. using different kinds of indirect methods that allow also to overcomeother experimental difficulties.

In this context, the first simple way to avoid the experimental problemsconsists in an extrapolation of the cross section down to astrophysical rele-vant energies. As already said, the S(E) factor is useful for an extrapolationfrom experimental data measured at higher energies because of its small en-ergy dependence. The standard procedure consists in fitting the high energydata using a proper theoretical function (in the simplest approximation, apolynomial). Then this is extrapolated to the astrophysical energies. Any-way, the presence of low-energy or subthreshold resonances (Rolf & Rodney,1988) and electron screening effects make the extrapolation not very reliable.In particular, there is a subthreshold resonance (see Figure 4.4) if the res-onance energy Er of an excited state of the compound nucleus does notexceed the Q-value for the reaction (Q = mx+mX −my−mY , where m arethe mass of involved particles). In this case the resonance peak lies below

46


Figure 4.4: A sub-threshold resonance and its influence of the behaviourof the astrophysical S(E)-factor.

the interaction energy where the tail of this resonance can influence the be-haviour of the S-factor and can be even dominant at astrophysical relevantenergies. Therefore precise information on the position, the strength and theFWHM (Full Width at Half Maximum) of the resonance are needed fromindependent experiments. As already advanced, a second relevant sourceof uncertainty in the extrapolation of the astrophysical factor down to zeroenergy is the enhancement of S(E) due to the electron screening effect. Upto now it was assumed that the interacting nuclei be completely strippedof electrons, so the Coulomb potential is typically expressed as in equation(4.1), being essentially bare nuclei. On the contrary, when nuclear reac-tions are studied in a laboratory, the projectile is usually in the form ofan ion and the target is usually a neutral atom or molecule surrounded bytheir electronic cloud (Assenbaum e al., 1987). The atomic electron cloudsurrounding the nucleus acts as a screening potential and consequently thetotal potential goes to zero outside the atomic radius (Ra)

Veff =Z1Z2e

2

Rn− Z1Z2e

2

Ra(4.20)

Then the projectile effectively sees a reduced Coulomb barrier. As a con-sequence, at low energies the cross section for screened nuclei, σs(E) (alsoshielded cross section), is enhanced, with respect to the cross section of thebare nucleus σb(E), by a factor:

f(E) =σsσb

∝ SsSb

∼ expo

(

πηUe

E

)

(4.21)

47


Figure 4.5: Schematic representation of the potential between chargedparticles. The potential is reduced at all distances and goes essentially tozero beyond the atomic radius Ra because of the presence of the electroncloud. The electron screening effects cause an enhancement of the S(E)-factor, increasing the penetrability through the barrier.

where Ue, representing the screening potential for the studied reaction, mustbe taken into account to determine the bare nucleus cross section. ForE/Ue > 1000 the electron screening effects are negligible so one essentiallymeasures σb(E), while if E/Ue < 100 one (Langanke et al., 1996) experimen-tally have an enhancement on the cross section, σs(E). The experimentalenhancement has been observed in several fusion reactions and it has beenseen that the lower is the interaction energy, the larger is this enhancingfactor. Because of the high temperature of stars, atoms are generally com-pletely ionized, and one can imagine that electron screening has no effecton nuclear reactions in stars. However nuclei are immersed in a sea of freeelectrons, the so-called plasma, resulting in an effect similar to the one dis-cussed above. In the condition of nearly perfect gas, therefore, when kBT ismuch larger than the Coulomb energy between the particles, the electronstend to cluster into spherical shells around the nuclei, with a Debye-Hunckelradius RD of:

RD =

(

kBT

4πe2ρNAξ

)1/2

(4.22)

whereNA is the Avogadro number and the quantity ξ is expressed by the

48

4.5. Indirect methods for nuclear astrophysics

equation:

ξ =∑

i

(

Z2i + Zi

) Xi

Ai(4.23)

Here the sum is performed over all positive ions and Xi is the mass fractionof the ith nucleus of charge Zi. The shielding effect reduces the Coulombpotential as in laboratory and it increases the reaction rate, or equivalentlythe cross section, by a factor g(E) according to the equation:

〈σv〉s = g(E) 〈σv〉b (4.24)

It is necessary to know the electron screening factor in the laboratory in orderto extract the bare nucleus cross section from the σs(E) using (4.21). Thenthe proper stellar screening factor should be applied to that (4.24). One ofthe most important uncertainties in experimental nuclear astrophysics de-rives from this procedure and, because of this, more exhaustive and precisedeterminations of σb are needed at energies as low as possible. In this con-text several indirect methods, for example the Coulomb dissociation (CD),the asymptotic normalization coefficient (ANC) method, and the Trojanhorse method (THM), have been proposed to overcome the specific difficul-ties of direct measurements. I will briefly present these new experimentalapproaches in the next section, and will describe with particular attentionthe Trojan horse method in next chapter.

4.5 Indirect methods for nuclear astrophysics

As already mentioned, both the Coulomb barrier penetration and the elec-tron screening effects represent problems that must be overcome in orderto get the cross-section for charged-particle-induced reactions in the energydomain relevant for astrophysics. For these reasons the indirect methodsmentioned above have been proposed. In particular, ANC and CD methodsprovide information about astrophysical relevant reactions involving pho-tons, while the THM is applied to reactions between charged particles. Theseindirect methods have been developed to extract cross sections relevant forastrophysics from other kind of experimental or theoretical approaches. Inthese complementary methods the cross-section for the relevant two-bodyreaction (transfer reaction, proton capture, photo-disintegration, etc.) isextracted by selecting a precise reaction mechanism in a suitably chosenthree-body reaction or through the application of some theoretical consid-erations. In this context, the most important steps consist in reproducingdirect data at high energies making use of data extracted from the indirectmethod and then in trying to go down at very low energies. Among indirectmethods, Coulomb dissociation allows to extract a precise radiative-capturecross section. The method, as proposed by Baur (1986), consist in studying

49

4.5. Indirect methods for nuclear astrophysics

the cross section for the reaction

x+X −→ y + γ (4.25)

through the use of inverse photo-dissociation reactions like:

y + γ −→ x+X (4.26)

This is done assuming a first-order perturbation theory for the electromag-netic excitation process and the principle of detailed balance. The secondindirect technique presented is the so-called ANC (Asymptotic Normaliza-tion Coefficient) (Mukhamedzhanov et al., 1990; Xu et al., 1994) method,which provides the normalization coefficients of the tails of the overlap func-tions, and determines S factors for direct capture reactions at astrophysicalenergies. The method can be applied for the analysis of the direct radiativecapture processes of type (4.1), where the binding energy of the capturedcharged particle is low. Moreover, the ANC technique turns out to be veryproductive for the analysis of the astrophysical process in presence of a sub-threshold state. Very recently, a work by (Johnson et al., 2006) developedANC techniques in order to determine the astrophysical factor also for re-actions different from radiative capture processes.

At the end, I mention the so-called Trojan horse method (hereafterTHM) (Baur, 1986; Spitaleri et al., 1999), which seems to be the best suitedfor investigation of low-energy charged-particle reactions relevant for nu-clear astrophysics. This method has already been used to derive indirectlythe cross section of a two-body reaction from the measurement of a suit-able three-body process to overcome the effects due to the entrance-channelCoulomb barrier. The measurement of such a cross section at energies aslow as possible is then necessary to gather more precise information aboutthe energy production and nucleosynthesis in astrophysical environments.In this paper we shall stress the importance of the THM as a complemen-tary tool to direct measurements in the study of 13C(α,n)16O reaction ofastrophysical interest.

50

CHAPTER

FIVE

MEASURE OF THE 13C(α,N)16O REACTION

THROUGH THE THM.

As already said in the third chapter nearly half of the heavy elements ob-served in the universe are produced by a sequence of slow neutron capturereactions, the so-called s-process nucleosynthesis (Busso et al., 1995). Thereaction 13C(α,n)16O is considered as the main neutron source for the maincomponent of the s process in low mass Asymptotic Giant Branch (AGB)stars. In this scenario, two factors can determinate the efficiency of thisreaction: the amount of 13C burnt and the cross section of the 13C(α,n)16Oreaction. An accurate knowledge of this reaction rate at relevant tempera-tures would eliminate an essential uncertainty regarding the overall neutronbalance and would allow for better tests of modern stellar models with re-spect to 13C production and burning in AGB stars.

Very recent observational constraints, like those by ?, and their inter-pretation, to which I have contributed as part of my work for this thesis(Maiorca et al., 2011b), suggest an enlarged 13C reservoir, that would in-duce higher neutron densities because part of the 13C would burn at aslightly higher temperature than before (up to 10keV, against 8keV thatwere standard before these works). Concerning the second aspect, a newaccurate measurement of the rate for the 13C(α,n)16O reaction might im-pose very restrictive constraints on the conditions in which 13C is burnt inthe ”pocket” during the AGB stage. Modern stellar models, run with theaccepted 13C(α,n)16Orate, show that the abundance of 13C produced in thepocket must burn locally in the radiative layers of the He intershell, beforea new convective pulse develops, in contrast with previous ideas that sug-gested carbon-13 combustion in a convective environment. An increase ofthe cross-section can have only small consequences in low and intermediatemass stars (above 1.5M⊙), because

13C would burn even faster than before,in the radiative intershell conditions, i.e. at low T and low neutron densities.Any marginal increase in the neutron density related to the increased rate

51

5.1. Theory of the Trojan Horse method.

would be cancelled by the subsequent operation of the 22Ne(α,n)25Mg re-action in the convective instability, where the neutron density is alreadyquite high (1010 n/cm3). Important effects would be instead seen in verylow mass stars, below 1.4 M⊙, where the 22Ne source is not active becausethe temperature in the pulses is not sufficient. In this case changes to the13C(α,n)16Orate would immediately affect the isotopic ratios of s-elementabundances in AGB stars. A possible reduction of the rate with respectto the present values by the NACRE compilation might instead imply that13C have insufficient time to burn in the interpulse phase. In this case itwould end up burning in the convective region, at a higher temperature andproducing energy, thus potentially modifying the whole energy budget ofthe star and the structure of the convective layer. All the above possibleastrophysical effects will be considered in a dedicated section (see Chapter6 for a more accurate discussion).

Direct measurements of the 13C(α,n)16O cross section have been per-formed down to 280 keV (Angulo et al., 1999), whereas in AGB stars thetemperatures at which α-captures on 13C occur are typically about (0.8 −1.0) × 108 K. The corresponding Gamow peak (Rolf & Rodney, 1988), inaccording with equations (4.14) and (4.16), is at Ecm = 190 ± 90 keV, sothat the direct data available stop at the right edge of Gamow window, whilethe Coulomb barrier is about

Ec =Z1Z2e

2

Rn=

Z1Z2e2

r0(A1/31 +A

1/32 )

∼ 3.7MeV (5.1)

where I use r0 = 1.3 fm. In this context, the study of the astrophysi-cal S-factor in the relevant region for astrophysics, where Coulomb-barrier-penetration and electron-screening effects are dominant, is highly desirable.

The indirect Trojan Horse Method permits to extend the measure belowthe current lower energy limit and to overcome both the cited difficulties.In this chapter, I first presented the theoretical approach for the TrojanHorse Method (THM) for the study of low-energy charged-particle reactions.Then I report on the application of this method in order to obtain indirectinformation about 13C(α,n)16O process at the low energy, starting from the13C(6Li,n16O)d reaction.

5.1 Theory of the Trojan Horse method.

The Trojan Horse method consists in investigating the three-body reaction,in the final state, between two charged (A and a) particles:

A+ a→ c+C + s (5.2)

in order to extract indirectly the cross section of a two-body sub-reaction ofastrophysical interest:

x+ a→ c+ C (5.3)

52

5.1. Theory of the Trojan Horse method.

A schematic representation of the (5.2) process, the so-called Trojan Horsereaction, is shown in Figure 5.1 through a pseudo-Feynman diagram. The

Figure 5.1: Pseudo-Feynman diagram for the break-up quasi-free processa(A,cC)s.

Trojan Horse approach, as suggested by Baur (1986), is based on the theoryof the quasi-free (hereafter QF) break-up mechanism in which the interactionbetween two nuclei produces the break of one particle in its constitutingnuclei. In particular, the starting point is to consider that the nucleus A iscomposed by two nucleon clusters x and s. The wave function for the targetnucleus A can be written in the following way

ψA = ψx(rx)ψs(rs)ψ(rx − rs) (5.4)

where the ψi are the internal wave functions of x and s, respectively, whileψ represents the relative motion wave function between the two clusters.

The interaction between target and projectile causes A, described by astructure such as A = x⊕ s, to break in the two clusters and the nucleus acan interact only with the transfer particle x. In practice the nucleus s doesnot participate to the reaction and it can be considered as a spectator forthe x(a, c)C reaction. After selecting appropriate kinematic conditions, thequasi-free process occurs if the cluster s maintains the same momentum ithad in the nucleus A before interacting. Figure 5.1 represents the two crucialmoments of the whole process: the upper pole is the break-up of the nucleusA into x and s, while the lower one describes the two-body interactionx(a, c)C. The A particle is called the Trojan Horse nucleus because, similarlyto what the wooden epic horse did for Ulysses and his comrades, it hides in its

53

5.2. Plane Wave Impulse Approximation.

interior the transferred, participating cluster x. In this way, the Trojan Horsereaction (5.2) can be performed at energies well above the corresponding Ec,so that the binary reaction cross section is not Coulomb-suppressed, as thebarrier has already been overcome in the entrance channel. For these reasonsthe THM has already been applied several times to reactions connectedwith fundamental astrophysical problems characterized by very low energies.Moreover, assuming that the beam energy can be compensated for by thex+ s binding energy and by the Fermi motion of x inside A, the two-bodyreaction can take place at very low a− x relative energy, inside the Gamowwindow.

5.2 Plane Wave Impulse Approximation.

The quasi-free break-up mechanism can be described by following differenttheoretical formalisms, such as the Distorted Wave Impulse Approxima-tion (DWIA) (Chant et al., 1977; Roos et al., 1977), the Distorted WaveBorn Approximation (DWBA) (Typel et al., 2000) and the Plane WaveImpulse Approximation (Jain et al., 1970; Slaus et al., 1977; Fallica et al.,1978). Distorted-wave approaches provide the most sophisticated and accu-rate formalisms, as it was established by several authors (Jacob et al., 1966;Roos et al., 1976). In these cases, the momentum distribution feels the ef-fects of the distortion due to the interaction between the interacting nuclei.Anyway, Roos et al. (1976), in a work on the 6Li cluster structure, concludesthat for recoil momenta of the spectator ks lower than 100 MeV/c both PWand DW approaches describe well the results about the experimental be-haviour of the momentum distribution. Hence, in the above limit of low ks,the various approaches show essentially the same results without introduc-ing significant systematic uncertainties. For these reasons, in this work Ishall focus on the simpler Plane Wave theoretical approach.

Quasi-free reactions (hereafter also QFR) can be easily described bymeans of the impulse approximation IA (Chew, 1950; Chew & Wick, 1952).Let us consider, as a typical case, the one of a particles striking a complexsystem A. The assumptions underlying the impulse approximation are thenthe following.

1. The incident particle never interacts strongly with the two con-stituents of the system at the same time.

2. The amplitude of the incident wave falling on each constituent isnearly the same as if that constituent were alone.

3. The binding forces between the constituents of the system are negli-gible during the decisive phase of the reaction.

Under these hypotheses, the incident particle a is considered as inter-acting only with a part (x) of the target nucleus A, whose wave function isassumed to have a large amplitude for the x⊕ s cluster configuration, while

54


the other part s behaves as a spectator to the process. In the THM thetransferred particle is virtual (off-energy-shell). However, here I neglect theoff-shell character of the transferred particle and use the on-shell approxima-tion. Moreover, with the plane wave (PW) approximation I am assuming, asmentioned above, that the incident and outgoing particles can be describedby plane waves without any distorting effects due to the Coulomb interactionbetween particles (Jain et al., 1970).

ψ(r) = 〈r|k〉 =(

1

2π

)3/2

exp(ik · r) (5.5)

Taking into account all these hypotheses, the Plain Wave Impulse Approxi-mation (Satchler, 1990) leads to a simple expression for the differential crosssection for the three body A(a, cC)s reaction (Jain et al., 1970)

d3σ

dEcdΩcdΩC∝ KF |Φ(ps)|2

(

dσaxdΩ

)

(5.6)

This is the cross section for the scattering of a particle c into the solid angledΩc with an energy between Ec and Ec + dEc and of particle C into thesolid angle dΩC . This can be factorized in three terms. Starting from theleft, the first term is the kinematical factor KF containing the final statephase-space factor; as I shall show later, it is a function of the masses,momenta and angles of the outgoing particles. This expression is derivedby assuming that the momentum of the spectator to the virtual two-bodyreaction is equal to the one before the reaction. The second term containsΦ(ps), which is the momentum distribution of the deuteron inside the 6Linucleus. In practice, this is the Fourier transform of the radial wave functionfor the x− s inter-cluster motion inside A, usually depending on the clusterconfiguration involved in the reaction:

|Φ(ps)| = (2π)3/2∫

∞

−∞

ψ(r)exp(−iKsr)dr (5.7)

The last term (dσ

dΩ) represents the differential cross section for the binary

reaction and it is the quantity to be determined through the Trojan Horseexperiment. In order to use equation (5.6), the starting point is the differ-ential cross section for a three-body (3B) final state with the momenta ofparticles c, C and s in the ranges d3kc,d

3kC and d3ks respectively

dσ =(2π)4

|vrel|d3kcd

3kCd3ksδ(Ki −Kf )δ(Ei − Ef )|t3Bfi |2 (5.8)

where the K and E values are the center-of-mass momenta and the totalenergy of the system in the initial and final states, and vrel = ka/Ea is the

55


relative velocity between the incident particle and the target. The variablet3Bfi , appearing in (5.8), is the three-body reduced matrix element and it isrelated to Tfi by the equation

Tfi = δ(Ki −Kf )t3Bfi (5.9)

This represents the transition matrix element for the three-body reactionthat I can write in the laboratory system for the break-up reaction as follow:

Tfi = 〈f |T |i〉 =⟨

kc,kC ,ks, ψc, ψC , ψs|T 3B |ψa, ψA,ka

⟩

(5.10)

T 3B is the complete T -operator for the reaction, while ψi is the wave func-tion that describes the internal generic particle i and ki the correspondingmomentum in the final state. At this point, the quasi-free process is takeninto account by introducing the Impulse Approximation (IA) through whichthe incident particle a interacts only with particle x in the nucleus A, whilethe residual cluster s is a spectator to the reaction. Therefore, T 3B canbe replaced by T 2B, the so-called T -operator for the two-body interactionbetween a and x. With this approximation, the transition matrix elementcan now be written as:

Tfi =⟨

kc,kC ,ks, ψc, ψC , ψs|T 2B |ψa, ψA,ka

⟩

=

∫

d3qx〈kc,kC , ψc, ψC |T 2B |ψa, ψx,ka,qx〉〈ψx, ψs,qx,ks|ψA〉 (5.11)

If ψA represents the intrinsic state of the target and qi is the momentum ofparticle in initial state, the correspondent momentum space wave functionis

〈qs,qx, ψs, ψx|ψA〉 = Φ(k)δ(qs + qx) (5.12)

and from the momentum conservation in the laboratory system where thetarget is at the rest one gets:

qx + qs = 0

qx = −qs (5.13)

qs = ks (5.14)

Hence, the final form of the equation for Tfi is

Tfi =

∫

d3qx〈kc,kC , ψc, ψC |T 2B |ψa, ψx,ka,qx〉Φ(k)δ(q2 + ks)

= 〈kc,kC , ψc, ψC |T 2B |ψa, ψx,ka,−ks〉Φ(−ks) (5.15)

One can note that the momentum of the cluster x before the collision, qx,is equal and opposite to the momentum of the outgoing residual nucleus ks,

56


a quantity that is experimentally measured. Substituting for Ki = ka andKf = kc + kC + ks and transforming to the c.m. and the relative momentafor the two-body system, I can write

Tfi = δ(ka − kc − kC − ks)⟨

kf |T 2B |ki

⟩

Φ(−ks)

= δ(ka − kc − kC − ks)t2Bfi Φ(−ks) (5.16)

At the end, the form of equation (5.8), using (5.9) and (5.16), becomes:

dσ =(2π)4

|vrel|k2cdkcdΩck

2CdkCdΩCd

3ksδ(Ei − Ef )δ(ka − kc − kC − ks)|

×φ(−ks)|2|t2Bfi |2 (5.17)

Integrating over dks, which in this reaction is unobserved, and over dkc andconsidering the two following conditions:

Ef = Ec + EC + Es = Ec +√

m2C + k2C

√

m2s + (ka − kc − kC)2 (5.18)

Ei = Ea +mA (5.19)

one obtains:

dσ

dEcdΩcdΩC=

(2π)4

ka

ECEs

kck2CEaEckCEs + EC [kC − kacosθC + kccos(θc − θC)]|

×φ(−ks)|2|t2Bfi |2 (5.20)

Here, θc and θC are the angles of the outgoing nuclei, measured with respectto the incident beam direction. The quantities in equation (5.20) are evalu-ated using the momentum conservation, ka = kc + kC + ks, and the energyconservation, Ef = Ei. It is important to note that the energy conservationis not the same as for the usual two-body system, because of the bindingenergy of cluster x in the target and of the recoil energy. At this point,introducing the two-body scattering cross section in the c.m. system of thetwo particles, the equation can be written as:

dσ

dEcdΩcdΩC=

(kck2CEsE

2c.m.

kaEx kCEs + EC [kC − kacosθc + kccos(θc − θC)]

×|φ(−ks)|2(

dσ

dΩ

)

c.m.

(5.21)

This is the required expression, already presented in equation (5.6). Onecan easily note that the three-body cross section for the A(a, cC)s reac-tion is strictly connected with the one corresponding to the two-body pro-cess a(x, c)C. The momentum distribution of the spectator in the TrojanHorse nucleus is also present in both equations, so that, while the remnant

57

5.3. Current measurement status

terms are defined as the kinematical factor, given by the following expression(Jain et al., 1970):

KF =(kck

2CEsE

2c.m.

kaEx kCEs + EC [kC − kacosθc + kccos(θc − θC)](5.22)

KF is made by terms measured or calculated and it depends only on thekinematic conditions of the process. Hence the only unknown variable isthe two body TH-cross section that I can easily obtain by (5.6) using thefollowing equation:

(

dσaxdΩ

)

=d3σ

dEcdΩcdΩC

(

KF |Φ(ps)|2)−1

(5.23)

In this way, I have formulated the THM two-body cross section; in order toget the direct one it is necessary to reintroduce the Coulomb-field effect, bymultiplying the (5.23) by the penetration factor Gl (Cherubini et al., 1996;Spitaleri et al., 1999). Performing an experiment where it is possible tomeasure the QF-contribution of the three-body reaction and knowing boththe kinematical factor and the momentum distribution for the relative s−xmotion inside the TH nucleus, makes it possible to extract the a(x, c)C crosssection by using the relation (5.6)

(

dσ

dΩ

)

cC

=∑

l

Gl

(

dσldΩ

)THM

cC

(5.24)

Here, Gl represents the transmission coefficient for the lth partial wave.Now, one can notice a very important point, which will be recalled in

the last chapter. Because of the factor Gl, the two-body cross section canonly be obtained with an arbitrary normalization but the essential energydependence can instead be extracted carefully. Absolute cross sections canbe obtained only after normalization to the directly-measured excitationfunction. This is the so-called validity test, for which we shall need to anchorour low-energy estimates to the high-energy measurements at energies abovethe Coulomb barrier. A comparison and an agreement between direct andTHM data over the already explored in the past is necessary in order toallow to extend the measurement at astrophysical energies using the THM-determined energy dependence. In this context, the Trojan Horse methodhas to be seen as a complementary tool in experimental nuclear astrophysics,because direct data are in any case needed at energies above the Coulombbarrier for normalization procedures.

5.3 Current measurement status

In the last fifty years, several investigations of the total cross section forthe 13C(α,n)16O reaction at low energies have been reported, motivated by

58


its importance as the main neutron source for the s-process in low massstars. However, many experimental difficulties did not allow experimental-ists to perform a direct measurement of the reaction-rate behavior at tem-peratures relevant for astrophysics. In the last years the most commonly-used rate was that presented in the European Compilation of ReactionsRates for Astrophysics (hereafter NACRE) by Angulo et al. (1999). There,the rate for 13C(α,n)16O is determined using experimental cross sectionsfrom Sekharan et al. (1967), Davids (1968), Bair et al. (1968), Drotleff et al.(1993) and Brune et al. (1993), covering the energy range between 0.28 and4.47 MeV.

Figure 5.2: Behavior of the astrophysical S-factor, the most useful param-eter for an extrapolation from experimental data measured at high energiesAngulo et al. (1999). An enhancement of the S-factor at low energies withrespect to previous recommendations was suggested because of the inclusionof a subthreshold resonance in the extrapolation.

Since direct measurements stop right at the limit of the Gamow win-dow (190 ± 90) several experiments using indirect methods have been per-formed. At temperatures of about 108 K the uncertainties are ∼ 300% dueto the prohibitively small reaction cross section at energy below 300 keV.For the lowest energy range, the one most relevant for neutron productionin AGB stars, the S-factor was usually extrapolated by fitting the data ofDrotleff et al. (1993). It was however shown that this extrapolation can bestrongly affected by the 1/2+ subthreshold resonance of 17O at an excitationenergy of 6.356 MeV, which is just 3 keV below the α+13C threshold.

As the above is a critical point, I present here a short review of differentexperimental approaches, both via direct measurements and via indirect

59


techniques, which tried to explore the low temperatures typical of stars(0.8 − 1.0 × 108 K) in order to have a more accurate knowledge of the13C(α,n)16O reaction rate in astrophysical conditions.

Before the nineties, the reaction rates of astrophysical important ther-monuclear reactions involving low-mass nuclei (1 ≤ Z ≤ 14), including also13C(α,n)16O, were collected in a huge paper by Caughlan et al. (1988). Thisprovided the numerical values and also gave analytic expressions for therates, updating previous publications from the same group.

Chronologically speaking, Drotleff et al. (1993) was the first to performa direct experiment in which differences from the previously accepted valuesfor the S(E) factor of 13C(α,n)16Oemerged. These authors measured theexcitation function reaching a sensitivity limit of 50 pico-barns in the bestcase. The new data covered the energy range 350keV ≤ E ≤ 1.4MeV , wherethe cross section varies over eight order of magnitude. The reaction rate wascalculated taking into account the subthreshold resonance described abovein according to the expression:

NA〈σv〉 =6.788 × 1015

T 29

exp

[

−33.093

T1/39

−(

T9330.271

)2]

×(

1 + 0.485T1/39 − 7.948T

2/39 + 10.725T9

)

+1016.988

T3/29

exp

[

−6.259

T9

]

+

+3.474 × 105

T3/29

exp

[

−8.430

T9

]

(5.25)

This relation is valid over the range 0.01 ≤ T9 ≤ 1.0 (T9 hereafter means thetemperature in units of 109 K). Because of the state in 17O just below theα-threshold and using their own measurements over the range 0-300 keV,Drotleff et al. (1993) suggested a low-energy increase in the reaction rateof 13C(α,n)16O with respect to previous investigations. Theoretical calcu-lations (Bach, 1992; Descouvemont, 1987) supported this indication, withan increase at low energies more rapid than expected. However, consideringthe errors, the result by Drotleff et al. (1993) would still be consistent witha constant, horizontally extrapolated, astrophysical factor.

A much lower rate for of the 13C(α,n)16O reaction (actually, the lowestreaction rate present in the literature) was subsequently suggested, throughthe direct α-transfer reaction by Kubono et al. (2003). In this work, thecontribution of the subthreshold state was found to be much smaller thanthe accepted prediction and the calculated reaction rate was parameterizedby the formula:

NA〈σv〉 = exp(

−36.90392 + 0.07784191T−19 − 48.15691T

−1/39

)

60


×exp(

+109.3879T1/39 − 21.95909T9 + 3.161556T

5/39 − 25.92545logT9

)

(5.26)Hence, the reaction rate is smaller than the NACRE recommended valueroughly by a factor 4 at T9 = 0.1. It is also smaller by a factor 3.5with respect to Drotleff et al. (1993), suggesting that the 13C(α,n)16O re-action would be slower at low temperatures (this rate is similar to theCaughlan et al., 1988, ’s value). For clarity, Table 5.1, showing the resultsof different experiments, allows us to have an immediate comparison of thedifferent suggested values of the 13C(α,n)16O rate for the relevant astrophys-ical range of energies (0.8− 1.0× 108 K). One can note that differences aresometimes very high and this is so because the total uncertainty at low tem-peratures includes mainly two components: one from the subthreshold statecontribution and the other from the extrapolation of the direct data. Forthis and for future applications, I use here the Maxwellian-averaged reactionrate NA〈σv〉, in analogy with (4.12) and (4.13), as follows:

NA〈σv〉 = NA

(

8

πµ

)1/2 1

(kbT )3/2

∫

∞

0

σ(E)Eexp

(

− E

kBT

)

dE (5.27)

Here, NA is the Avogadro number (∼ 6.022 × 1023mol−1), µ is the reducedmass of the system, kB the Boltzmann constant, T is the temperature, σis the cross section, v is the relative velocity and E is the energy in thecentre-of-mass system. The quantity NA〈σv〉 is in units of cm3 mol−1 s−1.

T9 Caughlan et al.(1988)

Drotleff et al. (1993) Angulo et al. (1999)

0.08 1.32 × 10−16 2.77 × 10−16 4.80× 10−16

0.09 2.25 × 10−15 4.18 × 10−15 6.99× 10−15

0.10 2.58 × 10−14 4.32 × 10−14 6.99× 10−14

T9 Kubono et al. (2003) Johnson et al. (2006) Pellegriti et al. (2008)

0.08 1.05 × 10−16 1.49 × 10−16 3.36× 10−16

0.09 1.77 × 10−15 2.41 × 10−15 5.41× 10−15

0.10 2.02 × 10−14 2.64 × 10−14 5.94× 10−14

Table 5.1: Table of reaction rates present in the literature for 13C(α,n)16O .T9 is the temperature in the interpulse phase expressed in units of GK.

Accurate measurements were then performed by Johnson et al. (2006)at the Tandem-LINAC facility of the Florida State University. In this work,we had the first case of the application of the asymptotic normalizationmethod (ANC) to the determination of the astrophysical S factor for the13C(α,n)16O reaction. Before, the ANC method had been applied only forradiative capture processes. In practice, the S(E) factor was determinedby measuring the ANC for the virtual synthesis α+13C →17O (6.356 MeV,

61

5.4. The Trojan Horse Method applied to the 13C(α,n)16O reaction.

1/2+) using the α-transfer reaction 6Li(13C,d). Measurements were per-formed at two different sub-Coulomb energies of 13C. For temperatures aboveT9 = 0.3 the calculated reaction rate turned out to be identical to the oneadopted in NACRE, while at temperatures useful for the s-process in AGBstars, T9 = 0.08− 0.1, the rate was smaller by a factor 3 with respect to theone by Angulo et al. (1999), but still inside the uncertainty band.

Another recent ANC determination of the S factor was performed byPellegriti et al. (2008). Their reaction rate at typical AGB temperatures isslightly lower than the value adopted in NACRE, but it is twice as largeas the one obtained in the previous ANC measurements (Johnson et al.,2006). The two ANC works show a difference by a factor as large as 5 inthe estimated contributions from the 1/2+ subthreshold resonance. This ishowever reduced to a factor of about 2.3 in the total reaction rate becauseof the role of the non-resonant term, which is dominant in Johnson et al.(2006).

The differences among the various articles and approaches show thatfurther work is necessary before drawing definite conclusions. Hence, verifi-cation of the results presented in the last two decades using an independentexperimental approach (e.g. the Trojan Horse technique) is highly desirable.Later in this chapter I therefore present the application of this kind of indi-rect method to measuring the cross section, or equivalently the S(E) factor,for the 13C(α,n)16O reaction.

5.4 The Trojan Horse Method applied to the 13C(α,n)16O re-

action.

The Trojan Horse method, as already said, is a powerful tool to extracta charged-particle binary reaction cross section at astrophysical energies,free of the Coulomb barrier effects thanks to a three-body starting processoccurring at energies well above the value of the Coulomb barrier itself.Hence, the THM appears to be very useful to study reactions in astrophysicalenvironments where energies are very low. The present work reports ona new investigation of the two-body 13C(α,n)16Oreaction by selecting theQF-contribution of the 13C(6Li,n16O)d three-body reaction (Q3B = 0.74128MeV), using a 6Li beam of energy ∼ 7.82 ± 0.05 MeV.

The theoretical approach described in section 5.2 is usually adopted forreactions for which break-up occurs in the target. In the present experiment,instead, the 6Li of the beam is the TH nucleus. The formalism remains thesame but obviously it has to be adapted at the projectile break-up case, usingappropriate system transforms. It is assumed that in the laboratory systemthe target is at rest while the projectile is connected with its fragments bythe relation:

kbeam = qx + qs (5.28)

62

5.4. The Trojan Horse Method applied to the 13C(α,n)16O reaction.

Figure 5.3: Pseudo-Feynman diagram for the 13C(6Li,n16O)d reaction.The 6Li projectile is considered as composed by x ⊕ s: the so-called THnucleus. After break-up (upper pole), the deuteron acts as a spectator.

where the ki is the three-momentum of the beam, the participant and thespectator particle, respectively. In order to compare Figure 5.3 and Figure5.1 the target made of 13C corresponds to nucleus a, the projectile A is the6Li nucleus, s the deuteron, x is the α particle, c is the neutron and C is16O.

Thanks to the high energy in the A+ a entrance channel, the two-bodyinteraction can be considered as taking place inside the nuclear field, so thatit does not experience neither Coulomb suppression nor electron screeningeffects. The A + a relative motion is compensated for by the x − s bind-ing energy EB , thus determining the so-called quasi-free (La Cognata et al.,2007) two-body energy (Eqf ), which is given by:

Eqf = Eax − EB =

=ma

mx +maEx − EB

=ma

mx +ma

mx

mAEA − EB (5.29)

Here, Ex represents the fraction of the beam energy EA corresponding tothe cluster x , Eax is the relative energy between a and x and mi is the massof particle. Thus, the relative energy of the fragments in the initial channela+ x of the binary reaction can be very low and even negative. In contrast,this condition is difficult or impossible to be satisfied in binary reactions,due to the Coulomb barrier.

63

5.5. Experimental setup.

In order to apply the Trojan Horse Method, which is based on a quasi-free break-up process, one needs to separate this contribution from all theothers which may occur between the same target and projectile, giving thesame particles in the exit channel: the so-called sequential process. A se-quential mechanism is a two steps interaction in which the final state isreached through an intermediate one, as shown in Figure 5.4. Only the pro-cess occurring through the QF-mechanism is of interest for the further THinvestigation. A detailed study of this level was needed In this context; it isclear that a detailed study and the discrimination of such mechanisms rep-resents an important stage of a TH-analysis. This kind of information canbe reached studying the relative energies between the particles in the exitchannel. In particular, the study of any two among the En16O , Ed160 andEdn relative energies allows to obtain information on the presence of excitedstates of 17O, 3H and 18F. Once this stage of analysis is confirmed, it willbe possible to apply the THM to the three-body data for the extraction ofthe two-body cross-section of interest.

5.5 Experimental setup.

The experiment was performed at ”The John D.Fox Superconducting Ac-celerator Laboratory” in the Florida State University by the ASFIN2 (inItalian, AStroFIsica Nucleare) group of the Laboratori Nazionali del Sud,Catania. In particular, I took part in the first phase of the experiment: theelectronic and mechanical assembly, the calibration runs and seven days ofon-beam data acquisition. The facility implied the use of the 9MV TAN-DEM to accelerate a beam made of 6Li, the isotope of litium characterizedby three protons and three neutrons. The spot size was about 1 mm andbeam intensities were around 1 - 4 nA. Then, the interaction between thebeam and the 13C target occurred in a vacuum chamber with a diameter of1 m placed in the second target room of the laboratory.

The acceleration took place in two stages: an ion source produced negatively-charged ions having a velocity of a few tenths percent of the velocity of light.Specifically, polarized ions of lithium were created by the optical pumpingtechnique. This is a process in which light is used to raise one or more elec-trons from their levels to more energetic states. Hence, sometimes bindingelectrons can be separated from their nuclei or molecules. The 9MV Tan-dem Van der Graff accelerator, 15,24 m long, provided the second stage ofvelocity increasing. In a tandem accelerator the same high voltage can beused twice if the charge of the particles can be reversed while they are insidethe terminal. At first negative ions coming from the source are acceleratedbecause they are attracted by the positive electrode and the beam, passingthrough a thin foil to strip off electrons inside the high voltage conductingterminal, become made of positive charges: the so-called stripping phase.

64


Figure 5.4: Possible three-body sequential processes resulting from theinteraction between 6Li and 13C, which gives in the exit channel the sameparticles (d,16O,n) through the formation and decay of intermediate statesof 17O∗, 3H and 18F∗, respectively.

65


The final result are positive ions that are accelerated again because they arerepelled by the positive terminal. One can observe the advantage of using atandem in the formula of the generated energy:

E = (1 + q)V (5.30)

Hence, one can obtain a large amount of energy if the beam particles havean elevated charge state (q) at a specific applied voltage V (the terminal canbe charged to a maximum potential of 9 to 10 million volts). The amount ofacceleration can be varied by changing the terminal voltage. This voltage ismaintained by continuously transferring charges using an endless insulatingbelt carrying positive charges between ground potential and the terminal.The beam then leaves the tandem and, thanks to focussing magnets, itarrives at the measurement chamber. A schematic draw (scale 1:7.5 cm)

Figure 5.5: Experimental setup of the 13C(6Li,n16O)d reaction discussed inthe text. The target is made of 13C while the beam is 6Li. PSD1, PSD2 andPSD3 are placed in the positive semi-plane at angles as specified in Table5.2, in order to detect the deuteron. PSD4 and PSD5 are used to observe160 in the negative semi-plane, while the third particle, the neutron, is notdetected. (scale 1:4)

of the experimental configuration is shown in Figure 5.5 where the zoneabove the beam track represents the positive semi-plane. Two 13C targetsof different thickness, 107 and 53 µg/cm2, were placed at 90 degree withrespect to the beam direction. In order to measure in coincidence the 16Oand deuteron particles the detection setup consisted of a set of five siliconposition-sensitive detectors (PSDs). Two telescopes, each of them composed

66


by a 20 µm thick ∆E silicon detector and a position-sensitive silicon (PSD2and PSD3), were used to reconstruct the experimental momentum distri-bution of the spectator in the positive semi-plane. Moreover, in the samesemi-plane the PSD1 covering about 3-13 deg was very useful to discrimi-nate the deuteron yield. This detector is covered by a 22.5 µm aluminiumfoil in order to suppress the elastic scattering contribution and the heavyfragments: only particles with A < 4 can pass it. Similarly, two position-sensitive detectors (PSD4 and PSD5) covering the scattering angles from17.33 degree up to 44.25 degrees on the other side of the chamber were usedto measure the yield of the Oxygen recoils. The thicknesses of PSDs, sum-

PSD Distance (cm) Central angle (deg) Angular range (deg ) Thickness (µm)

1 29.2 8.04 3.15 - 12.93 1000

2 28.0 23.05 17.95 - 28.15 500

3 24.5 37.93 32.11 - 43.75 500

4 25.5 22.93 17.33 - 28.53 500

5 21.3 37.56 30.87 - 44.25 500

Table 5.2: Experimental conditions for the 13C(6Li,n16O)d experiment:distances, angular positions, ranges covered and thickness of every PSD.

marized in Table 5.2, were chosen in order to cover the smallest angles, thatis the largest energies of the residual nuclei, with thick detectors. The thirdparticle, in our case the neutron, was not detected because neutral particlesare very difficult to study. The alignment of all detectors was checked byan optical system. The trigger for the event acquisition was given by coinci-dences between deuterons detected in the positive semi-plane and the signalof 16O coming from the other two PSDs. This allowed for the kinematicalidentification of our specific exit channel of reaction 13C(6Li,n16O)d. Whenenergy (E) and position (P ) signals were detected in each PSD, they hadto be elaborated and stored. The position signal was directly sent to theADC after a pre-amplification and an amplification stage. The E signal,after passing through the pre-amplifier, was instead split in two lines. Thefirst one was sent to a linear amplifier and then to the ADC, as for theP signal, while the second E line passed a quicker amplifier (Time FilterAmpifier) and then a discriminator module to have a logic signal before itwas sent to a TAC-SCA (Time to Amplitude Converter-Single Channel An-alyzer) in order to produce the coincidence event trigger. The start inputof TAC-SCA was given by a logical-or signal coming from PSD4 and PSD5,while the signal corresponding to the deuteron provided the stop. In thisway the coincidence between PSD1 or PSD2 or PSD3 and any one of theother detectors, placed on the opposite side (PSD4 or PSD5) was set.

In summary, a 6Li beam, previously accelerated by a tandem, interactedwith a 13C target producing deuteron and 16O detected in five PSDs. De-

67

5.6. Position Sensitive Detectors (PSDs).

tector signals were processed by standard electronic chains and sent to theacquisition system which allowed the on-line monitoring of the experimentand the data storage for off-line analysis.

5.6 Position Sensitive Detectors (PSDs).

The PSD, standing for Position Sensitive Detector, is a special kind of solidstate detector providing the information on position and energy of incidentcharged particles at the same time citepleo94,kno00. In practice, the detec-

Figure 5.6: Schematic view of a position-sensitive detector (PSD).

tor is a rectangular diode, usually made of n-type silicon with a p-type layerof boron, with a uniform, resistive electrode on the front face and a low-resistive back electrode. When a charged particle passes through the diode,a number of electron-hole pairs are produced and the charged deposited onthe contact will be proportional to the particle energy and to the properelectrode resistance. For clarity, Figure 5.6 shows a schematic diagram of aPSD. The signal of position P is extracted from the resistive layer becauseit acts as a charge divider and it depends on the hitting point. If one definesx as the distance between the grounded contact and the interaction pointof incident particles, while L is the total length of the resistive layer, theposition signal is proportional to the kinetic energy E in according to thefollowing expression:

P = Ex

L(5.31)

A second signal (the E signal), proportional to the total charge depositedin the detector, is derived from the normal conductive front electrode. As

68

5.6. Position Sensitive Detectors (PSDs).

Figure 5.7: Schematic draw of a position-sensitive detector and of itsholder. A grid with eighteen slits is placed in front of the holder to per-form the position calibration of the detector. The readout contacts are alsopresent, indicated by capital letters.

Figure 5.7 shows, a PSD presents three readout contacts:

1. The first contact on the left (A) is the one connected to the ground.It is usually closed through a resistor of the order of 1 kΩ, correspondingat about 20% of the total resistive layer, which ensures a measurable signalalso when the hit position is close to this end.

2. The one in the middle is connected to the cathode and provides theenergy signal E.

3. The contact on the right is connected to the resistive anode where thecharge fraction that provided the position signal P is collected.

One of the problems with this kind of detectors is to ensure linearity inthe position signal. This requires the semiconductor and the resistive layerto be highly uniform and homogeneous. The typical detector resolution canbe of the order of 0.5% FWHM at room temperature over active lengthsof 5 cm, corresponding to about 250 µm, for the position and also about

69

5.7. The position calibration.

0.5% for the energy. Every PSD is covered by a thin inactive layer, theso-called ”dead layer”, of thickness about 0.2 µm, made of aluminium. It isimportant to take it into account, because it induces a kinetic energy loss.Such detectors are usually rectangular, of 5 cm of length and 1 cm of width.Thicknesses, as reported in Table 5.2, were chosen considering that at lowerangles in the PSD particles are characterized by higher energies.

5.7 The position calibration.

The first phase of the data analysis for an experiment generally consists inthe calibration of the involved detectors. In order to extract the correctinformation for future analysis, it is important to convert both the E and Psignals, expressed in channels, into quantities of physical interest, expressedin physical units like MeV and degrees, respectively. The described pro-cedure must be repeated for every detector. A typical plot of the set ofposition data versus energy, expressed in channels (hereafter ”the matrix”),is shown in Figure 5.8 for PSD5. In order to perform off-line PSD positioncalibration, usually in the first part of the experimental run, a grid witheighteen equally spaced vertical slits was placed in front of each PSD (seeFigure 5.7).

(ch)5E0 1000 2000 3000 4000

(ch

)5

P

0

1000

2000

3000

4000Position-energy matrix

5

10

15

20

25

30

35

40

45

50

Figure 5.8: Position-energy two-dimension matrix for PSD5 for the cali-bration run with 6Li+12C. The eighteen slits and the linear loci are clearlyvisible.

The matrix, in most cases because of statistics and detector resolution,shows well separated lines corresponding to the various slits and almost-vertical highly populated zones, representing tracks left by two-body reac-

70

5.7. The position calibration.

tions in the final state: the so-called kinematical linear loci. In this kindof matrix, I selected the region showing a visible track by using a graphicalcut in order to get information about both energy and position. The datadistribution of the chosen region is typically a Gaussian curve, hence as rep-resentative values for position and energy we chose the mean values of theGaussian fits, while the errors were given by the corresponding σ. Thesepoints were used both for angular and energy calibration (Figure 5.9).

Energy and Position detection for a slit

(ch)PSD5E1900 2000 2100 2200 2300

Co

un

ts

0

200

400

600

800Energy

mean 2146.02

σ 19.68

(ch)PSD5P1100 1200 1300 1400 1500

Co

un

ts

0

200

400

600

800Position

mean 1262.22

σ 20.87

Figure 5.9: Energy and position spectrum for a singular slit. I used themean value and the σ of the Gaussian fit.

It is possible to establish a correspondence between each slit and anangular position with respect to the beam direction. In practice, the centralangular position of each detector θ0 was measured using a theodolite andthe angular position corresponding to each slit was calculated by means oftrigonometric identities. Recalling the expression (5.31) one can eliminatethe position dependence from the energy by introducing the variable:

x =P − P0

E − E0

(5.32)

where E0 and P0 are constants determined by using a fit for all slits. As aconsequence, the matrix shown in Figure 5.8 became made of straight hori-zontal lines, as in Figure 5.10. They are however still expressed in channels:it is the so-called rectification of the matrix. At this point, it is possible toobtain a relation in order to calculate the impact angle of each particle as afunction of the energy-independent linear position x:

θ = θ0 + arctg[c1(x − x0) + c2(x− x0)2] (5.33)

71

5.8. Energy calibration.

c1, c2 and x0 are the results of the best fit performed among all slits. Atthis point, I have a matrix with physical angles measured in degrees on they-axis. The angular resolution was found to be about 0.2 degree.

The calibration was performed by using a different kind of targets andsources: a 228Th alpha-source at six peaks was used because the energy decayis well known for each peak; elastic scattering on heavy 197Au nuclei and ona 12C target were used to measure both elastic and inelastic scattering at thebeam (6Li) energy Eb=7.82 MeV. In particular, this last run was performedin two different configurations to cover both small angles (high energies) andviceversa.

5.8 Energy calibration.

The energy calibration was performed by means of the same runs used forposition calibration. In order to convert the value of the energy comingfrom the acquisition system, expressed in channels, into a physical quantityin units of MeV, the adopted expression is:

EMeV = (a+ bEch)(1 + c1(θ − theta0) + c2(θ − θ0)2) (5.34)

Here, a, b, c1, c2 are constants resulting from the minimization procedure. Inorder to check for a possible dependence of the energy signal on the impactpoint, the procedure was performed for each slit. As a first approximationa simple linear relation between the PSD signal and the detected particleenergy can be sufficient but (5.34) includes further corrections due to theangular calibration. The overall energy resolutions were found to be about1%. The interaction between a 6Li beam of 7.82 MeV and a 12C targetcan produce different possible exit channels (excluding reactions producingneutrons or photons and those having three bodies in the final state, whichcannot be detected with our experimental setup):

1. 12C+6Li, the so-called elastic scattering (Q = 0 MeV)2. 14N+α (Q = 8.79805)3.16O + d (Q = 7.60641)

4. 17O + p (Q = 5.68764)In order to have a check of the procedure I plotted theoretical points

(black points) corresponding to the cited reactions over the matrix and Icompared the position between the two tracks. In particular I found agood agreement especially for the elastic scattering and for different excitedlevels of 14N + α as showed in Figure 5.10. The total kinetic energy of thedetected particles was reconstructed off-line taking into account the energyloss in the different layers passed through by the particle in the detector.This is a crucial stage of data analysis, because the measured energy is theone deposited in the detector but for the future application we are interestedin the reaction energy. The procedure of energy reconstruction requires the

72

5.9. Data Analysis and future work.

(MeV)5E0 5 10

(d

egre

e)5θ

40

45

50

55Calibrated position-energy matrix

5

10

15

20

25

30

35

40

45

50Li6C + 12

-4.43MeVLi6C + 12

0MeVHe4N + 14

2.35MeV

2.96MeV

3.69MeV

Figure 5.10: Calibrated position-energy matrix in the same case of Figure5.8. The theoretical kinematic linear loci for two different reactions are alsoshown. There is a good agreement.

identification of energy-loss functions (usually using one or more parameters)along the whole particle path and often depending on angles (see Figure5.11). In particular, it was conventionally assumed that reactions take placeat half target. Concerning 13C(α,n)16O, I calculated that deuteron can loseup to 1.8 MeV before arriving in PSD1, PSD2 or PSD3 while 16O loses atmost 900 keV. Finally, in Figure 5.10 the calibrated angle-energy matrix isplotted for PSD5.

5.9 Data Analysis and future work.

In the last section I introduced the notion of kinematical linear loci corre-sponding to a two-body reaction in the final state, confining the study to asingle detector or equivalently to a single matrix. At this point, I want tocheck instead coincidence events: a detector (PSD1, PSD2 or PSD3) detectsan outgoing particle, while a second one is detected by other detectors. Inorder to show it, in Figure 5.12 I show several points of different colors rep-resenting the theoretical tracks followed by particles in the angular range ofdetectors 1, 2 and 3 without any condition about the other outgoing parti-cles. On the contrary, Figure 5.13 shows the same cases presented in Figure5.12 but imposing the constraint that the other outgoing particle has to bedetected by PSD4 or PSD5.

In particular, with reference to Figure 5.14, (where the matrix represents

73


(MeV)AlDE

-210 -110 1

(MeV

)PS

D1E

-210

m di Al in PSD1µEnergy loss of d in 0.2

Fit function

[1]+[9]*exp([7]+[4]x))2([3]+[5]*x

)[6]

[2](1-exp([8]-[0]x)

/ ndf = 46.96 / 392χp0 0.02484± -4.652

p1 0.001597± 0.3429

p2 0.0002334± 0.01781

p3 0.01843± 0.5677

p4 0.1651± -3.691

p5 0.06253± 1.571

p6 0.02968± -3.829

p7 0.01945± 2.145

p8 0.001969± 0.04317

p9 0.002717± -0.0623

Figure 5.11: Energy loss function when a deuteron particle passes throughthe aluminium dead layer of PSD1 (thickness = 0.2µm). The analytic ex-pression with all parameters is also shown. This is angular independent, sothat it is the same for PSD2 and PSD3.

in the x-axis the energy and in the y-axis the position angle) I identifiedtwo linear loci as an example. In the upper panel of Figure there are thetwo tracks chosen, corresponding to the reaction 6Li + 13C→17O + d asdetected by PSD3. Theoretical (black) points are well in agreement with theexperimental data. In the lower panel, instead, I note that the correspondingtracks of the same reaction are at very low energies, where, because of noisesources and of difficult angular calibration, it is impossible to see clear tracksin the data. However the expected (theoretical) points fall inside the areaof the experimental data so that we are confident of the agreement.

After the calibration of the detectors, the next step of the data anal-ysis is the selection of the events corresponding to the process of interest:the 13C(6Li,n16O)d. This is accomplished first through a selection of thedeuteron locus in the dE-E two-dimension plot, as shown in Figure 5.15,where only hydrogen and helium loci are presented because heavier particlesare unable to pass the dE thickness. The expression of the average energyloss per unit length of charged particles other than electrons is known as theBethe-Bloch equation (?). I give here an approximate expression, which isenough for our purposes. If z is the charge of the particle, ρ the density ofthe medium, Z its atomic number and A its atomic mass, the equation is:

−dEdx

∝ ρZz2

A(5.35)

74


(MeV)reactionE0 5 10

(deg

ree)

θ

10

20

30

40

Two-body kinematics

PSD

1PS

D2

PSD

3

He4N+15

H3O+16

H2O+17

O+H18

Figure 5.12: Two-body kinematical calculation in the region of detector 1,2 and 3.

When a high-energy charged particle or a photon passes through matter,it loses energy that excites and ionizes the molecules of the material. Theenergy loss of relativistic charged particles more massive than electrons pass-ing through matter is due to its interaction with the atomic electrons. Theprocess results in a trail of ion-electron pairs along the path of the particle.In this context, particles with different charge z follow well-separated pathsand they can be identified and selected by a graphical cut as done in Figure5.15. Theoretically, one should be able to discriminate also the differentisotopes corresponding to a same charge value, but since the dependenceof dE/E on the atomic mass number A is only linear, this is usually notpossible.

The Q-value spectrum for the three-body reaction for the coincidenceevents is also a good variable to identify the right exit channel (where Q fotthe three body final state is given by the relation Q = E1+E2+E3−Ebeam).A well separated peak, usually of Gaussian form, has to be centered aroundthe theoretical value of 0.74128 MeV for each pair of detectors. The goodagreement between the experimental and the theoretical Q values confirmsthe identification of the reaction channel as well as the accuracy of thecalibration. Once the three-body reaction is selected, all the variables ofinterest can be calculated in order to perform the following steps of theanalysis and to be compared with the theoretical-simulated ones.

In particular, coincidence events are plotted as a function of relativeenergy Ec.m.

Ec.m. = E13C−α −Q2B (5.36)

75



(deg

ree)

θ

10

20

30

40

Two-body kinematics - Coincidences with PSD4 and PSD5

PSD

1PS

D2

PSD

3

He4N+15

H3O+16

H2O+17

O+H18

Figure 5.13: Two-body kinematical calculation in the region of detector 1,2 and 3, assuming coincidence events with PSD4 or PSD5.

restricting by a condition of a low spectator momentum (ks ≤ 40 MeV/c)and representing predominantly the case of a quasi-free process. Q2B is theQ-value for the two-body reaction. This energy spectrum represents thethree-body excitation functions and it will be used for the extraction of theS factor through the already discussed equation:

(

dσc.m.

dΩ

)

=d3σ

dEcdΩcdΩC

(

KF |Φ(ps)|2)−1

(5.37)

At this point of the analysis, an observable which turns out to be more sensi-tive to the reaction mechanism is the shape of the experimental momentumdistribution, usually expressed in arbitrary units.

A Monte Carlo calculation was then performed to extract theKF |Φ(ks)|2product. The momentum distribution entering the calculation is the ”Bakhadirfunction”, which describes the momentum behavior of a deuteron inside anα particle (Pizzone et al., 2009). Following the PWIA prescription, the two-body cross-section dσ/dΩc.m. was derived dividing the selected three-bodycoincidence yield by the result of the Monte Carlo calculation and usingequation (5.37). As already mentioned, since this approach provides theoff-energy-shell two-body cross section, it is necessary to perform the appro-priate validity tests for the adopted impulse approximation. Since the nextstep in the TH analysis provides for the normalization and then the compar-ison with the direct data, the effect of penetrability through the Coulomb

76



(deg

ree)

4θ

20

25

30

1

100 5 10

(deg

ree)

3θ

35

40

45

1

10

Two-body Kinematics - Coincidences PSD4 and PSD3

Figure 5.14: Coincidence events for the 6Li+13C→17O+d reaction. Theupper panel shows the matrix concerning PSD3 with two well evident kine-matical linear loci corresponding to the 17O+d exit channel. The agreementis good between theoretical points and experimental data. The lower panelis the PSD5 matrix, where the same reaction is plotted.

barrier must be introduced, calculating the penetrability Gl expressed as:

Gl(k13C−αR) =1

F 2l (k13C−αR) +H2

l (k13C−αR)(5.38)

where Fl and Hl are the regular and irregular Coulomb wave functions,while k13C−α and R are the relative wave number and the interaction radius,respectively (Spitaleri et al., 2001). Since equation (5.38) depends on thepartial waves involved in the behavior of the cross section and since theexcitation function can be actually expressed in terms of a coupling betweena non resonant and a resonant term, the penetrability and the relative weightof such contributions must be taken correctly into account. The extractedexcitation function is then calculated in form of the total astrophysical S-factor by equation (4.9).

Moreover, as already discussed in chapter 4, the Trojan Horse methodoffers the possibility to measure directly the bare nucleus astrophysical Sfactor and, by comparing the directly measured (screened) rate and thebare nucleus rate by THM, one can evaluate the screening potential Ue,following the relation (4.21).

The complete procedure described in this section is now under way. Wehave very good expectations for the results of the 13C(α,n)16O cross section,but the work is too long to be covered completely by a single Master thesis.

77


(ch)3DE0 1000 2000 3000 4000

(ch

)3

E

0

1000

2000

3000

4000

5

10

15

20

25

30

35

40

45

50

Graphical cut

3 vs E3DE

Figure 5.15: dE/E two-dimensional plot for the telescope at the PSD3position. The upper locus shows the charged particles with z=2 and thelower one is the region populated by z=1 particles where I expect to finddeuterons. The graphical cut shows the data used in the analysis.

For this reason the astrophysical consequences will be addressed in the nextChapter on the basis of a general overview of the possible changes in therate we can expect, rather than on the basis of actual data derived from ourmeasurements.

78

CHAPTER

SIX

ON THE ASTROPHYSICAL CONSEQUENCES OFCHANGES IN THE 13C(α,N)16O RATE.

6.1 General remarks

Although the data reduction of the measurements presented in this thesisis not yet complete (for indirect methods, as discussed previously, it is par-ticularly long and critical) one knows qualitatively, a priori, the merits andlimits of the indirect method adopted, hence the uncertainties that mightaffect the results.

In particular, on one side we hope that our experimental contributionwill permit a clear and unambiguous determination of the reaction rateratio at different (low) energies, in the region so far precluded; on the other,uncertainties will remain on the absolute normalization of the rate.

This can be understood with reference to Figure 5.2. In the figure, theexisting measurements are reported, down to energies of about 280 KeV.Our estimates will cover the lower energy range, below this value and acrossthe Gamow peak, which is what is really needed for stellar nucleosynthesis.In Figure 5.2 the efficiency of the reaction in this useful range had to beextrapolated theoretically and still waits for an experimental verification. Asour results provide relative measurements, we can obtain such a verificationfor what concerns the shape of the curve. This is of crucial importance, astheoretical extrapolations are very ambiguous, depending on the strengthand width of low-energy resonances, especially on sub-threshold ones. Asalready mentioned, in our case most of the uncertainties in the theoreticalestimates, which have raised so many controversies in the literature, descendfrom the presences of a resonance at -3KeV (in the center-of-mass energyscale), corresponding to an excited level of 17O at 6.356MeV. This is thefield in which our data will provide decisive clarifications.

For the absolute calibration, instead, we shall necessarily rely on thedata at high energies. According to the discussion of Chapter 5 this means

79

6.2. Effects of reducing the rate by a factor of three.

that we have actually repeated some measurements above 280 keV, whichwe shall over-impose to older values for obtaining an absolute scaling of ourmeasured energy dependence.

The problem in doing this normalization is that the uncertainties in therange from a few hundred keV to a few MeV are still essentially at the levelshown in the Figure 5.2 (up to a factor-of-three, at the 2σ level). This will betherefore also the expected uncertainty of our normalization. Now, for thenucleosynthesis of neutron-capture elements for which the neutron flux isgenerated by the 13C(α,n)16O reaction in He-burning conditions of evolvedstars, we are interested not only in knowing the ratio of the rate at variousenergies (all included in a small range below and above 10 keV), but also theabsolute value at each energy and especially at about 8 keV, which is thetypical temperature achieved, when shell-H burning restarts, immediatelybelow the stellar layer previously swept by the third dredge up (see Chapter3 for a discussion).

After our final data are available, further work will be needed, in a closecollaboration between stellar astrophysics and nuclear physics, for derivinga more precise absolute calibration from observational constraints. I havetherefore decided to anticipate here the basics of this work in a series of tests,performed with the help of the neutron-capture nucleosynthesis code of theastrophysics group operating the Department of Physics of the Universityof Perugia. For this scope I have personally modified that code to allow forreaction rate changes.

What I shall present in the next sections is therefore a discussion of thepossible expected effects of variations of this rate within a factor-of-threearound the value most commonly used in the calculations (i.e. the one fromDrotleff et al., 1993). In particular, in section 6.2 I shall discuss the effects ofreducing the rate by this factor, while in section 6.3 I shall consider the oppo-site, i.e. the effects of an increased rate. This series of tests will prepare thefuture work in the astrophysical field, mainly based on comparisons betweennucleosynthesis models using the new rate and observations of abundancesin both the solar system and AGB stars.

6.2 Effects of reducing the rate by a factor of three.

In case of a reduction of the rate with respect to the one so far adoptedin most s-process calculations, (i.e. the one from Drotleff et al., 1993), wecan expect two types of changes in neutron-capture nucleosynthesis calcu-lations. The first possible effect, whose consequences could be a priori themost dramatic, would be that of allowing part of the 13C nuclei available tosurvive the radiative interpulse stage and burn then convectively, at highertemperature, during the thermal instability. If the amount of 13C survivedis sufficiently large, we might expect that a remarkable amount of energy is

80


Pulse number Time scale for 13C combustion Duration of the interpulse stage

24 27261 yr 30500 yr

25 27890 yr 28780 yr

26 27261 yr 27580 yr

27 27890 yr 26270 yr

Table 6.1: Comparison between the time of combustion of 13C in radiativeconditions in the intershell and the one of interpulse for a star of 3 M⊙ andZ=0.006. Only the last four pulses are shown in table. It can be noted thatfor the last two pulses there is a certain amount of carbon (5× 10−7), whichremains unburnt in the radiative region and can burn during the convectiveinstability.

deposited in the convective layer. In this case stellar evolution models teachus that the convective shell might undergo a splitting in two sublayers, sep-arated from a radiative zone. If this occurs than the consequence is that theinner region, undergoing further α-capture burning and the activation ofthe 22Ne(α,n)25Mg source, and containing all the newly produced s-processnuclei, would remain separated by the upper layers where the next TDUepisode can occur. This would have the effect of preventing the pollutionof the envelope with s-process elements, with the abortion of the thermally-pulsing nucleosynthesis mechanism.

This possibility was verified with the help of the FRANEC evolution-ary code, whose use was granted by Sergio Cristallo, at the Observatory ofTeramo (INAF). We here thank him and his collaborators for this possibility.

By reducing the present 13C(α,n)16O rate by a factor of three we foundthat the time scale for 13C burning down to 1/100th of its initial abundancepasses from about 18000 years to about 27000 years (see Table 6.1), henceat least in the more massive stars of the LMS range, i.e. around 3 M⊙,and in the final stages (where the interpulse duration is relatively short)13C would actually end up burning partly in the thermal pulse. In ourtests this always occurred when its abundance had already been reducedremarkably, but independently of the amount entered into the convectiveregion, at the typical values of the temperature (1.5×108 K) and of thedensity (ρ = 104 g/cm3) and for the local abundance of helium (0.7), thetime scale for 13C burning is:

τburn = X(He)ρNA < σv >13,α= 4.5× 105sec (6.1)

(here the brackets indicate Maxwellian-averaging of the reaction rate). Bycontrast, the time required to carry 13C to the bottom of the convectivelayer, where the temperature is high (up to 3× 108 K) is:

τmix = ∆R/vconv = 495sec (6.2)

81


where ∆R (= 1.8×10−3 R⊙) is the distance to the bottom and vconv (=2.5×105 cm/sec) is the average convective velocity. In such conditions vir-tually any amount of 13C entering the pulse will burn only after reachingthe bottom and no shell splitting can occur. I conclude that reducing the13C rate has no effect on the development of the convective instabilities.Moreover, since the abundance left is very small, also the effects on theneutron density and on the s-element distribution are bound to be minimal.

A second important effect that can be expected concerns instead thelowest masses of our range (those with M in the range 1.2 − 1.4 M⊙).Here the temperature in the thermal pulses is insufficient to ignite the22Ne(α,n)25Mg reaction, so that the neutron density is limited to the low val-ues generated during radiative 13C burning. Reducing the rate for this burn-ing means also reducing the total neutron density. On most nuclei this willhave marginal effects, as the values commonly found with the Drotleff et al.(1993) rate were already very low (107 n/cm3). However, it is a priori possi-ble that for some special cases the further reduction of the neutron densitydue to the lower rate can be felt. We verified these effects with the al-ready mentioned neutron-capture code available to our group (Busso et al.,1999). In the mass range below 1.5 M⊙ we also applied our recent results(Maiorca et al., 2011a,b), where it was shown that the 13C-rich layer formedin such stars is much larger than for higher masses, due to the reverse de-pendence of the efficiency of proton mixing phenomena on the initial stellarmass. In particular, choosing as an example a star of 1.3 M⊙, with a metalcontent 1/3 solar (Z = 0.006) Figures 6.1 and 6.2 show the different efficien-cies of the old and new models in producing s elements, when the rate of the13C(α,n)16O reaction is left unchanged. The increased efficiency is evident.The difference in the assumptions for the 13C pocket between our currentmodels and previous calculations are summarized in Table 6.2.

Travaglio et al. (1999) Maiorca et al. (2011b)

Region Depth X(13C) Depth X(13C)

1 4.0 ×10−4 2.0× 10−3 2× 10−3 5× 10−3

2 5.3 ×10−4 4.25 × 10−3 2× 10−3 3× 10−3

Table 6.2: 13C pocket in the case of Travaglio et al. (1999) andMaiorca et al. (2011b) respectively.

Using the new models illustrated in Figure 6.2, we allowed the rate to de-crease by a factor of three, which roughly means to reproduce the suggestionsby Kubono et al. (2003). The results obtained for s process abundances inthe mentioned AGB star of 1.3 M⊙ and Z = 0.006 when the rate is reduced

82


Atomic Mass (A)50 100 150 200

)i/Xi

Log

(X

0

1

2

3

and Z=0.006s-process elements for M=1.3M

88

Legendr-only> 1%>20%>40%>60%>80%s-only

Figure 6.1: The distribution of production factors with respect to theinitial composition for elements above the iron-peak, for an AGB star of 1.3M⊙ with a metallicity about one third the solar one, undergoing neutroncapture nucleosynthesis with neutrons produced by the 13C(α,n)16O neutronsource. Nuclei whose production is attributed to the s process at variouspercentage levels are indicated by different symbols and colors, as describedin the label. Here the amount of 13C burnt per cycle is the same as inTravaglio et al. (1999).

in this way are shown in Figure 6.3 (lower panel) in terms of abundance ra-tios with respect to the results obtained with the presently-accepted rate. Inthe top panel the same ratio is shown for the old choice of the 13C-pocket. Itis clear that the only remarkable changes concern nuclei strongly affected byreaction branchings depending on the neutron density, like 96Zr and, in par-ticular, 86Kr and 87Rb. These last isotopes are close to the neutron-magicnumberN=50 (i.e. A= 88) and their abundances are drastically reduced, byroughly 35%. These are crucial nuclei, at the connection between the mainand weak s-process component, related to the complex reaction branchingat 85Kr illustrated in Figure 3.3. The case explored corresponds to such lowneutron densities (nn = 1×107 n/cm3, against nn = 1.3×107 n/cm3 whenusing the Drotleff et al. (1993) cross section) that the flux through 85Krpasses almost completely through the 85Rb-branch, so that the two men-tioned nuclei are not fed efficiently. As the abundances of these nuclei, andin particular 87Rb, are used as tests for the neutron density in current stellarobservations (see e.g. Abia et al. 2001), this effect would be very important.If the new measurements will point in the direction now explored, a moredetailed analysis should be done, considering also the possible activation of

83

6.3. Effects of increasing the rate by a factor of three.

Atomic Mass (A)50 100 150 200

)i/Xi

Log

(X

0

1

2

3

4

5 and Z=0.006 - News-process elements for M=1.3M

88

Legendr-only> 1%>20%>40%>60%>80%s-only

Figure 6.2: Same as Figure 6.1, but adopting the increased amount of13C burn per cycle suggested by our group in the paper Maiorca et al.(2011b) (Science, submitted).

the further neutron source 18O(α,n)21Ne, whose rate is uncertain and whoseactivation might become a proxy for the 22Ne(α,n)25Mg neutron source, notactivated because of the low temperature.

We can therefore conclude this section by saying that the most remark-able effect of a reduction of the 13C(α,n)16O rate would be seen in very lowmasses, and would affect the nuclei at the overlapping between the weakand the main component, modifying our ideas on the meaning of the obser-vational neutron-density tests for AGB stars.

6.3 Effects of increasing the rate by a factor ofthree.

If one artificially increases the rate for the 13C(α,n)16O reaction, 13C burnsmore efficiently and the neutrons are released in a shorter time interval,thus increasing the neutron density nn. As already mentioned, however, theneutron density due to the radiative 13C burning is much smaller than theone subsequently expected by the operation of the 22Ne(α,n)25Mg neutronsource in the convective thermal pulse. As a consequence, an increase ofnn in the radiative phase has only marginal effects on the ensuing elementdistribution. Measurable consequences are therefore limited, once again,to very low masses, where the 22Ne(α,n)25Mg reaction is not activated forthe too low ambient temperature. In population I stars (the stars of the

84


Atomic Mass (A)50 100 150 200

Frac

tion

0.6

0.7

0.8

0.9

1

1.1 and Z=0.006 - old(rate Drot/3)/(rate Drot) for M=1.3M

88

Atomic Mass (A)50 100 150 200

Frac

tion

0.6

0.7

0.8

0.9

1

1.1 and Z=0.006 - new(rate Drot/3)/(rate Drot) for M=1.3M

88

Figure 6.3: Ratios of the abundances for heavy elements obtained by re-ducing the 13C(α,n)16O reaction rate by a factor of three, with respect tothose shown in Figures 6.1 and 6.2, obtained with the rate by Drotleff et al.(1993). The upper panel is for the 13C reservoir by Travaglio et al. (1999),the lower panel for the choice by Maiorca et al. (2011b).

galactic disk) this corresponds to stars below 1.3 − 1.4 M⊙. Moreover,all the AGB stars presently observed in population II stellar systems (e.g.Globular Clusters, of low metallicity) should share this property, being of amass generally lower than solar.

An example of the effects induced in such low masses by an increase bya factor of three of the rate by Drotleff et al. (1993) is shown in Figure 6.4.An in Figure 6.3, the bottom panel shows the case of the new (extended)13C reservoir, the top panel that of the previously-accepted choice. For thissecond case the nuclei affected (all produced more efficiently) are mainly86Kr and 87Rb, which experience a change opposite to what was seen inthe previous section, with an increase between 25 and 30%. A few otherbranching-dependent nuclei show enhancements at a more limited level (upto 10%): they include 96Zr, 122Sn, 123Sb and 142Ce. The chart of the nuclidesaround these last three isotopes is shown in Figure 6.5 in order to illustratethe reason of the change. As the two panels show, the nuclei under analysisare always placed after an unstable isotope and the increase of the neutrondensity favors their production.

For the more recent choice of the 13C-pocket the nuclei affected are thesame but the changes are more remarkable. In particular, 86Kr, 87Rb, and142Ce show an increase by at least 30%. All these changes would be veryimportant in the understanding of the solar distribution of neutron-capture

85


Atomic Mass (A)50 100 150 200

Frac

tion

0.9

1

1.1

1.2

1.3

1.4 and Z=0.006 - old(rate Drot*3)/(rate Drot) for M=1.3M

88

Atomic Mass (A)50 100 150 200

Frac

tion

0.9

1

1.1

1.2

1.3

1.4 and Z=0.006 - new(rate Drot*3)/(rate Drot) for M=1.3M

88

Figure 6.4: Ratios of the abundances for heavy elements obtained by in-creasing the 13C(α,n)16O reaction rate by a factor of three, with respect tothose shown in Figures 6.1 and 6.2, obtained with the rate by Drotleff et al.(1993). The upper panel is for the 13C reservoir by Travaglio et al. (1999),the lower panel for the choice by Maiorca et al. (2011b).

nuclei. They would again modify our ideas on the neutron-density sensitiveobservational tests (like those based on the Rb/Sr ratio) and would be crit-ical in deducing predictions for the percentage of each nucleus that must beattributed to the r-process.

86


Figure 6.5: The reaction branchings involving tin and antimonium isotopes(left panel) and Ce isotopes (right). 122Sn, 123Sb, and 142Ce are placed afteran unstable nucleus and their abundance is a function of the neutron density,i.e. of the competition exerted by neutrons against β− decays along the s-process chain.

87


88

CHAPTER

SEVEN

CONCLUSIONS

This thesis was primarily dedicated to the new measurement, obtained withthe indirect method usually called ”of the Troian Horse” (THM), of the re-action rate for the reaction 13C(α,n)16O. The measurement wants to explorevery low energies (below 280 keV), not covered by traditional measurementsbut very important in stellar interiors.

In order to clarify the importance of the reaction chosen I outlined thephases of stellar evolution during which its activation is important and dis-cussed the processes of slow neutron capture that are started by the neutronsthat this reaction makes available.

I subsequently presented the idea (and an outline of the Quantum-Mechanics treatment) for the THM, based on a two-body reaction inducedat low energy by a virtual particle produced in a direct three-body reactionoccurring at higher energies. I also discussed why this technique is so im-portant for exploring the range in energy across the Gamow peak, which isof interest in stars.

I then illustrated my activity (in progress) on the long and complexdata reduction, which will be completed in about four-five months after thediscussion of this dissertation.

In order to know the possible astrophysical consequences of the measure-ment and prepare in advance the theoretical and observational tests that willbe required, I performed a parametric study, by varying the cross sectionaccepted today by a factor-of-three (in both directions) and performed s-process nucleosynthesis calculations putting in evidence the effects inducedby changes in the rate. In this way I identified the basic consequences thatcan be expected (concentrated either on nuclei at the overlapping of themain and weak s-process components, or near reaction branchings sensitiveto the neutron density).

During the mentioned tests I participated to the preparation of a pa-per containing several new ideas on s-process nucleosynthesis, which is now

89

undergoing referee’s scrutiny by the SCIENCE journal for publication.

90

LIST OF FIGURES

2.1 H-R diagram of a star of 1 M⊙ and Z=Z⊙. . . . . . . . . . . 132.2 Comparison of energy produced by pp-chain and CNO cycle. 14

2.3 Stellar structure of a star in the TP-AGB phase. . . . . . . . 162.4 Illustration of the structure of a TP-AGB star over time. . . 18

2.5 Observations of ls/Fe with respect to hs/Fe . . . . . . . . . . 23

3.1 Valley of β-stability. . . . . . . . . . . . . . . . . . . . . . . . 263.2 Behaviour of 〈σ(A)N(A)〉 as a function of mass number. . . . 28

3.3 The complex branching of 85Kr. . . . . . . . . . . . . . . . . . 303.4 Internal structure of a TP-AGB star as a function of time. . . 33

3.5 Successive thermal pulses for the 3 M⊙ model with Z = Z⊙. . 36

3.6 Schematic representation of the thermal pulse history. . . . . 37

4.1 Schematic representation of the total nuclear potential. . . . . 41

4.2 Cross section and astrophysical factor. . . . . . . . . . . . . . 434.3 The Gamow peak. . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Sub-threshold resonance. . . . . . . . . . . . . . . . . . . . . . 474.5 Representation of the potential between charged particles. . . 48

5.1 Pseudo-Feynman diagram for the break-up QF process. . . . 53

5.2 Behavior of the astrophysical S-factor in NACRE. . . . . . . 595.3 Pseudo-Feynman diagram for the 13C(6Li,n16O)d reaction. . . 63

5.4 Sequential processes for the interaction between 6Li and 13C. 655.5 Experimental setup of the 13C(6Li,n16O)d reaction. . . . . . . 66

5.6 Schematic view of a position-sensitive detector (PSD). . . . . 68

5.7 Schematic draw of a PSD and of its holder. . . . . . . . . . . 695.8 Position-energy two-dimension matrix. . . . . . . . . . . . . . 70

5.9 Energy and position spectrum for a singular slit. . . . . . . . 715.10 Calibrated position-energy matrix. . . . . . . . . . . . . . . . 73

5.11 Energy loss function of a deuteron in 0.2µm of Al. . . . . . . 745.12 Two-body kinematical calculation for detectors 1, 2 and 3. . . 75

5.13 Two-body kinematical calculation - coincidence events. . . . . 76

91

List of Figures

5.14 Coincidence events for the 6Li+13C→17O+d reaction. . . . . 775.15 dE/E two-dimensional plot for the telescope at PSD3 position. 78

6.1 Overabundances of s-elements for a star of 1.3M⊙and Z=0.006 836.2 Same as Figure 6.1 adopting Maiorca et al. (2011b)13C-pocket. 846.3 Reduced 13C(α,n)16O reaction rate - ratios of s-elements. . . 856.4 Increased 13C(α,n)16O reaction rate - ratios of s-elements. . . 866.5 Reaction branchings. . . . . . . . . . . . . . . . . . . . . . . . 87

A.1 Overview of the p-pI chain. . . . . . . . . . . . . . . . . . . . 104A.2 Overview of the CNO cycle. . . . . . . . . . . . . . . . . . . . 107A.3 Overview of the triple-alpha process. . . . . . . . . . . . . . . 107

92

LIST OF TABLES

5.1 Table of reaction rates for 13C(α,n)16O reaction. . . . . . . . 615.2 Experimental conditions for the 13C(6Li,n16O)d experiment. . 67

6.1 Comparison between 13C-burning time and time of interpulse 816.2 Old and new 13C pocket . . . . . . . . . . . . . . . . . . . . . 82

93

List of Tables

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100

CHAPTER

EIGHT

RINGRAZIAMENTI.

Se dovessi ringraziare adeguatamente tutte le persone che hanno contribuitoalla realizzazione di questa tesi di laurea, con tutta probabilita questa sezionesarebbe piu lunga di tutto il resto. Il primo pensiero va ai miei genitori per-che, se sono la persona che sono diventato, lo devo a loro. Piera e Umbertosono persone semplici e oneste e li ringrazio per la liberta che mi hanno sem-pre concesso nelle scelte e per l’amore incondizionato con cui mi ricoprono.

Un pensiero speciale va ai due relatori, Prof. Maurizio Busso e Prof.Claudio Spitaleri, due persone tanto diverse quanto simili per l’amore perquella disciplina meravigliosa che e la fisica. Grazie alla loro intraprendenzae stato infatti possibile realizzare un progetto decennale di congiunzione trai due gruppi, al quale sono stato onoratissimo di prendere parte. Spero sola-mente di essere stato all’altezza delle loro aspettative. Da loro ho imparatodavvero tanto: dall’infinitamente grande all’infinitamente piccolo.

Questa tesi non sarebbe mai venuta a compimento senza la collabo-razione di tante persone. Sara Palmerini e Enrico Maiorca hanno fattoveramente tanto per me mantenendosi sempre disponibili, dandomi semprei consigli giusti e insegnandomi tutto il possibile sull’astrofisica. Un grazieparticolare va a Marco La Cognata, del quale ho una stima immensa, perl’infinito aiuto e la pazienza concessami fin dal primo giorno nell’iniziarmialla fisica nucleare, che non e poi cosı male.

A Catania, all’ombra di Liotru, ho avuto anche l’onore di conoscereIolanda e Luca, miei compagni impagabili di tesi, Livio e Roberta, per-sone talmente gentili e disponibili che, malgrado una distanza di quasi millechilometri, mi hanno fatto sentire subito a casa. Tra Florida, Trojan Horse ecultura siciliana ho tante cose per cui esservi grato. Voglio inoltre ringraziarei restanti componenti del gruppo ASFIN2 cominciando dal Prof. Stefano Ro-mano e Aurora Tumino che mi hanno gentilmente offerto accoglienza nel loroufficio e continuando con Giuseppe, Letizia, Gianluca e Gabor. Se sono unpo sperimentale lo devo a tutti voi! Ovviamente, un caro saluto va anche a

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tutti i ragazzi del Laboratorio Nazionale del Sud.Tengo comunque a precisare che nulla sarebbe stato possibile senza il

supporto della mia famiglia che ha sempre avuto fiducia in me e nelle mieidee, dandomi la possibilita di raggiungere questo importante traguardo.

Mi ritengo, inoltre, un ragazzo davvero fortunato perche nel corso deglianni ho potuto contare su amici sinceri con i quali ne ho ”passate davverotante” e che, tra momenti belli e meno belli, mi sono sempre stati vicini.Voglio davvero bene a Simone, Matteo, i miei insostituibili compagni ditanti anni di scuola, Ale, Alessandro, Renzo, Fizia, Matteo e Riccardo.

Infine, voglio dedicare questa tesi di laurea alla mia dolce meta Alessia,che mi ha rubato il cuore e che e stata l’oggetto di ogni mio pensiero. Grazieper aver saputo aspettare e superare il passato: ”tu per me sei sempre l’unica,straordinaria, normalissima”. Ti amo con tutto me stesso e voglio che tu siail mio presente e il mio futuro.

Potrei davvero continuare all’infinito, ma con le lacrime agli occhi e ilmomento di concludere con un semplice GRAZIE.

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APPENDIX

A

MAIN THERMONUCLEAR REACTIONS IN PRE-AGBPHASES.

A.1 Hydrogen (H) burning.

As already discussed in previous chapters, thermonuclear reactions can occuronly if the temperature (or equivalently the kinetic energy) of the particlesis high enough to overcome their mutual electrostatic or Coulomb repulsion.For this reason and because of the large amount of hydrogen in the sunand in the universe, the first and most important nuclear reactions releasingenergy are those involving protons (?). This idea, coupled with the discoveryof the tunneling effect, was presented and discussed across the thirties andfourties. Atkinson and Houtermans were the first to suggest that, out offour protons and two electrons, a helium nucleus could be produced withthe release of large amount of energy (QTOT = 26.73 MeV). Starting fromBethe (1939), it was clear that two different sets of reactions could convertsufficient hydrogen into helium, to provide the energy needed for a star’sluminosity for the greater part of its life: the so-called proton-proton (p-p)chain and the CNO cycle.

A.1.1 pp-Chain.

The first step involves the fusion of two hydrogen nuclei H (protons) intodeuterium, releasing a positron and a neutrino, as one of the protons changesinto a neutron

H +H →2 H + e+ + νe (A.1)

This reaction provides 1.44 MeV of energy, if I consider that Q is the totalenergy released in the process including the subsequent annihilation of theemitted positron. A temperature of ten million degree, and equivalentlya stellar mass of about 0.8 M⊙, are needed in order to activate the (A.1)reaction. Since this process involves a weak interaction the cross section is

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A.1. Hydrogen (H) burning.

Figure A.1: Overview of the p-pI chain.

very small and the reaction is the slowest of the chain, so only a theoreticalvalue is available. After this, the deuteron produced in the first stage canfuse with another hydrogen to produce the lighter isotope of helium, 3He

2H +H →3 H + γ (A.2)

In order to created 4He, the newly formed 3He can be consumed by a numberof exothermic reactions through three different paths.

The first one takes place at low temperatures (less than 15× 106 K) andproceeds predominantly by the following fusion reaction

3He+3 He→4 He+ 2H (A.3)

This is the so-called p-pI chain(see Figure A.1). At this point the net result isthe fusion of four protons into an α particle, two positrons and two electronicneutrinos. (A.3) is considered as the crucial reaction also for driving aninversion of the molecular weight (µ), promoting readjustments in the star,hence mixing. In fact, it provides a reduction of the mean molecular weight.

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A.1. Hydrogen (H) burning.

In order to activate the p-pII chain a preliminary presence of 4He and atemperature included in the range 15 − 23 × 106K are necessary. The firstreaction of this process creates 7Be as follows:

3He+4 He→7 Be+ γ (A.4)

Then, 7Be decays to 7Li by capturing an electron from its own K shell (or,alternatively, from the stellar plasma):

7Be+ e− →7 Li+ ν (A.5)

and, after a proton capture, two nuclei of 4He are finally produced:

7Li+H →4 He+4 He+ γ (A.6)

The set of reactions from (A.7) to (A.10) is the so called p-pIII chain:

3He+4 He→7 Be+ γ (A.7)

7Be+ p→8 B + γ (A.8)

8B →8 Be+ e− + νe (A.9)

8Be→4 He+4 He+ γ (A.10)

The p-pIII chain is dominant if temperatures exceed 23 × 106K. It has anegligible importance from the energy-production point of view, especiallyin the Sun (0.11%), but is an important source for the solar neutrinos.

A.1.2 CNO-cycle.

The p-p chain is the main channel for 4He synthesis in the ancient stellarobjects, made of pure H and He, but for higher metallicity stars, formed froman ISM enriched in carbon (C), nitrogen (N), oxygen (O), other reactionscan contribute to the nuclear energy production on the Main Sequence. In1939, Bethe proposed the independent set of reactions called CNO cycle.In order to produce 4He starting from four protons, carbon, nitrogen, andoxygen nuclei are considered as catalysts: their individual abundances canchange, but not their sum and they are linked by an endless loop.

The CNO chain starts occurring at approximately 13 × 106 K, but itsenergy output rises much faster with increasing temperatures (see Figure2.1). At approximately 17 × 106 K, the CNO cycle becomes the dominantsource of energy. Hence, it is important especially in stars more massivethan the sun.

A reduced network, called CN cycle (because the only stable isotopesintervening are 12C, 13C, 14N and 15N) occurs for moderate temperatures.It contains the following reactions:

12C + p→13 N + γ (A.11)

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A.2. Helium (He) burning: triple-α process.

13N →13 C + e+ + νe (A.12)

13C + p→14 N + γ (A.13)

14N + p→15 O + γ (A.14)

15O →15 N + e+ + νe (A.15)

15N + p→16 O∗ →12 C + α (A.16)

where the 12C used in the (A.11) reaction is regenerated in the (A.16).The energy production for each reaction cycle is 26.77 MeV, 26.73 from theconversion of H into 4He and the rest from changes in the CNO isotopic mix.

At higher temperatures (higher than 2 × 106K) also 16O takes part inhydrogen burning, so that the cycle can be extended to:

15N + p→16 O∗ →16 O + γ (A.17)

16O + p→17 F + γ (A.18)

17F →17 O + e+ + νe (A.19)

17O + p→18 F ∗ →14 N + α (A.20)

17O + p→18 F ∗ →18 F + γ (A.21)

18F →18 O + e+ + νe (A.22)

This is the full CNO cycle (see Figure A.2), of which CN is only a part. Likethe carbon, nitrogen, and oxygen involved in the main branch, the fluorine(F) produced in the minor branch is merely catalytic and at steady state,does not accumulate in the star. Additional reactions can start from protoncaptures on 18O

18O + p→19 F ∗ →15 N + α (A.23)

18O + p→19 F ∗ →19 F + γ (A.24)

If 2× 107K ≤ T ≤ 7× 108K, the 18O(p,γ)19F rate is not negligible and thecycle is partially broken by the synthesis of an external nucleus of fluorine.

A.2 Helium (He) burning: triple-α process.

After hydrogen burning, helium is the most abundant element in the stellarcore, while the remaining hydrogen continues the combustion in a thin ex-ternal shell. All this happens during RGB phases that depend strongly onthe initial mass of the star (see section 2.2).

Generally speaking, the collapse of the stellar core brings the centraltemperature to near ∼ 100 × 106 K (8.6 keV). At this point helium nucleican fuse together at a rate high enough to rival the rate at which their prod-uct, 8Be, decays into two helium nuclei, so that some equilibrium berylliumremains:

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Figure A.2: Overview of the CNO cycle.

Figure A.3: Overview of the triple-alpha process.

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4He+4 He→8 Be (A.25)

This means that there are always a few 8Be nuclei in the core, which canfuse with yet another helium nucleus to form 12C:

8Be+4 He→12 C + γ (A.26)

This is the so-called triple-α process (see Figure A.3). The net energy releaseof the process is 7.275 MeV.

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