Download The 13C(α,n)16O reaction rate. Recent estimates, new

Università degli studi di Perugia
Facoltà di Scienze Matematiche, Fisiche e
Naturali
Tesi di Laurea Magistrale in
Fisica
The
13
C(α,n)16O reaction rate. Recent
estimates, new measurements through
the Trojan Horse Method and their
astrophysical consequences.
Relatore
Candidato
Prof. Busso Maurizio Maria
Trippella Oscar
Prof. Spitaleri Claudio
Anno Accademico 2010/2011
Università degli studi di Perugia
Facoltà di Scienze Matematiche, Fisiche e
Naturali
Tesi di Laurea Magistrale in
Fisica
The
13
C(α,n)16O reaction rate. Recent
estimates, new measurements through
the Trojan Horse Method and their
astrophysical consequences.
Relatore
Candidato
Prof. Busso Maurizio Maria
Oscar Trippella
Prof. Spitaleri Claudio
Anno Accademico 2010/2011
...We are stardust, we are golden
We are billion year old carbon...
Woodstock - Crosby, Stills, Nash and Young
3
CHAPTER
ONE
INTRODUCTION.
Historically, stars have been part of religious practices and used for 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 nucle5
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 0.8 - 3 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 106 − 1010 n/cm3 .
In order to set the stages for the nuclear astrophysics processes of interest,
I shall first discuss the typical evolutionary phases for a star of one solar mass
(assumed to represent a low mass star in general). A particular 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 ∼ 0.09−0.1×109 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
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 0.1 × 109 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 (Eres = 6.356 Mev or Ec.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.
Our experiment was performed by measuring the sub-Coulomb 13 C(α,n)16 O scattering within the interaction region via the THM, applied to the 13 C(6 Li,n16 O)d reaction 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
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
CONTENTS
1 Introduction.
5
2 Final evolutionary stages for low mass stars.
2.1 pre-AGB phases. . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Asymptotic Giant Branch (AGB) stars and Thermal Pulse.
2.3 The third dredge-up. . . . . . . . . . . . . . . . . . . . . . .
2.4 Nucleosynthesis and observations for AGB stars. . . . . . .
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11
12
15
19
21
3 s-Process nucleosynthesis in AGB stars.
3.1 Introduction. . . . . . . . . . . . . . . . .
3.2 The classical analysis of the s process. . .
3.3 Evolution and nucleosynthesis in the AGB
3.4 The neutron source 13 C(α,n)16 O. . . . . .
3.5 Possible future scenarios. . . . . . . . . .
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4 Cross sections of nuclear reactions at low energies.
4.1 Coulomb barrier and penetration factor. . . . . . . .
4.2 Cross section, astrophysical factor and reaction rate.
4.3 Gamow peak. . . . . . . . . . . . . . . . . . . . . . .
4.4 Direct measurements and experimental problems. .
4.5 Indirect methods for nuclear astrophysics . . . . . .
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40
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45
49
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stages. .
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5 Measure of the 13 C(α,n)16 O reaction through the THM.
51
5.1 Theory of the Trojan Horse method. . . . . . . . . . . . . . . 52
5.2 Plane Wave Impulse Approximation. . . . . . . . . . . . . . . 54
5.3 Current measurement status . . . . . . . . . . . . . . . . . . . 58
5.4 The Trojan Horse Method applied to the 13 C(α,n)16 O reaction. 62
5.5 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . 64
5.6 Position Sensitive Detectors (PSDs). . . . . . . . . . . . . . . 68
5.7 The position calibration. . . . . . . . . . . . . . . . . . . . . . 70
5.8 Energy calibration. . . . . . . . . . . . . . . . . . . . . . . . . 72
5.9 Data Analysis and future work. . . . . . . . . . . . . . . . . . 73
9
CONTENTS
6 On
6.1
6.2
6.3
the astrophysical consequences of changes in the 13 C(α,n)16 O rate. 79
General remarks . . . . . . . . . . . . . . . . . . . . . . . . . 79
Effects of reducing the rate by a factor of three. . . . . . . . . 80
Effects of increasing the rate by a factor of three. . . . . . . . 84
7 Conclusions
89
8 Ringraziamenti.
101
A Main thermonuclear reactions in pre-AGB phases.
A.1 Hydrogen (H) burning. . . . . . . . . . . . . . . . . .
A.1.1 pp-Chain. . . . . . . . . . . . . . . . . . . . .
A.1.2 CNO-cycle. . . . . . . . . . . . . . . . . . . .
A.2 Helium (He) burning: triple-α process. . . . . . . . .
10
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103
103
103
105
106
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 Mmin ∼ 0.08 M⊙ (M⊙ being the so-called solar
mass, corresponding to about 1.9891 × 1030 kg), the temperature is not high
enough to start hydrogen burning. In this work I shall limit my discussion to
stars belonging to the mass range 0.8 − 3 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 mp have a
very small mean free path l, to the point that they are almost in contact to
each other. This means that:
1/3 mp 1/3
µmH 1/3
1
=
=
l∼
n
ρ
ρ
11
(2.1)
2.1. pre-AGB phases.
has a numerical value close to the particle dimension, defined by the De
Broglie’s wavelength:
h̄
(2.2)
λ=
mp v
s
3kB T
where v indicates the thermal velocity v =
. Then:
mp
mp
ρ
1/3
from which I get ρ:
1/3
ρ
=
h̄
=
mp
r
mp
3kB T
5/6 √
mp
3kB T
h̄
√
3
3kB T
3/2 5/2
ρ=
T 3/2 m5/2
mp
p ∝T
h̄
(2.3)
(2.4)
(2.5)
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
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 Mbol versus
effective temperature Tef f .
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 (MJ ):
!
3/2
T
(2.6)
MJ = 2 · 1035 1/2
n
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
and in typical conditions of interstellar clouds corresponds to about 1000M⊙ .
Hence, if a cloud is more massive than this critical value the collapse can
13
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 Tef f 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 = 107 K, ρ = 100 g/cm3 )
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 = 26M eV (this
roughly corresponds to the Q-value resulting from the chain of reactions, see
Appendix A). For initial temperatures lower than about 18 × 106 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 T0 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
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, triple alpha reactions (4 He(2α,γ)12 C), which rapidly increase the
core luminosity, are ignited at the point of maximum temperature, but with
a degenerate equation of state. The temperature and density (∼ 108 K and
∼ 107 g/cm3 ) 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 oxy15
2.2. Asymptotic Giant Branch (AGB) stars and Thermal Pulse.
gen. The ascent of the AGB begins following the exhaustion of helium at 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 2500 − 6000 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
(above 105 g/cm3 ) and the C-O core degenerates and cools down with a huge
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 ∼ 104 (MCO − 0.5)
where L and MCO are measured in solar unities.
During the early phases (E-AGB), for all stars less massive than 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 (LHe /LH ∼
10−3 ). Late on the AGB, the stellar structure, schematically represented in
Figure 2.3, is characterized by a C-O core, two shells (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
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 < MCO /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 108 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
region is X(12 C) ∼ 0.25) are mixed over the whole intershell. 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
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
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 ∼ 0.9 × 108 K.
This neutron exposure lasts for about 10 - 20 thousand years with a very
low neutron density (106 to 107 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 3 × 108 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
neutron density up to 1010 n/cm−3 , depending on the maximum temperature
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 104 to 105 years, depending on the
core-mass.
2. The He-shell suddenly starts burning very strongly, producing luminosities up to ∼ 108 L⊙ . The energy deposited by these He-burning reactions
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 where they can be observed was found
19
2.3. The third dredge-up.
since the first numerical models for TP-AGB stages. In 1977, Iben 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:
∆MT DU
(2.7)
λ≡
∆MH
This is the ratio between the mass carried to the surface at each thermal
pulse, ∆MT DU , and the mass processed by the H-burning shell during the
interpulse phase, ∆MH . 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 (∆MT DU ) 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 kν , coupling
radiation to matter) and the value of the free parameter αP characterizing
the so-called mixing length lM treatment of convection. This last quantity
determines the mean free path of a convective eddy in units of the pressure
scale HP . One can use it to describe the transport of heat in convective conditions. In all evolutionary calculations for AGB stages, αP 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 αP 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 im20
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 0.8 − 0.9 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 ∼ 0.7 - 0.9), enriched in elements heavier than iron, contain the
unstable isotope 99 Tc (τ = 2 × 105 years) in their spectra. It was clear
that ongoing nucleosynthesis occurred in situ in their interior and that the
products were mixed to the surface. The fact that Tc is widespread in S stars
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 the 1960s (Sanders, 1967). AGB stars 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 hy21
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 C chain. Stellar model calculations (see e.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 (0.9 × 108 K,
or 8 keV), and the average neutron density never exceeds 1 × 107 n/cm3 .
As a consequence of neutron captures, a pocket of s-enhanced material 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). Such comparisons
confirm also for C(N) stars the existence of an intrinsic spread in the abun22
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
2.4. Nucleosynthesis and observations for AGB stars.
24
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
a more accurate knowledge of the 13 C(α,n)16 O reaction rate. In this 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 B2 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 bar25
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 B2 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 1023 n/cm3 ) allow the creation of very heavy (A 209) 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
106 to 1010 n/cm3 . 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
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 B2 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 Ns products between neutron-capture cross sections and s-process
abundances. The mentioned authors built the experimental distribution of
σNs 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 σNs curve, by computing analytically the s-process contributions
Ns to each isotope. As a consequence, the ratio (N (A) − Ns (A))/N (A)
yielded a prediction on the fractional abundances due to the more complex
r process. In a slow neutron-capture process, the abundance of an isotope
Ath varies in time through destruction and creation mechanisms:
dN (A)
= N (A − 1)nn hσ((A − 1), v)vi − N (A)nn hσ(A, v)vi
(3.1)
dt
where hσ(A, v)vi indicates the Maxwellian-averaged product of cross section
and relative velocity, and nn 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:
Z
τn = nn vT dt
(3.2)
27
3.2. The classical analysis of the s process.
This differential equation then becomes:
dN (A)
= N (A − 1)nn hσ((A − 1), v)vi − N (A)nn hσ(A, v)vi
(3.3)
dτ
In steady state conditions, production equals destruction, the time derivative
vanishes and hσ(A)N (A)i = const.
This simplified relation is rather well satisfied in the experimental solarsystem distribution of σNs 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 = 208 - 209, 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 hσ(A)N (A)i, 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
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 number of 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 /τ0 )
(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)Ns (A) = GN56 τ0
A
Y
Ai=56
1 + (σ(Ai )τ0 )−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 τ0 . Ns (A) represents the part
of the abundance NA due to the slow neutron capture.
Concerning the main component, the mean exposure τ0 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)
showed how the AGB phases of intermediate mass stars can indeed mimic 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
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 fβ can be defined by comparing the rates for β-decay and neutron capture, so that fβ = λβ /(λβ +λn ),
where λn = Nn hσib vT . Here hσib 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 σNs 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 accu30
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 18 × 106 K,
the CNO-cycle is the main energy source. This stage was the longest in stellar life, it was the so-called main sequence (MS). Core hydrogen goes on until
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 moves to the red and to increase luminosity, and then climbs a track
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 (E31
3.3. Evolution and nucleosynthesis in the AGB stages.
AGB) C-O core can increase and warm because of helium burning in shell. 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 (LHe /LH < 10−3 ), the
mass of the He intershell MH − MHe 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 in the case of degenerated core. In fact, these events are caused by
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 105 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 αP (the ratio of the mixing length l and the
pressure scale height HP ). 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 10-50 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 (MH ),
of the He-burning shell (MHe ) and of the bottom border of the convective
envelope (MCE ). 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
3.3. Evolution and nucleosynthesis in the AGB stages.
Figure 3.4: Plot of the internal structure of a TP-AGB star as a function of
time, for a 3 M⊙ model with Z = 0.02 (Straniero et al., 1997). The positions
in mass of the H-burning shell (MH ), of the He-burning shell (MHe ), and of
the bottom border of the convective envelope (MCE ) 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 26t 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
3.4. The neutron source
3.4
The neutron source
13
13 C(α,n)16 O.
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
Due to its natural occurrence, this neutron source was the first to be explored
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 1010 − 1012 n/cm3 and needs a temperature larger
than 3 − 3.2 × 108 K to be activated. The maximum temperature achieved
in LMS at the bottom of TPs barely reaches T = 3 × 108 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 =
0.8 − 1.0 × 108 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. The amount of 13 C thus obtained must burn through the 13 C(α,n)16 O reaction in layers where the temperature is low (T ≤ 0.8 − 1.0 × 108 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 en34
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
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
forming almost pure 13 C and hence to activate efficiently the 13 C(α,n)16 O reaction 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 3 × 108 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 (1010 n/cm3 ). 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
14 N
35
3.4. The neutron source
13 C(α,n)16 O.
Figure 3.5: Two successive thermal pulses (in particular, the 29th and
30th ) for the 3 M⊙ model with Z = Z⊙ are shown in their relative 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 1.5 × 108 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 107 n/cm3 . 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
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 sprocessed 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 20000 years, leaving several thousand
years before the growth of the next convective instability (at least 30000 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 = 0.8 − 1.0 × 108 K. Here s
processing takes place locally under radiative conditions generating an senhanced 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
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 nn 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.
On the basis of the scenario described above, it was shown by Travaglio et al.
(1999) that the chemical evolution of s-elements up to the solar 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 Cpocket 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
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 106 to
109 K. So, nuclear reactions take places at very low energies, of the order of
a few keV, because of the equation E = kB T , where kB 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 working in
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 in39
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:
EC =
Z1 Z2 e2
Rn
(4.1)
where Rn = R1 + R2 is the nuclear radius, Z1 and Z2 represent the integral
charges of the interacting nuclei. Classically, a reaction can occur only between particles with energies higher than Ec . Incident projectiles at lower
energies would reach the closest distance to the nucleus at the turning point
RC 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 < EC 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(Rn )|2
|χ(Rc )|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:
d2 χl 2µ
+ 2 [E − Vl (r)] = 0
dr 2
h̄
40
(4.3)
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 Vl (r)is the potential for the lth partial wave resulting when the centrifugal potential term is also present (Clayton et al., 1983)
Vl (r) =
l(l + 1)h̄2 Z1 Z2 e2
+
2µr 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 = Z1 Z2 e2 /h̄v, 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
4.2. Cross section, astrophysical factor and reaction rate.
for nucleon. More complex reactions lead to the production of the heavier
elements. These processes can take place through a slow (s) or a rapid (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 10−24 cm2 (or lower), for convection and
convenience, a new unit of area, the barn (b), equal to 10−28 m2 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
1
(4.7)
σ = πλ2DB
E
where λDB represents the De Broglie wavelength:
λDB =
h̄
mx + mX
mX
(2mx Ex )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 < Ec . Recalling equation (4.5) and (4.7), it is possible to factorize the cross section
as:
1
(4.9)
σ = S(E)e−2πη
E
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
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
down to EL , which is the lower limit for the beam energy at which experimental 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).
rxX =
1
Nx NX hσ(v)vi
1 + δxX
(4.10)
where the product Nx NX represents the total number of pairs of nonidentical nuclei X and x. For identical particles the Kronecker symbol δxX
is introduced, otherwise each pair would be counted twice. The bracketed
quantity hσ(v)vi 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
2πkB T
43
3/2
exp
−mv 2
2kT
(4.11)
4.3. Gamow peak.
1
by introducing the center of mass energy E = µv 2 with µ representing the
2
reduced mass of interacting particles, the reaction rate is then expressed as:
1/2
Z ∞
E
1
1
8
σ(E)Eexp −
rxX =
Nx NX
dE
1 + δxX
πµ
kB T
(kb T )3/2 0
(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 hσ(v)vi 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:
!
1/2
Z ∞
E
EG 1/2
1
8
S(E)exp −
−
dE (4.13)
hσvi =
µπ
kB T
E
(kB T )3/2 0
where the symbol EG 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 E0 :
kB T 3/2 1/2
E0 =
(4.14)
EG
2
The convolution of the two functions results into a peak, the so-called
Gamow peak, centered near the energy E0 and generally much larger than
kB T . The maximum value of the integrand will be:
2E0
(4.15)
Imax = exp
kB 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
∆ = √ (E0 kB T )1/2
3
44
(4.16)
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 E0 .
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 E0 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
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
particles collected (Nd ), the so-called signal events, on the detector according
to the equation
Nd ∝ σNi τ ∆Ω
(4.18)
where Ni 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 Nd :
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 Nd might increase by improving the experimental setup, this
number is affected by the background noise events Nb , 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:
Nd
≫1
(4.19)
Nb
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 Er of an excited state of the compound nucleus does not
exceed the Q-value for the reaction (Q = mx + mX − my − mY , where m are
the mass of involved particles). In this case the resonance peak lies below
46
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 (Ra )
Vef f =
Z1 Z2 e2 Z1 Z2 e2
−
Rn
Ra
(4.20)
Then the projectile effectively sees a reduced Coulomb barrier. As a consequence, at low energies the cross section for screened nuclei, σs (E) (also
shielded cross section), is enhanced, with respect to the cross section of the
bare nucleus σb (E), by a factor:
πηUe
Ss
σs
∝
∼ expo
(4.21)
f (E) =
σb
Sb
E
47
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 Ra 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 Ue , representing the screening potential for the studied reaction, must
be taken into account to determine the bare nucleus cross section. For
E/Ue > 1000 the electron screening effects are negligible so one essentially
measures σb (E), while if E/Ue < 100 one (Langanke et al., 1996) experimentally have an enhancement on the cross section, σs (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 kB 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 RD of:
RD =
kB T
4πe2 ρNA ξ
1/2
(4.22)
whereNA is the Avogadro number and the quantity ξ is expressed by the
48
4.5. Indirect methods for nuclear astrophysics
equation:
ξ=
X
Zi2 + Zi
i
Xi
Ai
(4.23)
Here the sum is performed over all positive ions and Xi is the mass fraction
of the ith nucleus of charge Zi . 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:
hσvis = g(E) hσvib
(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 σs (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 σb 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
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
CHAPTER
FIVE
MEASURE OF THE
13
C(α,N )16O 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
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 (1010 n/cm3 ). 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 (0.8 −
1.0) × 108 K. The corresponding Gamow peak (Rolf & Rodney, 1988), in
according with equations (4.14) and (4.16), is at Ecm = 190 ± 90 keV, so
that the direct data available stop at the right edge of Gamow window, while
the Coulomb barrier is about
Z1 Z2 e2
Z1 Z2 e2
∼ 3.7M eV
(5.1)
=
Ec =
1/3
1/3
Rn
r0 (A + A )
1
2
where I use r0 = 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,n16 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
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
ψA = ψx (rx )ψs (rs )ψ(rx − rs )
(5.4)
where the ψi 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
to what the wooden epic horse did for Ulysses and his comrades, it hides in its
53
5.2. Plane Wave Impulse Approximation.
interior the transferred, participating cluster x. In this way, the Trojan Horse
reaction (5.2) can be performed at energies well above the corresponding Ec ,
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 ks 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 ks ,
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
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) = hr|ki =
1
2π
3/2
exp(ik · r)
(5.5)
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)
d3 σ
2 dσax
∝ KF |Φ(ps )|
(5.6)
dEc dΩc dΩC
dΩ
This is the cross section for the scattering of a particle c into the solid angle
dΩc with an energy between Ec and Ec + dEc and of particle C into the
solid angle dΩC . 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(ps ), 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:
Z ∞
|Φ(ps )| = (2π)3/2
ψ(r)exp(−iKs r)dr
(5.7)
−∞
dσ
The last term ( ) 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 d3 kc ,d3 kC and d3 ks respectively
dσ =
(2π)4 3
2
d kc d3 kC d3 ks δ(Ki − Kf )δ(Ei − Ef )|t3B
fi |
|vrel |
(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 vrel = ka /Ea is the
55
5.2. Plane Wave Impulse Approximation.
relative velocity between the incident particle and the target. The variable
t3B
f i , appearing in (5.8), is the three-body reduced matrix element and it is
related to Tf i by the equation
Tf i = δ(Ki − Kf )t3B
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:
(5.10)
Tf i = hf |T |ii = kc , kC , ks , ψc , ψC , ψs |T 3B |ψa , ψA , ka
T 3B is the complete T -operator for the reaction, while ψi is the wave function that describes the internal generic particle i and ki 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:
Tf i = kc , kC , ks , ψc , ψC , ψs |T 2B |ψa , ψA , ka
Z
= d3 qx hkc , kC , ψc , ψC |T 2B |ψa , ψx , ka , qx ihψx , ψs , qx , ks |ψA i
(5.11)
If ψA represents the intrinsic state of the target and qi is the momentum of
particle in initial state, the correspondent momentum space wave function
is
hqs , qx , ψs , ψx |ψA i = Φ(k)δ(qs + qx )
(5.12)
and from the momentum conservation in the laboratory system where the
target is at the rest one gets:
qx + qs = 0
qx = −qs
(5.13)
qs = ks
(5.14)
Hence, the final form of the equation for Tf i is
Z
Tf i = d3 qx hkc , kC , ψc , ψC |T 2B |ψa , ψx , ka , qx iΦ(k)δ(q2 + ks )
= hkc , kC , ψc , ψC |T 2B |ψa , ψx , ka , −ks iΦ(−ks )
(5.15)
One can note that the momentum of the cluster x before the collision, qx ,
is equal and opposite to the momentum of the outgoing residual nucleus ks ,
56
5.2. Plane Wave Impulse Approximation.
a quantity that is experimentally measured. Substituting for Ki = ka and
Kf = kc + kC + ks and transforming to the c.m. and the relative momenta
for the two-body system, I can write
Tf i = δ(ka − kc − kC − ks ) kf |T 2B |ki Φ(−ks )
= δ(ka − kc − kC − ks )t2B
f i Φ(−ks )
(5.16)
At the end, the form of equation (5.8), using (5.9) and (5.16), becomes:
dσ =
(2π)4 2
2
k dkc dΩc kC
dkC dΩC d3 ks δ(Ei − Ef )δ(ka − kc − kC − ks )|
|vrel | c
2
×φ(−ks )|2 |t2B
fi |
(5.17)
Integrating over dks , which in this reaction is unobserved, and over dkc and
considering the two following conditions:
q
p
2
m2s + (ka − kc − kC )2 (5.18)
Ef = Ec + EC + Es = Ec + m2C + kC
Ei = Ea + mA
(5.19)
one obtains:
EC Es
(2π)4
dσ
=
2 E E k E + E [k − k cosθ + k cos(θ − θ )] |
dEc dΩc dΩC
ka kc kC
a c C s
C C
a
C
c
c
C
2
×φ(−ks )|2 |t2B
fi |
(5.20)
Here, θc and θC 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, ka = kc + kC + ks , and the energy
conservation, Ef = Ei . 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:
2 E E2
(kc kC
dσ
s c.m.
=
dEc dΩc dΩC
ka Ex {kC Es + EC [kC − ka cosθc + kc cos(θc − θC )]}
dσ
2
(5.21)
×|φ(−ks )|
dΩ c.m.
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
5.3. Current measurement status
terms are defined as the kinematical factor, given by the following expression
(Jain et al., 1970):
KF =
2 E E2
(kc kC
s c.m.
ka Ex {kC Es + EC [kC − ka cosθc + kc cos(θc − θC )]}
(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:
−1
dσax
d3 σ
KF |Φ(ps )|2
(5.23)
=
dΩ
dEc dΩc dΩC
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 Gl (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 inside the TH nucleus, makes it possible to extract the a(x, c)C cross
section by using the relation (5.6)
X dσl T HM
dσ
(5.24)
=
Gl
dΩ cC
dΩ cC
l
Here, Gl represents the transmission coefficient for the lth partial wave.
Now, one can notice a very important point, which will be recalled in
the last chapter. Because of the factor Gl , the two-body cross section 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
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
from Sekharan et al. (1967), Davids (1968), Bair et al. (1968), Drotleff et al.
(1993) and Brune et al. (1993), covering the energy range between 0.28 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 108 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+ subthreshold resonance of 17 O at an excitation
energy of 6.356 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
5.3. Current measurement status
techniques, which tried to explore the low temperatures typical of stars
(0.8 − 1.0 × 108 K) in order to have a more accurate knowledge of 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. The new data covered the energy range 350keV ≤ E ≤ 1.4M eV , 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:
"
2 #
T9
−33.093
6.788 × 1015
−
exp
NA hσvi =
1/3
330.271
T92
T9
1016.988
6.259
1/3
2/3
exp −
× 1 + 0.485T9 − 7.948T9 + 10.725T9 +
+
3/2
T9
T9
+
3.474 × 105
3/2
T9
8.430
exp −
T9
(5.25)
This relation is valid over the range 0.01 ≤ T9 ≤ 1.0 (T9 hereafter means the
temperature in units of 109 K). Because of the state in 17 O just below the
α-threshold and using their own measurements over the range 0-300 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:
−1/3
NA hσvi = exp −36.90392 + 0.07784191T9−1 − 48.15691T9
60
5.3. Current measurement status
1/3
5/3
×exp +109.3879T9 − 21.95909T9 + 3.161556T9 − 25.92545logT9
(5.26)
Hence, the reaction rate is smaller than the NACRE recommended value
roughly by a factor 4 at T9 = 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 (0.8 − 1.0 × 108 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 NA hσvi, in analogy with (4.12) and (4.13), as follows:
NA hσvi = NA
8
πµ
1/2
1
(kb T )3/2
Z
∞
0
E
σ(E)Eexp −
kB T
dE
(5.27)
Here, NA is the Avogadro number (∼ 6.022 × 1023 mol−1 ), µ is the reduced
mass of the system, kB 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 NA hσvi is in units of cm3 mol−1 s−1 .
T9
0.08
0.09
0.10
T9
0.08
0.09
0.10
Caughlan et al.
(1988)
1.32 × 10−16
2.25 × 10−15
2.58 × 10−14
Kubono et al. (2003)
1.05 × 10−16
1.77 × 10−15
2.02 × 10−14
Drotleff et al. (1993)
Angulo et al. (1999)
2.77 × 10−16
4.18 × 10−15
4.32 × 10−14
Johnson et al. (2006)
1.49 × 10−16
2.41 × 10−15
2.64 × 10−14
4.80 × 10−16
6.99 × 10−15
6.99 × 10−14
Pellegriti et al. (2008)
3.36 × 10−16
5.41 × 10−15
5.94 × 10−14
Table 5.1: Table of reaction rates present in the literature for 13 C(α,n)16 O .
T9 is the temperature in the interpulse phase expressed in units of GK.
Accurate measurements were then performed by Johnson et al. (2006)
at the Tandem-LINAC facility of the Florida State University. In this work,
we had the first case of the application of the asymptotic 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
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 performed at two different sub-Coulomb energies of 13 C. For temperatures above
T9 = 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, T9 = 0.08 − 0.1, 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
The Trojan Horse Method applied to the 13C(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,n16 O)d three-body reaction (Q3B = 0.74128
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
same but obviously it has to be adapted at the projectile break-up case, 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:
kbeam = qx + qs
(5.28)
62
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,n16 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 ki 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 EB , thus determining the so-called quasi-free (La Cognata et al.,
2007) two-body energy (Eqf ), which is given by:
Eqf = Eax − EB =
ma
Ex − EB
mx + ma
mx
ma
EA − EB
(5.29)
=
mx + ma mA
Here, Ex represents the fraction of the beam energy EA corresponding to
the cluster x , Eax is the relative energy between a and x and mi 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
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 En16 O , Ed16 0 and
Edn 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 TANDEM 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
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
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 setup of the 13 C(6 Li,n16 O)d reaction discussed in
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/cm2 , 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
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
17.33 degree up to 44.25 degrees on the other side of the chamber were used
to measure the yield of the Oxygen recoils. The thicknesses of PSDs, sumPSD
1
2
3
4
5
Distance (cm)
29.2
28.0
24.5
25.5
21.3
Central angle (deg)
8.04
23.05
37.93
22.93
37.56
Angular range (deg )
3.15 - 12.93
17.95 - 28.15
32.11 - 43.75
17.33 - 28.53
30.87 - 44.25
Thickness (µm)
1000
500
500
500
500
Table 5.2: Experimental conditions for the 13 C(6 Li,n16 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,n16 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. De67
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:
x
(5.31)
P =E
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
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
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
4000
50
45
40
3000
P5 (ch)
35
30
2000
25
20
15
1000
10
5
00
1000
2000
E5 (ch)
3000
4000
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 reac70
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
Energy
800
Position
mean 2146.02
σ
800
19.68
σ
20.87
Counts
600
Counts
600
mean 1262.22
400
400
200
200
0 1900 2000 2100 2200 2300
EPSD5 (ch)
0 1100 1200 1300 1400 1500
PPSD5 (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 θ0 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 − P0
E − E0
(5.32)
where E0 and P0 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 5.10. 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 = θ0 + arctg[c1(x − x0 ) + c2(x − x0 )2 ]
71
(5.33)
5.8. Energy calibration.
c1 , c2 and x0 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 Th alpha-source at six peaks was used because the energy decay
is well known for each peak; elastic scattering on heavy 197 Au nuclei and on
a 12 C target were used to measure both elastic and inelastic scattering at the
beam (6 Li) energy Eb =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:
EM eV = (a + bEch )(1 + c1 (θ − theta0 ) + c2 (θ − θ0 )2 )
(5.34)
Here, a, b, c1 , c2 are constants resulting from the minimization 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):
1. 12 C+6 Li, the so-called elastic scattering (Q = 0 MeV)
2. 14 N+α (Q = 8.79805)
3.16 O + d (Q = 7.60641)
4. 17 O + p (Q = 5.68764)
In order to have a check of the procedure I plotted theoretical points
(black points) corresponding to the cited reactions over the matrix and 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 5.10. 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
5.9. Data Analysis and future work.
Calibrated position-energy matrix
55
6
12
C + Li
-4.43MeV
12
6
C + Li
0MeV
50
14
N + 4He
2.35MeV 3.69MeV
45
2.96MeV
θ5 (degree)
40
50
35
30
25
45
20
15
40
10
5
0
5
E5 (MeV)
10
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 PSD5.
5.9
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
5.9. Data Analysis and future work.
Energy loss of d in 0.2 µ m di Al in PSD1
Fit function
[2](1-exp([8]-[0]x)
[6]
)
([3]+[5]*x 2 +[9]*exp([7]+[4]x))
[1]
χ2 / ndf = 46.96 / 39
EPSD1 (MeV)
p0
-2
10
10-2
10-1
DEAl (MeV)
-4.652 ± 0.02484
p1
0.3429 ± 0.001597
p2
0.01781 ± 0.0002334
p3
0.5677 ± 0.01843
p4
-3.691 ± 0.1651
p5
1.571 ± 0.06253
p6
-3.829 ± 0.02968
p7
2.145 ± 0.01945
p8
0.04317 ± 0.001969
p9
-0.0623 ± 0.002717
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,n16 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:
−
ρZz 2
dE
∝
dx
A
74
(5.35)
5.9. Data Analysis and future work.
30
20
PSD2
θ (degree)
40
PSD3
Two-body kinematics
15
N+4He
16
3
10
PSD1
O+ H
17
O+2 H
18
O+H
0
5
Ereaction (MeV)
10
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
5.15. 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
the three body final state is given by the relation Q = E1+E2+E3−Ebeam ).
A well separated peak, usually of Gaussian form, has to be centered around
the theoretical value of 0.74128 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 Ec.m.
Ec.m. = E13 C−α − Q2B
(5.36)
75
5.9. Data Analysis and future work.
30
20
PSD2
θ (degree)
40
PSD3
Two-body kinematics - Coincidences with PSD4 and PSD5
15
N+4He
16
3
10
PSD1
O+ H
17
O+2 H
18
O+H
0
5
Ereaction (MeV)
10
Figure 5.13: Two-body kinematical calculation in the region of detector 1,
2 and 3, assuming coincidence events with PSD4 or PSD5.
restricting by a condition of a low spectator momentum (ks ≤ 40 MeV/c)
and representing predominantly the case of a quasi-free process. Q2B is 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
=
−1
d3 σ
KF |Φ(ps )|2
dEc dΩc dΩC
(5.37)
At this point of the analysis, an observable which turns out to be more sensitive to the reaction mechanism is the shape of the experimental momentum
distribution, usually expressed in arbitrary units.
A Monte Carlo calculation was then performed to extract the KF |Φ(ks )|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/dΩc.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
5.9. Data Analysis and future work.
Two-body Kinematics - Coincidences PSD4 and PSD3
10
θ3 (degree)
45
40
35
0
5
10
1
10
θ4 (degree)
30
25
20
0
5
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
5.9. Data Analysis and future work.
DE3 vs E3
50
4000
45
40
3000
35
E3 (ch)
30
2000
25
20
15
1000
10
Graphical cut
5
00
1000
2000
DE3 (ch)
3000
4000
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
CHAPTER
SIX
ON THE ASTROPHYSICAL CONSEQUENCES OF
CHANGES IN THE 13C(α,N )16O 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
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
shown in the Figure 5.2 (up to a factor-of-three, at the 2σ level). This will 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
therefore decided to anticipate here the basics of this work in a series of tests,
performed with the help of the neutron-capture nucleosynthesis code of 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). In particular, in section 6.2 I shall discuss the effects of
reducing the rate by this factor, while in section 6.3 I shall consider the opposite, 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 reducing the rate by a factor 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
6.2. Effects of reducing the rate by a factor of three.
Pulse number
24
25
26
27
Time scale for 13 C combustion
27261 yr
27890 yr
27261 yr
27890 yr
Duration of the interpulse stage
30500 yr
28780 yr
27580 yr
26270 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=0.006. 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 (5 × 10−7 ), 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 18000 years to about 27000 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 (1.5×108 K) and of the
density (ρ = 104 g/cm3 ) and for the local abundance of helium (0.7), the
time scale for 13 C burning is:
τburn = X(He)ρNA < σv >13,α = 4.5 × 105 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 3× 108 K) is:
τmix = ∆R/vconv = 495sec
81
(6.2)
6.2. Effects of reducing the rate by a factor of three.
where ∆R (= 1.8×10−3 R⊙ ) is the distance to the bottom and vconv (=
2.5×105 cm/sec) is the average convective velocity. In such conditions 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 1.2 − 1.4 M⊙ ).
Here the temperature in the thermal pulses is insufficient to ignite the
22 Ne(α,n)25 Mg reaction, so that the neutron density is limited to the low values generated during radiative 13 C burning. Reducing the 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 (107 n/cm3 ). 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.
Region
1
2
Travaglio et al. (1999)
Depth
X(13 C)
4.0 ×10−4 2.0 × 10−3
5.3 ×10−4 4.25 × 10−3
Maiorca et al. (2011b)
Depth
X(13 C)
2 × 10−3
5 × 10−3
2 × 10−3
3 × 10−3
Table 6.2: 13 C pocket in the case of Travaglio et al. (1999) and
Maiorca et al. (2011b) respectively.
Using the new models illustrated in Figure 6.2, we allowed 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 = 0.006 when the rate is reduced
82
6.2. Effects of reducing the rate by a factor of three.
s-process elements for M=1.3M and Z=0.006
Log (Xi/Xi )
3
Legend
r-only
> 1%
>20%
>40%
>60%
>80%
s-only
2
1
0
50
88
100
150
200
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
capture nucleosynthesis with neutrons produced by the 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
number N =50 (i.e. A = 88) and their abundances are drastically reduced, 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 (nn = 1×107 n/cm3 , against nn = 1.3×107 n/cm3 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
6.3. Effects of increasing the rate by a factor of three.
s-process elements for M=1.3M and Z=0.006 - New
5
Log (Xi/Xi )
4
Legend
r-only
> 1%
>20%
>40%
>60%
>80%
s-only
3
2
1
0
50
88
100
150
200
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 nn . 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
nn 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
6.3. Effects of increasing the rate by a factor of three.
(rate Drot/3)/(rate Drot) for M=1.3M and Z=0.006 - old
1.1
Fraction
1
0.9
88
0.8
0.7
0.6
50
100
Atomic Mass (A)
150
200
(rate Drot/3)/(rate Drot) for M=1.3M and Z=0.006 - new
1.1
Fraction
1
0.9
88
0.8
0.7
0.6
50
100
Atomic Mass (A)
150
200
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 1.3 − 1.4 M⊙ . Moreover,
all the AGB stars presently observed in population II stellar systems (e.g.
Globular Clusters, of low metallicity) should share this property, being of 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
to 10%): they include 96 Zr, 122 Sn, 123 Sb and 142 Ce. The chart of the nuclides
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
6.3. Effects of increasing the rate by a factor of three.
(rate Drot*3)/(rate Drot) for M=1.3M and Z=0.006 - old
1.4
Fraction
1.3
88
1.2
1.1
1
0.9
50
100
Atomic Mass (A)
150
200
(rate Drot*3)/(rate Drot) for M=1.3M and Z=0.006 - new
1.4
Fraction
1.3
88
1.2
1.1
1
0.9
50
100
Atomic Mass (A)
150
200
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
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 sprocess chain.
87
6.3. Effects of increasing the rate by a factor of three.
88
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 QuantumMechanics 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 sprocess 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
undergoing referee’s scrutiny by the SCIENCE journal for publication.
90
LIST OF FIGURES
2.1
2.2
2.3
2.4
2.5
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 . . . . . . . . . .
13
14
16
18
23
3.1
3.2
3.3
3.4
3.5
3.6
Valley of β-stability. . . . . . . . . . . . . . . . . . . . . . .
Behaviour of hσ(A)N (A)i 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. . . .
.
.
.
.
.
.
26
28
30
33
36
37
4.1
4.2
4.3
4.4
4.5
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.
.
.
.
.
.
41
43
45
47
48
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
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,n16 O)d reaction. . .
Sequential processes for the interaction between 6 Li and 13 C.
Experimental setup of the 13 C(6 Li,n16 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 3. . .
Two-body kinematical calculation - coincidence events. . . . .
53
59
63
65
66
68
69
70
71
73
74
75
76
91
.
.
.
.
.
List of Figures
5.14 Coincidence events for the 6 Li+13 C→17 O+d reaction. . . . . 77
5.15 dE/E two-dimensional plot for the telescope at PSD3 position. 78
6.1
6.2
6.3
6.4
6.5
Overabundances of s-elements for a star of 1.3M⊙ and Z=0.006
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. . . . . . . . . . . . . . . . . . . . . . . .
83
84
85
86
87
A.1 Overview of the p-pI chain. . . . . . . . . . . . . . . . . . . . 104
A.2 Overview of the CNO cycle. . . . . . . . . . . . . . . . . . . . 107
A.3 Overview of the triple-alpha process. . . . . . . . . . . . . . . 107
92
LIST OF TABLES
5.1
5.2
Table of reaction rates for 13 C(α,n)16 O reaction. . . . . . . .
Experimental conditions for the 13 C(6 Li,n16 O)d experiment. .
61
67
6.1
6.2
Comparison between 13 C-burning time and time of interpulse
Old and new 13 C pocket . . . . . . . . . . . . . . . . . . . . .
81
82
93
List of Tables
94
BIBLIOGRAPHY
Abia C., Busso M.M., Gallino R., Dominguez I., Straniero O., Isern J., Ap.
J., 559, 1117, 2001.
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100
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 cosı̀ 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
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
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 (QT OT = 26.73 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+ + νe
(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
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)
In order to created 4 He, the newly formed 3 He can be consumed by a number
of exothermic reactions through three different paths.
The first one takes place at low temperatures (less than 15 × 106 K) and
proceeds predominantly by the following fusion reaction
3
He +3 He →4 He + 2H
(A.3)
This is the so-called p-pI chain(see Figure A.1). At this point the net result is
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
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 15 − 23 × 106 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)
Be + p →8 B + γ
(A.8)
B → Be + e + νe
(A.9)
Be → He + He + γ
(A.10)
7
8
8
8
−
4
4
The p-pIII chain is dominant if temperatures exceed 23 × 106 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, made of pure H and He, but for higher metallicity stars, formed from
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 13 × 106 K, but its
energy output rises much faster with increasing temperatures (see Figure
2.1). At approximately 17 × 106 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
105
(A.11)
A.2. Helium (He) burning: triple-α process.
13
N →13 C + e+ + νe
13
14
15
15
1 2C
C + p →14 N + γ
N + p →15 O + γ
O →15 N + e+ + νe
N +p→
16
∗
O →
12
C +α
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
where the
used in the (A.11) reaction is regenerated in the (A.16).
The energy production for each reaction cycle is 26.77 MeV, 26.73 from the
conversion of H into 4 He and the rest from changes in the CNO isotopic mix.
At higher temperatures (higher than 2 × 106 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
16
17
17
O + p →17 F + γ
F →
17
O+p→
17
+
O + e + νe
18
∗
(A.18)
(A.19)
N +α
(A.20)
O + p →18 F ∗ →18 F + γ
(A.21)
18
F →
14
(A.17)
F →18 O + e+ + νe
(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 2 × 107 K ≤ T ≤ 7 × 108 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 ∼ 100 × 106 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
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
A.2. Helium (He) burning: triple-α process.
4
He +4 He →8 Be
(A.25)
Be +4 He →12 C + γ
(A.26)
This means that there are always a few 8 Be nuclei in the core, which can
fuse with yet another helium nucleus to form 12 C:
8
This is the so-called triple-α process (see Figure A.3). The net energy release
of the process is 7.275 MeV.
108