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cross section, and has units of area. Neutron capture cross-sections are generally of greater use than proton capture cross sections, mainly because they are much larger. The reason for this is simply that a proton must overcome the repulsive coulomb forces to be captured, whereas a neutron, being neutral, is unaffected electrostatic forces. Neu- tron-capture cross sections are meas- ured in barns, which have units if 10 - (^24) cm 2 , and are denoted by σ. The physical cross-section of a typical nucleus (e.g., Ca) is of the order of 5 x 10 -25^ cm 2 , and increases somewhat with mass number (more precisely, R = r 0 A 1/3^ , where A is mass number and r 0 is the nuclear force radius, 1. x 10 -13^ cm). While many neutron cap- ture cross sections are of the order of 1 barn, they vary from 0 (for 4 He) to 10 5 for 157 Gd, and are not simple functions of nuclear mass (or size). They depend on nuclear structure, being for example, generally low at magic numbers of N. Capture cross- sections also dependent on the en- ergy of the neutron, the dependence varying from nuclide to nuclide.
Collective Model
A slightly more complex model is called the collective model. It is in- termediate between the liquid-drop and the shell models. It emphasizes the collective motion of nuclear mat- ter, particularly the vibrations and rotations, both quantized in energy, in which large groups of nucleons can participate. Even-even nuclides with Z or N close to magic numbers are particularly stable with nearly perfect spherical symmetry. Spherical nuclides cannot ro- tate because of a dictum of quantum mechanics that a rotation about an axis of symmetry is undetect- able, and in a sphere every axis is a symmetry axis. The excitation of such nuclei (that is, when their energy rises to some quantum level above the ground state) may be ascribed to the vibration of the nu- cleus as a whole. On the other hand, even-even nuclides far from magic numbers depart substantially from spherical symmetry and the excitation energies of their excited states may be ascribed to rotation of the nucleus as a whole.
R ADIOACTIVE D ECAY As we have seen, some combinations of protons and neutrons form nuclei that are only “metastable”. These ultimately transform to stable nuclei through the process called radioactive decay. Just as an atom can exist in any one of a number of excited states, so too can a nucleus have a set of discrete, quantized, excited nuclear states. The new nucleus produced by radioactive decay is often in one of these excited states. The behavior of nuclei in transforming to more stable states is somewhat similar to
Figure 1.06. Schematic of binding energy as a function of I, neutron excess number in the vicinity of N=50.
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atomic transformation from excited to more stable states, but there are some important differences. First, energy level spacing is much greater; second, the time an unstable nucleus spends in an excited state can range from 10 -14^ sec to 10 11 years, whereas atomic life times are usually about 10 -8^ sec; third, ex- cited atoms emit photons, but excited nuclei may emit photons or particles of non-zero rest mass. Nu- clear reactions must obey general physical laws, conservation of momentum, mass-energy, spin, etc. and conservation of nuclear particles. Nuclear decay takes place at a rate that follows the law of radioactive decay. There are two extremely interesting and important aspects of radioactive decay. First, the decay rate is dependent only on the energy state of the nuclide; it is independent of the history of the nucleus, and essentially independent of external influences such as temperature, pressure, etc. It is this property that makes radioactive de- cay so useful as a chronometer. Second, it is completely impossible to predict when a given nucleus will decay. We can, however, predict the probability of its decay in a given time interval. The probabil- ity of decay in some infinitesimally small time interval, dt is λ dt. Therefore, the rate of decay among some number, N , of nuclides is:
dN
dt
= −λ N 1.
The minus sign simply indicates N decreases. Equation 1.12 is a first-order rate law known as the basic equation of radioactive decay. Essentially all the significant equations of radiogenic isotope geochemistry and geochronology can be derived from this simple expression.
G AMMA D ECAY Gamma emission occurs when an excited nucleus decays to a more stable state. A gamma ray is sim- ply a high-energy photon (i.e. electromagnetic radiation). Its frequency is related to the energy dif- ference by:
ν = E u – El 1.
where E (^) u and E (^) l are simply the energies of the upper (excited) and lower (ground) states and is the re- duced Plank’s constant ( h /2π). The nuclear reaction is written as:
A Z* → A Z + γ 1.
A LPHA DECAY An α-particle is simply a helium nucleus. Since the helium nucleus is particularly stable, it is not surprising that such a group of particles might exist within the parent nucleus before α-decay. Emis- sion of an alpha particle decreases the mass of the nucleus by the mass of the alpha particle, and also by the kinetic energy of the alpha particle and the remaining nucleus (because of the conservation of mo- mentum, the remaining nucleus recoils from the decay reaction). The α particle may leave the nucleus with any of several discrete kinetic energy levels, as is illustrated in Figure 1.7. The escape of the α particle is a bit of a problem, because it must overcome a very substantial energy barrier, a combination of the strong force and the coulomb repulsion, to get out. For example, α parti- cles fired at in 238 U with energies below 8 Mev are scattered from the nucleus. However, during α de- cay of 238 U, the α particle emerges with an energy of only about 4 Mev. This is an example of an effect called tunneling and can be understood as follows. Essentially, the principle is that we can never know exactly where the α particle is (or any other particle, or you or I for that matter), we only the know the probability of its being in a particular place. This probability is given by the particle’s wave function, ψ (r). The wave is strongly attenuated through the potential energy barrier, but has a small but finite amplitude outside the nucleus, and hence a small but finite probability of its being located there. The escape of an alpha particle leaves a daughter nucleus with mass <A-4, the missing mass is the ki- netic energy of the alpha and remaining nucleus. The daughter may originally be in an excited state,
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energy. Thus in beta decay, a neutrino is also released and the sum of the kinetic energy of the beta and neutrino, plus the energy of any gamma, is constant. Beta decay involves the weak force, or weak interaction. The weak interaction transforms one flavor of quark into another and thereby a charged particle (e.g., a proton) into a one neutral (e.g., a neutron) and visa versa. Both the weak and the electromagnetic forces are thought to be simply a manifestation of one force, called electroweak, that accounts for all interactions involving charge (in the same sense that electric and magnetic forces are manifestations of electromagnetism). In β+^ decay, for example, a proton is converted to a neutron, giving up its +1 charge to a neutrino, which is converted to a positron. This process occurs through the intermediacy of the W+ particle in the same way that electromagnetic processes are mediated by photons. The photon, pion and W particles are members of a class of parti- cles called bosons which mediate forces between the basic constituents of matter. However the W par- ticles differs from photons in having a very substantial mass (around 80GeV or almost 2 orders of mag- nitude greater mass than the proton). Interestingly, Nature rejected the paper in which Fermi proposed the theory of beta decay involving the neutrino and the weak force in 1934!
E LECTRON C APTURE Another type of reaction is electron capture. This is sort of the reverse of beta decay and has the same effect, more or less, as β+^ decay. Interestingly, this is a process in which an electron is added to a nu- cleus to produce a nucleus with less mass than the parent! The missing mass is carried off as energy by an escaping neutrino, and in some cases by a γ. In some cases, a nucleus can decay by either electron capture, β-, or β+^ emission. An example is the decay of 40 K, which decays to 40 Ca by β-^ and 40 Ar to by β+ or electron capture. We should point out that electron capture is an exception to the environmental in- dependence of nuclear decay reactions in that it shows a very slight dependence on pressure. β decay and electron capture often leaves the daughter nucleus in an excited state. In this case, it will decay to its ground state (usually very quickly) by the emission of a γ-ray. Thus γ rays often ac- company β decay (as well as α decay). A change in charge of the nucleus necessitates a rearrangement of electrons in their orbits. As electrons jump down to lower orbits to occupy the orbital freed by the captured electron, they give off electromagnetic energy. This produces x-rays from electrons in the in- ner orbits.
S PONTANEOUS F ISSION This is a process in which a nucleus splits into two or more fairly heavy daughter nuclei. In nature, this is a very rare process, occurring only in the heaviest nuclei, 238 U, 235 U, and 232 Th (it is, however, most likely in 238 U). It also occurs in 244 Pu, an extinct radionuclide (we use the term ‘extinct radionuclide’ to refer to nuclides that once existed in the solar system, but which have subsequently decayed away en-
Figure 1.8. Proton and neutron occupation levels of boron 12, carbon 12 and nitrogen 12.
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tirely). This particular phenomenon is perhaps better explained by the liquid drop model than the shell model. Recall that in the liquid drop model, there are 4 contributions to total binding energy: volume energy, surface tension, excess neutron energy, and Coulomb energy. The surface tension tends to min- imize the surface area while the repulsive coulomb energy tends to increase it. We can visualize these nuclei as oscillating between various shapes. It may very rarely become so distorted by the repulsive force of 90 or so protons, that the surface tension cannot restore the shape. Surface tension is instead minimized by the splitting the nucleus entirely. Since there is a tendency for N/Z to increase with A for stable nuclei, the parent is much richer in neutrons than the daughters produced by fission (which may range from Z=30, zinc, to Z=65, terbium). Thus fission generally also produces some free neutrons in addition to two nuclear fragments (the daughters). The daughters are typically of unequal size, the exact mass of the two daughters being random. The average mass ratio of the high to the low mass fragment is about 1.45. Even though some free neutrons are created, the daughters tend to be too neutron-rich to be stable. As a result, they decay by β-^ to stable daughters. It is this decay of the daughters that results in radioactive fallout in bombs and radioactive waste in reactors (a secondary source of radioactivity is production of unstable nuclides by capture of the neutrons released). Some non-stable heavy nuclei and excited heavy nuclei are particularly unstable with respect to fis- sion. An important example is 236 U. Imagine a material rich in U. When 238 U undergoes fission, one of the released neutrons can be captured by 235 U nuclei, producing 236 U in an excited state. This 236 U then fissions producing more neutrons, etc. This is the basis of nuclear reactors and bombs (actually, the lat- ter more commonly use Pu). The concentration of U is not usually high enough in nature for this sort of thing to happen. But it apparently did once, 1.5 billion years ago in the Oklo U deposit in Africa. This deposit was found to have an anomalously high 238 U/^235 U ratio (227 vs. 137.88), indicating some of the 235 U had been 'burned' in a nuclear chain reaction. Could such a natural nuclear reactor happen again? Probably not, because there is a lot less 235 U around now than there was 1.7 billion years ago. With equations introduced in coming lectures, you should be able to calculate just how much less. Individual natural fission reactions are less rare. When fission occurs, there is a fair amount of kinetic energy produced (maximum about 200 MeV), the nuclear fragments literally flying apart. These frag- ments damage the crystal structure through which they pass, producing 'tracks', whose visibility can be enhanced by etching. This is the basis of fission-track dating. Natural fission also can produce variations in the isotopic abundance of elements among the natural, ultimate product. Xenon is an important product. Indeed, the critical evidence showing that a nuclear chain reaction had indeed occurred in the Oklo deposit was the discovery that fission product elements, such as Nd and Ru, had anomalous isotopic compositions. Analysis of the isotopic composition of an- other fission product, Sm, has led to a controversy over whether the fine scale constant, α, has changed over time. α is related to other fundamental constants as:
e
2
c
where e is the charge of the electron (and is the reduced Plank constant and c is the speed of light). A change in the fine scale constant thus raises the possibility of a change in c. The change, if it occurred, is quite small, less than 1 part in 10^7 , and could be consistent with some observations about quasars and the early universe.
NUCLEOSYNTHESIS A reasonable starting point for isotope geochemistry is a determination of the abundances of the naturally occurring nuclides. Indeed, this was the first task of isotope geochemists (although those en- gaged in this work would have referred to themselves simply as physicists). This began with Thomson, who built the first mass spectrometer and discovered that Ne consisted of 2 isotopes (actually, it con- sists of three, but one of them, 21 Ne is very much less abundant than the other two, and Thomson’s
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Various hints came from all three of the above observations. For example, it was noted that the most abundant nuclide of a given set of stable isobars tended to be the most neutron-rich one. We now un- derstand this to be a result of shielding from β-decay (see the discussion of the r-process). Another key piece of evidence regarding formation of the elements comes from looking back into the history of the cosmos. Astronomy is a bit like geology in that just as we learn about the evolution of the Earth by examining old rocks, we can learn about the evolution of the cosmos by looking at old stars. It turns out that old stars (such old stars are most abundant in the globular clusters outside the main disk of the Milky Way) are considerably poorer in heavy elements than are young stars. This suggests much of the heavy element inventory of the galaxy has been produced since these stars formed (some 10 Ga ago). On the other hand, they seem to have about the same He/H ratio as young stars. Indeed 4 He seems to have an abundance of 24-28% in all stars. Another key observation was the identification of technetium emissions in the spectra of some stars. Since the most stable isotope of this element has a half-life of about 100,000 years and for all intents and purposes it does not exist in the Earth, it must have been synthesized in those stars. Thus the observational evidence suggests (1) H and He are eve- rywhere uniform implying their creation and fixing of the He/H ratio in the Big Bang and (2) subse- quent creation of heavier elements (heavier than Li, as we shall see) by subsequent processes. As we mentioned, early attempts (~1930–1950) to understand nucleosynthesis focused on single mechanisms. Failure to find a single mechanism that could explain the observed abundance of nu- clides, even under varying conditions, led to the present view that relies on a number of mechanisms operating in different environments and at different times for creation of the elements in their observed abundances. This view, often called the polygenetic hypothesis, is based mainly on the work of Bur- bidge, Burbidge, Fowler and Hoyle. Their classic paper summarizing the theory, "Synthesis of the Ele- ments in Stars" was published in Reviews of Modern Physics in 1956. Interestingly, the abundance of trace elements and their isotopic compositions, were perhaps the most critical observations in devel- opment of the theory. An objection to this polygenetic scenario was the apparent uniformity of the iso- topic composition of the elements. But variations in the isotopic composition have now been demon- strated for many elements in some meteorites. Furthermore, there are quite significant compositional variations in heavier elements among stars. These observations provide strong support for this theory. To briefly summarize it, the polygenetic hypothesis proposes four phases of nucleosynthesis. Cos- mological nucleosynthesis occurred shortly after the universe began and is responsible for the cosmic in- ventory of H and He, and some of the Li. Helium is the main product of nucleosynthesis in the inte- riors of normal, or “main sequence” stars. The lighter elements, up to and including Si, but excluding Li and Be, and a fraction of the heavier elements may be synthesized in the interiors of larger stars dur- ing the final stages of their evolution ( stellar nucleosynthesis ). The synthesis of the remaining elements occurs as large stars exhaust the nuclear fuel in their interiors and explode in nature’s grandest specta- cle, the supernova ( explosive nucleosynthesis ). Finally, Li and Be are continually produced in interstellar space by interaction of cosmic rays with matter ( galactic nucleosynthesis ). In the following sections, we examine these nucleosynthetic processes as presently understood.
COSMOLOGICAL NUCLEOSYNTHESIS Immediately after the Big Bang, the universe was too hot for any matter to exist – there was only en- ergy. Some 10-^11 seconds later, the universe had expanded and cooled to the point where quarks and anti-quarks could condense from the energy. The quarks and anti-quarks, however, would also collide and annihilate each other. So a sort of thermal equilibrium existed between matter and energy. As things continued to cool, this equilibrium progressively favored matter over energy. Initially, there was an equal abundance of quarks and anti-quarks, but as time passed, the symmetry was broken and quarks came to dominate. The current theory is that the hyperweak force was responsible for an imbal- ance favoring matter over anti-matter. After 10-^4 seconds, things were cool enough for quarks to associ- ate with one another and form nucleons: protons and neutrons. After 10-^2 seconds, the universe has cooled to 10^11 K. Electrons and positrons were in equilibrium with photons, neutrinos and antineutri-
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nos were in equilibrium with photons, and antineutrinos combined with protons to form positrons and neutrons, and neutrinos combined with neutrons to form electrons and protons:
p +
ν → e +^ + n and
n + ν → e −^ + p
This equilibrium produced about an equal number of protons and neutrons. However, the neutron is unstable outside the nucleus and decays to a proton with a half-life of 17 minutes. So as time continued passed, protons became more abundant than neutrons. After a second or so, the universe had cooled to 10 10 K, which shut down the reactions above. Conse- quently, neutrons were no longer being created, but they were being destroyed as they decayed to pro- tons. At this point, protons were about 3 times as abundant as neutrons. It took another 3 minutes to for the universe to cool to 10^9 K, which is cool enough for 2 H, created by
p + n → 2 H + γ
to be stable. At about the same time, the following reactions could also occur:
2 H + 1 n → 3 H +γ; 2 H + 1 H → 3 H + γ
2 H + 1 H → 3 He + β+ +γ; 3 He + n → 4 He + γ
and 3 He + 4 He → 7 Be +γ; 7 Be + e–^ → 7 Li + γ
One significant aspect of this event is that it began to lock up neutrons in nuclei where they could no longer decay to protons. The timing of this event fixes the ratio of protons to neutrons at about 7:1. Be- cause of this dominance of protons, hydrogen is the dominant element in the universe. About 24% of the mass of the universe was converted to 4 He in this way; less than 0.01% was converted to 2 H, 3 He, and 7 Li (and there is good agreement between theory and observation). Formation of elements heavier than Li was inhibited by the instability of nuclei of masses 5 and 8. Shortly thereafter, the universe cooled below10^9 K and nuclear reactions were no longer possible. Thus, the Big Bang created H, He and a bit of Li (^7 Li/H < 10-^9 ). Some 500,000 years later, the universe had cooled to about 3000 K, cool enough for electrons to be bound to nuclei, forming atoms. It was at this time, called the “recombination era” that the universe first became transparent to radiation. Prior to that, photons were scattered by the free electrons, making the universe opaque. It is the radiation emitted during this recombination that makes up the cosmic microwave background radiation that we can still detect today. Discovery of this cosmic microwave background radiation, which has the exact spectra predicted by the Big Bang model, represents a major triumph for the model and is not easily ex- plained in any other way.
STELLAR NUCLEOSYNTHESIS
Astronomical Background
Before discussing nucleosynthesis in stars, it is useful to review a few basics of astronomy. Stars shine because of exothermic nuclear reactions occurring in their cores. The energy released by these processes results in thermal expansion that, in general, exactly balances gravitational collapse. Surface temperatures are very much cooler than temperatures in stellar cores. For example, the Sun, which is in almost every respect an average star, has a surface temperature of 5700 K and a core temperature thought to be 14,000,000 K. Stars are classified based on their color (and spectral absorption lines), which in turn is related to their temperature. From hot to cold, the classification is: O, B, F, G, K, M, with subclasses designated by numbers, e.g., F5. (The mnemonic is ' O Be a Fine Girl, Kiss Me!' ). The Sun is class G. Stars are also di- vided into Populations. Population I stars are second or later generation stars and have greater heavy element contents than Population II stars. Population I stars are generally located in the main disk of the galaxy, whereas the old first generation stars of Population II occur mainly in globular clusters that circle the main disk.
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than 8 solar masses die explosively, in supernovae (specifically, Type II supernovae). (Novae are en- tirely different events that occur in binary systems when mass from a main sequence star is pull by gravity onto a white dwarf companion). Supernovae are incredibly energetic events. The energy re- leased by a supernova can exceed that released by an entire galaxy (which, it will be recalled, consists of on the order of 10^9 stars) for a period of days or weeks!
Nucleosynthesis in Stellar Interiors
Hydrogen, Helium, and Carbon Burning in Main Sequence and Red Giant Stars
For quite some time after the Big Bang, the universe was a more or less homogeneous, hot gas. More or less turns out to be critical wording. Inevitably (according to fluid dynamics), inhomogeneities in the gas developed. These inhomogeneities enlarged in a sort of runaway process of gravitational at- traction and collapse. Thus were formed protogalaxies, thought to date to about 0.5-1.0 Ga after the big bang. Instabilities within the protogalaxies collapsed into stars. Once this collapse proceeds to the point where density reaches 6 g/cm and temperature reaches 10 to 20 million K, nucleosynthesis begins in the interior of stars, by hydrogen burning , or the pp process. There are three variants, PP I:
1 H + 1 H → 2 H +β+ + ν; 2 H + 1 H → 3 He + γ; and 3 He + 3 He → 4 He + 2 1 H + γ
PP II:
3 He + 4 He → 7 Be; 7 Be → β– + 7 Li + ν; 7 Li + 1 H → 24 He
and PP III:
7 Be + 1 H → 8 B + γ; 8 B → β+ + 8 Be + ν; 8 Be → 24 He
Which of these reactions dominates depends on temperature, but the net result of all is the production of 4 He and the consumption of H (and Li). All main sequence stars produce He, yet over the history of the cosmos, this has had little impact on the H/He ratio of the universe. This in part reflects the obser- vation that for small mass stars, the He produced remains hidden in their interiors or their white dwarf remnants and for large mass stars, later reactions consume the He produced in the main sequence stage. Once some carbon had been produced by the first generation of stars and supernovae, second and subsequent generation stars could He by another process as well, the CNO cycle :
12 C(p,γ) 13 N(β+,γ) 13 C(p,γ) 14 N(p,γ) 15 O(β+,ν) 15 N(p,α) 12 C‡
It was subsequently realized that this reaction cycle is just part of a larger reaction cycle, which is il- lustrated in Figure 1.11. Since the process is cyclic, the net effect is consumption of 4 protons and two positrons to produce a neutrino, some energy, and a 4 He nucleus. Thus to a first approximation, carbon acts as a kind of nuclear catalyst in this cycle: it is neither produced nor consumed. When we consider these reaction's in more detail, not all of them operate at the same rate, resulting in some production and some consumption of these heavier nuclides. The net production of a nuclide can be expressed as:
dN
dt
= ( creation rate − destruction rate ) 1.
Reaction rates are such that some nuclides in this cycle are created more rapidly than they are con- sumed, while for others the opposite is true. The slowest of the reactions in Cycle I is 14 N(p,γ)^15 O. As a result, there is a production of 14 N in the cycle and net consumption of C and O. The CNO cycle will
‡ (^) Here we are using a notation commonly used in nuclear physics. The reaction:
12 C(p,γ) 13 N
is equivalent to: 12 C + p → 13 N + γ
