The Physics of Energy, page 70
The neutron-induced fission cross section of requires close examination. It is reproduced in Figure 18.5 along with the induced fission cross section for . Note that the -induced fission cross section is huge for very low-energy (e.g. thermal) neutrons (highlighted in red). The thermal-neutron-induced fission cross section for is far smaller. From Table 18.3, we see that the fission barrier in the compound nucleus * (5.8 MeV) is about 1 MeV higher than the energy delivered by a thermal neutron (4.8 MeV), so thermal neutrons are not effective at inducing to fission. On the other hand, neutrons with energy greater than ~1 MeV may induce to fission. From Figure 18.5 we see that this is indeed the case: the fission cross section of rises by over four orders of magnitude as the neutron energy grows past 1 MeV (highlighted in red). Thus, the that accompanies the fissile isotope in a typical nuclear reactor can contribute in two ways to energy production: first, a nucleus can radiatively capture a neutron, producing the fissile nuclide ; and second, a fast neutron from fission can sometimes induce a nucleus to fission. A pure chain reaction is impossible because the neutrons from fission of fission are not energetic enough to induce further fission. A chain reaction, however, can be sustained by created when absorbs a fast neutron. Fast breeder reactors exploit the neutron absorption properties of to produce more fissile material than they consume (see §19.3).
Figure 18.5 Neutron-induced fission cross sections for (a) and (b) for neutron energies from eV to ~10 MeV. Note: the vertical scales differ by many orders of magnitude. Large cross sections are highlighted in red. From [70].
18.3.5 Fission Products and Energy Release
When a heavy nucleus fissions, many different pairs of neutron-rich nuclei can be produced along with extra neutrons. Here are a few common fission reactions,
(18.3)
Table 18.4 lists some of the most common fission fragments.
Table 18.4 Fission products produced with greater than 1% probability in thermal-neutron-induced fission of . The yield is the percentage of fission events that lead quickly to this isotope (see Example 18.1). From [88].
* is stable, but neutron captures to which β-decays with y.
Figure 18.6 Atomic mass distribution of fragments in fission. After [86].
Fission is typically asymmetric: one of the fragments from fission generally has atomic mass in the range –105 and the other in the range –150. The reasons for this asymmetry are complex and do not follow directly from the liquid drop model. The distribution is shown in Figure 18.6. As explained earlier, fission fragments are usually excessively neutron-rich. Typically, the fragments are unstable to β-decay, with lifetimes that range from fractions of a second up to years. Often the resulting decay products are themselves radioactive, leading to a complex decay chain (§17.4.5) with many branches before reaching a stable nuclide (see Example 18.1 and Problem 18.7). Some of the radioactive decay products have serious human health implications, particularly strontium-90, cesium-137, and iodine-131. We discuss these issues further in §20.
Neutron Reactions with
Thermal neutrons are more than eight orders of magnitude less likely to induce fusion in than in . Fast neutrons (1/2 MeV) most often scatter elastically from . The cross section for neutron-induced fission of increases dramatically for neutrons with energies MeV, though the cross section is still much smaller than the thermal-neutron fission cross section of . has a large but rapidly varying (in energy) cross section for capturing neutrons with keV, leading quickly to the formation of .
Fission Poisons Among fission fragments are certain nuclides with extremely large neutron absorption cross sections. These nuclides are known as fission poisons. In practice, the most important fission poison is . Although direct production of in fission is not too common (0.3% of all slow-neutron-induced fissions lead directly to ), is produced in the β-decay of , one of the most common fission products (see Table 18.4). decays to with a half-life of 6.46 hours, and itself β-decays to with a half-life of 9.14 hours ( has a lifetime exceeding a million years). So slowly builds up to an equilibrium level in material where a fission chain reaction is occurring. has a huge absorption cross section, b, for thermal neutrons. Slow neutrons are essential to the perpetuation of a fission chain reaction, so as builds up in a reactor and it absorbs an increasing fraction of the available thermal neutrons, it becomes harder and harder to maintain the chain reaction. This affects both the operation and safety of nuclear reactors, as described in §19.1.7.
Prompt and Delayed Neutrons When undergoes the process of thermal-neutron-induced fission, on average a total of about 2.5 neutrons are given off. To maintain a chain reaction, at least one of those neutrons must be able to induce another fission. Almost all these neutrons are emitted within seconds of the fission event. These are known as prompt neutrons. A tiny fraction are emitted later, over time scales ranging from fractions of a second to minutes. These delayed neutrons are emitted with small probabilities during the β-decays of certain fission fragments. For example, the fission fragment β-decays with a half-life of 55 s. 98% of the time it decays to the ground state of , but 2% of the time it decays to plus an additional neutron. The β-decays of fission fragments that can yield delayed neutrons all have half-lives of less than a minute.
Example 18.1 What Happens when Fissions?
As an example of the sequence of events following slow-neutron-induced fission, we follow the history of one particular fission reaction,
The energy released promptly is
Most of this energy appears as kinetic energy on the fission fragments; a few MeV is carried by the neutrons and by γ-rays. All of this energy is quickly shared with other atoms in the reactor contributing to the thermal distribution of energies characteristic of the reactor temperature.
Both and decay by a sequence of β-decays:
where the β-decay electrons and antineutrinos have been omitted. Notice the long lived isotopes of cerium and strontium (in red). If released into the environment, they represent potential radiation hazards. Strontium-90 is a particularly insidious hazard since it can find its way into milk and substitute for calcium in bones.
The decays that precede the formation of these long lived nuclides occur over the time scale of minutes. These rapid decays contribute and of additional energy (some of which is lost on the antineutrinos) to the energy released in the fission reaction.
The subsequent decays of and contribute an additional over their lifetimes. If the original reaction takes place in a fission reactor, this energy continues to heat the reactor fuel long after the fuel is removed from the reactor.
Instability of Fission Fragments
Two consequences of the instability of fission fragments are (1) even after the chain reaction is turned off the material undergoing the chain reaction continues to emit considerable power, and (2) material that has undergone a fission reaction contains fission fragments that are very radioactive and remain so for many years.
In fission about 0.65% of all neutrons are delayed, and the average delay time is seconds. As described in §19 , these delayed neutrons turn out to be critical to the control of a fission power reactor. The fraction of delayed neutrons in fission is smaller, only about 0.20%, making control of a plutonium-fueled reactor more difficult.
Energy Release in Fission The exact amount of energy liberated in slow-neutron-induced fission depends on the specific reaction involved. On average, about 203 MeV is released when fissions. Most of this energy is liberated promptly and appears as kinetic energy of the fission fragments and neutrons, and as γ-rays. A significant fraction is emitted later as electrons, neutrinos, and more γ-rays when fission fragments β-decay. The average energy budget for fission is summarized in Table 18.5. The emitted (fast) neutrons have energies in the MeV range. The mean neutron energy is just less than 2 MeV. The neutron energy distribution is sketched in Figure 18.7. Table 18.5 shows that about 4.3% of the energy is lost to neutrinos and 6.3% of the energy liberated in fission comes from the radioactive decay of fission fragments. If the fission event takes place in matter, then subsequent radiative capture of some of the prompt neutrons can contribute significantly to the total energy released in fission. In a thermal-neutron reactor this additional energy amounts to about 9 MeV, roughly balancing the energy lost to neutrinos.
Table 18.5 Average energy budget for fission. From [87].
Prompt emission Energy (MeV)
Fission fragments 169.1
Fission neutrons 4.8
γ (and associated electrons) 7.0
Delayed emission (radioactivity)
β-decay electrons 6.5
β-decay neutrinos (lost) 8.8
γ-emission 6.3
Subtotal 202.5
Neutron capture* 8.8
Total 211.3
* Although not part of the fission reaction itself, energy released when fission neutrons are later radiatively captured contributes to the total budget of energy released when fissions in a reactor.
18.3.6 Moderators
The energy spectrum of the neutrons emitted in fission (Figure 18.7) is dominated by fast neutrons ( MeV), which have only a small probability of inducing a subsequent fission when absorbed by or . Thermal neutrons (E ~0.02–0.05 eV), on the other hand, have the best chance of inducing another fissile nucleus to fission. So it is a priority to slow the neutrons down quickly, before they can be captured.
Figure 18.7 The neutron energy spectrum in slow-neutron-induced fission.
The mechanism for slowing neutrons down is simple and general. It makes use of the kinetics of two-body collisions between the neutrons and the nuclei of another substance known as a moderator. A moderator must be able to slow neutrons down in collisions but must not have a large probability of absorbing them.
First we investigate how a collision of a neutron (mass m) with a stationary nucleus (mass ) degrades the neutron energy. In the laboratory reference frame, the neutron approaches the target nucleus with energy E, scatters through an angle χ and emerges from the collision with energy (Figure 18.16(a)). Though is a relatively complicated function of E and χ, the relation between and E is quite simple when expressed in terms of the center-of-mass scattering angle θ (see Figure 18.8(b) and Problem 18.10).
(18.4)
Summary: Slow-neutron-induced Fission
When a nucleus undergoes slow-neutron-induced fission in a reactor, several things happen: About 180 MeV is promptly liberated in a useful form. On average about 2.5 extra neutrons are liberated. Another MeV is generated by radioactivity of the fission fragments and radiative capture of some neutrons. About 30% of this additional energy is lost on neutrinos, which do not interact. Rarely, a delayed neutron is emitted by one of the fission fragments.
Figure 18.8 Elastic scattering of a neutron from a stationary nucleus, before (black) and after (red) the collision: (a) in the rest frame of the nucleus; (b) in the center-of-mass frame.
Nuclear physicists use a measure of the moderating ability of a nucleus with atomic mass A known as the logarithmic energy decrement ξ. This quantity is defined as the average value of the decrease in the logarithm of the energy in one collision,
(18.5)
Assuming that the scattering is isotropic in the center-of-mass frame and averaging over θ, we find (Problem 18.11),
(18.6)
(18.7)
The logarithmic energy decrement turns out to be the most useful measure of neutron slowing when discussing moderation.
A good moderator should have a large value of ξ, corresponding to large fractional energy loss per collision. ξ has a maximum value of one, realized for , since colliding with a proton () is the most efficient way for a neutron to lose energy. On the other hand, ξ is very small when , reflecting the fact that when a neutron collides with a heavy nucleus the neutron simply bounces off, losing little energy. Clearly, light nuclei make better moderators.
* Irrelevant for neutron absorbers Table 18.6 Neutron energy loss and absorption for selected nuclei, including possible moderators and fission poisons.ξ is the logarithmic energy decrement (18.5). N is an estimate of the number of collisions needed to thermalize a neutronthat begins with 2 MeV (Problem 18.12). is the thermal neutron absorption cross section. Source [70, 77].
A good moderator also must not absorb the neutrons that collide with it. Thus light nuclei with small thermal-neutron absorption cross sections make the best moderators. The thermal-neutron absorption cross sections and other relevant data are given in Table 18.6 for some potential moderators and a few other nuclei of interest. Hydrogen, with the lightest nucleus, is a good, but not ideal moderator, because it has a relatively large neutron absorption cross section. Water moderates like hydrogen – – because the oxygen nucleus is neither a particularly good moderator nor a strong absorber of neutrons. Because water is abundant and can also function as a heat transfer fluid in a reactor, it is the moderator most commonly used in commercial nuclear reactors.
Deuterium () is both light and a poor neutron absorber. Heavy water, DO, is therefore an excellent moderator, with . Deuterated water HDO occurs naturally in water on Earth at an abundance of approximately 1 molecule in 3200. HDO can be separated from ordinary water, for example by successive distillation, and then further processed either physically or chemically to obtain DO. The process is energy intensive and costly, but the fact that a reactor fueled with natural uranium and moderated with heavy water can maintain a chain reaction has led to the use of heavy water as a moderator in some power reactors (see §19.4.1). Carbon is less efficient at slowing neutrons down, but has a small neutron absorption cross section and is used extensively as a moderator. Boron, although a light nuclide, is a poor moderator because , which accounts for nearly 20% of naturally occurring boron, has a huge neutron absorption cross section, 11 000 times that of hydrogen. Instead, boron is a classic example of a neutron absorber used to control a nuclear reactor (§19).
Moderators
Moderators slow down fast neutrons before they are captured by fertile nuclei such as . Once slowed to thermal speeds, neutrons have a high probability of inducing fissile nuclei such as to fission. Carbon, hydrogen, water, and deuterium are all good moderators. Deuterium is best since it has both a low mass and a small neutron absorption cross section. Water (effectively hydrogen) and carbon are cheap and common moderators.
With this survey of moderators we have finished introducing all the ingredients necessary to understand energy production by controlled nuclear fission reactions. We return to the subject in §19, where we describe how these ingredients, especially the production, moderation, and control of neutrons from fission, can be combined to make a stable nuclear fission power reactor.
18.4Physics of Nuclear Fusion
From Figure 17.6 it is clear that energy will be given off if light nuclei can be brought to fuse. Light nuclei do not fuse spontaneously, however, because the Coulomb barrier
(18.8)
prevents two nuclei with charges from coming close enough to reach the domain where nuclear forces dominate and lead to fusion. For two protons, the potential energy at the top of the Coulomb barrier is about MeV (Problem 18.13), which is the average kinetic energy in a gas at a temperature of K. The temperature in the core of the Sun is only K, which is far too low to provide the thermal energy necessary to overcome this Coulomb barrier. Yet nuclear fusion takes place spontaneously in stars. The goal of this section is to explain how two key concepts, quantum tunneling and the Boltzmann distribution, combine to make this possible. The full set of fusion reactions that power the Sun and other stars are described in more detail in the context of solar energy in §22. In closing this section we survey the nuclear reactions that are candidates for controlled nuclear fusion here on Earth. Attempts to harness fusion power are described in the following chapter.
18.4.1 Fusion in a Gas at Temperature T
The fusion rate in a gas at temperature T is given in principle by an integral over energy of the product of three factors: (a) the probability of finding two particles with energy E in their center-of-mass frame; (b) the probability that two particles with center-of-mass energy E get close enough to interact; and (c) a reaction rate factor describing the relative likelihood of a fusion event given (a) and (b). The first factor is determined by the Boltzmann distribution, derived in §8; the second factor is given by the barrier penetration factor of §15; and the third factor varies depending on whether the fusion reaction is strong, electromagnetic, or weak. Note that the cross section for a particular fusion process is given by the product of (b) the tunneling probability and (c) the reaction rate factor.
The energy distribution in a hot gas The Boltzmann distribution, which determines the probability of finding a particle in a state with energy in a gas at temperature T, was derived in §8
(18.9)
For a hot gas the energy levels are very close together and we can consider the energy as a continuous variable. The sum over states then becomes an integral, giving a probability per unit energy known as the Maxwell–Boltzmann distribution
