The Physics of Energy, page 66
In Figure 17.3, we plot the negative of the binding energy per nucleon, with on the z-axis and Z and N on the x- and y-axes respectively. The more bound the nuclide, the lower its height in the graph. The resulting shape is picturesquely called the valley of stability, with stable nuclei and very long lived nuclei along a line that we refer to as the valley floor, and progressively less stable nuclei up the sides. Note that the valley itself, like a river valley, is not exactly flat. As A increases, the floor of the valley, which corresponds to the most tightly bound nucleus for each A, gradually rises.
Figure 17.3 A 3D view of the valley of stability. is plotted as a function of Z and N for and . The pairing energy term, which would modify this smooth shape by small local fluctuations depending upon the parities of N and Z, has been dropped for clarity. Contours of equal binding energy per nucleon are marked. The most tightly bound nuclei correspond to the bottom of the valley. Nuclei at both low A and large A are less bound, as are nuclei that depart from the value of N/Z that yields the tightest binding for fixed A.
Another perspective on the valley of stability is given in Figure 17.4, which looks down on the valley of stability from above. Since the depth of the valley measures the binding energy per nucleon, the lowest point in the valley (where water would accumulate were this geography rather than nuclear physics) is the most stable nuclide. This turns out to be very close to , , the most common isotope of iron. Iron (and its neighbors, nickel and cobalt) therefore have a special place in our universe: nuclear processes, both nuclear fusion and nuclear fission, transform nuclei with small and large A toward iron. Energy can be extracted from nuclei by fusing light nuclei or by fissioning heavy nuclei. To study these processes more deeply we need to get a quantitative understanding of the valley of stability.
Figure 17.4 A view of the valley of stability from above. The most stable nuclides, the black squares, lie along the floor of the valley. Less stable nuclides, denoted with colors ranging from dark blue (relatively stable) to yellow (very short lived), line the sides of the valley. From [82].
Iron, Fusion, and Fission
The most stable nucleus is iron, . Lighter nuclides (smaller ) can fuse, combining together with protons or other light nuclei, while heavier nuclides (larger ) can fission into smaller nuclides, often releasing additional neutrons. In each case, energy is released as the nuclei approach the stable iron nuclide.
17.3.2 The Valley of Stability and the Systematics of Nuclear Binding
Armed with the SEMF, we take a closer and more quantitative look at the systematics of nuclear stability. To display general trends, we ignore the pairing energy term, thereby essentially averaging over nearby nuclides that differ from one another by only one proton or neutron. We can thus treat Z and A as continuous variables, with respect to which we can differentiate to find extrema of the nuclear binding energy.
The Path Along the Valley Floor The most stable nuclide for a given A can be found by setting the derivative to zero and solving for . The result is (Problem 17.10)
(17.20)
When A is small, is very close to . But as A increases to mass numbers on the order of 100, Z falls significantly below . is plotted as a function of A in Figure 17.5. The SEMF estimate of agrees well with data (Problem 17.9).
Figure 17.5 The charge of the most stable nuclide (solid) as a function of A, compared to (dashed).
Nuclear decay processes allow less stable nuclei to transform into more stable ones. Picturesquely, they can roll down the valley of stability, eventually ending up as a stable or very long-lived nucleus near the valley floor. In particular, as described further below, nuclei with fixed A can decay into one another by -decay and related processes (§17.4.3), moving toward at fixed A unless they first decay by -particle emission. We examine decays of unstable nuclei in detail in §17.4. Equation 17.20 is a useful result because it suggests the types of nuclear species – Z as a function of A – that are most likely to be found in our world, and that could be fuels for nuclear energy processes.
Binding Energy Along the Valley Floor If we substitute from eq. (17.20) back into , we obtain the binding energy of the most-bound nuclear species with each A, . It is most convenient to study the binding energy per nucleon, , because this quantity appears in the criteria for nuclear power reactions. After a little simplifying algebra, we obtain
(17.21)
This prediction is compared with data in Figure 17.6 for values of A that occur in nature. The accuracy of the SEMF is confirmed by the fact that agrees well with the binding energy of the most tightly bound nuclide as a function of A. Figure 17.6 also illustrates the fact, mentioned at the end of the last section, that the binding energy per nucleon reaches a maximum near , with both lighter and heavier nuclei being less bound.
Figure 17.6 (a) The measured binding energy per nucleon of all long-lived nuclides is plotted (increasing downward) versus A. The valley of stability has been flattened into a one-dimensional plot whose lower edge traces the floor of the valley. The SEMF prediction (17.21) for the binding energy per nucleon of the most bound nuclide at each value of A is plotted in red. (b) The low-A region of the left-hand plot, with several anomalously bound nuclides indicated.
Some Special Light and Heavy Nuclei A closer look at data on , as depicted in Figure 17.6(b), reveals several significant binding energy anomalies for small A. The semi-empirical mass formula does a pretty good job, even for the lightest nuclei, with the very significant exception of . The assumptions that went into the derivation are not valid for nuclei with very small N and Z, however, so we need to look at these cases individually. The only nuclide with , , is the deuteron (§14). The deuteron is quite weakly bound, with a binding energy of 2.2 MeV. In contrast, is quite tightly bound for a light nucleus, with a binding energy of over 28 MeV. Both nuclides with – , the triton (the nucleus of the tritium atom, an unstable, long-lived ( y), isotope of hydrogen), and , the rare, but stable isotope of helium – are rather weakly bound, as are both isotopes of lithium, and . 12C and 16O and a few even heavier nuclei such as are anomalously tightly bound. These anomalies are accounted for by the filling of shells in the nuclear mean field mentioned in §17.2.6.
Light Nuclei
Light nuclei, particularly those with fewer than 10 nucleons, have binding energies that depart significantly from the SEMF. Some of these nuclei, particularly and the isotopes of hydrogen, and , are particularly important in nuclear energy physics.
The Limit of Nuclear Stability Even though the binding energy per nucleon is still very large around , the plot of measured nuclear binding energies in Figure 17.6 ends abruptly there. There are no stable, or even long-lived nuclides with values of A larger than about 250. In fact, the heaviest nuclide with a lifetime longer than a year is . (Californium is an artificially created nuclide. This isotope has a lifetime of about 900 years.) The SEMF, however, predicts positive binding energy for nuclei anywhere up to (see Problem 17.12).
Why are these bound nuclei not found in nature? The answer is that they are classically unstable against deformations that lead to fission. In constructing the SEMF we noted that the nucleus behaves like an incompressible (constant density) fluid and that the surface energy (§17.2.4) is minimized when a nucleus is spherical. A volume-preserving deformation away from the spherical shape can, however, decrease the Coulomb energy of the nucleus. For small A, surface energy dominates because , and nuclei remain roughly spherical. As A increases the Coulomb repulsion grows like and becomes progressively more important. Eventually, for large enough A, Coulomb repulsion wins and an initially spherical nucleus would quickly deform and fission into two smaller, stable nuclei (see Figure 17.7). A nucleus with has no position of stable equilibrium. If it could somehow be created, such a nucleus would almost instantly fall apart. To distinguish such instability from other types of fission, we refer to this as instantaneous fission. A simple estimate (Problem 17.13) suggests that the most recently discovered nuclide, with and , is close to the limit of stability allowed by instantaneous fission.
Figure 17.7 The nuclear surface energy (purple) and Coulomb energy (blue) and their sum (red) as a function of a small deformation away from spherical shape (see Problem 17.12). (a) For nuclei with low to medium A, the increase in surface energy dominates over the decrease in Coulomb energy and the nucleus is stable against deformation. (b) For large values of A, the Coulomb energy dominates over the surface energy and the nucleus is classically unstable, so it instantaneously fissions as indicated by the arrows.
The End of the Table of Nuclides
The table of nuclei that live long enough to be observed ends at about because larger nuclei would instantaneously fission as the Coulomb repulsion among protons overwhelms the surface energy.
Instantaneous fission is not the kind of fission that generates power in a nuclear reactor. Controlled nuclear fission occurs when certain very long lived nuclides such as are induced to fission by absorbing a neutron (see §18.3).
17.4Nuclear Decays
Unstable nuclides, whether naturally occurring or created in nuclear fission or fusion reactors, almost always decay in one of three ways: -particle emission, -decay (and closely related processes), and (rarely) by spontaneous fission (§18.3.1, not to be confused with instantaneous fission).
These basic types of decays are often accompanied by emission of electromagnetic radiation as -rays. These decays are responsible for the radioactivity that accompanies the production of nuclear power. - and -decays are discussed in this section, while -radiation is discussed in §17.4.6. Spontaneous fission is addressed in §18.
The naturally occurring or easily produced nuclides of interest to us have A and Z values close to the minimum of the valley of stability. Thus we can use the SEMF, whose parameters were fit to nuclei near the minimum of the valley of stability, to study the decays of the nuclides that are of interest.
The most massive nuclei decay in a sequence of steps known as a decay chain (§17.4.5). Along a decay chain, -decays (§17.4.2) decrease the mass of the nucleus by lowering A by four at each step. At various steps along the way, -decays (§17.4.3) change the ratio of protons to neutrons to optimize stability at a given value of A. In this section we explore the details of these processes.
17.4.1 Particle Emission in General
The most obvious way that a nucleus can decay is to break into two pieces. We are often interested in a situation where a nucleus with charge Z and baryon number A emits a small nucleus (or even a proton or neutron) with charge z and baryon number a. Fission can be regarded as a limiting class of such particle emission processes in which the “particle” and the residual nucleus are of comparable size, though fission processes are often accompanied by neutron emission to keep the decay products near the valley floor. If an emission reaction is exothermic, the decay can occur. This condition can be expressed as an inequality in terms of binding energies or mass excesses. We adopt a notation for a nucleus with Z protons and neutrons. Then the decay
(17.22)
is energetically allowed if the products are more bound than the initial nucleus,
(17.23)
or equivalently, if the mass excess of the products is less than that of the initial nucleus,
(17.24)
The stability of any given nucleus with respect to such decay processes can be determined by consulting nuclear data tables and checking this inequality (Problem 17.14).
The SEMF enables us to approximate the terms in eq. (17.23) and obtain an estimate of where in the plane a particular kind of particle emission decay is energetically allowed to occur. Since we are interested in naturally occurring or easily created nuclides that cluster around the floor of the valley of stability, we may replace Z by and use . This approximation ignores the fact that emission of typically does not leave the residual nucleus on the floor of the valley of stability. The error is small, however, when a is small, since along the floor of the valley. If the mass number a of the emitted nuclide is very small and A is large, then the difference in binding energies in eq. (17.23) can be approximated by the derivative, , leading to
(17.25)
as the condition for an allowed decay. It is convenient to re-write this condition in terms of binding energies per nucleon. Substituting and , eq. (17.25) becomes
(17.26)
The left-hand side of this inequality can be obtained from eq. (17.21), and is plotted in Figure 17.8. Particle emission is energetically allowed when the curve in Figure 17.8 lies below the binding energy per nucleon of the emitted nuclide .
Figure 17.8 The solid curve shows the left-hand side of eq. (17.26) plotted versus A. The horizontal dashed lines give the binding energy per nucleon for a nucleon, tritium (), and an -particle (). When the solid curve lies below the dashed line, emission of an particle is allowed.
Nucleon emission Hydrogen and the neutron n have zero binding energy by definition, so a nucleus could emit a single nucleon only if the curve in Figure 17.8 went below zero. This does not occur for any nucleus along the floor of the valley of stability all the way up to where stable nuclei end. So no nuclei of interest to us decay by nucleon emission.6 The same reasoning applies to emission of or . Their binding energy per nucleon is ~3 MeV, below the curve for all relevant A values.
α-emission The nucleus with the smallest a of interest for emission is the alpha-particle, . Its binding energy per nucleon is anomalously large for such a light nucleus (see Figure 17.6(b)), MeV, and this crosses the curve in Figure 17.8 near . So the SEMF predicts that it is energetically possible for a nucleus near the floor of the valley of stability to emit an -particle if its mass number exceeds about 155.
The lowest-A naturally occurring nuclide that decays by -emission is an isotope of neodymium, , which accounts for 24% of naturally occurring neodymium and has a half-life of about years (a long time!). Below , -emission is possible only for very unstable isotopes far from the valley of stability. Above -emitters become progressively more common. Thus the SEMF provides a good guide to the occurrence of -decay.
Other nuclide emission Figure 17.8 suggests that emission of other relatively light nuclei may also be possible because their binding energy per nucleon is even greater than the -particle. With a couple of minor exceptions (Problems 17.14 and 17.15), however, virtually no nuclei near the minimum of the valley of stability decay by emitting light nuclei heavier than an -particle. Note, however, that even larger nuclei can be “emitted” in spontaneous fission processes where the nucleus breaks up into two parts of relatively comparable size (§18.3.1).
Decay by Particle Emission
Many nuclei decay by emitting an -particle, the nucleus of . The SEMF predicts that nuclides near the floor of the valley of stability become unstable to emission of -particles for . For A just above 155 only sporadic nuclides emit -particles. As A increases, more and more nuclides are -emitters, and for values of A in excess of 200 nearly every nuclide is unstable to -decay unless it decays in some other way first.
17.4.2 -particle Emission
-particle emission is the principal way that large nuclei reduce their mass number. It is a primary source of both natural and manmade radioactivity. It is responsible for a significant fraction of geothermal energy, and it is the reason why people worry about long-term sequestration of nuclear reactor waste and about radon in their basements. It is even the source of all the helium that is used to cool superconductors and to fill party balloons.
The lifetimes of nuclei that decay by -emission vary from tiny fractions of a second up to many orders of magnitude greater than the age of the universe. (Recall , with a lifetime greater than years.) Where does this huge variability come from? Due to the relatively strong binding of the -particle, a nucleus typically contains significant -particle-like correlations of two protons and two neutrons. Roughly speaking, several -particles are rattling around inside a heavy nucleus such as at any time. The time scale for their rattling is about seconds, approximately the time it would take an -particle moving at about 1/100th the speed of light to cross a nucleus. Yet it takes an -particle on average years to find its way out of . That is more than trips across the nucleus! This huge suppression for -decay of arises because -particle emission is a classically forbidden process, and the decay occurs through the quantum mechanical magic of tunneling.
α-decay Through Quantum Tunneling
-decay is a classically forbidden process, made possible by quantum mechanical tunneling. The exponential suppression of the barrier penetration factor is responsible for the huge range of -decay lifetimes.
If -decay were not suppressed so much by tunneling, all emitters would have decayed long ago and no fissionable elements would exist on Earth: nuclear weapons and nuclear fission energy would not be possible. This is the first time in this book that tunneling has played such an important role in a physical situation, so it bears careful examination.
The first step in understanding the vast variability of -decay lifetimes is the observation that both the -particle and the nucleus are positively charged. Therefore they repel one another when separated. The potential energy of an -particle in the vicinity of a heavy nucleus with charge Z is sketched in Figure 17.9. The two objects repel at large distances according to Coulomb’s Law,
