Nuclear-binding energies

Let us consider a nucleus made of Z protons and N neutrons (A=Z+N), and with mass M(Z, N). The binding energy B(Z, N) of this nucleus is defined as the energy required to break it up into the A individual nucleons. Using Einstein relation of mass-energy equivalence, it is defined by

(1)$$M(Z,N)c^2 = Nm_{\rm n} c^2 + Zm_{\rm p} c^2 - B(Z,N),$$

where m _n and m _p are the neutron and proton mass, respectively (m _nc ² = 939.57 MeV and m _pc ² = 938.27 MeV). The nuclear force makes the mass of the nucleus smaller than the sum of the individual nucleon masses. The difference is the binding energy that is positive for bound nuclei.

The binding energy per nucleon B(Z, N)/A is displayed in Fig. 1. This graph illustrates many important properties of nuclei. Iron is the most tightly bound nucleus, as its binding energy is B(Z, N)/A = 8.8 MeV for the ⁵⁶Fe isotope. In the low-mass region, ⁴He is strongly bound, and almost behaves as an elementary particle. It is often referred to as the α particle. This high binding energy also explains why the α + p and α + α systems (⁵Li and ⁸Be) are unstable: they immediately breakup.

Fig. 1. Binding energy per nucleon B(Z, N)/A as a function of the mass number (http://www.dlt.ncssm.edu/tiger).

The behaviour of the nuclear binding energy with A in Fig. 1 shows that, for A<56, energy is released by increasing the mass or, in other words, by capturing a nucleon or an α particle. This is the origin of fusion reactions occurring in stars and in fusion reactors. In contrast, for masses A>56, nuclei increase their binding energy (or, equivalently, reduce their mass) by emitting particles. In this mass region, many nuclei are unstable by α emission. Spontaneous fission occurs in the uranium region (A ≈ 200 and above).

In nature, there are 330 stable isotopes of 82 elements, from hydrogen (H, Z=1) to lead (Pb, Z = 82), with the exceptions of Technetium (Tc, Z = 43, T _1/2 ∼ 10⁶ years for ⁹⁸Tc) and Promethium (Pm, Z = 61, T _1/2 ∼ 17 years for ¹⁴⁵Pm) and 9 unstable isotopes with Z>82. There are no stable elements with A = 5 and 8, due to the strong binding energy of the α particle (A = 4).

Most visible matter in Universe is made of atomic nuclei comprising protons and neutrons. Bound nuclei are those with mass smaller than the sum of masses of their free constituents. There are about 3000 known bound nuclei, which decay by weak interaction emitting electrons (e⁻) or positrons (e⁺) and neutrinos. This process is known as the β radioactivity. Heavy bound nuclei can decay by α emission and there are some examples of exotic proton decays.

In Fig. 2, we show the distribution of isotopic abundances in the solar system. The isotopic abundances vary on a scale of 10¹², H and He being the most abundant elements with 75% for H and 25% for ⁴He. The most stable element, iron, corresponds to a peak in the abundance distribution. Other signatures of nuclear physics properties are present in this diagram. Peaks above A = 100 are associated with magic numbers, corresponding to nuclei with high excitation and breakup energies. These nuclei are therefore more abundant in stars. The same comment can be given when comparing even and odd nuclei. It is well known in nuclear physics that odd nuclei are less bound than even nuclei, and are therefore more fragile.

Fig. 2. Isotopic abundances of the solar system, as a function of the nuclear mass number A (reprinted from José & Iliadis Reference José and Iliadis2011 with permission from IOP).

The experimental data on chemical composition are collected not only from the composition of the Earth and of the Moon but also from meteorites and from absorption lines of the photosphere of the Sun. It is verified that the bulk isotopic composition of the solar system is homogeneous. The chemical composition of distant stars is obtained from the electromagnetic spectra emitted by their atoms and molecules.

Interaction potential between nuclei

The potential V(r) between two nuclei depends on their relative coordinate r, and involves nuclear V _N(r) and Coulomb V _C(r) contributions. Both are presented schematically in Fig. 3, where we use standard units of nuclear physics (lengths are expressed in fm = 10⁻¹⁵ m, and energies in MeV = 10⁶ eV). The nuclear potential is not known precisely, but is attractive at short distances, and tends rapidly to zero when r increases. In contrast V _C(r) is repulsive and extends to large distances.

Fig. 3. Typical nuclear (V _N) and Coulomb (V _C) potentials. In the lower panel, V _B is the Coulomb barrier and the arrow shows the energy range of astrophysical interest.

A typical energy in nucleus–nucleus collisions is the Coulomb barrier. Its height V _B represents the maximum of the potential. At relative energies E lower than V _B, any reaction would be classically forbidden. However, quantum mechanics allows tunnelling effects, and the fusion of two nuclei is possible even for E < V _B. This probability, also called penetration factor P(E), presents a fast energy dependence as

(2)$$P(E)\sim {\rm exp}( - 2{\rm \pi} {\kern 1pt} {\rm \eta} ),$$

where η is the Sommerfeld parameter, which provides a ‘measurement’ of Coulomb effects. It is defined by

(3)$${\rm \eta} = Z_1 Z_2 e^2 /\hbar\! v,$$

$v$ being the relative velocity and Z ₁, Z ₂ the charges of the colliding nuclei. At low energies, reaction cross sections between charged particles follow this energy dependence, and therefore drop to very low values when E≪V _B. The situation is different for neutron induced reactions, where η = 0. However, the short lifetime of the neutron (∼15 min) limits their role in astrophysics to specific processes.

The nucleus–nucleus potential V(r) also contains a centrifugal term ħ ²L(L + 1)/2μr ², depending on the angular momentum L. This additional repulsive contribution increases the height of the Coulomb barrier, and therefore still reduces the penetration factor. At low energies (E≪V _B), L = 0 is dominant.

Astrophysical S-factor

The main characteristic of a reaction is the cross section σ(E), which has the dimension of a surface (1 barn = 10⁻²⁸ m²), and depends on energy. As mentioned before, the cross section at stellar energies is governed by Coulomb effects. The astrophysical S-factor is defined as

(4)$$S(E) ={\rm \sigma} (E)E\;{\rm exp}(2{\rm \pi} {\kern 1pt} {\rm \eta} ).$$

It removes the fast Coulomb dependence and is mainly sensitive to the nuclear effects.

An example is shown in Fig. 4 with the ³He(α, γ)⁷Be reaction. The cross section has a strong variation with energy (it follows the exp(−2πη) dependence) and it drops quickly to very low values (∼10⁻¹⁰ barns), making impossible the extrapolation to lower energies. In contrast, the S-factor puts the nuclear effects in evidence and presents a weak energy dependence, which turns simple the extrapolation to low energies. The relevant astrophysical energy is around 23 keV for this reaction.

Fig. 4. ³He(α, γ)⁷Be cross section (lower panel) and S-factor (upper panel), as a function of the energy.

Reaction rates and stellar energies

The important quantity in stellar models is the reaction rate 〈σv〉, which represents the averaged value of the cross section times the relative velocity. The study of stellar evolution requires a large number of reaction rates for different reactions (Clayton Reference Clayton1983; Rolfs & Rodney Reference Rolfs and Rodney1988; Iliadis Reference Iliadis2007). The star is considered as a gas in equilibrium, where the energy distribution is given by the Maxwell–Boltzmann function N(E, T). The reaction rate therefore depends on the temperature T of the star.

According to the energy dependence (2) of the non-resonant cross section, the reaction rate is obtained by integrating over energy

(5)$${\rm \sigma} (E) v N(E)\sim {\rm exp}( - 2{\rm \pi} {\kern 1pt} {\rm \eta} ) \times {\rm exp}( - E/k_{\rm B} T),$$

where k _B is the Boltzmann constant. The former term stems from the penetration factor, and the latter from the Maxwell–Boltzmann distribution. Both terms present very different energy dependences and this product can be approximated by

(6)$$\exp ( - 2{\rm \pi} {\kern 1pt} {\rm \eta} ) \times {\rm exp}( - E/k_{\rm B} T) \sim {\rm exp}( - ((E - E_{\rm G} )/2\Delta )^2 ),$$

where E _G and Δ define the Gamow peak. These quantities can be easily determined from the properties of the system (masses and charges) and from the temperature (see Clayton Reference Clayton1983 for details). The Gamow peak defines the energy range where the cross section must be known to derive the reaction rate. In practice E _G is much lower than the Coulomb barrier, and typical stellar energies are very low at the nuclear scale. For example the Gamow energy of the important reaction ¹²C(α, γ)¹⁶O is E _G = 300 keV (at the typical Helium burning temperature T = 2 × 10⁸ K), whereas the Coulomb barrier is approximately V _B∼3 MeV.

A general problem in nuclear astrophysics is that the cross sections at the Gamow energy are too small to be measured in the laboratory. Recently underground laboratories have been developed (as for example LUNA in Italy; Broggini et al. Reference Broggini2010) to reduce background effects coming from cosmic rays. This technique allows very low counting rates, but of course raises technological and practical problems. Until now the ³He+³He → α + 2p (important in the Sun) is the only reaction where the cross section has been measured in the Gamow peak. For many reactions, theoretical models are necessary to extrapolate the available data or to predict the cross sections.

Let us also mention that many reactions present resonances, in particular for heavy elements. In that case the reaction rate contains a resonant contribution, in addition to the non-resonant term discussed above. The resonant term depends on the properties of the resonances, such as their spin, energy and width. We refer the reader to Clayton (Reference Clayton1983) for more details.

Radiative capture

In the radiative-capture process, two nuclei, A and B, fuse to the excited final nucleus C, with emission of γ-rays. This reaction is denoted as A+B→γ+C, or A(B, γ)C. The (p, γ) reactions of the capture of protons, or the alpha capture (α, γ) are the most important, and are among the most common reactions occurring in stellar environments. The neutron capture (n, γ) is important in the formation of heavy nuclei (s- and r-processes).

The capture cross sections are usually small since they occur through the electromagnetic interaction, which is weaker than the nuclear force. Due to the small cross section they can put limit to the nuclear processing. The final nucleus C can be excited in several final states, necessary to be detected in order to obtain the total reaction cross section.

At energies where the cross section is not too small, capture cross sections can be studied in laboratory through a variety of methods: the detection of the recoiling C nucleus, the direct detection of the γ-rays and delayed decay measurements. Typical examples are ²H(p, γ)³He which is the first capture reaction in hydrogen burning, ⁷Be(p, γ)⁸B which determines the solar-neutrino spectrum at high energies, or ¹²C(α, γ)¹⁶O which influences the ¹²C/¹⁶O ratio in stars. Many capture reactions, essentially involving protons or α particles, play a key role in stellar evolution.

Transfer reactions

In transfer reactions, nucleons are exchanged between the target and the projectile. They are denoted as A + B→C + D or A(B, D)C, and also play an important role in nuclear astrophysics. Typical examples are ³H(d, n)⁴He, ⁶Li(p, α)³He or ¹³C(α, n)¹⁶O. In this notation, A is the target, B the projectile and D the detected particle.

The process occurs through the nuclear interaction and the cross sections σ_t are much larger than those of radiative-capture reactions σ_C for the same entrance channel. This can be seen in Fig. 5, where the S-factors of the ⁶Li(p, α)³He and ⁶Li(p, γ)⁷Be reactions are compared as a function of energy. They have the same entrance channel ⁶Li + p, and therefore present similar energy dependences. However, there is a factor of 10⁴ between the absolute values, in favour of the transfer process. This factor is typical of the ratio between the strengths of the nuclear (strong) and electromagnetic interactions. As a general statement, if the transfer channel is open, the radiative capture can be neglected since σ_C≪σ_t. Another consequence is that transfer cross sections can be measured at lower energies than capture cross sections.

Fig. 5. Comparison of the S-factors for the ⁶Li(p, α)³He and ⁶Li(p, γ)⁷Be reactions.

Weak capture reactions

These reactions occur through the weak interaction, and the corresponding cross sections are far below the experimental possibilities. The p(p, e⁺ν)d is, however, very important, since it is the first reaction of the hydrogen burning pp chain. It presents a tiny cross section and there are no experimental measurements. Stellar codes make use of theoretical calculations, which are expected to be very precise since the pp interaction at low energies is well understood. Owing to the very small cross section of this reaction, hydrogen burning is a slow process, our Sun has a long life time, and life could develop on Earth.

The weak-capture reaction ³He(p, e⁺ν)⁴He plays a role in the neutrino spectrum, since it produces high-energy neutrinos that can be detected in terrestrial experiments. However, the cross section is made even smaller due to nuclear physics properties of this reaction. Only theoretical estimates are available.

Timescale of events

The Big Bang occurred 13.7 billion years ago (13.7 Gyr). It was an extremely dense and hot initial state, followed by rapid expansion and cooling. In the first stage relativistic particle–antiparticle pairs were created and destroyed in collisions continuously, constituting what we call today quark–gluon plasma.

At t = 10⁻⁶ s, quarks and gluons combine to form protons and neutrons. At the existing very high temperature particles (electrons, neutrinos, protons and neutrons) and photons were in equilibrium. At t = 10 s and T ∼ 3 GK the decay of free neutrons into protons with half-life T _1/2 = 614 s became the dominant weak interaction (José & Iliadis Reference José and Iliadis2011), decreasing the neutron-to-proton number ratio n _n/n _p. At this temperature both the processes of creation (p + n→d + γ) and destruction (d + γ → p+n) of the deuteron were possible.

However, due to cooling, at t = 250 s and T = 0.9 GK, even the high energy tail of the Planck distribution has a photon energy that dropped below 2.2 MeV, the destruction of deuteron by photons could not occur anymore and the primordial nucleosynthesis, through successive nuclear reactions, started. At that time, the decay of free neutrons gave rise to a neutron-to-proton number ratio n _n/n _p∼1/7.

Due to cooling, at t = 20 min the Coulomb barrier between the existing nuclei suppressed all nuclear reactions and the Big Bang nucleosynthesis stopped. Afterwards, until the formation of the first stars, no new nuclides were created or destroyed (except ³H and ⁷Be that are unstable and decay by β emission).

At t ∼ 380 000 years and T − 3000 K, nuclei recombined with electrons to form neutral atoms. Consequently, the mean-free path of photons increased, and they could travel without interacting with matter. At that time, photons were decoupled from baryons and the Universe became transparent to electromagnetic radiation.

Today, after t − 13.7 Gyr, due to cooling and expansion, photons have continuously lost energy; photons have a black body spectrum of 2.725 K, isotropic with fluctuations of 10⁻⁵. The WMAP has measured the temperature distribution of this microwave spectrum in all directions of the sky. The WMAP observation and the reliability of cosmological parameters deduced from these measurements is a key piece of evidence for the Big Bang cosmological model.

⁴He production

After t = 250 s, the subsequent nuclear reactions were relatively fast and nearly all neutrons were incorporated into the tightly bound element ⁴He, with a small fraction of other nuclides. A simple counting argument allows estimating the primordial ⁴He abundance: for a neutron-to-proton ratio n _n/n _p∼1/7, in each eight nucleons, one proton and one neutron will end up bound in ⁴He. Consequently the primordial ⁴He mass fraction becomes X _α^pred ∼ 2/8 = 0.25.

Observations of ⁴He (in clouds of ionized HII in dwarf galaxies), extrapolated to zero-metallicity, give the value X _α^obs=0.248 ± 0.003 (Peimbert et al. Reference Peimbert, Luridiana and Peimbert2007). Very recently, Izotov & Thuan (Reference Izotov and Thuan2010) presented a new determination of the primordial mass fraction of helium, based on 93 spectra of 86 low-metallicity extragalactic HII regions. The new value is 0.2565±0.0010(stat)±0.0050(syst). This value is higher, at the 2σ level, than calculations using the Standard Big Bang Nucleosynthesis (SBBN). The authors claim that this disagreement demands for modifications in the SBBN model.

From this reasoning, it becomes clear that the ⁴He abundance depends on the free neutron half-life, on the weak interaction cross sections, on the neutron–proton mass difference and on the expansion rate, but it is insensitive to the baryon density and to the nuclear reaction cross sections. The destruction of deuterium occurred by the d(d, n)³He, d(d, p)t and by the t(d, n)⁴He reactions. Tritium was mainly produced by the d(d, p)t and ³He(n, p)t reactions, and destroyed by t(d, n)⁴He. The ⁴He nuclide was mainly produced by the t(d, n)⁴He and destroyed by the ⁴He(t, γ)⁷Li reactions. The ⁷Be nuclide was mainly produced by ³He(α, γ)⁷Be, which decays to ⁷Li, with a half-life of T _1/2 = 53.3 days. The pro-duction of ⁷Li was mainly via ⁷Be, but also through ⁷Be(n,p)⁷Li and the destruction through the ⁷Li(p, α)⁴He reaction. Direct cross section measurements exist for most of the above reactions at the relevant energies (Descouvemont et al. Reference Descouvemont2004).

In Fig. 6, the primordial abundances of elements produced shortly after the Big Bang (250 s to 20 min) are shown, as a function of the baryon density η, also called the baryon-to-photon ratio. The above-mentioned insensitivity of the ⁴He abundance can be seen in the figure, as well as the fairly strong dependence of the other nuclides, as Deuterium, ³He and ⁷Li on η. The vertical line corresponds to the WMAP baryon density.

Fig. 6. Primordial abundances of light elements as a function of the baryon density η. Ref.: NASA/WMAP Science Team.

Primordial abundances are calculated using reaction network codes (Wagoner et al. Reference Wagoner, Fowler and Hoyle1967) and using all existing information on production and destruction reactions. The only free parameter for the primordial nucleosynthesis in the standard cosmological model is the baryon density η or baryon-to-photon ratio. The rates of nuclear reactions depend on this parameter and the final abundances also. If the WMAP value of η = (6.2 ± 0.2) × 10⁻¹⁰ is adopted, the SBBN becomes a parameter-free model.

The primordial abundance of ⁴He using the WMAP baryon density gives X _α^pred = 0.2486±0.0002 (Cyburt et al. Reference Cyburt, Fields and Olive2008), in very good agreement with older observational values given above, but at a 2σ level disagreement with more recent measurements.

Deuterium and lithium production

There is no alternative to the Big Bang for synthesizing deuterium: processes in stars destroy it rather than produce it. The predicted number abundance ratio of deuterium to hydrogen, using the WMAP η value is (D/H)^pred = (2.5 ± 0.2) × 10⁻⁵ (Cyburt et al. Reference Cyburt, Fields and Olive2008). Observation of absorption lines (Lyman-α) in low-metallicity gas clouds yields (D/H)^obs = (2.8 ± 0.2) × 10⁻⁵ (Pettini et al. Reference Pettini2008). As the deuterium abundance depends strongly on η, this agreement is a key piece of evidence in favour of SBBN.

There is a large disagreement between the predicted and observed values and the primordial Lithium abundance is the central unresolved problem of the SBBN model. The predicted lithium to hydrogen abundance ratio is (⁷Li/H)^pred = (5.2±0.7) × 10⁻¹⁰ (Cyburt et al. Reference Cyburt, Fields and Olive2008). The measurement of the ⁷Li primordial abundance is a challenge, and carries a large systematic error; the most recent value is (⁷Li/H)^obs = (1.1−2.3) × 10⁻¹⁰ (Ryan et al. Reference Ryan2000; Asplund et al. Reference Asplund2006; Bonifacio et al. Reference Bonifacio2007).

Beryllium and boron production

The lithium, beryllium and boron elements have very low abundances as we can see in Fig. 2. Due to the non-existence of stable A = 5 and 8 isotopes, the Big Bang nucleosynthesis almost stopped at A = 7. In the nuclear burning in stars, they are by-passed by the triple α capture that goes directly from ⁴He to ¹²C. These fragile and rare nuclides are mostly destroyed in stellar interiors. Their production in an alternative model, called inhomogeneous BBN (IBBN) was proposed (Malaney & Fowler Reference Malaney and Fowler1988; Kajino et al. Reference Kajino, Mathews and Fuller1990; Lara et al. Reference Lara, Kajino and Mathews2006).

Hydrogen burning

Hydrogen is the most abundant element in the Universe. In the Hydrogen burning, four protons fuse through different reactions, with a ⁴He nucleus as final product. The total energy release is 26.731 MeV. This process can occur in two different ways: the pp chains and the CNO cycle. Both are briefly discussed below.

The pp chains correspond to different paths to synthesize helium from hydrogen. These paths are given in Table 1. The first two reactions (p(p, e⁺ν)d and d(p, γ)³He) are common to the three chains. The pp1 chain is the most important since it involves the ³He(³He, 2p)α transfer reaction which has a high cross section. This cross section has been measured down to 15 keV, which correspond to the central temperature of the Sun (15 × 10⁶ K).

Table 1. Reactions (or α and β decays) involved in the pp chains

The pp2 and pp3 chains involve capture and β decays. The pp3 chain plays a minor role in the nucleosynthesis, but is important for the solar neutrino problem. The ⁷Be(p, γ)⁸B reaction produces high-energy neutrinos in the subsequent β decay of ⁸B. This reaction has been recently studied by different groups.

If the star contains a small fraction of ¹²C, this nucleus can act as a catalyst. This process is known as the CNO cycle, and also converts four protons into an α particle. The main CNO cycle is given in Table 2. All reactions involved in the CNO cycle are fairly well known. The last reaction ¹⁵N(p, α)¹²C could in principle compete with the ¹⁵N(p, γ)¹⁶O reaction. However, the (p, α) cross section is much larger than the (p, γ) cross section (see Fig. 5), and the initial ¹²C nucleus is reformed at the end of the cycle. As for the pp chains, the end result of each process is the transformation of four protons in an α particle, i.e. 4p→⁴He + 2e⁺ + 2ν.

Table 2. Reactions (or β decays) involved in the CNO cycle

Other variants of the CNO cycle exist, but play a role in explosive burning only. For example, if the temperature is high enough, the ¹³N(p, γ)¹⁴O reaction is faster than the ¹³N β decay. This leads to the ‘hot’ CNO cycle, and other reactions must be considered. This variant of the CNO cycle is initiated by the ¹³N(p, γ)¹⁴O reaction, which involves the radioactive nucleus ¹³N (T _1/2≈10 min). In those conditions, traditional experiments, using a proton beam, cannot be used, since the lifetime of ¹³N is too short to be considered as a target. Significant progresses have been carried out with the availability of radioactive beams (see ‘Primordial nucleosynthesis’ section).

Helium burning

Hydrogen burning is the longest phase in the evolution of a star. When hydrogen is consumed in the core, the star contracts and its central temperature increases. In those conditions, helium burning can start. However, the α + p and α + α reactions are impossible since ⁵Li and ⁸Be are unbound, and immediately break up. Helium burning therefore proceeds through the triple α process, where ⁸Be can exist for a short time, in equilibrium with α + α, and captures a third α particle by the ⁸Be(α, γ)¹²C reaction.

The second step of the 3α process, i.e. the ⁸Be(α, γ)¹²C reaction, provides a striking example of the interplay between astrophysics and nuclear physics. The observed abundance of ¹²C in the Universe can only be explained if ¹²C presents a 0⁺ resonance in the Gamow energy range. This resonance was predicted by Hoyle in 1954 from anthropic arguments (a carbon-based life), and discovered experimentally a few years later. This 0⁺ state of ¹²C is known as the ‘Hoyle state’, and its properties are now firmly established.

The ¹²C(α, γ)¹⁶O follows the 3α process. This reaction determines the ¹²C/¹⁶O ratio after helium burning, and is crucial in many stellar models. As for ⁸Be(α, γ)¹²C the cross section is enhanced by specific nuclear properties of ¹⁶O. Owing to the high Coulomb barrier, the cross section cannot be directly measured at stellar energies. A theoretical support is therefore necessary to extrapolate the available data to the Gamow energy range.

In principle, Helium burning could continue with the ¹⁶O(α, γ)²⁰Ne reaction. However, ²⁰Ne does not present any resonance at stellar energies, and the cross section is therefore quite small in standard conditions (temperature and density). In general, helium burning ends at ¹⁶O. The formation of heavier elements involves various processes. Beyond iron, elements are essentially produced by neutron-capture reactions since the Coulomb barrier increases with the charge of the elements, and proton captures are impossible.