The 13 C(α,n) 16 O reaction rate. Recent estimates, new measurements through the Trojan Horse Method and their astrophysical consequences.

Transcription

1 Università degli studi di Perugia Facoltà di Scienze Matematiche, Fisiche e Naturali Tesi di Laurea Magistrale in Fisica The 13 C(α,n) 16 O reaction rate. Recent estimates, new measurements through the Trojan Horse Method and their astrophysical consequences. Relatore Prof. Busso Maurizio Maria Candidato Trippella Oscar Prof. Spitaleri Claudio Anno Accademico 2010/2011

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3 Università degli studi di Perugia Facoltà di Scienze Matematiche, Fisiche e Naturali Tesi di Laurea Magistrale in Fisica The 13 C(α,n) 16 O reaction rate. Recent estimates, new measurements through the Trojan Horse Method and their astrophysical consequences. Relatore Prof. Busso Maurizio Maria Candidato Oscar Trippella Prof. Spitaleri Claudio Anno Accademico 2010/2011

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5 ...We are stardust, we are golden We are billion year old carbon... Woodstock - Crosby, Stills, Nash and Young 3

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7 CHAPTER ONE INTRODUCTION. Historically, stars have been part of religious practices and used for celestial navigation and orientation: there are examples of astronomical studies all around the world from Egypt to Greece, from the Maya population to the Chinese one. However, it was only due to European researchers during and after the XVIth century that astronomy assumed its modern role as a science. A special role was obviously played by the introduction of the telescope by Galileo Galilei in the XVIIth century; the subsequent search for physical explanations 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. Important conceptual progresses on the physical behaviour of stars occurred during the twentieth century because of new theoretical approaches, the application of modern physics and the advent of more accurate photometric and spectroscopic measurements. During the first decades of the XXth century, results from nuclear physics research, in particular the discovery of the enormous energy stored in the nuclei, led astrophysicists to guess that reactions among nuclear species were the source of the stellar power (Rolf & Rodney, 1988; Eddington et al., 1920). Since then, nuclear astrophysics has played a key role in providing the interpretation of astrophysical observations. In this sense, using the observational evidence coming from stellar atmospheres and the experimental evidence coming from nuclear experiments aimed at studying specific nuclear reactions, nuclear astrophysics can determine how the processes of nuclear fusion drive the structural changes and promotes stellar evolution. This thesis is a particular example of the role played by nuclear astrophysics, as it covers the steps from the nuclear measurement of a reaction rate of astrophysical interest (the 13 C(α,n) 16 O reaction) up to the study of the stellar consequences implied by a reaction rate change. These consequences concern the release of neutrons and the ensuing n-capture nucle- 5

8 osynthesis in low mass stars. The above mentioned reaction is important because it is considered as the dominant neutron source active in stars with a mass included in the range M, which actively contribute to the nucleosynthesis of heavy nuclei through neutron capture processes. Roughly a half of all elements heavier than iron in the universe were produced in this way, in the so-called s (slow) process (Burbidge et al., 1957), which basically includes neutron-induced capture reactions and beta decays. The term slow, used to distinguish this mechanism from a rapid one (r-process, occurring in supernovae), refers to the fact that the neutroncapture timescale is in general longer than for the decay of unstable nuclei, which fact requires typical neutron densities of about n/cm 3. In order to set the stages for the nuclear astrophysics processes of interest, Ishallfirstdiscussthetypicalevolutionaryphasesforastarofonesolarmass (assumed to represent a low mass star in general). A particular emphasis will be dedicated to the Asymptotic Giant Branch (AGB) stage when, after the exhaustion of helium at the center, the representative point in the H-R diagram 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 burning alternatively. There is a helium rich intershell region between the two shells that becomes almost completely convective at intervals, while the temperature suddenly increases: it is the so-called thermal pulse (TP). The thermal pulse is repeated many times (from 5 to 50 cycles) before the envelope is completely eroded by mass loss, so nucleosynthesis products manufactured by He burning and the s-process at its bottom are carried to the surface. In the intershell region 12 C is abundant. The existence, now proven, of mixing episodes carrying protons downward from the envelope yields the formation of a p- and 12 C-rich layer after each thermal pulse. There, after the ignition of the H shell, p-captures generate the so-called 13 C pocket. In this context I shall discuss how neutrons are released thanks to the 13 C(α,n) 16 O reaction and s-processing occurs in AGB stars, in the radiative inter-pulse phases. The typical stellar environment in which our reaction takes place corresponds to T K. In such conditions, the other main neutron source, the 22 Ne(α,n) 25 Mg reaction, is switched off, as it needs higher temperatures to be activated. In the above conditions big problems affecting our knowledge of reaction rates are related to the effects of the Coulomb barrier for the chargedparticle-induced reactions and to electron screening. The presence of the barrier implies an exponential suppression for the cross section and does not allow a direct measurement at the energies of astrophysical interest. Cross section measurements at such low energies must also cope with a low signalto-noise ratio, which can be improved only in underground experimental facilities, such as LUNA at the Gran Sasso National Laboratories. 6

9 At present, existing direct measurements for the reaction 13 C(α,n) 16 O, collected in the NACRE compilation by Angulo et al. (1999), stop at the minimum value of 280 kev (Drotleff et al., 1993), whereas the region of astrophysical interest, the so-called Gamow window, corresponds to 190 ± 90 kev at a temperature of K. Below the limit reached be measurements only a theoretical extrapolation is possible. Various types of approaches have been tried over the years to extend the measurement of the cross section into the region of astrophysical interest. The main aim of these efforts is to improve the accuracy of the measurement, reducing the uncertainty, which sometimes exceeds 300%. The major source of error is the presence of a subthreshold resonance corresponding to the excited state of 17 O (E res = Mev or E c.m. = 3 kev). The most recent works in the literature are oriented towards a substantial lowering of the reaction rate, because it is believed that the role of the resonance mentioned above was overestimated in the past. In this context I participated to a new experiment at the Florida State University, made by the ASFIN2 collaboration (centered at Laboratorio Nazionale del Sud) applying an indirect technique called Trojan Horse Method. The THM is based on a quasi-free break-up process and allows to extract 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 the spectator of the virtual 2-body reaction of interest for astrophysics. Ourexperimentwasperformedbymeasuringthesub-Coulomb 13 C(α,n) 16 OscatteringwithintheinteractionregionviatheTHM,appliedtothe 13 C( 6 Li,n 16 O)dreaction in the quasi-free kinematics regime. However, the final result deriving by the Trojan Horse method is not complete yet, because data analysis is still under development and will be finalized in the next months. Since the result derived from the THM is not yet applicable, it was decided to check what would be the consequences for n-capture nucleosynthesis 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 correspond approximately to the alternative indications by Kubono et al. (2003). I shall show that a result in this direction would imply substantial changes in the operation of the crucial s-process branching at 85 Kr with respect to what is assumed 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

10 by the same factor of 3, noting that the changes would be more widespread over the s-process path and would introduce remarkable changes in our ideas on the solar abundance distribution. These results encourage a deeper study of the 13 C(α,n) 16 O reaction. This thesis would not have been possible without the help of the Dipartimento di Fisica di Perugia and of INFN, in particular of the Laboratori Nazionali del Sud and of the Perugia and Catania Sections. Thanks are due to INFN for providing me with a fellowship covering the expenses of the stages in Catania and in Tallahassee (Florida). 8

11 CONTENTS 1 Introduction. 5 2 Final evolutionary stages for low mass stars pre-agb phases Asymptotic Giant Branch (AGB) stars and Thermal Pulse The third dredge-up Nucleosynthesis and observations for AGB stars s-process nucleosynthesis in AGB stars Introduction The classical analysis of the s process Evolution and nucleosynthesis in the AGB stages The neutron source 13 C(α,n) 16 O Possible future scenarios Cross sections of nuclear reactions at low energies Coulomb barrier and penetration factor Cross section, astrophysical factor and reaction rate Gamow peak Direct measurements and experimental problems Indirect methods for nuclear astrophysics Measure of the 13 C(α,n) 16 O reaction through the THM Theory of the Trojan Horse method Plane Wave Impulse Approximation Current measurement status The Trojan Horse Method applied to the 13 C(α,n) 16 O reaction Experimental setup Position Sensitive Detectors (PSDs) The position calibration Energy calibration Data Analysis and future work

12 CONTENTS 6 On the astrophysical consequences of changes in the 13 C(α,n) 16 O rate General remarks Effects of reducing the rate by a factor of three Effects of increasing the rate by a factor of three Conclusions 89 8 Ringraziamenti. 101 A Main thermonuclear reactions in pre-agb phases. 103 A.1 Hydrogen (H) burning A.1.1 pp-chain A.1.2 CNO-cycle A.2 Helium (He) burning: triple-α process

13 CHAPTER TWO FINAL EVOLUTIONARY STAGES FOR LOW MASS STARS. Stars, like for example the Sun, are gaseous objects that shine of proper light because of thermonuclear fusion reactions occurring in their interior producing electromagnetic energy and neutrinos. They are considered as the forges of universe because the whole set of elements (excluding initial abundances of nuclei lighter than 12 C, which are created during the first minutes after the Big Bang) are produced in stars. The main cause of heating, contraction and density increase in stars is the total gravitational energy of the stellar mass. Generally speaking, the larger is the mass, the higher is the central temperature allowing reactions among heavier elements. Theoretical 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 of a star is less than about M min 0.08 M (M being the so-called solar mass, corresponding to about kg), the temperature is not high enough to start hydrogenburning. Inthis workishall limit my discussion to stars belonging to the mass range M, the so-called Low Mass Stars (hereafter LMS). They experience only hydrogen and helium burning before electron degeneracy in a C-O core stops the proceeding of stellar evolution. Concerning this concept of electron degeneracy, it is the state in which matter has such high values of density ρ and pressure P that electrons become a Fermi condensate, whose pressure effectively stops the slow gravitational contraction of the star, thus preventing the appropriate conditions to start thermonuclear reactions. In practice, particles of mass m p have a very small mean free path l, to the point that they are almost in contact to each other. This means that: l ( ) 1 1/3 = n ( ) 1/3 µmh = ρ 11 ( mp ρ ) 1/3 (2.1)

14 2.1. pre-agb phases. has a numerical value close to the particle dimension, defined by the De Broglie s wavelength: λ = h (2.2) m p v 3k B T where v indicates the thermal velocity v =. Then: m p ( mp ρ ) 1/3 = h m p mp 3k B T (2.3) from which I get ρ: ρ = ρ 1/3 = m5/6 p 3kB T (2.4) h ( ) 3 3kB T T 3/2 m 5/2 p T 3/2 m 5/2 p (2.5) h This is the critical density at which particles begin to degenerate and cannot be described any more by a Maxwell-Boltzmann distribution. Such a critical density is lower when the particle mass is lower: hence, electrons degenerate before atomic nuclei. The occurrence of electron degeneracy depends on the stellar temperature and initial mass, in the sense that lower masses degenerate more easily having a lower internal temperature. Let s briefly discuss the main evolutionary stages of a typical low-mass star making use of a schematic view of the track followed by the stellar representative point in the Hertzsprung-Russell diagram (hereafter H-R diagram). This is a plot reporting the absolute magnitudes or luminosities of stars versus their spectral types or effective temperatures and is a very useful tool, providing important information about stellar structure and evolution. In particular, I shall concentrate on the structure of the so-called asymptotic giant branch (AGB) stars. These stars are climbing for the second time along the red giant branch; here they experience thermal instabilities, or pulses, from the He shell activating on the border of the degenerate C-O core. Following a pulse, AGB stars provide to mix to the surface fresh carbon (which is the main product of incomplete helium burning) and s-process isotopes. 2.1 pre-agb phases. At first, I discuss the pre-agb evolution adopting a typical model of a 1 M star, introducing the required terminology and physics when necessary. For clarity, I present in Figure 2.1 the track followed by the stellar representative point in the H-R diagram. Stars are born from gas clouds in the interstellar medium (ISM) thanks to the gravitational collapse of a massive 12

15 2.1. pre-agb phases. Figure 2.1: Schematic evolution in the H-R diagram of a 1 M stellar model and solar metallicity. All the major evolutionary phases discussed in the text are indicated. The plot reports bolometric magnitude M bol versus effective temperature T eff. fragment of a cloud. The ISM, in the physical conditions just described, is mainly composed of atoms and molecules of hydrogen and heavy elements. Sir James Jeans, in the twenties, laid down the quantitative circumstances allowing a cold gas cloud in the ISM to become gravitationally unstable and to condense into a proto-star. Starting from the Virial theorem and assuming a spherical mass, he deduced the so-called Jeans mass (M J ): ( ) M J = T3/2 n 1/2 (2.6) In equation (2.6) I indicate the cloud temperature with T, while n corresponds to the particle number density in the same zone. The numerical value of the Jeans mass, expressed in grams, depends on temperature and density andintypicalconditionsofinterstellarcloudscorrespondstoabout1000m. Hence, if a cloud is more massive than this critical value the collapse can 13

16 2.1. pre-agb phases. occur. After the gravitational collapse, the representative point of a star in the H-R diagram moves along a line called Hayashi track, from the name of the Japanese physicist who derived it, characterized by heat transport occurring through convention. The luminosity decreases while the surface temperature T eff is almost constant because of the decreasing radius. Then, the representative point moves to a track of increasing temperatures(henyey track), until it stops on the Main Sequence (hereafter MS) that corresponds to reaching central temperatures and densities (T = 10 7 K, ρ = 100 g/cm 3 ) sufficient to start hydrogen fusion. Core hydrogen burning starts on the socalled zero age main sequence (ZAMS) and the star remains near this zone for 80-90% of its life. The main effect is the transformation of four protons into a nucleus of 4 He, with a release of energy of about Q = 26MeV (this roughly corresponds to the Q-value resulting from the chain of reactions, see Appendix A). For initial temperatures lower than about K, corresponding to an initial mass of about 1.3 M, reactions proceed through direct fusions of protons (the so-called pp-chain); for higher temperatures the CNO cycle prevails. This last process needs non-zero initial abundances of carbon, nitrogen and oxygen (CNO), which act as catalysts for the conversion of hydrogen into helium. Figure 2.2 shows the relative efficiency of the 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 temperature, for the pp-chain and the CNO cycle. For stars with M > 1.3 M the CNO cycle prevails in the energy production. The vertical line shows the temperature T 0 at which the energy production is the same for the two mechanisms. interest, during the whole main sequence the stellar structure consists in a 14

17 2.2. Asymptotic Giant Branch (AGB) stars and Thermal Pulse. H-burning core, a large He-rich inert buffer and a relatively thin convective envelope. When, because of hydrogen exhaustion, the nuclear processes fail to contrast the gravitational pressure, the hydrostatic equilibrium is broken and the core starts to contract. At this stage stars leave the main sequence while the central He core becomes electron degenerate and nuclear burning is established in a shell surrounding this core. Simultaneously, the star expands and the outer layers become convective. Convection extends quite deeply 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 recently affected by p-captures and this determines a modification of the chemical abundances; in particular, a significant depletion of 12 C and 15 N and an increase of 4 He, 13 C and 14 N occur. Oxygen isotopes experience changes too, with an increase in 17 O and a depletion in 18 O (Boothroyd & Sackmann, 1999; Charbonnel, 1994). The activation of the H-burning shell increases the stellar luminosity and the star leaves the MS toward the RGB on the H-R diagram. Here, the Hecore continues to contract and heat. Neutrino energy losses from the center cause the temperature maximum to move outward, as shown in Figure 2.1. Eventually, triplealphareactions ( 4 He(2α,γ) 12 C),whichrapidlyincreasethe core luminosity, are ignited at the point of maximum temperature, but with a degenerate equation of state. The temperature and density ( 10 8 K and 10 7 g/cm 3 ) are decoupled, as the equilibrium of a degenerate gas does not depend on T. In such a case He-burning ignition can occur only in an explosive way (the He-flash). Following this, the star quickly moves to the Horizontal Branch, where it burns 4 He gently in a convective core, and H in a shell (which provides most of the luminosity). Helium burning increases the mass fraction of 12 C and 16 O (the latter through the further reaction 12 C(α,γ) 16 O) and the outer regions of the convective core become stable to the Schwarzschild s criterion for convection. It is however unstable to the Ledoux s stability rule. This situation is referred to as semi-convection. At core He exhaustion, the star shrinks again and has to carry out the excess energy, generated by gravitational contraction of the C-O core and by He burning in a shell. The representative point in the H-R diagram, for lowmass stars, asymptotically approaches the RGB track and is therefore known as 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 giant branch star with an electron-degenerate core composed of carbon and oxy- 15

18 2.2. Asymptotic Giant Branch (AGB) stars and Thermal Pulse. gen. Theascent of the AGB begins following theexhaustion of heliumat the center. The phenomenon was discovered by Schwarzschild & Härm (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-R diagram, all with surface temperatures in the range K, in a region near the RGB track. At core He exhaustion, the star, whose mass has been reduced by stellar winds by up to 10%, starts to be powered by He burning in a shell and partly by the release of potential energy from the gravitationally contracting C-O core. The central density rapidly increases (above10 5 g/cm 3 ) andthec-o coredegenerates andcools downwithahuge energy loss by plasma neutrinos. In LMS core burning is completely prevented by degeneracy and one can note that there exists a relation between the luminosity and the mass of the degenerate core: L 10 4 (M CO 0.5) where L and M CO are measured in solar unities. During the early phases (E-AGB), for all stars less massive than about 3 M, the energy output from the He shell forces the star to expand ad cool so that the H shell remains substantially inactive. When the E-AGB phase is terminated, the H shell is reignited, and from then on it dominates the energy production, whereas the He shell is almost inactive (L He /L H 10 3 ). Late on the AGB, the stellar structure, schematically represented in Figure 2.3, is characterized bya C-O core, twoshells (an inner of helium and an outer of hydrogen) burning alternatively, separated by a thin He-rich layer in radiative equilibrium, ( 10 2 M, the so-called intershell region) and an extended 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

19 2.2. Asymptotic Giant Branch (AGB) stars and Thermal Pulse. He synthesized by the H shell is high enough to be compressed and heated as requested for its re-ignition. The first thermal pulse determines the end of 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, with C-O cores of mass 0.5 < M CO /M < 0.6, they are brighter than the tip of the red giant branch (logl 3.3). During the quiescent hydrogen-burning phase, the temperatures and densities in the helium-rich layers below the burning shell increase together with the mass of these same layers. Once the mass of the helium-rich region exceeds a critical value, the rate at which energy is emitted by helium burning becomes larger than the rate at which it can escape via radiative losses, and a thermonuclear runaway ensues. Although the degree of electron degeneracy of the He-rich material is weak, this thermonuclear runaway occurs because the thermodynamic time scale needed to locally expand the gas is much longer than the nuclear burning time scale of the 3α-reactions. The power generated blows up to 10 8 L (most of which being spent to expand the structure); radiative mechanisms cannot transmit all this energy and the intershell region from radiative becomes convective. Then, the freshly synthesized products of He burning (such as 12 C, whose resulting mass fraction in the top layers of the intershell regionis X( 12 C) 0.25) aremixedoverthewholeintershell. Afterwards,the star readjusts its structure and the thermal instability pushes outward the layers of material located above the He-burning shell. The temperature and the density at the base of the H-rich envelope decrease and the H-burning shell is quenched. As a consequence, the intershell region becomes radiative again. The above process is repeated many times (from about 5 to 50) until the envelope is completely eroded by mass loss, which strongly affects the AGB phase. The illustration (see Figure 2.4) shows the structure of a TP-AGB star over time, showing with thick black lines the base of the convective envelope, the H-burning shell, and the He-burning shell. The region between the H and He shells is the helium intershell. Horizontal gray bars represent zones where protons can partially penetrate the He layers, because the convective eddies do not stop abruptly at the convective border, but have a decreasing profile of temperatures. When H burning in the shell starts, these protons build fresh 13 C through the 12 C(p,γ) 13 N(β + ν) 13 C reaction. This subsequently undergoes alpha captures through (α,n) 16 O, releasing neutrons. In current models, 13 C is naturally burned under radiative conditions before being ingested in the convective zone of the following thermal pulse. Note that proton penetration into the He-rich layers cannot occur in other ways. In particular, the convective thermal pulse does not reach the H-burning shell, despite it can extend very close to it. An entropy barrier is present, during the thermal instability, between the intershell region and the base of the stellar envelope, preventing the direct penetration of convection from the He-rich layers into the H shell. After the expansion and cooling of the envelope, the stellar structure 17

20 2.2. Asymptotic Giant Branch (AGB) stars and Thermal Pulse. Figure 2.4: Illustration of the structure of a thermally pulsing-asymptotic giant branch star over time. shrinks. Because of the low density, the ratio of the gas pressure to the radiation pressure decreases and the local temperature gradient increases. The adiabatic temperature gradient approaches its minimum allowed value for a fully ionized gas plus radiation and convection from the envelope penetrates below the H-He discontinuity, beyond the former position of the now inactive H shell. He-shell burning continues radiatively for another few thousand years, 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 is inactive, the mentioned penetration of the convective envelope reaches down to regions of the He intershell previously affected by the TP so that newly synthesized 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 during the E-AGB phase). However, the occurrence of TDU is much faster and it is expected to repeat many times. The star undergoes recurrent TDU episodes, whose efficiency depends on the physics of the convective borders. The TDU is influenced by the parameters affecting the H-burning rate, such as the metallicity, the mass of the H-exhausted core, and the mass of the envelope, which in turn depends on the effectiveness of mass loss by stellar winds [see the discussion in Straniero et al. (2006)]. During the TP-AGB phase, the envelope becomes progressively enriched in primary 12 C and in s-process elements (the s process will be discussed in the third chapter). As mentioned, a few protons penetrate into the top layers of the He intershell at TDU. At hydrogen re-ignition, these protons 18

21 2.3. The third dredge-up. are captured by the abundant 12 C forming 13 C in a thin region of the He intershell ( 13 C pocket). Hence, neutrons are released in the pocket under radiative conditions by the 13 C(α,n) 16 O reaction at about T K. This neutron exposure lasts for about thousand years with a very low neutron density (10 6 to 10 7 n/cm 3 ). The pocket, strongly enriched in s-process elements, is then engulfed by the subsequent convective TP. At the maximum extension of the convective TP, when the temperature at the base of the convective zone exceeds K, a second neutron burst is powered for a few years by the marginal activation of the 22 Ne(α,n) 25 Mg reaction. This neutron burst is characterized by a low neutron exposure and a high neutrondensityupto10 10 n/cm 3, dependingonthemaximumtemperature reached at the bottom of the thermal pulse. Summing up, the main characteristics of the He-burning shell in AGB stars, from the point of view of the nuclear processes occurring, are related to the development of thermal instabilities called shell flashes or thermal pulses. The four phases of such a thermal pulse can be summarized essentially as follows. 1. During the first stage almost all of the surface luminosity is provided by the H-shell. This phase lasts for 10 4 to 10 5 years, depending on the core-mass. 2. The He-shell suddenly starts burning very strongly, producing luminosities upto 10 8 L. Theenergy deposited bythese He-burningreactions is too large to be transported by radiative processes and a convective shell develops, which extends from the He-shell almost to the H-shell. This convective zone includes mostly He (about 72-75%) and 12 C (about 22-25%), and lasts for about 200 years. 3. During the so-called power-down phase, were the He shell begins to die out and the convection is shut-off, the previously released energy drives a substantial expansion, pushing the H-shell to such low temperatures and densities that it is extinguished. 4. The dredge-up phase follows, where the convective envelope, in response 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) and can even penetrate the flash-driven convective zone which was produced by the He-shell. This phenomenon allows ashes from both He and H burning to be mixed to the surface. This is the so-called TDU, accounting for the existence of carbon stars enriched in s-process elements in the late stages of the 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 wherethey can be observed was found 19

22 2.3. The third dredge-up. since the first numerical models for TP-AGB stages. In 1977, Iben drew attention to the fact that the direct penetration of convention, associated to a thermal pulse, into the H-shell is inhibited by an entropy barrier placed between the He-intershell and the envelope. For this reason, hydrogen can t approach zones where He is burning until the entropy excess is carried out, causing expansion and cooling of the envelope. The stellar structure shrinks and the base of the convective envelope sinks below the interface between the two shells. This event, as already mentioned, is know as the third dredge-up or TDU. The depth and efficiency of the dredge-up phenomenon typically grows from pulse to pulse; it is measured through the so-called dredge-up parameter λ, defined as: λ M TDU M H (2.7) This is the ratio between the mass carried to the surface at each thermal pulse, M TDU, and the mass processed by the H-burning shell during the interpulse phase, M H. Generally speaking, the whole TP-AGB evolution depends 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 as the metallicity, the core and the envelope mass. In particular, there is a strong dependence of the evolutionary properties of AGB stars on the initial metallicity (Z) and the value of λ increases when Z decreases. The amount of material dredged-up in a single episode ( M TDU ) initially increases when the core mass increases, then decreases, when the mass loss erodes a substantial fraction of the envelope. Mass loss also determines the number of thermal pulses: the higher the stellar mass is, the larger is the number of thermal pulses. A lot of problems still affect the determination of the TDU efficiency. They include in particular the opacity tables (that give the k ν, coupling radiation to matter) and the value of the free parameter α P characterizing the so-called mixing length l M treatment of convection. This last quantity determines the mean free path of a convective eddy in units of the pressure scale H P. One can use it to describe the transport of heat in convective conditions. In all evolutionary calculations for AGB stages, α P is maintained constant to a value calibrated on the solar model. At first, TDU was easily discovered in models of stars belonging to Population II(low metallicity) and in intermediate mass stars (IMS) with massive envelopes. Then Lattanzio (1989) and subsequently Straniero et al. (1995), using the Schwarzschild criterion for convention and values of the α P parameter in excess of 1.5 (the value accepted today is 2.1), succeeded in finding TDU also in LMS of Population I, thus explaining the existence of carbon stars of low luminosity in the solar neighborhoods. Subsequently, new opacity tables stimulated a number of calculations of AGB models by various groups (Vassiliadis & Wood, 1993; Straniero et al., 1995; Forestini & Charbonnel, 1997; Frost et al., 1998). Since these im- 20

23 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 established as a self-consistent process agreed upon by researchers. However, the complexities of the AGB structure, involving extreme contrasts in local matter properties, the use of the mixing-length theory for describing convective transport, and the short duration of the interpulse phases available for mixing, continue to make it difficult to address the problem from first principles. In summary, concerning TDU events, only most recent stellar models confirmed numerically its existence for initial masses as low as about 1.5 M, in typical solar conditions. In fact, AGB stars belonging to Galactic Globular Clusters, whose initial mass are of the order of M, do not show the enhancement of carbon and s elements, which is the signature of the TDU. Moreover, depending on stellar physical parameters, there is a minimum envelope mass for which TDU takes place. The efficiency of TDU is connected with the chemical composition; for given values of the core and envelope masses, it is deeper in low metallicity stars, where H burning is less efficient. Actually, the propagation of the convective instability is selfsustained due to the increase of the local opacity that occurs because fresh hydrogen (high opacity) is brought by convection into the He-rich layers (low opacity). In general, TDU occurs only after some initial, less intense thermal pulses and ends when the envelope mass becomes smaller than about 0.4 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 the nucleosynthesis of heavy elements that was demonstrated observationally to occur there. Several years before stellar model could address the problem, Merril (1952) discovered that the chemically peculiar S stars (characterized by C/O ), enriched in elements heavier than iron, contain the unstable isotope 99 Tc (τ = years) in their spectra. It was clear that ongoing nucleosynthesis occurred in situ in their interior and that the productsweremixedtothesurface. Thefactthat TciswidespreadinSstars and also in the more evolved C stars (C/O > 1) was subsequently confirmed by many workers on a quantitative basis. It is therefore not surprising that red giants in the TP-AGB phase were suggested as the site for the s processes as early as in the1960s (Sanders, 1967). AGBstars are well known as the main site where the s-process occurs, i.e. where the slow addition of neutrons proceeding along the valley of β-stability generates about 50% of nuclei 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 the 13 C(α,n) 16 O reaction, whose activation however depends on still uncertain mixing mechanisms for protons. In this case they must inject hy- 21

24 2.4. Nucleosynthesis and observations for AGB stars. drogen from the envelope into the He-rich region, during the TDU phenomenon. Here protons react on the abundant 12 C, producing 13 C through the 12 C(p,γ) 13 N(β + ν) 13 Cchain. Stellarmodelcalculations(seee.g. Gallino et al., 1998; Straniero et al., 1997) showed that any 13 C produced in the radiative He-rich layers at dredge-up burns locally before a convective pulse develops. The temperature is rather low for He-burning conditions ( K, or 8 kev), and the average neutron density never exceeds n/cm 3. As a consequence of neutron captures, a pocket of s-enhanced material is formed and subsequently engulfed into the next pulse. Here s-elements are mixed over the whole He intershell by convection and are slightly modified by the marginal activation of the 22 Ne source. They are then brought to the surface during the following episode of TDU. The 22 Ne source provides only a small contribution in low mass stars, which is nevertheless significant, because it occurs at higher temperature and neutron densities, which can therefore explain several details of s-process branching reactions depending on the environment conditions. AGB stars are important manufacturing sites also for other elements and isotopes. I can broadly divide them into two groups: the H-burning products (mainly coming from regions across and above the H-burning layers) and He-burning products (mainly coming from He-rich zones, above the degenerate C-O core). Several such nuclei of both groups are suitable for direct observational tests in either evolved stars or in their descendants and the diffuse Planetary Nebulae generated by their mass loss. Over the years several studies provided the observational basis for neutroncapture nucleosynthesis models in AGB stars, in particular for discriminating between the competing neutron sources. Coupling of high-resolution spectroscopic observations with sophisticated stellar atmosphere models allowed the determination of heavy-element abundances in AGB stars (see Gustaffson, 1989, for a discussion). In particular, (Smith & Lambert, 1985, 1986, 1990) and Plez et al. (1992) revealed that MS and S stars show an increased concentration of s-process elements. Despite large observational uncertainties, this was recognized to apply also to C stars, characterized by a 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 now aviable (Abia et al., 2001, 2002), based on high-resolution spectra. This has lead to strong revisions in the quantitative s-element abundances. N stars were 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 significantly lower than estimated by Utsumi and is more similar to S star abundances (Smith & Lambert, 1990; Busso et al., 2001). This revision allowed the extension to C(N) stars of the generally good agreement between observed s-process abundances and theoretical predictions of s-process nucleosynthesis in AGB stars (Gallino et al., 1998; Busso et al., 1999). Suchcomparisons confirm also for C(N) stars the existence of an intrinsic spread in the abun- 22

25 2.4. Nucleosynthesis and observations for AGB stars. dance of 13 C burnt, and allow us to place observed AGBs with different s-process and carbon enrichment along simple evolutionary sequences (see Figure 2.5). Figure 2.5: Observations of the logarithmic ratios [ls/fe] of light s elements (Y, Zr) with respect to the logarithmic ratios between heavy (Ba, La, Nd, Sm) and light (Y, Zr) slow neutron capture (s) elements. Symbols refer to different types of s-enriched stars. Stars with the higher s-element enrichments are C-rich (adapted from Busso et al., 1995). Direct information on AGB nucleosynthesis can also be derived spectroscopically from stars belonging to the post-agb phase and evolving to the blue (see Figure 2.1) after envelope ejection (Gonzalez & Wallerstein, 1992; Waelkens et al., 1991; Decin et al., 1998). Since the pioneering work of McClure et al. (1980) and McClure (1984), another source of information has come from the observation of surface abundances for the binary relatives of AGB stars, that is, for the various classes of binary sources whose enhanced concentrations of n-rich elements are caused by mass transfer in a binary system (Pilachowski et al., 1998; Wallerstein et al., 1997). In summary, direct observations contain compelling evidence that AGB stars are the main astrophysical site for the s process and provide abundant constraints on its occurrence: its neutron exposure, correlation with 12 C production, inferred masses of parent stars, etc... 23

26 2.4. Nucleosynthesis and observations for AGB stars. 24

27 CHAPTER THREE S-PROCESS NUCLEOSYNTHESIS IN AGB STARS. In this section of my thesis I present a discussion of nucleosynthesis processes occurring in the final evolutionary stages of stars with moderate mass, when they climb for the second time along the red giant branch (the so-called Asymptotic Giant Branch, or AGB, phase), with particular attention for slow neutron captures. I dedicate most of the space to low mass stars (0.8 3 M ) where the dominant neutron source is the reaction 13 C(α,n) 16 O as they are now recognized as the most important contributors to the s process. I also present a short review of the researches on s-process nucleosynthesis, starting from the first hypotheses of a release of neutron in convective layers, and summarizing the improvements that subsequently led to a crisis in the traditional ideas and to a new scenario in which slow-neutron capture in AGB stars occurs in radiative interpulse phases. In particular, I underline the fact that, in order to understand quantitatively the complexity of s-process nucleosynthesis in the galaxy, we still need amoreaccurate knowledgeof the 13 C(α,n) 16 Oreaction rate. Inthis context, a new measurement of this cross section, performed with the Trojan Horse Method, 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 thermonuclear reactions in stellar environments. A fundamental paper on stellar nucleosynthesis, now recognized as the basis of any subsequent study, was written by E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle in 1957 (Burbidge et al., 1957, often referred as B 2 FH). These authors described the processes of hydrogen and helium fusion, the burning of elements with an intermediate mass (from carbon to silicon) and the production of heavier elements above iron through neutron captures. The Coulomb bar- 25

28 3.1. Introduction. rier of iron is too high to be overcome by charged particle interactions so to create elements heavier than Fe only reactions involving neutrons can be at play. Following B 2 FH, neutron addiction reactions can be divided according to their time scale, as compared with those for competing β-decays of unstable nuclei encountered along the neutron capture path. Following the usual definition, I call r(rapid) process the set of neutronaddition reactions that occur on time scales so short to prevail over the decay times (τ) of unstable nuclei τ (σφ) 1 even when they are rather far from the valley of beta stability (see Figure 3.1). This sets the typical time scales to be smaller than a few seconds. In the previous expression I used σ for the cross section and φ to indicate the neutron flux in the burning region of a star. The r process can occur in supernovae, where huge neutron fluxes (about n/cm 3 ) allow the creation of very heavy (A209) and very neutron-rich elements. In such conditions a stable nucleus can capture many neutrons before it decays. On the other hand, in hydrostatic evolutionary Figure 3.1: The valley of β-stability. Illustration of the neutron-capture path, followed by processes responsible for the formation of 50% of the nuclei between iron and the actinides. stages one meets less extreme conditions of temperature and neutron density, so that the neutron flow proceeds along the valley of beta stability, where the lifetime of unstable nuclei is generally shorter than the neutron capture time scale. Typical neutron densities in this case range from about 10 6 to n/cm 3. The corresponding neutron-capture nucleosynthesis is then called s(slow) process; in it, elements are produced through a series of subsequent neutron captures on stable nuclei followed by a β-decay when an unstable nucleus is encountered. In the s process only rarely neutron 26

29 3.2. The classical analysis of the s process. captures can compete in time scale with weak interactions. However, these few cases are important, as the flow encounters a branching point where the abundances of the nearby nuclei inform us on the physical conditions (neutron density, temperature, etc...). About half of all elements heavier than iron are produced in a stellar environment through s processes. Many improvements on the first ideas by Burbidge et al. (1957) were soon presented, thanks to increased precision in the measurements of isotope abundances from meteorites and of neutron capture cross sections. Various reviews dealing with the s process, and with connected stellar and nuclear issues have been published over the years, especially for the asymptotic giant branch (AGB) stars where neutron-rich elements are produced in the inner regions and then carried to the surface by a series of mixing phenomena known 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 starting from the B 2 FH article that opened the road for the modern theories of heavy element production in stars. Clayton et al. (1961) and Seeger et al. (1965) provided the mathematical tools that outlined the so-called phenomenological approach or classical analysis of the process, i.e. an analytical formulation based only on nuclear properties and abundance systematics. The starting point of this analysis was the study of the distribution of the σ N s products between neutron-capture cross sections and s-process abundances. The mentioned authors built the experimental distribution of σn s values, using data on the neutron-capture cross sections then available and on the solar system isotopic composition. This was then compared with a model σn s curve, by computing analytically the s-process contributions N s to each isotope. As a consequence, the ratio (N(A) N s (A))/N(A) yielded a prediction on the fractional abundances due to the more complex r process. In a slow neutron-capture process, the abundance of an isotope A th varies in time through destruction and creation mechanisms: dn(a) dt = N(A 1)n n σ((a 1),v)v N(A)n n σ(a,v)v (3.1) where σ(a, v)v indicates the Maxwellian-averaged product of cross section and relative velocity, and n n is the neutron density. In the simple expression of equation (3.1) only stable nuclei of atomic mass number A 1 and A, affected only by neutron captures are considered, without branchings. It is convenient to replaces time with the time-integrated neutron flux, or neutron exposure τ, through the substitution: τ n = n n v T dt (3.2) 27

30 3.2. The classical analysis of the s process. This differential equation then becomes: dn(a) = N(A 1)n n σ((a 1),v)v N(A)n n σ(a,v)v (3.3) dτ In steady state conditions, production equals destruction, the time derivative vanishes and σ(a)n(a) = const. This simplified relation is rather well satisfied in the experimental solarsystem distribution of σn s values for heavy nuclei, over large intervals of the atomic mass number A. A modern version of this curve is presented in Figure 3.2, (taken from Käppeler et al., 2011). The curve appears often smooth, but is interrupted by steep drops at nuclei where a neutron shell closure occurs, (their number of neutrons are then called magic neutron numbers, 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 = , at the end of the stable nuclei distribution, and involves 208 Pb and 209 Bi. The solar abundances show s- Figure 3.2: The characteristic product of cross section times s-process abundance σ(a)n(a), plotted as a function of mass number. The thick solid line represents the main component obtained by means of the classical model, and the thin line corresponds to the weak component in massive stars (see text). Symbols denote the empirical products for the s-only nuclei. Some important branchings of the neutron-capture chain are indicated as well. 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 co workers derived two main conclusions: 28

31 3.2. The classical analysis of the s process. 1. the whole distribution of s-element abundances above Fe in the solar system requires more than one s-process mechanism (or component) occurring in separated astrophysical environments in order to bypass the bottlenecks introduced by neutron magic nuclei. One of the components of the process 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 main component). A third (strong) component was also initially assumed for producing roughly 50% of 208 Pb that was missing. This was subsequently proven to be simply due to low metallicity AGB stars with high neutron exposures (Busso et al., 1995; Gallino et al., 1998). In this paper I concentrate my attention on the behaviour of elements from Sr to Pb, i.e. the main component. 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 of repeated neutron irradiations with a continuous distribution of decreasing neutron fluxes, in which many nuclei capture a relatively small number of neutrons and few nuclei capture a large numberof them. The reason for this approximation is that it can expressed by a continuous function (a powerlaw or an exponentially-decreasing function) yielding simplified solutions. In particular, they adopted a distribution of neutron exposures ρ(τ) = Kexp( τ/τ 0 ) (3.4) where ρ(τ)dτ represents the number of seed nuclei (mainly 56 Fe) exposed to an integrated flux between τ and τ + dτ. Their choice soon became very popular, because it allows an exact analytic solution for the set of equations: σ(a)n s (A) = GN 56 τ 0 A A i=56 [ 1+(σ(Ai )τ 0 ) 1] 1 (3.5) where the only degrees of freedom are: 1. the fraction G of solar Fe nuclei irradiated, and 2. the mean neutron exposure τ 0. N s (A) represents the part of the abundance N A due to the slow neutron capture. Concerning the main component, the mean exposure τ 0 was originally estimated to be around 0.2 mbarn 1, but was updated over the years with the improvements in the nuclear data, up to around 0.3 mbarn 1 (at 30 kev). The success of the exponential distribution of neutron exposure was a result of its mathematical convenience and also of the fact that Ulrich (1973) showedhowthe AGBphases of intermediatemass stars can indeedmimic an exponential form, under the assumption that neutrons are released during the convective instabilities of He-shell. He showed that the exponential distribution derives from the overlap factor r between subsequent convective pulses, if a constant exposure τ is produced in every pulse. 29

32 3.2. The classical analysis of the s process. In fact, after N pulses the fraction of material experiencing an exposure τ = N τ is r N = r r/ r. This is an exact solution if the neutron density and the temperature don t change during the s-process. The classical analysis rapidly became a technique sophisticated enough to account for reaction branchings along the s-path, contrary to the simple assumptions implied by equation (3.1). Even at the low neutron densities characterizing the s process, the competition between captures and decays has still to be considered for a number of crucial unstable isotopes, like 79 Se, 85 Kr, 148 Pm and 151 Sm. For them, the probability of a neutron capture is high enough to compete with the beta decay. Application of the branching analysis to specific ramifications of the process was since then used for inferring the stellar parameters (average neutron density, temperature, electron density). It was also shown by Ward & Newman (1978) that the branchings held information on the pulsed nature of the neutron flux. For each branching, a branching ratio f β can be defined by comparingthe rates for β-decay and neutroncapture, so that f β = λ β /(λ β +λ n ), where λ n = N n σ b v T. Here σ b is the Maxwellian averaged (n,γ) cross section of the nucleus at the branching point. In the case of a branching, the curve describing the product σn s is divides in two ramifications and each branch is studied separately. Also the existence of metastable isomeric states of nuclei, for example of 85 Kr, pointed to that result. The method briefly described so far was continuously Figure 3.3: The complex branching of 85 Kr. updated over the past three decades, to take into account progresses in neutron capture cross-sections measured along the s path. The level of accu- 30

33 3.3. Evolution and nucleosynthesis in the AGB stages. racy reached today in cross-section measurements has finally demonstrated that the phenomenological approach, based on an exponential distribution of exposures, can no longer be seen as an acceptable approximation of the s process. 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 (for thermonuclear reaction) of low and intermediate mass stars. Even below 8 M the AGB evolutionary scenario and related nucleosynthesis significantly change with the mass of the star. In the following I review the properties of AGB for stars of low mass. The quantitative results have been derived from recently published AGB models computed by several authors, in particular: Straniero et al. (1997) For clarity, I first discuss the previous phases of stellar evolution before the representative point of a star in H-R diagram goes to 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 inferior mass 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 the metallicity. The lower limit, instead, corresponds to the mass value to reach the inner temperature of about 10 million of degree (measured in Kelvin) necessary to start hydrogen combustion. Hydrogen burning follows the reactions of pp-chain but, if temperature in star is bigger than about K, the CNO-cycle is the main energy source. This stage was the longest in stellar life, it wastheso-called mainsequence(ms). Corehydrogengoes onuntil H is exhausted in the core over a mass fraction is close to 10%. A schematic view of track followed by the stellar representative point is given by the H-R diagram (see Figure 2.1). Then the He core shrinks, while the stellar radius increase to carry out the energy produced by the H-burning shell. As consequence of envelope expansion, the stellar representative point in the H-R diagram movesto theredandto increaseluminosity, and thenclimbsatrack called the red giant branch (RGB). While the envelope expands outward, convection penetrates into region that had already experienced partial C-N processing or proton captures and it carried to surface part of them. At helium core exhaustion, star become powered by He burning in a shell, so the large energy output pushes the representative point in a track that, for low mass star, asymptotically approaches the former RGB and is therefore known as the AGB. The AGB stage is characterized by a degenerate core made of C-O whose pressure is mainly provided by degenerate electrons, by two shells (of H and He), and by an extended convective envelope and it can be divided in two stages: E-AGB and TP-AGB. During the early phases (E- 31

34 3.3. Evolution and nucleosynthesis in the AGB stages. AGB)C-Ocorecanincreaseandwarmbecauseofheliumburninginshell. In star with M > 2 msb a second dredge-up can occur delivering some elements from hydrogen shell to surface. After E-AGB the two shells are separated by a thin layer in radiative equilibrium: the so-called He-intershell. As shell H burning proceeds while the He shell is inactive (L He /L H < 10 3 ), the mass of the He intershell M H M He increases (owing to sinking of newly formed He) and attains higher densities and temperatures. This results in a dramatic increase of the He-burning rate for short period of time: the socalled Thermal Pulse (hereafter TP). Thermal pulses are real thermonuclear flashes repeating at regular time lapse (the so-called interpulse during which He-shell remains inactive) and during which He burns in semi-explosive conditions, as inthecaseof degenerated core. Infact, theseeventsarecausedby combination of two main factors: intrinsic instability of thin shells and the partial degeneration. Since the unstable thermal configuration the emission of energy due to He-shell begin to oscillate with increasing amplitude until a thermal pulse is created with a typical power of about 10 5 L. The radiative state of the He intershell is thereby interrupted, and the shell then becomes almost completely convective. This results in a mixing process called third dredge-up (hereafter TDU), which carries processing material to surface. In this way it is possible to study internal process, so the discovery of 99 Tc by Merril in 1952 was a proof to affirm that also heavy elements are created in stars. From the structural point of view, the TDU is very similar to the second dredge-up however, its occurrence is much faster and is expected to repeat many times. Modelling TDU was always very difficult; it was related to the choice of the opacity tables and, in the framework of the mixinglength theory, to the value of α P (the ratio of the mixing length l and the pressure scale height H P ). Now the main energy source is helium and star has to readjust its structure expanding too radiate the energy surplus. The process is repeated many times (about cycles) before the envelope is completely eroded by mass loss. This evolutionary phase is usually referred to as the TP-AGB (Thermally-Pulsing AGB). The Figure 3.4 shows the internal structure of a thermal-pulse-asymptotic giant branch star as a function of time. One can easily looks at the alternate motion (in mass) of the two shells following the position in mass of the H-burning shell (M H ), of the He-burning shell (M He ) and of the bottom border of the convective envelope (M CE ). During the whole AGB stage a star loses a big part of its convective envelupe. Then one of the most severe uncertainties still affecting AGB models concerns mass loss. The duration of the AGB and the number of TPs, the amount of mass dredged up, the impact of stellar winds on interstellar abundances and many other important predictions depend on the assumed mass loss rate. The available data indicate that this rate ranges between 10 8 and 10 4 M / yr (Loup et al., 1993). Studies of Mira and semi-regular variables show that mass loss is not a monotonically increasing function of time, and the star certainly encounters variations in its mass 32

35 3.3. Evolution and nucleosynthesis in the AGB stages. Figure 3.4: Plot of the internal structureof atp-agb star as a functionof time, for a 3 M model with Z = 0.02 (Straniero et al., 1997). The positions in mass of the H-burning shell (M H ), of the He-burning shell (M He ), and of the bottom border of the convective envelope (M CE ) are shown. Convective pulses(shown in Figure 2.4) occupy almost the whole intershell region during the sudden advancement in mass of the He shell. The periodic penetration of the envelope into the He intershell (third dredge-up) is clearly visible. This model reaches the C star phase (C/O > 1) at the 26 t h pulse. Pulses from 17 to 32 are shown. loss efficiency, until a final violent (perhaps dynamical) envelope ejection occurs. The pressure radiation in envelope, increasing after helium burning, is the responsible of solar wind injection. In this phase AGB star pumps in interstellar medium about or most than 70% of their whole mass in the form of dust and gas until it is completely expelled leaving the naked core. This is the post-agb stage. The representative point of core nebula describes a big excursion in temperature. It goes toward the blue zone because it shows the internal and hotter zones and the warm coming from stellar surface is enough to ionize the material. A star now is surrounded by a brilliant zone, the so-called planetary nebula. In the main time luminosity decrease very quickly because mass loss extinguishes the thermonuclear reactions in two shell H and He then star came under the track of main sequence. This is the white dwarfs stage, the final phase of life of a low mass star, where it radiates its residual energy travelling along a diagonal line, the so-called cooling sequence. 33

36 3.4. The neutron source 13 C(α,n) 16 O. 3.4 The neutron source 13 C(α,n) 16 O. There are two important neutron sources in typical AGB conditions: the 13 C(α,n) 16 O reaction, originally introduced by Cameron et al. (1954) and the 22 Ne(α,n) 25 Mg reaction; also this one was suggested by Cameron et al. (1960). 22 Ne is naturally produced in the He intershell starting from the original CNO nuclei present in the star at its birth and transformed mainly into 14 N by the operation of the H-burning shell. In He-rich layers 14 N is consumed through the chain: 14 N(α,γ) 18 F(β + ν) 18 O(α,γ) 22 Ne Duetoitsnaturaloccurrence, thisneutronsourcewasthefirsttobeexplored in stellar models to describe s-process, mostly for stars in mass range 4-8 M, known as Intermediate Mass Stars (IMS). This source produces a high neutron density of about n/cm 3 and needs a temperature larger than K to be activated. The maximum temperature achieved in LMS at the bottom of TPs barely reaches T = K, hence the 22 Ne source is only marginally at play. At the beginning of the eighties, this fact pushed some authors to reanalyze the conditions for the activation of the alternative 13 C(α,n) 16 O source that had been previously largely ignored. This second reaction is activated at relatively low temperatures (T = K) and can therefore easily explain why the abundances of s-elements are highly enhanced in low mass AGB stars, where the temperature is low. The idea was confirmed by further observations, including the abundance trends of heavy s-elements in not evolved stars of both the galactic halo and the disk. In order to allow the 13 C(α,n) 16 O reaction to be the main neutron source for s-processing at low temperatures, two conditions must be met. 1. A mechanism for injecting protons into the He-rich region must be found, so that interacting with the abundant 12 C they can produce 13 C in He intershell. 2. Theamountof 13 Cthusobtainedmustburnthroughthe 13 C(α,n) 16 Oreaction in layers where the temperature is low (T k) to maintain the neutron density low. The reaction 13 C(α,n) 16 O is considered to be the main source of neutrons for the s-process in low mass stars during the asymptotic giant branch phase. However, producing neutrons through 1 3C-burning is more difficult than through 22 Ne burning, mainly because one needs some mixing process suitable to bring protons into the He intershell: indeed, the amount of 13 C naturally left behind by H burning is by far insufficient to drive significant neutron captures. In the He-rich layers of AGB stars one has then to start from a 13 C abundance built locally at H-reignition, through small amounts of protons diffused down from the envelope into the intershell region. The direct en- 34

37 3.4. The neutron source 13 C(α,n) 16 O. gulfment of protons from the H shell when convective instabilities develop is instead inhibited by an entropy barrier at the H shell. Since the occurrence of the third dredge-up forces the hydrogen-rich and the carbon-rich layers to establish a contact, this will naturally produce some mixing at the H/He interface: by chemical diffusion during the interpulse phase (for which it is difficult to define a quantitative approach) or by hydrodynamical effects induced by convective overshooting, or even from buoyancy in magnetic fields. The assumption that proton mixing occurs during the third dredge-up, forming a 13 C-pocket whose mass was left as a free parameter proved to be a fruitful approach (Gallino et al., 1998). Subsequently, observations and chemical evolution models for the galaxy guided the research, indicating that the average efficiency of the mixing processes at TDU must be such that the reservoir of 13 C reaches a mass of a few 10 4 M (Travaglio et al., 1999; Busso et al., 2001). Afterwards, possible physical mechanisms for producing a 13 C pocket of the suitable mass and with the suitable abundance distribution have been extensively investigated by different authors, in order to find a more secure basis for s-process nucleosynthesis in stars. In order to provide a suitable site for s-processing the 13 C reservoir must be formed through a limited number of protons captures by the chain of reactions: 12 C(p,γ) 13 N(β + ν) 13 C Too efficient proton captures, indeed, activate a full CN cycling, leading to 14 N production through the 13 C(p,γ) 14 N reaction, and 14 N is a very efficient absorber for neutrons, which would inhibit the captures on heavier nuclei. In general, one expects a zone close to H-He interface, where more protons are expected and where the subsequent burning produces mainly 14 N: this region is not useful for s-processing, but will manufacture a lot of 15 N from neutron captures on 14 N. Here the subsequent convective instability of the He-shell produces abundant 19 F, from 15 N(α,γ) reactions. Below this region the decaying abundance of protons creates the conditions suitable for formingalmostpure 13 Candhencetoactivate efficiently the 13 C(α,n) 16 Oreaction and the neutron capture nucleosynthesis processes. Later, when the convective instability of the He-shell develops and attains its maximum strength, the temperature reaches value of typically K, the 22 Ne source is marginally activated, providing a small neutron burst of higher peak neutron density. This second neutron burst was recognized as being able to explain several details of the solar s-process abundance distribution, for nuclei after reaction branchings requiring a relatively high neutron density (10 10 n/cm 3 ). An important point concerns the time scale of 13 C burning. Actually, the first models (Käppeler et al., 1990) assumed that the locally-produced 13 C could remain essentially inactive until the next convective instability, when it would be ingested and burned at the typical 35

38 3.4. The neutron source 13 C(α,n) 16 O. Figure 3.5: Two successive thermal pulses (in particular, the 29 th and 30 th ) for the 3 M model with Z = Z are shown in their relative positions as calculated from the stellar model. The shaded zone is the 13 C pocket, in which protons are captured by 12 C. In the figure on the left, ingestion and burning of 13 C in a pulse is based on the older models. 13 C(α,n) 16 O is first burned convective, producing the major neutron exposure, followed by a small exposure from the 22 Ne(α,n) 25 Mg neutron source in the pulse. The newer model, as shown in the second illustration, states that 13 C burns in the thin radiative layer where it is produced, releasing neutrons locally. After ingestion into the convective intershell region, this is then followed by a second small neutron exposure from the marginal activation of the 22 Ne source. temperature of K, characteristic of the first phases of a thermal pulse. Subsequently, it was understood Straniero et al. (1995, 1997) that the neutron release by 13 C burning starts very early, before the convective instability develops. It therefore occurs in radiative and not in convective conditions and at very low temperatures, as mentioned. All 13 C nuclei available below the H shell were found by Straniero et al. (1997) to be consumed by the 13 C(α,n) 16 O reaction before the growth of the next instability. The neutron density in each layer scales with the local 13 C abundance, reaching at most 10 7 n/cm 3. The thermal velocity is close to 8 kev. The convective pulse driven by each thermal instability simply dilutes the s-process products over the whole intershell zone and exposes it to the new neutron flux from 22 Ne burning. The seed material in the next 13 C-pocket is therefore a combination of nuclei present in the H burning ashes from the upper intershell, and of the s-processed material left behind in the lower part of 36

39 3.4. The neutron source 13 C(α,n) 16 O. the intershell zone at the quanching of the previous convective instability. The thermal pulse history is represented schematically in Figure 3.5. The thin zone q indicates the position of the 13 C-pocket where neutrons are released. The fraction r of the mass of the convective He shell contains s- processed material from the previous pulses; the fraction 1 r contains the H-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 13 C consumption, including the effects of some delayed neutron recycling by the 12 C(n,γ) 13 C(α,n) 16 O chain, is about years, leaving several thousand years before the growth of the next convective instability (at least yr in 2 M stars). However, the reaction rate for (α,n) captures on 13 C is very uncertain at the very low energies at play. I shall discuss extensively the implications of this in the rest of this thesis. Based on the above analysis, Figure 3.6: Schematic representation of the thermal pulse history and of s-processing in the interpulse periods. s-process nucleosynthesis in AGB stages can be summarized as occurring in different phases: 1. penetration of a small amount of protons into the top layers of the cool He intershell (to form a proton pocket); 2. formation of a 13 C pocket at H reignition; 3. release of neutrons by the 13 C(α,n) 16 O reaction when the region is subsequently compressed and heated to T = K. Here s processing takes place locally under radiative conditions generating an s- enhanced pocket; 4. ingestion into the convective thermal pulse, where the s-enhanced pocket is mixed with H-burning ashes from below the H shell (Fe seeds, 37

40 3.5. Possible future scenarios. 14 N) and with material s-processed in the previous pulses; 5. exposure to a small neutron irradiation at high n n by the 22 Ne 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 12 C-rich material is mixed into the envelope; 7. repetition of the above cycle until the TP phase is over. 3.5 Possible future scenarios. Onthebasisofthescenariodescribedabove, it wasshownbytravaglio et al. (1999) that the chemical evolution of s-elements up to the solar formation age could be well reproduced. Very recently, however, observations of open clusters by our group D Orazi et al. (2009);? revealed that the above picture is insufficient to account for the s-element enrichment in the more recent galactic disk, where an s-process enhancement larger than in the Sun exists. This indicates that AGB stars of very small mass (M < 1.5M ), contributing in the Galaxy only after the solar formation, must produce s-elements more efficiently than more massive stars. They should therefore have more extended 13 C pockets. These enlarged 13 C reservoirs would cover regions of the star where a higher temperature (10 kev) is present and would induce higher n-densities. Due to this new scenario and to the warnings already presented on the uncertainty in in the present rate for the 13 C(α,n) 16 O reaction, there is now a 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 so low that a possible increase of the rate would have minimal effects, unless it is larger than a factor of 3-5. More relevant would be a possible reduction of the rate with respect to the values indicated by NACRE. This is a real possibility, if the rate is less affected than so far assumed by the contribution of a sub-threshold resonance. In such a case, 13 C might have insufficient time to burn in the interpulse phase, and would end up burning, at least partially, in the convective thermal pulse. Here the extra energy generated would be crucial and might induce phenomena like a shell- splitting, with strong changes in the neutron density and large modifications in our present picture of the s-process. 2. For masses below 1.5 M, both an increase and a decrease of the rate might be critical, as the slightly higher temperature spanned by the 13 C- pocket would emphasize the effects on the otherwise low n-density. Again, some 13 C in the cooler layers might remain unburned, with the same destabilizing effects described at point 1. 38

41 CHAPTER FOUR CROSS SECTIONS OF NUCLEAR REACTIONS AT LOW ENERGIES. Nuclear reactions have a fundamental role in many astrophysical environments because they provide the energy to sustain their luminosity over their lifetimes and also because they are responsible of nucleosynthesis of elements in stars. Usually one can refer to these as thermonuclear reactions because the star contracts converting gravitational energy into thermal energy, until the temperature and density become high enough to ignite them. In a stellar environment at thermodynamic equilibrium, velocities and energies of interacting nuclei follow the Maxwell-Boltzmann distribution with typical temperatures depending on stellar mass and evolution stage: from 10 6 to 10 9 K. So, nuclear reactions take places at very low energies, of the order of a few kev, because of the equation E = k B T, where k B is the Boltzmann constant. An accurate knowledge at typical astrophysical energies of the reaction rates and therefore of the cross sections is highly desirable because they affect the different stellar evolutionary phases as well as the estimates of the chemical element abundances. Uncertainties of reaction rates are usually high in stellar conditions because of difficulties to implement experiments at such low energies. As I have already said in previous chapters, in the low mass star AGB phases the region between the H shell and the He shell (He-intershell) is affected by brief convective instabilities, (thermal pulses), due to the sudden ignition of He burning in the He shell. In these conditions the 13 C(α,n) 16 O reaction is the main neutron source for the s process workingin radiative conditions in a thin layer at the top of the intershell ( 13 C-pocket) during the interpulse periods. The rate for α captures on 13 C is measured at high energy only, while for stellar energies its values are deduced by extrapolation. It is therefore necessary, and this is the main goal of this thesis, to determine the 13 C(α,n) 16 O reaction rate in the unexplored energy zone. In order to set the stage for this task, in this chapter I will preliminary in- 39

42 4.1. Coulomb barrier and penetration factor. troduce some concepts and a general discussion of the theoretical problems involved in the study of the nuclear processes in astrophysics as reported in detail in Rolf & Rodney (1988). 4.1 Coulomb barrier and penetration factor. In stars, nuclear reactions take place between charged particles because the atoms are in most cases completely stripped of their atomic electrons. It is assumed that they are almost completely ionized because of the typical high temperature conditions (around a few kev at least). This high temperature is on the other hand needed to permit the reactions, because nuclei are positively charged and repel each other with a Coulomb force proportional to their nuclear charge. Nuclear reactions are therefore inhibited by the Coulomb barrier whose height (in MeV) is given, in CGS units, by: E C = Z 1Z 2 e 2 R n (4.1) where R n = R 1 +R 2 is the nuclear radius, Z 1 and Z 2 represent the integral charges of the interacting nuclei. Classically, a reaction can occur only between particles with energies higher than E c. Incident projectiles at lower energies would reach the closest distance to the nucleus at the turning point R C and would not penetrate the Coulomb barrier. Figure 4.1 represents the schematic view of the effective potential resulting when one combines the very strong and attractive nuclear potential with the electromagnetic potential. Consequently, if this is the case, the fractions of particles whose energies exceeds the Coulomb barrier is negligible and it seems necessary an higher stellar temperature. This obstacle was removed when Gamow (1928) showed that, in according to the quantum mechanics, there is a small but finite probability for the particles with energies E < E C to penetrate the barrier: this is the so-called tunnel effect. One might define the penetration factor, which is the basis of the tunnel effect, through the following ratio (Clayton et al., 1983): T = χ(r n) 2 χ(r c ) 2 (4.2) where the upper quantity represents the probability of finding the particles at the interaction radius, and the other one at the classical turning point of the Coulomb barrier. It can be calculated by solving the radial part of the Schrödinger equation: d 2 χ l dr 2 + 2µ h 2 [E V l(r)] = 0 (4.3) 40

43 4.2. Cross section, astrophysical factor and reaction rate. Figure 4.1: Schematic representation of the combined nuclear and Coulomb potentials. The plot reports the total potential V(r) versus the relative distance r between the two interacting particle. where V l (r)is the potential for the l th partial wave resulting when the centrifugal potential term is also present (Clayton et al., 1983) V l (r) = l(l+1) h2 2µr 2 + Z 1Z 2 e 2 r (4.4) At low energies or, equivalently, where the classical turning point is much larger than the nuclear radius, equation (4.2) can be approximated by the simpler expression giving the so-called Gamow factor: T = e 2πη (4.5) with the Sommerfeld parameter, η = Z 1 Z 2 e 2 / hv, depending only on the relative velocity of the two interacting particles and their charges. At low energy, below the Coulomb barrier, tunneling probability has an approximate expression that drops exponentially with (4.5). 4.2 Cross section, astrophysical factor and reaction rate. In stellar objects, the production of nuclear energy and the synthesis of elements proceeds through fusion reactions until all light nuclei are converted to iron (A 60), corresponding to the maximum binding energy 41

44 4.2. Cross section, astrophysical factor and reaction rate. for nucleon. More complex reactions lead to the production of the heavier elements. Theseprocesses cantakeplacethroughaslow(s) orarapid(r)sequence of neutron captures with respect to the rate of β-decays of the nuclei just formed. The rate of each fusion reactions depends on the astrophysical conditions and it can vary by several orders of magnitude for different temperature and density. I now briefly present the formalism adopted in order to derive the astrophysical rates of charged-particle-induced reactions. In general, 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 constitute the exit channel. In order to have a description for the nuclear process, in astrophysical environments we require the introduction of a cross section. The cross section is defined as the probability that a given nuclear reaction will take place. It is used to determine how many reactions occur per unit time and unit volume providing important information on energy production in stars. Classically, this cross section σ depends only on the combined geometrical area of the projectile and the target nucleus. Since all nuclear cross sections are of the order of cm 2 (or lower), for convection and convenience, a new unit of area, the barn (b), equal to m 2 has been defined for cross sections. In reality, since nuclear reactions are governed by the laws of quantum mechanics, the cross section must be described by the energy-depend quantity σ = πλ 2 1 DB (4.7) E where λ DB represents the De Broglie wavelength: λ DB = m x +m X m X h (2m x E x ) 2 (4.8) For charged-particle nuclear reaction the cross section is strongly suppressed by Coulomb and centrifugal barriers and it drops rapidly for E < E c. Recalling equation (4.5) and (4.7), it is possible to factorize the cross section as: σ = 1 E S(E)e 2πη (4.9) where S(E) is the so-called nuclear or astrophysical factor and contains all nuclear effects. The astrophysical factor is a much more useful quantity because 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 than the cross section and it allows an easier procedure for extrapolating the energy behaviour at astrophysical energies. As just discussed, nuclear cross sections are in general energy-dependent or, equivalently, velocity-dependent, 42

45 4.2. Cross section, astrophysical factor and reaction rate. Figure 4.2: In the upper panel, the cross section σ(e) of a charged-particleinduced nuclear reaction in shown. There is a rapid exponential decrease downtoe L,whichisthelowerlimitforthebeamenergyatwhichexperimental measurements can be made. So, as the lower panel shows, extrapolation to lower energies is more reliable if one uses the S(E) factor. σ = σ(v), where v represents the relative velocity between projectile and the target nucleus. Starting from cross section, I can introduce another important quantity, the so-called reaction rate, to describe the nuclear process in astrophysical scenarios. The reaction rate is defined as the number of given reactions per unit volume per unit times (this gives an idea of the velocity of the reaction). r xx = 1 1+δ xx N x N X σ(v)v (4.10) where the product N x N X represents the total number of pairs of nonidentical nuclei X and x. For identical particles the Kronecker symbol δ xx is introduced, otherwise each pair would be counted twice. The bracketed quantity σ(v)v is referred to as the reaction rate per particles pair and it is the mean of the product σ(v)v over all the possible energies, weighted over the Maxwell-Boltzmann distribution: φ(v) = 4πv 2 ( m 2πk B T ) 3/2 ) mv 2 exp( 2kT (4.11) 43

46 4.3. Gamow peak. by introducing the center of mass energy E = 1 2 µv2 with µ representing the reduced mass of interacting particles, the reaction rate is then expressed as: r xx = ( ) 1 8 1/2 1 N x N X 1+δ xx πµ (k b T) 3/2 0 ( σ(e)eexp E ) de k B T (4.12) Here r is expressed in units of reactions per cubic centimeter per second. Its equation characterizes the reaction rate at a given stellar temperature T and, during stellar evolution, its temperature changes. Then it s important to have 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 the same, analytic expression for σ(v)v in terms of temperature T. 4.3 Gamow peak. Starting from the mathematical expression for the nuclear reaction rate found in the previous section (4.12), one can easily calculate the theoretical best condition for reactions taking place in stellar environments. Then if equation (4.9) is inserted into equation (4.12), one obtains: σv = ( ) 8 1/2 1 µπ (k B T) 3/2 0 S(E)exp ( E k B T ( EG E ) 1/2 ) de (4.13) where the symbol E G is the so-called Gamow energy. The integrand of the equation (4.13), because of the limited dependence of S(E) from E, is governed by the combination of two exponential terms: the first represents the Maxwell-Boltzmann distribution and the second one is the probability of tunneling through the Coulomb barrier. The maximum of the integrand is reached at an energy E 0 : ( ) kb T 3/2 E 0 = E 1/2 2 G (4.14) The convolution of the two functions results into a peak, the so-called Gamow peak, centered near the energy E 0 and generally much larger than k B T. The maximum value of the integrand will be: ( ) 2E0 I max = exp (4.15) k B T which depends strongly on the Coulomb barrier. As it can see from Figure 4.3 the Gamow peak has an effective width, which is referred as Gamow window, wherein most of reactions take place: = 4 3 (E 0 k B T) 1/2 (4.16) 44

47 4.4. Direct measurements and experimental problems. Figure 4.3: The Gamow peak is the result of the convolution of two functions: the Maxwell-Boltzmann distribution and the quantum mechanical tunneling function through the Coulomb barrier. The energetic region relevant for the astrophysical investigation (the zone with gray lines) is around the value E 0. Usually the effective energy for thermonuclear reactions ranges from few kev to about a hundred kev depending on both the reaction and the astrophysical site in which the reaction occurs. However, the nuclear processes of astrophysical interest occur at energies that in general are too low for direct measurement in laboratory, as discussed in the next chapters. These difficulties are related to different problems and usually the standard solution is to measure the cross section or, equivalently, the S-factor over a wide range of energies and to the lowest energies possible and then to extrapolate the data downward to E 0 with the help of theoretical arguments and other methods. 4.4 Direct measurements and experimental problems. The direct measurement of the cross sections in the low-energy conditions under which thermonuclear reactions take place between charged particles in stars is a hard task. First of all, the Coulomb barrier between the interacting particles is usually of the order of 1 MeV while reactions, mainly induced around Gamow peak, are often centered in the range from 1 kev to a few hundred kev. The cross section then is strongly suppressed by an exponential 45

48 4.4. Direct measurements and experimental problems. factor (4.) σ(e) exp( 2πη) (4.17) At the energy corresponding to the Gamow peak, the cross section is of the order of nano or pico barns. A low cross section means a low number of particlescollected (N d ), theso-called signalevents,onthedetectoraccording to the equation N d σn i τ Ω (4.18) where N i is the number of incident particles τ is the thickness of the target and Ω the solid angle covered by detector. So, different ways are possible in order to increase N d : 1. the use of detectors with large solid angles; 2. the use of a more intense beam current (with cautions, not to damage the target); 3. the use of thicker target that however implies a worse energy resolution. Even if N d might increase by improving the experimental setup, this number is affected by the background noise events N b, coming from the cosmic rays, from natural radiation or from the electronic noise introduced by the experimental setup. For a successful experiment it is important to reach the condition: N d N b 1 (4.19) This ratio can be adjusted by increasing the detected particles or by reducing the background noise, with the following techniques: 1. using of very low-noise electronics. 2. performing nuclear astrophysics experiments in underground laboratories, as the Laboratori Nazionali del Gran Sasso. 3. using different kinds of indirect methods that allow also to overcome other experimental difficulties. In this context, the first simple way to avoid the experimental problems consists in an extrapolation of the cross section down to astrophysical relevant energies. As already said, the S(E) factor is useful for an extrapolation from experimental data measured at higher energies because of its small energy dependence. The standard procedure consists in fitting the high energy data using a proper theoretical function (in the simplest approximation, a polynomial). Then this is extrapolated to the astrophysical energies. Anyway, 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 resonance energy E r of an excited state of the compound nucleus does not exceed the Q-value for the reaction (Q = m x +m X m y m Y, where m are the mass of involved particles). In this case the resonance peak lies below 46

49 4.4. Direct measurements and experimental problems. Figure 4.4: A sub-threshold resonance and its influence of the behaviour of the astrophysical S(E)-factor. the interaction energy where the tail of this resonance can influence the behaviour of the S-factor and can be even dominant at astrophysical relevant energies. Therefore precise information on the position, the strength and the FWHM (Full Width at Half Maximum) of the resonance are needed from independent experiments. As already advanced, a second relevant source of uncertainty in the extrapolation of the astrophysical factor down to zero energy is the enhancement of S(E) due to the electron screening effect. Up to now it was assumed that the interacting nuclei be completely stripped of electrons, so the Coulomb potential is typically expressed as in equation (4.1), being essentially bare nuclei. On the contrary, when nuclear reactions are studied in a laboratory, the projectile is usually in the form of an ion and the target is usually a neutral atom or molecule surrounded by their electronic cloud (Assenbaum e al., 1987). The atomic electron cloud surrounding the nucleus acts as a screening potential and consequently the total potential goes to zero outside the atomic radius (R a ) V eff = Z 1Z 2 e 2 R n Z 1Z 2 e 2 R a (4.20) Then the projectile effectively sees a reduced Coulomb barrier. As a consequence, at low energies the cross section for screened nuclei, σ s (E) (also shielded cross section), is enhanced, with respect to the cross section of the bare nucleus σ b (E), by a factor: f(e) = σ s S ( ) s πηue expo (4.21) σ b S b E 47

50 4.4. Direct measurements and experimental problems. Figure 4.5: Schematic representation of the potential between charged particles. The potential is reduced at all distances and goes essentially to zero beyond the atomic radius R a because of the presence of the electron cloud. The electron screening effects cause an enhancement of the S(E)- factor, increasing the penetrability through the barrier. where U e, representing the screening potential for the studied reaction, must be taken into account to determine the bare nucleus cross section. For E/U e > 1000 the electron screening effects are negligible so one essentially measuresσ b (E), whileife/u e < 100one(Langanke et al.,1996)experimentally have an enhancement on the cross section, σ s (E). The experimental enhancement has been observed in several fusion reactions and it has been seen that the lower is the interaction energy, the larger is this enhancing factor. Because of the high temperature of stars, atoms are generally completely ionized, and one can imagine that electron screening has no effect on nuclear reactions in stars. However nuclei are immersed in a sea of free electrons, the so-called plasma, resulting in an effect similar to the one discussed above. In the condition of nearly perfect gas, therefore, when k B T is much larger than the Coulomb energy between the particles, the electrons tend to cluster into spherical shells around the nuclei, with a Debye-Hunckel radius R D of: R D = ( ) kb T 1/2 4πe 2 (4.22) ρn A ξ wheren A is the Avogadro number and the quantity ξ is expressed by the 48

51 4.5. Indirect methods for nuclear astrophysics equation: ξ = i ( Z 2 i +Z i ) X i A i (4.23) Here the sum is performed over all positive ions and X i is the mass fraction of the i th nucleus of charge Z i. The shielding effect reduces the Coulomb potential as in laboratory and it increases the reaction rate, or equivalently the 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 order to extract the bare nucleus cross section from the σ s (E) using (4.21). Then the proper stellar screening factor should be applied to that (4.24). One of the most important uncertainties in experimental nuclear astrophysics derives from this procedure and, because of this, more exhaustive and precise determinations of σ b are needed at energies as low as possible. In this context several indirect methods, for example the Coulomb dissociation (CD), the asymptotic normalization coefficient (ANC) method, and the Trojan horse method (THM), have been proposed to overcome the specific difficulties of direct measurements. I will briefly present these new experimental approaches in the next section, and will describe with particular attention the Trojan horse method in next chapter. 4.5 Indirect methods for nuclear astrophysics As already mentioned, both the Coulomb barrier penetration and the electron screening effects represent problems that must be overcome in order to get the cross-section for charged-particle-induced reactions in the energy domain relevant for astrophysics. For these reasons the indirect methods mentioned above have been proposed. In particular, ANC and CD methods provide information about astrophysical relevant reactions involving photons, while the THM is applied to reactions between charged particles. These indirect methods have been developed to extract cross sections relevant for astrophysics from other kind of experimental or theoretical approaches. In these complementary methods the cross-section for the relevant two-body reaction (transfer reaction, proton capture, photo-disintegration, etc.) is extracted by selecting a precise reaction mechanism in a suitably chosen three-body reaction or through the application of some theoretical considerations. In this context, the most important steps consist in reproducing direct data at high energies making use of data extracted from the indirect method and then in trying to go down at very low energies. Among indirect methods, Coulomb dissociation allows to extract a precise radiative-capture cross section. The method, as proposed by Baur (1986), consist in studying 49

52 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 electromagnetic excitation process and the principle of detailed balance. The second indirect technique presented is the so-called ANC (Asymptotic Normalization Coefficient) (Mukhamedzhanov et al., 1990; Xu et al., 1994) method, which provides the normalization coefficients of the tails of the overlap functions, and determines S factors for direct capture reactions at astrophysical energies. The method can be applied for the analysis of the direct radiative capture processes of type (4.1), where the binding energy of the captured charged particle is low. Moreover, the ANC technique turns out to be very productive for the analysis of the astrophysical process in presence of a subthreshold state. Very recently, a work by (Johnson et al., 2006) developed ANC techniques in order to determine the astrophysical factor also for reactions different from radiative capture processes. At the end, I mention the so-called Trojan horse method (hereafter THM) (Baur, 1986; Spitaleri et al., 1999), which seems to be the best suited for investigation of low-energy charged-particle reactions relevant for nuclear astrophysics. This method has already been used to derive indirectly the cross section of a two-body reaction from the measurement of a suitable three-body process to overcome the effects due to the entrance-channel Coulomb barrier. The measurement of such a cross section at energies as low as possible is then necessary to gather more precise information about the energy production and nucleosynthesis in astrophysical environments. In this paper we shall stress the importance of the THM as a complementary tool to direct measurements in the study of 13 C(α,n) 16 O reaction of astrophysical interest. 50

53 CHAPTER FIVE MEASURE OF THE 13 C(α,N) 16 O REACTION THROUGH THE THM. As already said in the third chapter nearly half of the heavy elements observed in the universe are produced by a sequence of slow neutron capture reactions, the so-called s-process nucleosynthesis (Busso et al., 1995). The reaction 13 C(α,n) 16 O is considered as the main neutron source for the main component of the s process in low mass Asymptotic Giant Branch (AGB) stars. In this scenario, two factors can determinate the efficiency of this reaction: the amount of 13 C burnt and the cross section of the 13 C(α,n) 16 O reaction. An accurate knowledge of this reaction rate at relevant temperatures would eliminate an essential uncertainty regarding the overall neutron balance and would allow for better tests of modern stellar models with respect to 13 C production and burning in AGB stars. Very recent observational constraints, like those by?, and their interpretation, to which I have contributed as part of my work for this thesis (Maiorca et al., 2011b), suggest an enlarged 13 C reservoir, that would induce higher neutron densities because part of the 13 C would burn at a slightly higher temperature than before (up to 10keV, against 8keV that were standard before these works). Concerning the second aspect, a new accurate measurement of the rate for the 13 C(α,n) 16 O reaction might impose very restrictive constraints on the conditions in which 13 C is burnt in the pocket during the AGB stage. Modern stellar models, run with the accepted 13 C(α,n) 16 Orate, show that the abundance of 13 C produced in the pocket must burn locally in the radiative layers of the He intershell, before a new convective pulse develops, in contrast with previous ideas that suggested carbon-13 combustion in a convective environment. An increase of the cross-section can have only small consequences in low and intermediate mass stars (above 1.5 M ), because 13 C 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

54 5.1. Theory of the Trojan Horse method. would be cancelled by the subsequent operation of the 22 Ne(α,n) 25 Mg reaction in the convective instability, where the neutron density is already quite high (10 10 n/cm 3 ). Important effects would be instead seen in very low mass stars, below 1.4 M, where the 22 Ne source is not active because the temperature in the pulses is not sufficient. In this case changes to the 13 C(α,n) 16 Orate would immediately affect the isotopic ratios of s-element abundances in AGB stars. A possible reduction of the rate with respect to the present values by the NACRE compilation might instead imply that 13 C have insufficient time to burn in the interpulse phase. In this case it would end up burning in the convective region, at a higher temperature and producing energy, thus potentially modifying the whole energy budget of the star and the structure of the convective layer. All the above possible astrophysical effects will be considered in a dedicated section (see Chapter 6 for a more accurate discussion). Direct measurements of the 13 C(α,n) 16 O cross section have been performed down to 280 kev (Angulo et al., 1999), whereas in AGB stars the temperatures at which α-captures on 13 C occur are typically about ( ) 10 8 K. The corresponding Gamow peak (Rolf & Rodney, 1988), in according with equations (4.14) and (4.16), is at E cm = 190 ± 90 kev, so that thedirectdata available stop at therightedge of Gamow window, while the Coulomb barrier is about E c = Z 1Z 2 e 2 R n = Z 1 Z 2 e 2 r 0 (A 1/3 1 +A 1/3 2 ) 3.7MeV (5.1) where I use r 0 = 1.3 fm. In this context, the study of the astrophysical S-factor in the relevant region for astrophysics, where Coulomb-barrierpenetration and electron-screening effects are dominant, is highly desirable. The indirect Trojan Horse Method permits to extend the measure below the current lower energy limit and to overcome both the cited difficulties. In this chapter, I first presented the theoretical approach for the Trojan Horse Method (THM) for the study of low-energy charged-particle reactions. Then I report on the application of this method in order to obtain indirect information about 13 C(α,n) 16 O process at the low energy, starting from the 13 C( 6 Li,n 16 O)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 of astrophysical interest: x+a c+c (5.3) 52

55 5.1. Theory of the Trojan Horse method. A schematic representation of the (5.2) process, the so-called Trojan Horse reaction, is shown in Figure 5.1 through a pseudo-feynman diagram. The Figure 5.1: Pseudo-Feynman diagram for the break-up quasi-free process a(a,cc)s. Trojan Horse approach, as suggested by Baur (1986), is based on the theory of the quasi-free(hereafter QF) break-up mechanism in which the interaction between two nuclei produces the break of one particle in its constituting nuclei. In particular, the starting point is to consider that the nucleus A is composed by two nucleon clusters x and s. The wave function for the target nucleus A can be written in the following way ψ A = ψ x (r x )ψ s (r s )ψ(r x r s ) (5.4) where the ψ i are the internal wave functions of x and s, respectively, while ψ represents the relative motion wave function between the two clusters. The interaction between target and projectile causes A, described by a structure such as A = x s, to break in the two clusters and the nucleus a can interact only with the transfer particle x. In practice the nucleus s does not participate to the reaction and it can be considered as a spectator for the x(a, c)c reaction. After selecting appropriate kinematic conditions, the quasi-free process occurs if the cluster s maintains the same momentum it had in the nucleus A before interacting. Figure 5.1 represents the two crucial moments of the whole process: the upper pole is the break-up of the nucleus A into x and s, while the lower one describes the two-body interaction x(a, c)c. The A particle is called the Trojan Horse nucleus because, similarly towhatthewoodenepichorsedidforulyssesandhiscomrades, ithidesinits 53

56 5.2. Plane Wave Impulse Approximation. interior the transferred, participating cluster x. In this way, the Trojan Horse reaction (5.2) can be performed at energies well above the corresponding E c, so that the binary reaction cross section is not Coulomb-suppressed, as the barrier has already been overcome in the entrance channel. For these reasons the THM has already been applied several times to reactions connected with fundamental astrophysical problems characterized by very low energies. Moreover, assuming that the beam energy can be compensated for by the x+s binding energy and by the Fermi motion of x inside A, the two-body reaction can take place at very low a x relative energy, inside the Gamow window. 5.2 Plane Wave Impulse Approximation. The quasi-free break-up mechanism can be described by following different theoretical formalisms, such as the Distorted Wave Impulse Approximation (DWIA) (Chant et al., 1977; Roos et al., 1977), the Distorted Wave Born Approximation (DWBA) (Typel et al., 2000) and the Plane Wave Impulse Approximation (Jain et al., 1970; Slaus et al., 1977; Fallica et al., 1978). Distorted-wave approaches provide the most sophisticated and accurate formalisms, as it was established by several authors (Jacob et al., 1966; Roos et al., 1976). In these cases, the momentum distribution feels the effects of the distortion due to the interaction between the interacting nuclei. Anyway, Roos et al. (1976), in a work on the 6 Li cluster structure, concludes that for recoil momenta of the spectator k s lower than 100 MeV/c both PW and DW approaches describe well the results about the experimental behaviour of the momentum distribution. Hence, in the above limit of low k s, the various approaches show essentially the same results without introducing significant systematic uncertainties. For these reasons, in this work I shall focus on the simpler Plane Wave theoretical approach. Quasi-free reactions (hereafter also QFR) can be easily described by means of the impulse approximation IA (Chew, 1950; Chew & Wick, 1952). Let us consider, as a typical case, the one of a particles striking a complex system A. The assumptions underlying the impulse approximation are then the following. 1. The incident particle never interacts strongly with the two constituents of the system at the same time. 2. The amplitude of the incident wave falling on each constituent is nearly the same as if that constituent were alone. 3. The binding forces between the constituents of the system are negligible during the decisive phase of the reaction. Under these hypotheses, the incident particle a is considered as interacting only with a part (x) of the target nucleus A, whose wave function is assumed to have a large amplitude for the x s cluster configuration, while 54

57 5.2. Plane Wave Impulse Approximation. the other part s behaves as a spectator to the process. In the THM the transferred particle is virtual (off-energy-shell). However, here I neglect the off-shell character of the transferred particle and use the on-shell approximation. Moreover, with the plane wave(pw) approximation I am assuming, as mentioned above, that the incident and outgoing particles can be described by plane waves without any distorting effects due to the Coulomb interaction between particles (Jain et al., 1970). ψ(r) = r k = ( ) 1 3/2 exp(ik r) (5.5) 2π Taking into account all these hypotheses, the Plain Wave Impulse Approximation (Satchler, 1990) leads to a simple expression for the differential cross section for the three body A(a,cC)s reaction (Jain et al., 1970) d 3 σ de c dω c dω C KF Φ(p s ) 2 ( dσax dω ) (5.6) This is the cross section for the scattering of a particle c into the solid angle dω c with an energy between E c and E c + de c and of particle C into the solid angle dω C. This can be factorized in three terms. Starting from the left, the first term is the kinematical factor KF containing the final state phase-space factor; as I shall show later, it is a function of the masses, momenta and angles of the outgoing particles. This expression is derived by assuming that the momentum of the spectator to the virtual two-body reaction is equal to the one before the reaction. The second term contains Φ(p s ), which is the momentum distribution of the deuteron inside the 6 Li nucleus. In practice, this is the Fourier transform of the radial wave function for the x s inter-cluster motion inside A, usually depending on the cluster configuration involved in the reaction: Φ(p s ) = (2π) 3/2 ψ(r)exp( ik s r)dr (5.7) The last term ( dσ ) represents the differential cross section for the binary dω reaction and it is the quantity to be determined through the Trojan Horse experiment. In order to use equation (5.6), the starting point is the differential cross section for a three-body (3B) final state with the momenta of particles c,c and s in the ranges d 3 k c,d 3 k C and d 3 k s respectively dσ = (2π)4 v rel d3 k c d 3 k C d 3 k s δ(k i K f )δ(e i E f ) t 3B fi 2 (5.8) where the K and E values are the center-of-mass momenta and the total energy of the system in the initial and final states, and v rel = k a /E a is the 55

58 5.2. Plane Wave Impulse Approximation. relative velocity between the incident particle and the target. The variable t 3B fi, appearing in (5.8), is the three-body reduced matrix element and it is related to T fi by the equation T fi = δ(k i K f )t 3B fi (5.9) This represents the transition matrix element for the three-body reaction that I can write in the laboratory system for the break-up reaction as follow: T fi = f T i = k c,k C,k s,ψ c,ψ C,ψ s T 3B ψ a,ψ A,k a (5.10) T 3B is the complete T-operator for the reaction, while ψ i is the wave function that describes the internal generic particle i and k i the corresponding momentum in the final state. At this point, the quasi-free process is taken into account by introducing the Impulse Approximation (IA) through which the incident particle a interacts only with particle x in the nucleus A, while the residual cluster s is a spectator to the reaction. Therefore, T 3B can be replaced by T 2B, the so-called T-operator for the two-body interaction between a and x. With this approximation, the transition matrix element can now be written as: T fi = k c,k C,k s,ψ c,ψ C,ψ s T 2B ψ a,ψ A,k a = d 3 q x k c,k C,ψ c,ψ C T 2B ψ a,ψ x,k a,q x ψ x,ψ s,q x,k s ψ A (5.11) If ψ A represents the intrinsic state of the target and q i is the momentum of particle in initial state, the correspondent momentum space wave function is q s,q x,ψ s,ψ x ψ A = Φ(k)δ(q s +q x ) (5.12) and from the momentum conservation in the laboratory system where the target is at the rest one gets: q x +q s = 0 q x = q s (5.13) q s = k s (5.14) Hence, the final form of the equation for T fi is T fi = d 3 q x k c,k C,ψ c,ψ C T 2B ψ a,ψ x,k a,q x Φ(k)δ(q 2 +k s ) = k c,k C,ψ c,ψ C T 2B ψ a,ψ x,k a, k s Φ( k s ) (5.15) One can note that the momentum of the cluster x before the collision, q x, is equal and opposite to the momentum of the outgoing residual nucleus k s, 56

59 5.2. Plane Wave Impulse Approximation. a quantity that is experimentally measured. Substituting for K i = k a and K f = k c +k C +k s and transforming to the c.m. and the relative momenta for the two-body system, I can write T fi = δ(k a k c k C k s ) k f T 2B k i Φ( ks ) = δ(k a k c k C k s )t 2B fi Φ( k s) (5.16) At the end, the form of equation (5.8), using (5.9) and (5.16), becomes: dσ = (2π)4 v rel k2 c dk cdω c k 2 C dk CdΩ C d 3 k s δ(e i E f )δ(k a k c k C k s ) φ( k s ) 2 t 2B fi 2 (5.17) Integrating over dk s, which in this reaction is unobserved, and over dk c and considering the two following conditions: E f = E c +E C +E s = E c + m 2 C C +k2 m 2 s +(k a k c k C ) 2 (5.18) one obtains: E i = E a +m A (5.19) dσ = (2π)4 E C E s de c dω c dω C k a k c kc 2E ae c k C E s +E C [k C k a cosθ C +k c cos(θ c θ C )] φ( k s ) 2 t 2B fi 2 (5.20) Here, θ c and θ C are the angles of the outgoing nuclei, measured with respect to the incident beam direction. The quantities in equation (5.20) are evaluated using the momentum conservation, k a = k c +k C +k s, and the energy conservation, E f = E i. It is important to note that the energy conservation is not the same as for the usual two-body system, because of the binding energy 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 the two particles, the equation can be written as: dσ de c dω c dω C = (k c kc 2E sec.m. 2 k a E x {k C E s +E C [k C k a cosθ c +k c cos(θ c θ C )]} ( ) dσ φ( k s ) 2 (5.21) dω This is the required expression, already presented in equation (5.6). One can easily note that the three-body cross section for the A(a, cc)s reaction is strictly connected with the one corresponding to the two-body process a(x, c)c. The momentum distribution of the spectator in the Trojan Horse nucleus is also present in both equations, so that, while the remnant 57 c.m.

60 5.3. Current measurement status terms are defined as the kinematical factor, given by the following expression (Jain et al., 1970): KF = (k c k 2 C E se 2 c.m. k a E x {k C E s +E C [k C k a cosθ c +k c cos(θ c θ C )]} (5.22) KF is made by terms measured or calculated and it depends only on the kinematic conditions of the process. Hence the only unknown variable is the two body TH-cross section that I can easily obtain by (5.6) using the following equation: ( dσax dω ) = d 3 σ de c dω c dω C ( KF Φ(ps ) 2) 1 (5.23) In this way, I have formulated the THM two-body cross section; in order to get the direct one it is necessary to reintroduce the Coulomb-field effect, by multiplying the (5.23) by the penetration factor G l (Cherubini et al., 1996; Spitaleri et al., 1999). Performing an experiment where it is possible to measure the QF-contribution of the three-body reaction and knowing both the kinematical factor and the momentum distribution for the relative s x motion insidethe TH nucleus, makes it possible to extract the a(x,c)c cross section by using the relation (5.6) ( ) dσ = dω cc l ( ) THM dσl G l dω cc (5.24) Here, G l represents the transmission coefficient for the l th partial wave. Now, one can notice a very important point, which will be recalled in the last chapter. Because of the factor G l, the two-body cross section can only be obtained with an arbitrary normalization but the essential energy dependence can instead be extracted carefully. Absolute cross sections can be obtained only after normalization to the directly-measured excitation function. This is the so-called validity test, for which we shall need to anchor our low-energy estimates to the high-energy measurements at energies above the Coulomb barrier. A comparison and an agreement between direct and THM data over the already explored in the past is necessary in order to allow to extend the measurement at astrophysical energies using the THMdetermined energy dependence. In this context, the Trojan Horse method has to be seen as a complementary tool in experimental nuclear astrophysics, because direct data are in any case needed at energies above the Coulomb barrier for normalization procedures. 5.3 Current measurement status In the last fifty years, several investigations of the total cross section for the 13 C(α,n) 16 O reaction at low energies have been reported, motivated by 58

61 5.3. Current measurement status its importance as the main neutron source for the s-process in low mass stars. However, many experimental difficulties did not allow experimentalists to perform a direct measurement of the reaction-rate behavior at temperatures relevant for astrophysics. In the last years the most commonlyused rate was that presented in the European Compilation of Reactions Rates for Astrophysics (hereafter NACRE) by Angulo et al. (1999). There, the rate for 13 C(α,n) 16 O is determined using experimental cross sections fromsekharan 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 and 4.47 MeV. Figure 5.2: Behavior of the astrophysical S-factor, the most useful parameter for an extrapolation from experimental data measured at high energies Angulo et al. (1999). An enhancement of the S-factor at low energies with respect to previous recommendations was suggested because of the inclusion of a subthreshold resonance in the extrapolation. Since direct measurements stop right at the limit of the Gamow window (190 ± 90) several experiments using indirect methods have been performed. At temperatures of about 10 8 K the uncertainties are 300% due to the prohibitively small reaction cross section at energy below 300 kev. For the lowest energy range, the one most relevant for neutron production in AGB stars, the S-factor was usually extrapolated by fitting the data of Drotleff et al. (1993). It was however shown that this extrapolation can be strongly affected by the 1/2 + subthresholdresonance of 17 O at an excitation energy of MeV, which is just 3 kev below the α+ 13 C threshold. As the above is a critical point, I present here a short review of different experimental approaches, both via direct measurements and via indirect 59

62 5.3. Current measurement status techniques, which tried to explore the low temperatures typical of stars ( K) in order to have a more accurate knowledge of the 13 C(α,n) 16 O reaction rate in astrophysical conditions. Before the nineties, the reaction rates of astrophysical important thermonuclear reactions involving low-mass nuclei (1 Z 14), including also 13 C(α,n) 16 O, were collected in a huge paper by Caughlan et al. (1988). This provided the numerical values and also gave analytic expressions for the rates, updating previous publications from the same group. Chronologically speaking, Drotleff et al. (1993) was the first to perform a direct experiment in which differences from the previously accepted values for the S(E) factor of 13 C(α,n) 16 Oemerged. These authors measured the excitation function reaching a sensitivity limit of 50 pico-barns in the best case. Thenewdatacoveredtheenergyrange350keV E 1.4MeV,where the cross section varies over eight order of magnitude. The reaction rate was calculated taking into account the subthreshold resonance described above in according to the expression: [ ( ) ] N A σv = T9 T9 2 exp T 1/ ( ) T 1/ T 2/ T 9 [ T 3/2 exp T [ T 3/2 exp ] + T 9 9 ] (5.25) This relation is valid over the range 0.01 T (T 9 hereafter means the temperature in units of 10 9 K). Because of the state in 17 O just below the α-threshold and using their own measurements over the range kev, Drotleff et al. (1993) suggested a low-energy increase in the reaction rate of 13 C(α,n) 16 O with respect to previous investigations. Theoretical calculations (Bach, 1992; Descouvemont, 1987) supported this indication, with an increase at low energies more rapid than expected. However, considering the errors, the result by Drotleff et al. (1993) would still be consistent with a constant, horizontally extrapolated, astrophysical factor. A much lower rate for of the 13 C(α,n) 16 O reaction (actually, the lowest reaction rate present in the literature) was subsequently suggested, through the direct α-transfer reaction by Kubono et al. (2003). In this work, the contribution of the subthreshold state was found to be much smaller than the accepted prediction and the calculated reaction rate was parameterized by the formula: ( ) N A σv = exp T T 1/3 9 60

63 5.3. Current measurement status ( ) exp T 1/ T T 5/ logT 9 (5.26) Hence, the reaction rate is smaller than the NACRE recommended value roughly by a factor 4 at T 9 = 0.1. It is also smaller by a factor 3.5 with respect to Drotleff et al. (1993), suggesting that the 13 C(α,n) 16 O reaction would be slower at low temperatures (this rate is similar to the Caughlan et al., 1988, s value). For clarity, Table 5.1, showing the results of different experiments, allows us to have an immediate comparison of the different suggested values of the 13 C(α,n) 16 O rate for the relevant astrophysical range of energies ( K). One can note that differences are sometimes very high and this is so because the total uncertainty at low temperatures includes mainly two components: one from the subthreshold state contribution and the other from the extrapolation of the direct data. For this and for future applications, I use here the Maxwellian-averaged reaction rate N A σv, in analogy with (4.12) and (4.13), as follows: ( ) 8 1/2 1 N A σv = N A πµ (k b T) 3/2 0 ( σ(e)eexp E ) de (5.27) k B T Here, N A is the Avogadro number ( mol 1 ), µ is the reduced mass of the system, k B the Boltzmann constant, T is the temperature, σ is the cross section, v is the relative velocity and E is the energy in the centre-of-mass system. The quantity N A σv is in units of cm 3 mol 1 s 1. T 9 Caughlan et al. (1988) Drotleff et al. (1993) Angulo et al. (1999) T 9 Kubono et al. (2003) Johnson et al. (2006) Pellegriti et al. (2008) Table 5.1: Table of reaction rates presentin theliteraturefor 13 C(α,n) 16 O. T 9 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 normalization method (ANC) to the determination of the astrophysical S factor for the 13 C(α,n) 16 O reaction. Before, the ANC method had been applied only for radiative capture processes. In practice, the S(E) factor was determined by measuring the ANC for the virtual synthesis α+ 13 C 17 O (6.356 MeV, 61

64 5.4. The Trojan Horse Method applied to the 13 C(α,n) 16 O reaction. 1/2 + ) using the α-transfer reaction 6 Li( 13 C,d). Measurements were performedattwodifferentsub-coulombenergiesof 13 C.Fortemperaturesabove T 9 = 0.3 the calculated reaction rate turned out to be identical to the one adopted in NACRE, while at temperatures useful for the s-process in AGB stars, T 9 = , the rate was smaller by a factor 3 with respect to the one by Angulo et al. (1999), but still inside the uncertainty band. Another recent ANC determination of the S factor was performed by Pellegriti et al. (2008). Their reaction rate at typical AGB temperatures is slightly lower than the value adopted in NACRE, but it is twice as large as 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 in the estimated contributions from the 1/2 + subthreshold resonance. This is however reduced to a factor of about 2.3 in the total reaction rate because of the role of the non-resonant term, which is dominant in Johnson et al. (2006). The differences among the various articles and approaches show that further work is necessary before drawing definite conclusions. Hence, verification of the results presented in the last two decades using an independent experimental approach (e.g. the Trojan Horse technique) is highly desirable. Later in this chapter I therefore present the application of this kind of indirect method to measuring the cross section, or equivalently the S(E) factor, for the 13 C(α,n) 16 O reaction. 5.4 TheTrojanHorse Methodapplied to the 13 C(α,n) 16 O reaction. The Trojan Horse method, as already said, is a powerful tool to extract a charged-particle binary reaction cross section at astrophysical energies, free of the Coulomb barrier effects thanks to a three-body starting process occurring at energies well above the value of the Coulomb barrier itself. Hence, the THM appears to be very useful to study reactions in astrophysical environments where energies are very low. The present work reports on a new investigation of the two-body 13 C(α,n) 16 Oreaction by selecting the QF-contribution of the 13 C( 6 Li,n 16 O)d three-body reaction (Q 3B = MeV), using a 6 Li beam of energy 7.82±0.05 MeV. The theoretical approach described in section 5.2 is usually adopted for reactions for which break-up occurs in the target. In the present experiment, instead, the 6 Li of the beam is the TH nucleus. The formalism remains the samebutobviouslyithastobeadaptedattheprojectilebreak-upcase, using appropriate system transforms. It is assumed that in the laboratory system the target is at rest while the projectile is connected with its fragments by the relation: k beam = q x +q s (5.28) 62

65 5.4. The Trojan Horse Method applied to the 13 C(α,n) 16 O reaction. Figure 5.3: Pseudo-Feynman diagram for the 13 C( 6 Li,n 16 O)d reaction. The 6 Li projectile is considered as composed by x s: the so-called TH nucleus. After break-up (upper pole), the deuteron acts as a spectator. where the k i is the three-momentum of the beam, the participant and the spectator particle, respectively. In order to compare Figure 5.3 and Figure 5.1 the target made of 13 C corresponds to nucleus a, the projectile A is the 6 Li nucleus, s the deuteron, x is the α particle, c is the neutron and C is 16 O. Thanks to the high energy in the A+a entrance channel, the two-body interaction can be considered as taking place inside the nuclear field, so that it does not experience neither Coulomb suppression nor electron screening effects. The A + a relative motion is compensated for by the x s binding energy E B, thus determining the so-called quasi-free (La Cognata et al., 2007) two-body energy (E qf ), which is given by: E qf = E ax E B = m a = E x E B m x +m a m a m x = E A E B (5.29) m x +m a m A Here, E x represents the fraction of the beam energy E A corresponding to the cluster x, E ax is the relative energy between a and x and m i is the mass of particle. Thus, the relative energy of the fragments in the initial channel a+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

66 5.5. Experimental setup. In order to apply the Trojan Horse Method, which is based on a quasifree break-up process, one needs to separate this contribution from all the others which may occur between the same target and projectile, giving the same particles in the exit channel: the so-called sequential process. A sequential mechanism is a two steps interaction in which the final state is reached through an intermediate one, as shown in Figure 5.4. Only the process occurring through the QF-mechanism is of interest for the further TH investigation. A detailed study of this level was needed In this context; it is clear that a detailed study and the discrimination of such mechanisms represents an important stage of a TH-analysis. This kind of information can be reached studying the relative energies between the particles in the exit channel. In particular, the study of any two among the E n 16 O, E d 16 0 and E dn relative energies allows to obtain information on the presence of excited states of 17 O, 3 H and 18 F. Once this stage of analysis is confirmed, it will be possible to apply the THM to the three-body data for the extraction of the two-body cross-section of interest. 5.5 Experimental setup. The experiment was performed at The John D.Fox Superconducting Accelerator Laboratory in the Florida State University by the ASFIN2 (in Italian, AStroFIsica Nucleare) group of the Laboratori Nazionali del Sud, Catania. In particular, I took part in the first phase of the experiment: the electronic and mechanical assembly, the calibration runs and seven days of on-beam data acquisition. The facility implied the use of the 9MV TAN- DEM to accelerate a beam made of 6 Li, the isotope of litium characterized by three protons and three neutrons. The spot size was about 1 mm and beam intensities were around 1-4 na. Then, the interaction between the beam and the 13 C target occurred in a vacuum chamber with a diameter of 1 m placed in the second target room of the laboratory. The acceleration took place in two stages: an ion source produced negativelycharged ions having a velocity of a few tenths percent of the velocity of light. Specifically, polarized ions of lithium were created by the optical pumping technique. This is a process in which light is used to raise one or more electrons from their levels to more energetic states. Hence, sometimes binding electrons can be separated from their nuclei or molecules. The 9MV Tandem Van der Graff accelerator, 15,24 m long, provided the second stage of velocity increasing. In a tandem accelerator the same high voltage can be used twice if the charge of the particles can be reversed while they are inside the terminal. At first negative ions coming from the source are accelerated because they are attracted by the positive electrode and the beam, passing through a thin foil to strip off electrons inside the high voltage conducting terminal, become made of positive charges: the so-called stripping phase. 64

67 5.5. Experimental setup. Figure 5.4: Possible three-body sequential processes resulting from the interaction between 6 Li and 13 C, which gives in the exit channel the same particles (d, 1 6O,n) through the formation and decay of intermediate states of 17 O, 3 H and 18 F, respectively. 65

68 5.5. Experimental setup. The final result are positive ions that are accelerated again because they are repelled by the positive terminal. One can observe the advantage of using a tandem 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 have an elevated charge state (q) at a specific applied voltage V (the terminal can be charged to a maximum potential of 9 to 10 million volts). The amount of acceleration can be varied by changing the terminal voltage. This voltage is maintained by continuously transferring charges using an endless insulating belt carrying positive charges between ground potential and the terminal. The beam then leaves the tandem and, thanks to focussing magnets, it arrives at the measurement chamber. A schematic draw (scale 1:7.5 cm) Figure 5.5: Experimental setupof the 13 C( 6 Li,n 16 O)d reaction discussedin the text. The target is made of 13 C while the beam is 6 Li. PSD1, PSD2 and PSD3 are placed in the positive semi-plane at angles as specified in Table 5.2, in order to detect the deuteron. PSD4 and PSD5 are used to observe 16 0 in the negative semi-plane, while the third particle, the neutron, is not detected. (scale 1:4) of the experimental configuration is shown in Figure 5.5 where the zone above the beam track represents the positive semi-plane. Two 13 C targets of different thickness, 107 and 53 µg/cm 2, were placed at 90 degree with respect to the beam direction. In order to measure in coincidence the 16 O and deuteron particles the detection setup consisted of a set of five silicon position-sensitive detectors (PSDs). Two telescopes, each of them composed 66

69 5.5. Experimental setup. by a 20 µm thick E silicon detector and a position-sensitive silicon (PSD2 and PSD3), were used to reconstruct the experimental momentum distribution of the spectator in the positive semi-plane. Moreover, in the same semi-plane the PSD1 covering about 3-13 deg was very useful to discriminate the deuteron yield. This detector is covered by a 22.5 µm aluminium foil in order to suppress the elastic scattering contribution and the heavy fragments: only particles with A < 4 can pass it. Similarly, two positionsensitive detectors (PSD4 and PSD5) covering the scattering angles from degree up to degrees on the other side of the chamber were used to measure the yield of the Oxygen recoils. The thicknesses of PSDs, sum- PSD Distance (cm) Central angle (deg) Angular range (deg ) Thickness (µm) Table 5.2: Experimental conditions for the 13 C( 6 Li,n 16 O)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, that is the largest energies of the residual nuclei, with thick detectors. The third particle, in our case the neutron, was not detected because neutral particles are very difficult to study. The alignment of all detectors was checked by an optical system. The trigger for the event acquisition was given by coincidences between deuterons detected in the positive semi-plane and the signal of 16 O coming from the other two PSDs. This allowed for the kinematical identification of our specific exit channel of reaction 13 C( 6 Li,n 16 O)d. When energy (E) and position (P) signals were detected in each PSD, they had to be elaborated and stored. The position signal was directly sent to the ADC after a pre-amplification and an amplification stage. The E signal, after passing through the pre-amplifier, was instead split in two lines. The first one was sent to a linear amplifier and then to the ADC, as for the P signal, while the second E line passed a quicker amplifier (Time Filter Ampifier) and then a discriminator module to have a logic signal before it was sent to a TAC-SCA (Time to Amplitude Converter-Single Channel Analyzer) in order to produce the coincidence event trigger. The start input of 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 this way the coincidence between PSD1 or PSD2 or PSD3 and any one of the other detectors, placed on the opposite side (PSD4 or PSD5) was set. In summary, a 6 Li beam, previously accelerated by a tandem, interacted with a 13 C target producing deuteron and 16 O detected in five PSDs. De- 67

70 5.6. Position Sensitive Detectors (PSDs). tector signals were processed by standard electronic chains and sent to the acquisition system which allowed the on-line monitoring of the experiment and 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 solid state detector providing the information on position and energy of incident charged 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 layer of boron, with a uniform, resistive electrode on the front face and a lowresistive back electrode. When a charged particle passes through the diode, a number of electron-hole pairs are produced and the charged deposited on the contact will be proportional to the particle energy and to the proper electrode resistance. For clarity, Figure 5.6 shows a schematic diagram of a PSD. The signal of position P is extracted from the resistive layer because it acts as a charge divider and it depends on the hitting point. If one defines x as the distance between the grounded contact and the interaction point of incident particles, while L is the total length of the resistive layer, the position signal is proportional to the kinetic energy E in according to the following expression: P = E x (5.31) L A second signal (the E signal), proportional to the total charge deposited in the detector, is derived from the normal conductive front electrode. As 68

71 5.6. Position Sensitive Detectors (PSDs). Figure 5.7: Schematic draw of a position-sensitive detector and of its holder. A grid with eighteen slits is placed in front of the holder to perform the position calibration of the detector. The readout contacts are also present, 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, corresponding at about 20% of the total resistive layer, which ensures a measurable signal also when the hit position is close to this end. 2. The one in the middle is connected to the cathode and provides the energy signal E. 3. The contact on the right is connected to the resistive anode where the charge fraction that provided the position signal P is collected. One of the problems with this kind of detectors is to ensure linearity in the position signal. This requires the semiconductor and the resistive layer to be highly uniform and homogeneous. The typical detector resolution can be of the order of 0.5% FWHM at room temperature over active lengths of 5 cm, corresponding to about 250 µm, for the position and also about 69

72 5.7. The position calibration. 0.5% for the energy. Every PSD is covered by a thin inactive layer, the so-called dead layer, of thickness about 0.2 µm, made of aluminium. It is important 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 lower angles 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 in the calibration of the involved detectors. In order to extract the correct information for future analysis, it is important to convert both the E and P signals, expressed in channels, into quantities of physical interest, expressed in physical units like MeV and degrees, respectively. The described procedure must be repeated for every detector. A typical plot of the set of position data versus energy, expressed in channels (hereafter the matrix ), is shown in Figure 5.8 for PSD5. In order to perform off-line PSD position calibration, usually in the first part of the experimental run, a grid with eighteen equally spaced vertical slits was placed in front of each PSD (see Figure 5.7) Position-energy matrix (ch) P E 5 (ch) Figure 5.8: Position-energy two-dimension matrix for PSD5 for the calibration run with 6 Li+ 12 C. The eighteen slits and the linear loci are clearly visible. The matrix, in most cases because of statistics and detector resolution, shows well separated lines corresponding to the various slits and almostvertical highly populated zones, representing tracks left by two-body reac- 70

73 5.7. The position calibration. tions in the final state: the so-called kinematical linear loci. In this kind of matrix, I selected the region showing a visible track by using a graphical cut in order to get information about both energy and position. The data distribution of the chosen region is typically a Gaussian curve, hence as representative values for position and energy we chose the mean values of the Gaussian fits, while the errors were given by the corresponding σ. These points were used both for angular and energy calibration (Figure 5.9). Energy and Position detection for a slit 800 Energy mean σ Position mean σ Counts 400 Counts E PSD5 (ch) P PSD5 (ch) Figure 5.9: Energy and position spectrum for a singular slit. I used the mean value and the σ of the Gaussian fit. It is possible to establish a correspondence between each slit and an angular position with respect to the beam direction. In practice, the central angular position of each detector θ 0 was measured using a theodolite and the angular position corresponding to each slit was calculated by means of trigonometric identities. Recalling the expression (5.31) one can eliminate the position dependence from the energy by introducing the variable: x = P P 0 E E 0 (5.32) where E 0 and P 0 are constants determined by using a fit for all slits. As a consequence, the matrix shown in Figure 5.8 became made of straight horizontal lines, as in Figure They are however still expressed in channels: it is the so-called rectification of the matrix. At this point, it is possible to obtain a relation in order to calculate the impact angle of each particle as a function of the energy-independent linear position x: θ = θ 0 +arctg[c1(x x 0 )+c2(x x 0 ) 2 ] (5.33) 71

74 5.8. Energy calibration. c 1, c 2 and x 0 are the results of the best fit performed among all slits. At this point, I have a matrix with physical angles measured in degrees on the y-axis. The angular resolution was found to be about 0.2 degree. The calibration was performed by using a different kind of targets and sources: a 228 Thalpha-sourceatsixpeakswasusedbecausetheenergydecay is well known for each peak; elastic scattering on heavy 197 Au nuclei and on a 12 C target wereused to measure both elastic and inelastic scattering at the beam ( 6 Li) energy E b =7.82 MeV. In particular, this last run was performed in two different configurations to cover both small angles (high energies) and viceversa. 5.8 Energy calibration. The energy calibration was performed by means of the same runs used for position calibration. In order to convert the value of the energy coming from the acquisition system, expressed in channels, into a physical quantity in units of MeV, the adopted expression is: E MeV = (a+be ch )(1+c 1 (θ theta 0 )+c 2 (θ θ 0 ) 2 ) (5.34) Here, a, b, c 1, c 2 areconstantsresultingfromtheminimization procedure. In order to check for a possible dependence of the energy signal on the impact point, the procedure was performed for each slit. As a first approximation a simple linear relation between the PSD signal and the detected particle energy can be sufficient but (5.34) includes further corrections due to the angular calibration. The overall energy resolutions were found to be about 1%. The interaction between a 6 Li beam of 7.82 MeV and a 12 C target can produce different possible exit channels (excluding reactions producing neutrons or photons and those having three bodies in the final state, which cannot be detected with our experimental setup): C+ 6 Li, the so-called elastic scattering (Q = 0 MeV) N+α (Q = ) O + d (Q = ) O + p (Q = ) In order to have a check of the procedure I plotted theoretical points (black points) corresponding to the cited reactions over the matrix and I compared the position between the two tracks. In particular I found a good agreement especially for the elastic scattering and for different excited levels of 14 N + α as showed in Figure The total kinetic energy of the detected particles was reconstructed off-line taking into account the energy loss 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 the one deposited in the detector but for the future application we are interested in the reaction energy. The procedure of energy reconstruction requires the 72

75 5.9. Data Analysis and future work. (degree) θ C + Li -4.43MeV Calibrated position-energy matrix (MeV) E C + Li 0MeV 14 4 N + He 2.35MeV 3.69MeV 2.96MeV Figure 5.10: Calibrated position-energy matrix in the same case of Figure 5.8. The theoretical kinematic linear loci for two different reactions are also shown. 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 Figure 5.11). In particular, it was conventionally assumed that reactions take place at half target. Concerning 13 C(α,n) 16 O, I calculated that deuteron can lose up to 1.8 MeV before arriving in PSD1, PSD2 or PSD3 while 16 O loses at most 900 kev. Finally, in Figure 5.10 the calibrated angle-energy matrix is plotted for PSD Data Analysis and future work. In the last section I introduced the notion of kinematical linear loci corresponding to a two-body reaction in the final state, confining the study to a single detector or equivalently to a single matrix. At this point, I want to check instead coincidence events: a detector (PSD1, PSD2 or PSD3) detects an outgoing particle, while a second one is detected by other detectors. In order to show it, in Figure 5.12 I show several points of different colors representing the theoretical tracks followed by particles in the angular range of detectors 1, 2 and 3 without any condition about the other outgoing particles. On the contrary, Figure 5.13 shows the same cases presented in Figure 5.12 but imposing the constraint that the other outgoing particle has to be detected by PSD4 or PSD5. In particular, with reference to Figure 5.14, (where the matrix represents 73

76 5.9. Data Analysis and future work. Energy loss of d in 0.2 µm di Al in PSD1 Fit function [6] [2](1-exp([8]-[0]x) ) 2 [1] ([3]+[5]*x +[9]*exp([7]+[4]x)) χ 2 / ndf = / 39 p ± p ± p ± E PSD1 (MeV) p ± p ± p ± p ± p ± p ± p ± (MeV) DE Al -1 Figure 5.11: Energy loss function when a deuteron particle passes through the aluminium dead layer of PSD1 (thickness = 0.2µm). The analytic expression with all parameters is also shown. This is angular independent, so that it is the same for PSD2 and PSD3. in the x-axis the energy and in the y-axis the position angle) I identified two linear loci as an example. In the upper panel of Figure there are the two tracks chosen, corresponding to the reaction 6 Li + 13 C 17 O + d as detected by PSD3. Theoretical (black) points are well in agreement with the experimental data. In the lower panel, instead, I note that the corresponding tracks of the same reaction are at very low energies, where, because of noise sources and of difficult angular calibration, it is impossible to see clear tracks in the data. However the expected (theoretical) points fall inside the area of the experimental data so that we are confident of the agreement. After the calibration of the detectors, the next step of the data analysis is the selection of the events corresponding to the process of interest: the 13 C( 6 Li,n 16 O)d. This is accomplished first through a selection of the deuteron locus in the de-e two-dimension plot, as shown in Figure 5.15, where only hydrogen and helium loci are presented because heavier particles are unable to pass the de thickness. The expression of the average energy loss per unit length of charged particles other than electrons is known as the Bethe-Bloch equation (?). I give here an approximate expression, which is enough for our purposes. If z is the charge of the particle, ρ the density of the medium, Z its atomic number and A its atomic mass, the equation is: de dx ρzz2 A (5.35) 74

77 5.9. Data Analysis and future work. Two-body kinematics θ (degree) PSD1 PSD2 PSD N+ He 16 3 O+ H 17 2 O+ H 18 O+H E reaction (MeV) 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. The energy loss of relativistic charged particles more massive than electrons passing through matter is due to its interaction with the atomic electrons. The process 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 paths and they can be identified and selected by a graphical cut as done in Figure Theoretically, one should be able to discriminate also the different isotopes corresponding to a same charge value, but since the dependence of de/e on the atomic mass number A is only linear, this is usually not possible. The Q-value spectrum for the three-body reaction for the coincidence events is also a good variable to identify the right exit channel (where Q fot thethreebodyfinalstate isgiven bytherelation Q = E1+E2+E3 E beam ). A well separated peak, usually of Gaussian form, has to be centered around the theoretical value of MeV for each pair of detectors. The good agreement between the experimental and the theoretical Q values confirms the identification of the reaction channel as well as the accuracy of the calibration. Once the three-body reaction is selected, all the variables of interest can be calculated in order to perform the following steps of the analysis and to be compared with the theoretical-simulated ones. In particular, coincidence events are plotted as a function of relative energy E c.m. E c.m. = E13C α Q 2B (5.36) 75

78 5.9. Data Analysis and future work. Two-body kinematics - Coincidences with PSD4 and PSD5 θ (degree) PSD1 PSD2 PSD N+ He 16 3 O+ H 17 2 O+ H 18 O+H E reaction (MeV) 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 (k s 40 MeV/c) and representing predominantly the case of a quasi-free process. Q 2B is the Q-value for the two-body reaction. This energy spectrum represents the three-body excitation functions and it will be used for the extraction of the S factor through the already discussed equation: ( dσc.m. dω ) = d 3 σ de c dω c dω C ( KF Φ(ps ) 2) 1 (5.37) At this point of theanalysis, an observablewhich turnsout to be more sensitive to the reaction mechanism is the shape of the experimental momentum distribution, usually expressed in arbitrary units. AMonteCarlocalculationwasthenperformedtoextracttheKF Φ(k s ) 2 product. The momentum distribution entering the calculation is the Bakhadir function, which describes the momentum behavior of a deuteron inside an α particle (Pizzone et al., 2009). Following the PWIA prescription, the twobody cross-section dσ/dω c.m. was derived dividing the selected three-body coincidence yield by the result of the Monte Carlo calculation and using equation (5.37). As already mentioned, since this approach provides the off-energy-shell two-body cross section, it is necessary to perform the appropriate validity tests for the adopted impulse approximation. Since the next step in the TH analysis provides for the normalization and then the comparison with the direct data, the effect of penetrability through the Coulomb 76

79 5.9. Data Analysis and future work. Two-body Kinematics - Coincidences PSD4 and PSD3 10 θ3 (degree) θ4 (degree) Ereaction (MeV) 10 1 Figure 5.14: Coincidence events for the 6 Li+13 C 17 O+d reaction. The upper panel shows the matrix concerning PSD3 with two well evident kinematical linear loci corresponding to the 17 O+d exit channel. The agreement is good between theoretical points and experimental data. The lower panel is the PSD5 matrix, where the same reaction is plotted. barrier must be introduced, calculating the penetrability Gl expressed as: Gl (k13 C α R) = 1 Fl2 (k13 C α R) + Hl2 (k13 C α R) (5.38) where Fl and Hl are the regular and irregular Coulomb wave functions, while k13 C α and R are the relative wave number and the interaction radius, respectively (Spitaleri et al., 2001). Since equation (5.38) depends on the partial waves involved in the behavior of the cross section and since the excitation function can be actually expressed in terms of a coupling between a non resonant and a resonant term, the penetrability and the relative weight of such contributions must be taken correctly into account. The extracted excitation function is then calculated in form of the total astrophysical Sfactor by equation (4.9). Moreover, as already discussed in chapter 4, the Trojan Horse method offers the possibility to measure directly the bare nucleus astrophysical S factor and, by comparing the directly measured (screened) rate and the bare 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. We have very good expectations for the results of the 13 C(α,n)16 O cross section, but the work is too long to be covered completely by a single Master thesis. 77

80 5.9. Data Analysis and future work vs E 3 DE (ch) E Graphical cut DE 3 (ch) Figure 5.15: de/e two-dimensional plot for the telescope at the PSD3 position. The upper locus shows the charged particles with z=2 and the lower one is the region populated by z=1 particles where I expect to find deuterons. The graphical cut shows the data used in the analysis. For this reason the astrophysical consequences will be addressed in the next Chapter on the basis of a general overview of the possible changes in the rate we can expect, rather than on the basis of actual data derived from our measurements. 78

81 CHAPTER SIX ON THE ASTROPHYSICAL CONSEQUENCES OF CHANGES IN THE 13 C(α,N) 16 O RATE. 6.1 General remarks Although the data reduction of the measurements presented in this thesis is not yet complete (for indirect methods, as discussed previously, it is particularly long and critical) one knows qualitatively, a priori, the merits and limits of the indirect method adopted, hence the uncertainties that might affect the results. In particular, on one side we hope that our experimental contribution will permit a clear and unambiguous determination of the reaction rate ratio 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, the existing measurements are reported, down to energies of about 280 KeV. Our estimates will cover the lower energy range, below this value and across the 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 be extrapolated theoretically and still waits for an experimental verification. As our results provide relative measurements, we can obtain such a verification for what concerns the shape of the curve. This is of crucial importance, as theoretical extrapolations are very ambiguous, depending on the strength and width of low-energy resonances, especially on sub-threshold ones. As already mentioned, in our case most of the uncertainties in the theoretical estimates, which have raised so many controversies in the literature, descend from the presences of a resonance at -3KeV (in the center-of-mass energy scale), corresponding to an excited level of 17 O at 6.356MeV. This is the field in which our data will provide decisive clarifications. For the absolute calibration, instead, we shall necessarily rely on the data at high energies. According to the discussion of Chapter 5 this means 79

82 6.2. Effects of reducing the rate by a factor of three. that we have actually repeated some measurements above 280 kev, which we shall over-impose to older values for obtaining an absolute scaling of our measured energy dependence. The problem in doing this normalization is that the uncertainties in the range from a few hundred kev to a few MeV are still essentially at the level shownin thefigure5.2 (upto afactor-of-three, at the2σ level). Thiswill be therefore also the expected uncertainty of our normalization. Now, for the nucleosynthesis of neutron-capture elements for which the neutron flux is generated by the 13 C(α,n) 16 O reaction in He-burning conditions of evolved stars, we are interested not only in knowing the ratio of the rate at various energies (all included in a small range below and above 10 kev), but also the absolute value at each energy and especially at about 8 kev, which is the typical temperature achieved, when shell-h burning restarts, immediately below the stellar layer previously swept by the third dredge up (see Chapter 3 for a discussion). After our final data are available, further work will be needed, in a close collaboration between stellar astrophysics and nuclear physics, for deriving a more precise absolute calibration from observational constraints. I have thereforedecidedtoanticipate herethebasicsof thisworkinaseriesof tests, performed with the help of the neutron-capture nucleosynthesis code of the astrophysics group operating the Department of Physics of the University of Perugia. For this scope I have personally modified that code to allow for reaction rate changes. What I shall present in the next sections is therefore a discussion of the possible expected effects of variations of this rate within a factor-of-three around the value most commonly used in the calculations (i.e. the one from Drotleff et al., 1993). Inparticular, insection 6.2 Ishall discusstheeffects of reducing theratebythisfactor, whileinsection 6.3 Ishall consider theopposite, i.e. the effects of an increased rate. This series of tests will prepare the future work in the astrophysical field, mainly based on comparisons between nucleosynthesis models using the new rate and observations of abundances in both the solar system and AGB stars. 6.2 Effects of reducingtherateby afactor of three. In case of a reduction of the rate with respect to the one so far adopted in most s-process calculations, (i.e. the one from Drotleff et al., 1993), we can expect two types of changes in neutron-capture nucleosynthesis calculations. The first possible effect, whose consequences could be a priori the most dramatic, would be that of allowing part of the 13 C nuclei available to survive the radiative interpulse stage and burn then convectively, at higher temperature, during the thermal instability. If the amount of 13 C survived is sufficiently large, we might expect that a remarkable amount of energy is 80

83 6.2. Effects of reducing the rate by a factor of three. Pulse number Time scale for 13 C combustion Duration of the interpulse stage yr yr yr yr yr yr yr yr Table 6.1: Comparison between the time of combustion of 13 C in radiative conditions in the intershell and the one of interpulse for a star of 3 M and Z= Only the last four pulses are shown in table. It can be noted that for the last two pulses there is a certain amount of carbon ( ), which remains unburnt in the radiative region and can burn during the convective instability. deposited in the convective layer. In this case stellar evolution models teach us that the convective shell might undergo a splitting in two sublayers, separated from a radiative zone. If this occurs than the consequence is that the inner region, undergoing further α-capture burning and the activation of the 22 Ne(α,n) 25 Mg source, and containing all the newly produced s-process nuclei, would remain separated by the upper layers where the next TDU episode can occur. This would have the effect of preventing the pollution of the envelope with s-process elements, with the abortion of the thermallypulsing nucleosynthesis mechanism. This possibility was verified with the help of the FRANEC evolutionary code, whose use was granted by Sergio Cristallo, at the Observatory of Teramo(INAF). We here thank him and his collaborators for this possibility. By reducing the present 13 C(α,n) 16 O rate by a factor of three we found that the time scale for 13 C burning down to 1/100th of its initial abundance passes from about years to about years (see Table 6.1), hence at 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) 13 C would actually end up burning partly in the thermal pulse. In our tests this always occurred when its abundance had already been reduced remarkably, but independently of the amount entered into the convective region, at the typical values of the temperature ( K) and of the density (ρ = 10 4 g/cm 3 ) and for the local abundance of helium (0.7), the time scale for 13 C burning is: τ burn = X(He)ρN A < σv > 13,α = sec (6.1) (here the brackets indicate Maxwellian-averaging of the reaction rate). By contrast, the time required to carry 13 C to the bottom of the convective layer, where the temperature is high (up to K) is: τ mix = R/v conv = 495sec (6.2) 81

84 6.2. Effects of reducing the rate by a factor of three. where R (= R ) is the distance to the bottom and v conv (= cm/sec) is the average convective velocity. In such conditions virtually any amount of 13 C entering the pulse will burn only after reaching the bottom and no shell splitting can occur. I conclude that reducing the 13 C rate has no effect on the development of the convective instabilities. Moreover, since the abundance left is very small, also the effects on the neutron density and on the s-element distribution are bound to be minimal. A second important effect that can be expected concerns instead the lowest masses of our range (those with M in the range M ). Here the temperature in the thermal pulses is insufficient to ignite the 22 Ne(α,n) 25 Mgreaction, sothattheneutrondensityislimitedtothelowvalues generated duringradiative 13 C burning. Reducingthe rate for this burning means also reducing the total neutron density. On most nuclei this will have marginal effects, as the values commonly found with the Drotleff et al. (1993) rate were already very low (10 7 n/cm 3 ). However, it is a priori possible that for some special cases the further reduction of the neutron density due to the lower rate can be felt. We verified these effects with the already 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 13 C-rich layer formed in such stars is much larger than for higher masses, due to the reverse dependence of the efficiency of proton mixing phenomena on the initial stellar mass. In particular, choosing as an example a star of 1.3 M, with a metal content 1/3 solar (Z = 0.006) Figures 6.1 and 6.2 show the different efficiencies of the old and new models in producing s elements, when the rate of the 13 C(α,n) 16 O reaction is left unchanged. The increased efficiency is evident. The difference in the assumptions for the 13 C pocket between our current models and previous calculations are summarized in Table 6.2. Travaglio et al. (1999) Maiorca et al. (2011b) Region Depth X( 13 C) Depth X( 13 C) Table 6.2: 13 C pocket in the case of Travaglio et al. (1999) and Maiorca et al. (2011b) respectively. Using thenew models illustrated in Figure 6.2, weallowed the rate to decrease by a factor of three, which roughly means to reproduce the suggestions by Kubono et al. (2003). The results obtained for s process abundances in the mentioned AGB star of 1.3 M and Z = when the rate is reduced 82

85 6.2. Effects of reducing the rate by a factor of three. s-process elements for M=1.3M and Z=0.006 ) i /X i Log (X Legend r-only > 1% >20% >40% >60% >80% s-only Atomic Mass (A) Figure 6.1: The distribution of production factors with respect to the initial composition for elements above the iron-peak, for an AGB star of 1.3 M with a metallicity about one third the solar one, undergoing neutron capturenucleosynthesiswithneutronsproducedbythe 13 C(α,n) 16 O neutron source. Nuclei whose production is attributed to the s process at various percentage levels are indicated by different symbols and colors, as described in the label. Here the amount of 13 C burnt per cycle is the same as in Travaglio et al. (1999). in this way are shown in Figure 6.3 (lower panel) in terms of abundance ratios with respect to the results obtained with the presently-accepted rate. In the top panel the same ratio is shown for the old choice of the 13 C-pocket. It is clear that the only remarkable changes concern nuclei strongly affected by reaction branchings depending on the neutron density, like 96 Zr and, in particular, 86 Kr and 87 Rb. These last isotopes are close to the neutron-magic numbern=50(i.e. A=88)andtheirabundancesaredrasticallyreduced, by roughly 35%. These are crucial nuclei, at the connection between the main and weak s-process component, related to the complex reaction branching at 85 Kr illustrated in Figure 3.3. The case explored corresponds to such low neutron densities (n n = n/cm 3, against n n = n/cm 3 when using the Drotleff et al. (1993) cross section) that the flux through 85 Kr passes almost completely through the 85 Rb-branch, so that the two mentioned nuclei are not fed efficiently. As the abundances of these nuclei, and in particular 87 Rb, are used as tests for the neutron density in current stellar observations (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 more detailed analysis should be done, considering also the possible activation of 83

86 6.3. Effects of increasing the rate by a factor of three. 5 s-process elements for M=1.3M and Z= New ) i /X i Log (X Legend r-only > 1% >20% >40% >60% >80% s-only Atomic Mass (A) Figure 6.2: Same as Figure 6.1, but adopting the increased amount of 13 C burn per cycle suggested by our group in the paper Maiorca et al. (2011b) (Science, submitted). the further neutron source 18 O(α,n) 21 Ne, whose rate is uncertain and whose activation might become a proxy for the 22 Ne(α,n) 25 Mg neutron source, not activated because of the low temperature. We can therefore conclude this section by saying that the most remarkable effect of a reduction of the 13 C(α,n) 16 O rate would be seen in very low masses, and would affect the nuclei at the overlapping between the weak and the main component, modifying our ideas on the meaning of the observational neutron-density tests for AGB stars. 6.3 Effects of increasing the rate by a factor of three. If one artificially increases the rate for the 13 C(α,n) 16 O reaction, 13 C burns more efficiently and the neutrons are released in a shorter time interval, thus increasing the neutron density n n. As already mentioned, however, the neutron density due to the radiative 13 C burning is much smaller than the one subsequently expected by the operation of the 22 Ne(α,n) 25 Mg neutron source in the convective thermal pulse. As a consequence, an increase of n n in the radiative phase has only marginal effects on the ensuing element distribution. Measurable consequences are therefore limited, once again, to very low masses, where the 22 Ne(α,n) 25 Mg reaction is not activated for the too low ambient temperature. In population I stars (the stars of the 84

87 6.3. Effects of increasing the rate by a factor of three. 1.1 (rate Drot/3)/(rate Drot) for M=1.3M and Z= old 1 Fraction Atomic Mass (A) (rate Drot/3)/(rate Drot) for M=1.3M and Z= new Fraction Atomic Mass (A) Figure 6.3: Ratios of the abundances for heavy elements obtained by reducing the 13 C(α,n) 16 O reaction rate by a factor of three, with respect to those shown in Figures 6.1 and 6.2, obtained with the rate by Drotleff et al. (1993). The upper panel is for the 13 C reservoir by Travaglio et al. (1999), the lower panel for the choice by Maiorca et al. (2011b). galactic disk) this corresponds to stars below 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 a mass generally lower than solar. An example of the effects induced in such low masses by an increase by a 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) 13 C reservoir, the top panel that of the previously-accepted choice. For this second case the nuclei affected (all produced more efficiently) are mainly 86 Kr and 87 Rb, which experience a change opposite to what was seen in the previous section, with an increase between 25 and 30%. A few other branching-dependent nuclei show enhancements at a more limited level (up to10%): they include 96 Zr, 122 Sn, 123 Sband 142 Ce. Thechartofthenuclides around these last three isotopes is shown in Figure 6.5 in order to illustrate the reason of the change. As the two panels show, the nuclei under analysis are always placed after an unstable isotope and the increase of the neutron density favors their production. For the more recent choice of the 13 C-pocket the nuclei affected are the same but the changes are more remarkable. In particular, 86 Kr, 87 Rb, and 142 Ce show an increase by at least 30%. All these changes would be very important in the understanding of the solar distribution of neutron-capture 85

88 6.3. Effects of increasing the rate by a factor of three. 1.4 (rate Drot*3)/(rate Drot) for M=1.3M and Z= old Fraction Atomic Mass (A) (rate Drot*3)/(rate Drot) for M=1.3M and Z= new Fraction Atomic Mass (A) Figure 6.4: Ratios of the abundances for heavy elements obtained by increasing the 13 C(α,n) 16 O reaction rate by a factor of three, with respect to those shown in Figures 6.1 and 6.2, obtained with the rate by Drotleff et al. (1993). The upper panel is for the 13 C 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 sensitive observational tests (like those based on the Rb/Sr ratio) and would be critical in deducing predictions for the percentage of each nucleus that must be attributed to the r-process. 86

89 6.3. Effects of increasing the rate by a factor of three. Figure 6.5: The reaction branchings involving tin and antimonium isotopes (left panel) and Ce isotopes (right). 122 Sn, 123 Sb, and 142 Ce are placed after an 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

90 6.3. Effects of increasing the rate by a factor of three. 88

91 CHAPTER SEVEN CONCLUSIONS This thesis was primarily dedicated to the new measurement, obtained with the indirect method usually called of the Troian Horse (THM), of the reaction rate for the reaction 13 C(α,n) 16 O. The measurement wants to explore very low energies (below 280 kev), not covered by traditional measurements but very important in stellar interiors. In order to clarify the importance of the reaction chosen I outlined the phases of stellar evolution during which its activation is important and discussed the processes of slow neutron capture that are started by the neutrons that 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 induced at low energy by a virtual particle produced in a direct three-body reaction occurring at higher energies. I also discussed why this technique is so important for exploring the range in energy across the Gamow peak, which is of interest in stars. I then illustrated my activity (in progress) on the long and complex data reduction, which will be completed in about four-five months after the discussion of this dissertation. In order to know the possible astrophysical consequences of the measurement and prepare in advance the theoretical and observational tests that will be required, I performed a parametric study, by varying the cross section accepted today by a factor-of-three (in both directions) and performed s- process nucleosynthesis calculations putting in evidence the effects induced by changes in the rate. In this way I identified the basic consequences that can be expected (concentrated either on nuclei at the overlapping of the main and weak s-process components, or near reaction branchings sensitive to the neutron density). During the mentioned tests I participated to the preparation of a paper containing several new ideas on s-process nucleosynthesis, which is now 89

92 undergoing referee s scrutiny by the SCIENCE journal for publication. 90

93 LIST OF FIGURES 2.1 H-R diagram of a star of 1 M and Z=Z Comparison of energy produced by pp-chain and CNO cycle Stellar structure of a star in the TP-AGB phase Illustration of the structure of a TP-AGB star over time Observations of ls/fe with respect to hs/fe Valley of β-stability Behaviour of σ(a)n(a) as a function of mass number The complex branching of 85 Kr Internal structure of a TP-AGB star as a function of time Successive thermal pulses for the 3 M model with Z = Z Schematic representation of the thermal pulse history Schematic representation of the total nuclear potential Cross section and astrophysical factor The Gamow peak Sub-threshold resonance Representation of the potential between charged particles Pseudo-Feynman diagram for the break-up QF process Behavior of the astrophysical S-factor in NACRE Pseudo-Feynman diagram for the 13 C( 6 Li,n 16 O)d reaction Sequential processes for the interaction between 6 Li and 13 C Experimental setup of the 13 C( 6 Li,n 16 O)d reaction Schematic view of a position-sensitive detector (PSD) Schematic draw of a PSD and of its holder Position-energy two-dimension matrix Energy and position spectrum for a singular slit Calibrated position-energy matrix Energy loss function of a deuteron in 0.2µm of Al Two-body kinematical calculation for detectors 1, 2 and Two-body kinematical calculation - coincidence events

94 List of Figures 5.14 Coincidence events for the 6 Li+ 13 C 17 O+d reaction de/e two-dimensional plot for the telescope at PSD3 position Overabundances of s-elements for a star of 1.3M and Z= Same as Figure 6.1 adopting Maiorca et al. (2011b) 13 C-pocket Reduced 13 C(α,n) 16 O reaction rate - ratios of s-elements Increased 13 C(α,n) 16 O reaction rate - ratios of s-elements Reaction branchings A.1 Overview of the p-pi chain A.2 Overview of the CNO cycle A.3 Overview of the triple-alpha process

95 LIST OF TABLES 5.1 Table of reaction rates for 13 C(α,n) 16 O reaction Experimental conditions for the 13 C( 6 Li,n 16 O)d experiment Comparison between 13 C-burning time and time of interpulse Old and new 13 C pocket

96 94 List of Tables

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102 100 BIBLIOGRAPHY

103 CHAPTER EIGHT RINGRAZIAMENTI. Se dovessi ringraziare adeguatamente tutte le persone che hanno contribuito alla realizzazione di questa tesi di laurea, con tutta probabilità questa sezione sarebbe più lunga di tutto il resto. Il primo pensiero va ai miei genitori perchè, se sono la persona che sono diventato, lo devo a loro. Piera e Umberto sono persone semplici e oneste e li ringrazio per la libertà che mi hanno sempre 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 per quella disciplina meravigliosa che è la fisica. Grazie alla loro intraprendenza è stato infatti possibile realizzare un progetto decennale di congiunzione tra i due gruppi, al quale sono stato onoratissimo di prendere parte. Spero solamente di essere stato all altezza delle loro aspettative. Da loro ho imparato davvero tanto: dall infinitamente grande all infinitamente piccolo. Questa tesi non sarebbe mai venuta a compimento senza la collaborazione di tante persone. Sara Palmerini e Enrico Maiorca hanno fatto veramente tanto per me mantenendosi sempre disponibili, dandomi sempre i consigli giusti e insegnandomi tutto il possibile sull astrofisica. Un grazie particolare va a Marco La Cognata, del quale ho una stima immensa, per l infinito aiuto e la pazienza concessami fin dal primo giorno nell iniziarmi alla fisica nucleare, che non è poi così male. A Catania, all ombra di Liotru, ho avuto anche l onore di conoscere Iolanda e Luca, miei compagni impagabili di tesi, Livio e Roberta, persone talmente gentili e disponibili che, malgrado una distanza di quasi mille chilometri, mi hanno fatto sentire subito a casa. Tra Florida, Trojan Horse e cultura siciliana ho tante cose per cui esservi grato. Voglio inoltre ringraziare i restanti componenti del gruppo ASFIN2 cominciando dal Prof. Stefano Romano e Aurora Tumino che mi hanno gentilmente offerto accoglienza nel loro ufficio e continuando con Giuseppe, Letizia, Gianluca e Gabor. Se sono un pò sperimentale lo devo a tutti voi! Ovviamente, un caro saluto va anche a 101

104 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 mie idee, dandomi la possibilità di raggiungere questo importante traguardo. Mi ritengo, inoltre, un ragazzo davvero fortunato perchè nel corso degli anni ho potuto contare su amici sinceri con i quali ne ho passate davvero tante e che, tra momenti belli e meno belli, mi sono sempre stati vicini. Voglio davvero bene a Simone, Matteo, i miei insostituibili compagni di tanti anni di scuola, Ale, Alessandro, Renzo, Fizia, Matteo e Riccardo. Infine, voglio dedicare questa tesi di laurea alla mia dolce metà Alessia, che mi ha rubato il cuore e che è stata l oggetto di ogni mio pensiero. Grazie per 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 sia il mio presente e il mio futuro. Potrei davvero continuare all infinito, ma con le lacrime agli occhi è il momento di concludere con un semplice GRAZIE. 102

105 APPENDIX A MAIN THERMONUCLEAR REACTIONS IN PRE-AGB PHASES. A.1 Hydrogen (H) burning. As already discussed in previous chapters, thermonuclear reactions can occur only if the temperature (or equivalently the kinetic energy) of the particles is high enough to overcome their mutual electrostatic or Coulomb repulsion. For this reason and because of the large amount of hydrogen in the sun and in the universe, the first and most important nuclear reactions releasing energy are those involving protons(?). This idea, coupled with the discovery of the tunneling effect, was presented and discussed across the thirties and fourties. Atkinson and Houtermans were the first to suggest that, out of four protons and two electrons, a helium nucleus could be produced with the release of large amount of energy (Q TOT = MeV). Starting from Bethe (1939), it was clear that two different sets of reactions could convert sufficient hydrogen into helium, to provide the energy needed for a star s luminosity 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) into deuterium, releasing a positron and a neutrino, as one of the protons changes into a neutron H +H 2 H +e + +ν e (A.1) This reaction provides 1.44 MeV of energy, if I consider that Q is the total energy released in the process including the subsequent annihilation of the emitted positron. A temperature of ten million degree, and equivalently a 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 103

106 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 theoretical value is available. After this, the deuteron produced in the first stage can fuse with another hydrogen to produce the lighter isotope of helium, 3 He 2 H +H 3 H +γ (A.2) Inordertocreated 4 He, thenewlyformed 3 Hecanbeconsumedbyanumber of exothermic reactions through three different paths. The first one takes place at low temperatures (less than K) and proceeds predominantly by the following fusion reaction 3 He+ 3 He 4 He+2H (A.3) Thisistheso-called p-pichain(seefigurea.1). Atthispointthenetresultis the fusion of four protons into an α particle, two positrons and two electronic neutrinos. (A.3) is considered as the crucial reaction also for driving an inversion of the molecular weight (µ), promoting readjustments in the star, hence mixing. In fact, it provides a reduction of the mean molecular weight. 104

107 A.1. Hydrogen (H) burning. In order to activate the p-pii chain a preliminary presence of 4 He and a temperature included in the range K are necessary. The first reaction of this process creates 7 Be as follows: 3 He+ 4 He 7 Be+γ (A.4) Then, 7 Be decays to 7 Li by capturing an electron from its own K shell (or, alternatively, from the stellar plasma): 7 Be+e 7 Li+ν (A.5) and, after a proton capture, two nuclei of 4 He are finally produced: 7 Li+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: 3 He+ 4 He 7 Be+γ (A.7) 7 Be+p 8 B +γ (A.8) 8 B 8 Be+e +ν e (A.9) 8 Be 4 He+ 4 He+γ (A.10) The p-piii chain is dominant if temperatures exceed K. It has a negligible importance from the energy-production point of view, especially in 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 4 He synthesis in the ancient stellar objects, madeofpurehandhe, butforhighermetallicity stars, formedfrom an ISM enriched in carbon (C), nitrogen (N), oxygen (O), other reactions can contribute to the nuclear energy production on the Main Sequence. In 1939, Bethe proposed the independent set of reactions called CNO cycle. In order to produce 4 He starting from four protons, carbon, nitrogen, and oxygen nuclei are considered as catalysts: their individual abundances can change, but not their sum and they are linked by an endless loop. The CNO chain starts occurring at approximately K, but its energy output rises much faster with increasing temperatures (see Figure 2.1). At approximately K, the CNO cycle becomes the dominant source of energy. Hence, it is important especially in stars more massive than the sun. A reduced network, called CN cycle (because the only stable isotopes intervening are 12 C, 13 C, 14 N and 15 N) occurs for moderate temperatures. It contains the following reactions: 12 C +p 13 N +γ (A.11) 105

108 A.2. Helium (He) burning: triple-α process. 13 N 13 C +e + +ν e (A.12) 13 C +p 14 N +γ (A.13) 14 N +p 15 O+γ (A.14) 15 O 15 N +e + +ν e (A.15) 15 N +p 16 O 12 C +α (A.16) where the 1 2C used in the (A.11) reaction is regenerated in the (A.16). The energy production for each reaction cycle is MeV, from the conversion of H into 4 He and the rest from changes in the CNO isotopic mix. At higher temperatures (higher than K) also 16 O takes part in hydrogen burning, so that the cycle can be extended to: 15 N +p 16 O 16 O+γ (A.17) 16 O+p 17 F +γ (A.18) 17 F 17 O+e + +ν e (A.19) 17 O+p 18 F 14 N +α (A.20) 17 O+p 18 F 18 F +γ (A.21) 18 F 18 O+e + +ν e (A.22) This is the full CNO cycle (see Figure A.2), of which CN is only a part. Like the 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 proton captures on 18 O 18 O+p 19 F 15 N +α (A.23) 18 O+p 19 F 19 F +γ (A.24) If K T K, the 18 O(p,γ) 19 F rate is not negligible and the cycle 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 stellar core, while the remaining hydrogen continues the combustion in a thin external shell. All this happens during RGB phases that depend strongly on the initial mass of the star (see section 2.2). Generally speaking, the collapse of the stellar core brings the central temperature to near K (8.6 kev). At this point helium nuclei can fuse together at a rate high enough to rival the rate at which their product, 8 Be, decays into two helium nuclei, so that some equilibrium beryllium remains: 106

109 A.2. Helium (He) burning: triple-α process. Figure A.2: Overview of the CNO cycle. Figure A.3: Overview of the triple-alpha process. 107