1.2 Nucleosynthesis

A realistic model for the nucleosynthesis of the elements must be based on empirical data for their ‘cosmic abundance’. True cosmic abundances can be derived from stellar spectroscopy or by chemical analysis of galactic cosmic rays. However, such data are difficult to measure at high precision, so cosmic abundances are normally approximated by solar-system abundances. These can be determined by solar spectroscopy or by direct analysis of the most ‘primitive’ meteorites, carbonaceous chondrites. A comparison of the latter two sources of data (Ross and Aller, 1976) demonstrates good agreement for most elements (Fig. 1.3). Exceptions are the volatile elements, which have been lost from meteorites, and the Li)Be)B group, which are unstable in stars.

Fig. 1.3. Comparison of solar-system abundances (relative to silicon) determined by solar spectroscopy and by analysis of carbonaceous chondrites. After Ringwood (1979).

It is widely believed (e.g. Weinberg, 1977) that about 30 minutes after the ‘hot big bang’,the matter of the universe (in the form of protons and neutrons) consisted mostly of ¹H and 22%)28% by mass of ⁴He, along with traces of ²H (deuterium) and ³He. Hydrogen is still by far the most abundant element in the universe (88.6% of all nuclei) and with helium, makes up 99% of its mass, but naturally occurring heavy nuclides now exist up to atomic weight 254 or beyond (Fig. 1.1). These heavier nuclei must have been produced by nucleosynthetic processes in stars, rather than the big bang, because stars of different ages have different compositions which can be detected spectroscopically. Furthermore, stars at particular evolutionary stages may have compositional abnormalities, such as the presence of ²⁵⁴Cf in supernovae. If nucleosynthesis of the heavy elements had occurred in the big bang then their distribution would be uniform throughout the universe.

1.2.1 Stellar evolution

Present day models of stellar nucleosynthesis are based heavily on a classic review paper by Burbidge et al. (1957), in which eight element-building processes were identified (hydrogen burning, helium burning, ", e, s, r, x and p). Different processes were invoked to explain the abundance patterns of different groups of elements. These processes are, in turn, linked to different stages of stellar evolution. It is therefore appropriate at this point to summarise the life-history of some typical stars (e.g. Iben, 1967). The length of this life-history depends directly on the stellar mass, and can be traced on a plot of absolute magnitude (brightness) against spectral class (colour), referred to as the Hertzsprung-Russell or H)R diagram (Fig. 1.4).

Fig. 1.4. Plot of absolute magnitude against spectral class of stars. Hatched areas show distributions of the three main star groups. The postulated evolutionary path of a star of solar mass is shown.

Gravitational accretion of a star of solar mass from cold primordial hydrogen and helium would probably take about 10⁶ yr to raise the core temperature to ca. 10⁷ K, when nuclear fusion of hydrogen to helium can begin (Atkinson and Houtermans, 1929). This process is also called ‘hydrogen burning’. The star spends most of its life at this stage, as a ‘Main Sequence’ star, where equilibrium is set up between energy supply by fusion and energy loss in the form of radiation. For the Sun, this stage will probably last ca. 10¹⁰ yr, but a very large star with 15 times the Sun’s mass may remain in the Main Sequence for only 10⁷ yr.

When the bulk of hydrogen in a small star has been converted into ⁴He, inward density-driven forces exceed outward radiation pressure, causing gravitational contraction. However, the resulting rise in core temperature causes expansion of the outer hydrogen-rich layer of the star. This forms a huge low-density envelope whose surface temperature may fall to ca. 4000 K, observed as a ‘Red Giant’. This stage lasts only one tenth as long as the Main Sequence stage. When core temperatures reach 1.5 H 10⁷ K, a more efficient hydrogen- burning reaction becomes possible if the star contains traces of carbon, nitrogen and oxygen inherited from older generations of stars. This form of hydrogen burning is called the C)N)O cycle (Bethe, 1939).

At some point during the Red Giant stage, core temperatures may reach 10⁸ K, when He fusion to carbon is ignited (the ‘helium flash’). Further core contraction, yielding a temperature of ca. 10⁹ K, follows as helium becomes exhausted. At these temperatures an endothermic process of "-particle emission can occur, allowing the building of heavier nuclides up to mass 40. However, this quickly expends the remaining burnable fuel of the star, which then cools to a White Dwarf.

More massive stars (of several solar masses) have a different life-history. In these stars, the greater gravitationally induced pressure)temperature conditions allow the fusion of helium to begin early in the Red Giant stage. This is followed by further contraction and heating, allowing the fusion of carbon and successively heavier elements. However, as lighter elements become exhausted, gravitationally induced contraction and heating occur at an ever increasing pace (Fig. 1.5), until the implosion is stopped by the attainment of neutron-star density. The resulting shock wave causes a supernova explosion which ends the star’s life (e.g. Burrows, 2000).

In the minutes before explosion, when temperatures exceed 3 H 10⁹ K, very rapid nuclear interactions occur. Energetic equilibrium is established between nuclei and free protons and neutrons, synthesising elements like Fe by the so-called e-process. The supernova explosion itself lasts only a few seconds, but is characterised by colossal neutron fluxes. These very rapidly synthesise heavier elements, terminating at ²⁵⁴Cf, which undergoes spontaneous fission. Products of the supernova explosion are distributed through space and later incorporated in a new generation of stars.

Fig. 1.5. Schematic diagram of the evolution of a large star showing the nucleosynthetic processes that occur along its accelerating life-history in response to increasing temperature. Note that time is measured backwards from the end of the star’s life on the right. After Burbidge et al. (1957).

1.2.2 Stages in the nucleosynthesis of heavy elements

A schematic diagram of the cosmic abundance chart is given in Fig. 1.6. We will now see how different nucleosynthetic processes are invoked to account for its form.

Fig. 1.6. Schematic diagram of the cosmic abundances of the elements, highlighting the nucleosynthetic processes responsible for forming different groups of nuclides. After Burbidge et al. (1957).

The element building process begins with the fusion of four protons to one ⁴He nucleus, which occurs in three stages:

¹H + ¹H 6 ²D + e⁺ + < (Q = +1.44 MeV, t_1/2 = 1.4 H 10¹⁰ yr)

²D + ¹H 6 ³He + ( (Q = +5.49 MeV, t_1/2 = 0.6 s)

³He + ³He 6 ⁴He + 2 ¹H + ( (Q = +12.86 MeV, t_1/2 = 10⁶ yr),

where Q is the energy output and t_1/2 is the reaction time of each stage (the time necessary to consume one-half of the reactants) for the centre of the sun. The long reaction time for the first step explains the long duration of the hydrogen-burning (Main Sequence) stage for small stars like the Sun. The overall reaction converts four protons into one helium nucleus, two positrons and two neutrinos, plus a large output of energy in the form of high frequency photons. Hence the reaction is very strongly exothermic. Although deuterium and ³He are generated in the first two reactions above, their consumption in the third accounts for their much lower cosmic abundance than ⁴He.

If heavier elements are present in a star (e.g. carbon and nitrogen) then the catalytic C)N)O sequence of reactions can occur, which also combines four protons to make one helium nucleus:

¹²C + ¹H 6 ¹³N + ( (Q = +1.95 MeV, t_1/2 = 1.3 H 10⁷ yr)

¹³N 6 ¹³C + e⁺ + < (Q = +2.22 MeV, t_1/2 = 7 min)

¹³C + ¹H 6 ¹⁴N + ( (Q = +7.54 MeV, t_1/2 = 3 H 10⁶ yr)

¹⁴N + ¹H 6 ¹⁵O + ( (Q = +7.35 MeV, t_1/2 = 3 H 10⁵ yr)

¹⁵O 6 ¹⁵N + e⁺ + < (Q = +2.70 MeV, t_1/2 = 82 s)

¹⁵N + ¹H 6 ¹²C + ⁴He (Q = +4.96 MeV, t_1/2 = 10⁵ yr).

The C)N)O elements have greater potential energy barriers to fusion than hydrogen, so these reactions require higher temperatures to operate than the simple proton)proton (p)p) reaction. However, the reaction times are much shorter than for the p)p reaction. Therefore the C)N)O reaction contributes less than 10% of hydrogen-burning reactions in a small star like the Sun, but is overwhelmingly dominant in large stars. This explains their much shorter lifespan in the Main Sequence.

Helium burning also occurs in stages:

⁴He + ⁴He X ⁸Be (Q = +0.09 MeV)

⁸Be + ⁴He X ¹²C^* (Q = !0.37 MeV)

¹²C^* 6 ¹²C + ( (Q = +7.65 MeV)

The ⁸Be nucleus is very unstable (t_1/2 < 10^!15 s) and in the core of a Red Giant the Be/He equilibrium ratio is estimated at 10^!9. However its life is just long enough to allow the possibility of collision with another helium nucleus. (Instantaneous 3-particle collisions are very rare). The energy yield of the first stage is small, and the second is actually endothermic, but the decay of excited ¹²C^* to the ground state is strongly exothermic, driving the equilibria to the right.

The elements Li, Be and B have low nuclear binding energies, so that they are unstable at the temperatures of 10⁷ K and above found at the centres of stars. They are therefore bypassed by stellar nucleosynthetic reactions, leading to low cosmic abundances (Fig. 1.6). The fact that the five stable isotopes ⁶Li, ⁷Li, ⁹Be, ¹⁰B and ¹¹B exist at all has been attributed to fragmentation effects (spallation) of heavy cosmic rays (atomic nuclei travelling through the galaxy at relativistic speeds) as they hit interstellar gas atoms (Reeves, 1974). This is termed the x-process.

Problems have been recognised in the x-process model for generating the light elements Li, Be and B, since cosmic ray spallation cannot explain the observed isotope ratios of these elements in Solar System materials. However, Casse et al. (1995) proposed that carbon and oxygen nuclei ejected from supernovae can generate these nuclides by collision with hydrogen and helium in the surrounding gas cloud. This process is believed to occur in regions such the Orion nebula. The combination of supernova production with spallation of galactic cosmic rays can explain observed Solar System abundances of Li, Be and B.

Following the synthesis of carbon, further helium-burning reactions are possible, to produce heavier nuclei:

¹²C + ⁴He 6 ¹⁶O + ( (Q = +7.15 MeV)

¹⁶O + ⁴He 6 ²⁰Ne + ( (Q = +4.75 MeV)

²⁰Ne + ⁴He 6 ²⁴Mg + ( (Q = +9.31 MeV).

Intervening nuclei such as ¹³N can be produced by adding protons to these species, but are themselves consumed in the process of catalytic hydrogen burning mentioned above.

In old Red Giant stars, carbon-burning reactions can occur:

¹²C + ¹²C 6 ²⁴Mg + ( (Q = +13.85 MeV)

6 ²³Na + ¹H (Q = +2.23 MeV)

6 ²⁰Ne + ⁴He (Q = +4.62 MeV).

The hydrogen and helium nuclei regenerated in these processes allow further reactions which help to fill in gaps between masses 12 and 24.

When a small star reaches its maximum core temperature of 10⁹ K the endothermic "-process can occur:

²⁰Ne + ( 6 ¹⁶O + ⁴He (Q = !4.75 MeV).

The energy consumption of this process is compensated by strongly exothermic reactions such as:

²⁰Ne + ⁴He 6 ²⁴Mg + ( (Q = +9.31 MeV),

so that the overall reaction generates a positive energy budget. The process resembles helium burning, but is distinguished by the different source of ⁴He. The "-process can build up from ²⁴Mg through the sequence ²⁸Si, ³²S, ³⁶Ar and ⁴⁰Ca, where it terminates, owing to the instability of ⁴⁴Ti.

The maximum temperatures reached in the core of a small star do not allow substantial heavy element production. However, in the final stages of the evolution of larger stars, before a supernova explosion, the core temperature exceeds 3 H 10⁹ K. This allows energetic equilibrium to be established by very rapid nuclear reactions between the various nuclei and free protons and neutrons (the e-process). Because ⁵⁶Fe is at the peak of the nuclear binding-energy curve, this element is most favoured by the e-process (Fig. 1.6). However, the other first-series transition elements V, Cr, Mn, Co and Ni in the mass range 50 to 62 are also attributed to this process.

During the last few million years of a Red Giant’s life, a slow process of neutron addition with emission of ( rays (the s-process) can synthesise many additional nuclides up to mass 209 (see Fig. 1.7). Two possible neutron sources are:

¹³C + ⁴He 6 ¹⁶O + n + (

²¹Ne + ⁴He 6 ²⁴Mg + n + (.

The ¹³C and ²¹Ne parents can be produced by proton bombardment of the common ¹²C and ²⁰Ne nuclides.

Because neutron capture in the s-process is relatively slow, unstable neutron-rich nuclides generated in this process have time to decay by $ emission before further neutron addition. Hence the nucleosynthetic path of the s-process climbs in many small steps up the path of greatest stability of proton/neutron ratio (Fig. 1.7) and is finally terminated by the " decay of ²¹⁰Po back to ²⁰⁶Pb and ²⁰⁹Bi back to ²⁰⁵Tl.

The ‘neutron capture cross-section’ of a nuclide expresses how readily it can absorb incoming thermal neutrons, and therefore determines how likely it is to be converted to a higher atomic mass species by neutron bombardment. Nuclides with certain neutron numbers (e.g. 50, 82 and 126) have unusually small neutron capture cross sections, making them particularly resistant to further reaction, and giving rise to local peaks in abundance at masses 90, 138 and 208. Hence, N = 50, 82 and 126 are empirically referred to as neutron ‘magic numbers’.

In contrast to the s-process, which may occur over periods of millions of years in Red Giants, r-process neutrons are added in very rapid succession to a nucleus before $-decay is possible. The nuclei are therefore rapidly driven to the neutron-rich side of the stability line, until they reach a new equilibrium between neutron addition and $ decay, represented by the hatched zone in Fig. 1.7. Nuclides move along this r-process pathway until they reach a configuration with low neutron capture cross-section (a neutron magic number). At these points a ‘cascade’ of alternating $ decays and single neutron additions occurs, indicated by the notched ladders in Fig. 1.7. Nuclides climb these ladders until they reach the next segment of the r-process pathway.

Fig. 1.7. Neutron capture paths of the s-process and r-process shown on the chart of the nuclides. Hatched zone indicates the r-process nucleosynthetic pathway for a plausible neutron flux. Neutron ‘magic numbers’ are indicated by vertical lines, and mass numbers of nuclide abundance peaks are marked. After Seeger et al. (1965).

Nuclides with neutron magic numbers build to excess abundances, as with the s-process, but they occur at proton-deficient compositions relative to the s-process stability path. Therefore, when the neutron flux falls off and nuclides on the ladders undergo $ decay back to the stability line, the r-process local abundance peaks are displaced about 6 ) 12 mass units below the s-process peaks (Fig. 1.6).

The r-process is terminated by neutron-induced fission at mass 254, and nuclear matter is fed back into the element-building processes at masses of ca. 108 and 146. Thus, cycling of nuclear reactions occurs above mass 108. Because of the extreme neutron flux postulated for the r-process, its occurrence is probably limited to supernovae.

The effects of r- and s-process synthesis of typical heavy elements may be demonstrated by an examination of the chart of the nuclides in the region of the light rare earths (Fig. 1.8). The step by step building of the s-process contrasts with the ‘rain of nuclides’ produced by $ decay of r-process products. Some nuclides, such as those from ¹⁴³Nd to ¹⁴⁶Nd, are produced by both r- and s-processes. Some, such as ¹⁴²Nd are s-only nuclides ‘shielded’ from the decay products of the r-process by intervening nuclides. Others, such as ¹⁴⁸Nd and ¹⁵⁰Nd are r-only nuclides which lie off the s-process production pathway.

Fig. 1.8. Part of the chart of the nuclides in the area of the light rare earths to show p-, r- and s-process product nuclides. After O’Nions et al. (1979).

Several heavy nuclides from ⁷⁴Se to ¹⁹⁶Hg lie isolated on the proton-rich side of the s-process growth path (e.g. ¹⁴⁴Sm in Fig. 1.8), and are also shielded from r-process production. In order to explain the existence of these nuclides it is necessary to postulate a p-process by which normal r- and s-process nuclei are bombarded by protons at very high temperature (> 2 H 10⁹K), probably in the outer envelope of a supernova.

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