Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

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8.3 Stellar nucleosynthesis 373

8.3 Stellar nucleosynthesis

One of the great triumphs of nuclear physics has been its ability to provide

semiquantitative understanding of the observed abundances of the elements

and their isotopes. Most attention has focused on “solar-system abundances”

that mostly reﬂect the initial composition of the pre-solar cloud that con-

densed 4.5 million years ago to form the Sun, the planets, and meteorites.

About 98% of the solar system mass consists of

Hand

He and most of

these two nuclei were produced in the primordial Universe when the cosmo-

logical temperature was kT ∼ 60keV. The processes leading to the formation

of these two elements will be discussed in Chap. 9. The remaining 2% of the

solar system mass consists of heavier elements that are believed to have been

produced in stars. Prior to the formation of the solar system, these elements

were dispersed into the interstellar medium either by continuous mass loss or

by supernovae-like events. This pollution of the primordial mixture of helium

and hydrogen was essential for the formation of Earth-like planets and the

emergence of life.

8.3.1 Solar-system abundances

The estimated solar-system abundances are shown in Fig. 8.9 [79]. The dis-

tribution falls with increasing A with peaks at the α nuclei

He,

Ne,

Mg,

Si,

Ar and

Ca that all consist of an integer number of

He nuclei. A prominent peak is also seen at

Fe which is believed to be the

result of the decay of the α nuclei

Ni produced in the last stage of stellar

nuclear burning.

Beyond the iron peak, the distribution continues to fall with increasing A

but shows peaks at A =80, 87, 130, 138, 195, 208. We will see that these peaks

are due to the systematics of neutron captures responsible for the production

of heavy elements.

The solar-system abundances are derived from a variety of sources. Only

the Sun and the giant planets could be expected to have started with an en-

tirely representative sample of material since gravitational attraction was the

dominant factor in the formation of these bodies. The small planets and me-

teorites condensed out via processes that depended on chemistry and caution

must therefore be exercised in using elemental abundances derived from these

bodies. While chemical separation is important, the formation of small bod-

ies would not have been expected to result in isotopic separation (with a few

exceptions). Therefore, isotopic abundance ratios for the Earth or meteorites

can generally be taken as representative of the Solar System.

Since most of the mass of the solar system is in the Sun, it would seem

best to use “photospheric abundances” from the absorption lines that appear

in the Sun’s continuous spectrum. Reliable elemental abundances can be de-

rived for most elements. It should be emphasized that photosphere estimates

depend on detailed models of the Solar atmosphere since the importance of

374 8. Nuclear Astrophysics

180

208

138

232

238

235

Log

nuclei

α−

−10

20 30 40 50 60 70

16080 100 120 140 180 200 220 24

−14

−4

−12

−10

−8

−6

iron peak

s peaks

r peaks

p−nuclei

Te,Xe

Ir,Pt

−8

−4

−2

−6

Fig. 8.9. The solar system abundances ρ(A, Z)/ρ

tot

[79]. The ﬁlled circles corre-

spond to even-even nuclei. For A<70, the distribution is visually dominated by

cosmogenic

H and

He and by “iron-peak” elements near A = 56 Between these

two features, the distribution is dominated by “α-nuclei” comprised of an integer

number of

He nuclei. For A>60, the distribution has peaks corresponding to

neutron magic numbers that are produced by the s-process. The r-process produces

peaks shifted to lower A after neutron-rich magic-N nuclei β-decay to the bottom

of the valley of stability. Rare elements are produced by the p-process.

8.3 Stellar nucleosynthesis 375

a particular line is highly dependent on the photosphere temperature and

density which determines the populations of atomic levels. In this regard, it

should be noted that one of the most prominent lines in the solar spectrum is

due to calcium, a rather rare element. This caused a great deal of confusion

before the development of quantum mechanics allowed one to understand the

physics behind the creation of absorption lines. A more fundamental prob-

lem with the use of photosphere abundances is that they give only elemental

abundances since isotopic splittings of lines are generally narrower than the

thermally determined line widths. Exceptions are the isotopic abundances

of carbon and oxygen where the vibrational and rotational lines of the CO

molecule are directly determined by the atomic weights of the constituent

atoms.

The most accurate elemental abundances for most elements come from the

analysis of “carbonaceous chondritic meteorites” that are thought to have a

representative sample of elements with the important exceptions of hydro-

gen, carbon, nitrogen, oxygen and the noble gases. While the formation of

meteorites was a complicated process involving chemical separation, this type

contains three representative phases (silicate, sulﬁde, and metallic) that give

consistent results. With a few exceptions, agreement with photospheric abun-

dances is to within ±10%.

The most important elemental abundance that is accurately determined

neither by photosphere nor meteor abundances is that of

He. This nuclide

was not retained in the formation of small bodies. In fact, the majority of

terrestrial

He is believed to be due to α decay of heavy elements after the

formation of the Earth. In the Sun, helium lines are seen only in the chromo-

sphere where its abundance may not be entirely representative. In fact, the

most reliable estimation of the helium abundance appears to be that derived

from solar models where the initial helium abundance is a free parameter that

is adjusted so as to predict that correct solar radius and luminosity [73, 74].

The derived helium abundance is conﬁrmed by measurements of the helio-

seismological oscillation frequencies [73]. These frequencies depend directly

on the sound speed v

∼ kT/µ where µ is the mean atomic weight. The

temperature proﬁle is well determined in solar models so the sound speed

determines µ. Since the Sun is essentially made of hydrogen and helium, this

then determines the abundance of helium.

Once the elemental abundances are determined, the nuclear abundances

are generally found by multiplying elemental abundances by the terrestrial

isotopic abundance. Once again, this does not work for the noble gases. An

extreme example is argon where the atomic weight listed in the periodic table

is 39.948 reﬂecting the fact that

Ar is the dominant terrestrial isotope. In

reality, most terrestrial argon comes from

K decay (Fig. 5.1) while the α

nucleus

Ar dominates in the Sun. The isotopic abundance of this element

is therefore best determined from the solar wind.

376 8. Nuclear Astrophysics

8.3.2 Production of A<60 nuclei

In previous sections we have seen how a succession of stellar burning stages

produces elements from

He up to the iron-peak. This process favors α-nuclei

since their high binding energy involves them in many exothermic reactions.

The distribution shown in Fig. 8.9 has peaks at these elements so it is rea-

sonable to suppose that they were produced in stars and then later dispersed

into the interstellar medium.

One possible dispersion mechanism would be core-collapse supernovae. As

illustrated in Figs. 8.7, the iron core that will collapse to a neutron star is

surrounded by concentric shells of the ashes of the diﬀerent stages of nuclear

burning. These shells will be blown oﬀ after the core-collapse.

To see whether this mechanism can account for the observed abundances,

we note that about 2M



O would be dispersed per supernova. The rate

for core-collapse supernova explosions in Milky Way-type galaxies is about

2 per century so there have been about 10

core-collapse supernova over the

∼ 10

yr life of our galaxy, about half of which occurred before the formation

of the solar system. This gives about 2 × 10



O or about 1% of

the total mass of the Milky Way. This nicely matches the observed solar-

system abundance of 0.9% ! While this comparison can hardly be considered

quantitative, it seems reasonable to say that a signiﬁcant fraction of the

observed intermediate mass nuclei were dispersed in supernovae.

Core-collapse supernovae are not very eﬃcient in dispersing iron-peak nu-

clei since it is these nuclei that collapsed to form a neutron star. On the other

hand, we saw in Sect. 8.2.3 that type Ia supernovae lead to the destruction

of a ∼ 1.4M



carbon–oxygen white dwarf after its explosive burning to

Ni.

After β-decay to

Fe there would be about 0.7 M



Fe dispersed into

the interstellar medium. The rate for type Ia supernovae is a about 1/4 the

core-collapse rate so the total amount of

Fe produced is not far from the

0.1% observed in the solar system.

8.3.3 A>60: the s-, r- and p-processes

The production of elements with A>56 is paradoxical in the sense that

thermal equilibrium requires the dominance of either iron-peak elements (at

low temperatures) or of

He and nucleons (at high temperatures). In spite of

this, captures of single protons or neutrons by nuclei near the stability line

are always exothermic, so it is possible to create heavy nuclei from a non-

equilibrium mixture of nucleons and iron-peak nuclei. Thermal equilibrium

could then be approached by later ﬁssioning of the heavy nuclei, though the

time scale for spontaneous ﬁssion is so long that equilibrium is reached only

after time scales that are enormous even for astrophysics.

The modern theory of the nucleosynthesis of elements beyond the iron

peak was spelled out in a classic paper by Burbidge, Burbidge, Fowler and

Hoyle [80]. These authors realized the importance of neutron captures in

8.3 Stellar nucleosynthesis 377

the production of heavy elements. While neutrons are present in only tiny

numbers in most stars, neutron capture has the advantage not having the

Coulomb barrier associated with proton captures. Its importance is imme-

diately suggested by the fact that for values of A>56 with more than one

β-stable isobar, the most neutron-rich isobar is the most abundant.

Of course the problem with neutron captures is that normally very few

neutrons are present in stars. Important neutron-producing exothermic reac-

tions are (α,n) reactions on the relatively rare nuclei

Cand

C → n

O Q=3.00 MeV (8.69)

Ne → n

Mn Q=0.30 MeV . (8.70)

The slow build-up of heavy nuclei by capture of such neutrons is called the

“s-process” for “slow” neutron capture.

Neutrons would also be expected to be present in large numbers in explo-

sive events like supernovae or neutron-star collisions. This type of nucleosyn-

thesis is called the “r-process” for “rapid” neutron capture. In fact, we will

see that the existence of both the s- and r- process is necessary to explain

the observed abundances.

The workings of the s and r processes are illustrated in Fig. 8.10 which

shows how heavy nuclei can be constructed by neutron captures on

Fe.

After ﬁrst reaching the stable nuclei

and

,theβ-unstable nu-

cleus

1/2

=44.5 day) is produced. If the rate of neutron captures

is slow compared to the β-decay rate, the

decays “immediately” to

. The next neutron capture produces

which immediately de-

cays to

and so on. This is the s-process where neutron capture is

slow compared to β-decay. The s-process produces nuclei by following a well-

deﬁned path along the bottom of the valley of stability. The path sometimes

bifurcates (at

in Fig. 8.10) only to quickly merge (at

On the other hand, if the rate of neutron capture is higher than typical

decay rates, it is possible to produce nuclei far from the bottom of the valley

of stability. Referring to Fig. 8.10, starting with

, the r-process can

sequentially produce

Fe,

Fe and so on until the addition of a

further neutron yields an unstable nucleus that decays by neutron emission,

Fe → n

A−1

Fe. In fact, the r-process is expected to take place in at high

temperature in the presence of many photons so the limiting reaction is most

likely photon dissociation γ

Fe → n

A−1

Fe. In Fig. 8.10 we show this happen-

ing at

, though this is uncertain because nuclei so far from the bottom

of the stability valley have not been studied in the laboratory.

Fe has a

very short half-life, 0.4s, for β-decay producing

Co. This nucleus can then

captures neutrons continuing the r-process.

Events producing the r-process are expected to last a very short time

(about 10 sec for a type II supernova) and at the end of the event nuclei

would have been produced along the southern slope of the stability valley

378 8. Nuclear Astrophysics

6.6 13.7 2.6 1.8 0.8 0.3 −0.4 −0.4 −0.3 −1.0 −0.8

7.4 6.8 8.2 3.8 2.0 1.4 −0.5 0.1 −0.6 −0.4 −0.7 −0.6 −0.8 −0.7

12.4 9.5 4.0 5.3 1.3 1.3 1.1 0.3 0.3

1.9 3.2 4.1 2.8 4.7 2.5 5.3 1.5 2.2 0.7 1.3 0.8 0.6

2.2 1.9 4.5 3.4 7.3 3.5 2.2 5.2 1.4 2.0

−0.8 −0.9 1.5 2.2 3.0 4.5 5.5 3.6 3.1 4.7 4.2 2.7 2.1

−1.0 1.8 1.5 3.9 3.1 7.4 5.1 6.0 3.7

30 3

33 3

35 36 37 38 39

41 42 4

Fig. 8.10. Nucleosynthesis by neutron capture starting at

Fe. The decimal loga-

rithm of the half-life in seconds is shown for β-unstable nuclei. If the neutron ﬂux is

small, β-decay occurs “immediately” after neutron absorption and the path follows

the nuclei at the bottom of the stability valley indicated by the arrows. This is the

s-process. On the other hand, if the neutron ﬂux is suﬃciently high, the r-process

is operative where nuclei can absorb many neutrons before β-decaying so the path

may ascend the sides of the valley until the nuclei are either photo-dissociated or β

decay. Along the Fe-line, this is shown as happening at

. After the neutron

ﬂux is turned oﬀ, the nuclei on the slopes of the valley β-decay down to the bottom.

Note that the neutron capture path in a nuclear reactor (Fig. 6.12) is intermediate

between the astrophysical s- and r- processes since the time for neutron absorption

is typically a month or so.

as shown in Fig. 8.11. After the neutron ﬂux is turned oﬀ, the nuclei then

β-decay to the bottom of the valley.

The systematics of the s- and r-process explain the peaks in the abun-

dances of A>80 nuclei shown in Fig. 8.9. The s-process produces an ac-

cumulation of nuclei with magic N where subsequent neutron captures are

diﬃcult because of the low cross-section. The s-process peaks would occur at

N =50, 82, 126 corresponding to A = 90, 138, and 208.

On the other hand, the r-process path in Fig. 8.11 follows the lower edge of

the valley but takes a northern direction at the neutron magic numbers 50, 82

and 126. This is because the addition of a neutron to a closed shell is inhibited

both because of a small cross-section for neutron capture and because, once

captured, the extra neutron is weakly bound and easily ejected by a photon.

We can therefore expect that at the end of the r-process event there is an

accumulation of nuclei along the segments of the path following the magic

N lines. After the subsequent β-decays, this will lead to an accumulation

of nuclei at A ∼ 80, 130 and 195. Such r-process peaks are observed in the

solar-system abundances (Fig. 8.9).

We note that the details of the r-process peaks depend on the extent

to which the shell structure and their magic numbers is maintained for very

8.3 Stellar nucleosynthesis 379

Z=20

Z=28

Z=50

Z=82

N=20

N=28

N=50N=50

N=82

N=126

A=208

A=195

N=126

A=138

A=130

N=82

A=90

A=80

stable

Fig. 8.11. The paths for nucleosynthesis by slow and fast neutron capture. In the

s-process, the prompt β-decays insure that all produced nuclei are near the bottom

the of the stability valley. In the r-process, rapid neutron absorption during an

intense pulse of neutrons leads to nuclei distributed along the thick arrow on the

southern slope of the valley. After the pulse is turned oﬀ, the nuclei β-decay down

to the bottom of the valley, as indicated by the thin arrows.

380 8. Nuclear Astrophysics

neutron-rich nuclei. This is an important area of investigation in experimental

nuclear physics.

26.5 11.7

48.2

12.5 12.8 24.1 12.2 28.7 7.5

4.3 95.7

1.0 0.6 0.3 14.5 7.7 24.2 8.6 32.6 4.6 5.8

57.4 42.6

2.4 6.6 7.9 11.2 71.7

0.1 99.9

0.2 0.2 88.5 11.1

100.

27.1 12.2 23.8 8.3 17.2 5.8 5.6

3.1 15.0 11.3 13.8 7.4 26.7 22.7

N=61

N=78

62 63 64 65 66 67 68 69 70 71 72 73 74 75

79 80 81 82 83 84 85 86 87 88 89 90 91 92

s−only

r−only

p−only

r−only

s−only

p−only

Fig. 8.12. Isotopic abundances and the s-process path for 46 ≤ Z ≤ 51 and for

56 ≤ Z ≤ 62. Nuclei below the s-process path are labeled “r-only” and can only

be produced by rapid neutron capture in, for instance, supernovae. Nuclei that are

shielded from the r-process by a r-only nucleus are labeled “s-only.” Nuclei above

the s-process path have low isotopic abundances. The can only be produced by the

“p-process,” due, for instance, to (p, γ)or(γ, n) reactions.

Further indication that both s- and r-processes are needed to explain the

solar-system abundances comes from the fact that there are nuclei that can

only be produced by the s-process and also nuclei that can only be produced

by the r-process. The s-process is clearly incapable of producing elements

heavier than

209

Bi since the β-stable elements for A = 210, 211 are the short-

lived α-emitters

210

Po (t

1/2

= 138.376 day) and

211

Bi (t

1/2

=2.14 m). The

existence of natural Uranium and Thorium therefore necessitates the exis-

tence of the r-process.

8.4 Nuclear astronomy 381

Selected r-only and s-only nuclei are shown in Fig. 8.12. This ﬁgure show

the solar-system isotopic abundances of the stable nuclei and the s-process

path. We can see in Fig. 8.12 that

110

Pd,

122

Sn, and

124

Sn cannot be produced

bythes-process.Thesameistrueof

148

Nd,

150

Nd,

152

Sm, and

154

Sm. On the

other hand,

110

Cd can only be produced by the s-process since it is “shielded”

by the β-stable nucleus

110

Pd from the β-decay of neutron-rich nuclei at the

end of an r-process event. In Fig. 8.12, we see that

148

Sm and

150

Sm are

shielded from the r-process by

148

Nd and

150

Nd.

Comparing the abundances of the s-only Sm isotopes with those of the

r-only isotopes, we see that the two processes produce comparable quantities

of nuclei. This is surprising in view of the quite diﬀerent nature of the two

processes.

In Fig. 8.12 we also see a number of proton-rich nuclei that can be

produced neither by the s- nor the r-process. By deﬁnition, these nuclei

are created by the “p-process.” These nuclei all have very low solar-system

abundances indicating that the p-process is less important than the s- and

r-processes. Originally, it was thought that these nuclei would be created

by proton capture, but more recent work has indicated that (γ,n) reactions

(photo-ejection of a neutron) in explosive environments may be the dominant

process.

8.4 Nuclear astronomy

While the nuclear processes discussed in this chapter explain much about en-

ergy production in stars and nuclear abundances, direct evidence for nuclear

reactions in astrophysical conditions is extremely diﬃcult to obtain. Most

reactions take place deep inside stars so the only products that escape are

neutrinos produced in hydrogen burning and in stellar collapse. The cross-

sections for reactions that can be used to to detect these neutrinos are, unfor-

tunately, so small that astronomical neutrinos have only been observed from

the Sun and from the nearby (∼ 50 kpc) core-collapse supernova SN1987A in

the Large Magellanic Cloud.

The other observable reaction product are γ-rays from regions of space

that are suﬃciently transparent to MeV-photons. One source is long-lived

radioactive nuclei that have been dispersed into the interstellar medium by

supernovae. Much more abundant are continuous spectra of γ-rays associated

with a variety of mechanisms, often related to the acceleration and interac-

tions of cosmic-rays. All these photons have a small probability of penetrating

the Earth’s atmosphere so are better observed by γ-telescopes in orbit about

the Earth.

382 8. Nuclear Astrophysics

8.4.1 Solar Neutrinos

Neutrinos are necessarily produced in hydrogen burning stars by the weak-

interaction processes that transform protons to neutrons. Most of the 27 MeV

liberated in the hydrogen-helium transformation is thermalized in the Sun

and eventually escapes from the surface as photons. (On average, only about

500 keV immediately escapes with the neutrinos.) Therefore, the total neu-

trino production rate is simply determined by the observed solar luminosity,



∼



27 MeV

=2×10

Bq. (8.71)

(This formula assumes that the sun is in a steady state, an assumption that

is justiﬁed by detailed solar models.) At Earth, this gives a solar neutrino

ﬂux

=6×10

−2

−1

. (8.72)

While the total neutrino ﬂux is easy to calculate, the energy spectrum

is not since it depends on the nature of the nuclear reactions that create

them. As discussed in Sect. 8.2.1, energy production in the Sun is believed to

be dominated by the PPI, PPII, and PPIII cycles, which produce neutrinos

through the reactions:

H →

Q = 420 keV PPI, II, III (8.73)

− 7

Be →

Li ν

Q = 862 keV PPII (8.74)

B →

Be e

Q=16MeV PPIII (8.75)

The Q’s determine the energies of the produced neutrinos. Neutrinos from

reaction (8.73) share the Q with the ﬁnal-state positron, so they have a

continuous spectrum with an endpoint of 420 keV. Neutrinos produced in

this reaction are often called “ν

.” In PPI, two ν

are produced.

In PPII, one ν

is produced along with one “ν

” in the electron capture

(8.74). Being produced in a two body reaction, the ν

is monochromatic

with an energy of 861 keV (branching ratio 90 percent) or 383 keV (branching

ratio 10 percent with the production of an excited state of

Li). In PPIII,

one ν

is produced along with one “ν

”intheβ-decay (8.75). The ν

have

a β-spectrum with an endpoint of 16 MeV.

Additional neutrinos come from the CNO decays

N →

Q=1.20 MeV , (8.76)

O →

Q=1.73 MeV , (8.77)

F →

Q=1.74 MeV , (8.78)

and to the “pep” reaction