10.4 Nuclear Reactions

Alpha Decay

Heavy unstable nuclei emit $\alpha$ radiation. In $\alpha$-particle decay (or alpha decay), the nucleus loses two protons and two neutrons, so the atomic number decreases by two, whereas its mass number decreases by four. Before the decay, the nucleus is called the parent nucleus. The nucleus or nuclei produced in the decay are referred to as the daughter nucleus or daughter nuclei. We represent an $\alpha$ decay symbolically by

where ${}_Z^A\text{X}$ is the parent nucleus, ${}_{Z-2}^{A-4}\text{X}$ is the daughter nucleus, and ${}_2^4\text{H}\text{e}$ is the $\alpha$ particle. In $\alpha$ decay, a nucleus of atomic number Z decays into a nucleus of atomic number $Z-2$ and atomic mass $A-4.$ Interestingly, the dream of the ancient alchemists to turn other metals into gold is scientifically feasible through the alpha-decay process. The efforts of the alchemists failed because they relied on chemical interactions rather than nuclear interactions.

An example of alpha decay is uranium-238:

The atomic number has dropped from 92 to 90. The chemical element with $Z=90$ is thorium. Hence, Uranium-238 has decayed to Thorium-234 by the emission of an $\alpha$ particle, written

Subsequently, ${}_{90}^{234}\text{T}\text{h}$ decays by $\beta$ emission with a half-life of 24 days. The energy released in this alpha decay takes the form of kinetic energies of the thorium and helium nuclei, although the kinetic energy of thorium is smaller than helium due to its heavier mass and smaller velocity.

Beta Decay

In most $\beta$ particle decays (or beta decay), either an electron (${\beta }^{-}$) or positron (${\beta }^{+}$) is emitted by a nucleus. A positron has the same mass as the electron, but its charge is $\text{+}e$. For this reason, a positron is sometimes called an antielectron. How does $\beta$ decay occur? A possible explanation is the electron (positron) is confined to the nucleus prior to the decay and somehow escapes. To obtain a rough estimate of the escape energy, consider a simplified model of an electron trapped in a box (or in the terminology of quantum mechanics, a one-dimensional square well) that has the width of a typical nucleus (${10}^{\mathrm{-14}}\text{m}$). According to the Heisenberg uncertainty principle in Quantum Mechanics, the uncertainty of the momentum of the electron is:

Taking this momentum value (an underestimate) to be the “true value,” the kinetic energy of the electron on escape is approximately

Experimentally, the electrons emitted in ${\beta }^{-}$ decay are found to have kinetic energies of the order of only a few MeV. We therefore conclude that the electron is somehow produced in the decay rather than escaping the nucleus. Particle production (annihilation) is described by theories that combine quantum mechanics and relativity, a subject of a more advanced course in physics.

Nuclear beta decay involves the conversion of one nucleon into another. For example, a neutron can decay to a proton by the emission of an electron (${\beta }^{-}$) and a nearly massless particle called an antineutrino ($\bar{\nu }$):

The notation ${}_{\mathrm{-1}}^0\text{e}$ is used to designate the electron. Its mass number is 0 because it is not a nucleon, and its atomic number is $\mathrm{-1}$ to signify that it has a charge of $\text{−}e$. The proton is represented by ${}_1^1\text{p}$ because its mass number and atomic number are 1. When this occurs within an atomic nucleus, we have the following equation for beta decay:

Enrico Fermi proposed a theory of beta decay in 1934, but his work was initially rejected. Other physicists' experiments to prove it were unsuccessful, casting further doubt on the theory. Chinese-born physicist Chien-Shiung Wu, who had developed a number of processes critical to the Manhattan Project and related research, identified a number of flaws in the earlier experimental methods and materials. She designed a new method, verified Fermi's theory, and later went on to establish the core principles of beta decay. As discussed in another chapter, this process occurs due to the weak nuclear force.

As an example, the isotope ${}_{90}^{234}\text{T}\text{h}$ is unstable and decays by ${\beta }^{-}$ emission with a half-life of 24 days. Its decay can be represented as

Since the chemical element with atomic number 91 is protactinium (Pa), we can write the ${\beta }^{-}$ decay of thorium as

The reverse process is also possible: A proton can decay to a neutron by the emission of a positron (${\beta }^{+}$) and a nearly massless particle called a neutrino (v). This reaction is written as ${}_1^1\text{p}\to {}_0^1\text{n}+{}_{\text{+}1}^0\text{e}\text{+}v.$

The positron ${}_{\text{+}1}^0\text{e}$ is emitted with the neutrino v, and the neutron remains in the nucleus. (Like ${\beta }^{-}$ decay, the positron does not precede the decay but is produced in the decay.) For an isolated proton, this process is impossible because the neutron is heavier than the proton. However, this process is possible within the nucleus because the proton can receive energy from other nucleons for the transition. As an example, the isotope of aluminum ${}_{13}^{26}\text{A}\text{l}$ decays by ${\beta }^{+}$ emission with a half-life of $7.40\times {10}^5\text{y}.$ The decay is written as

The atomic number 12 corresponds to magnesium. Hence,

As a nuclear reaction, positron emission can be written as

The neutrino was not detected in the early experiments on $\beta$ decay. However, the laws of energy and momentum seemed to require such a particle. Later, neutrinos were detected through their interactions with nuclei.

Gamma Decay

A nucleus in an excited state can decay to a lower-level state by the emission of a “gamma-ray” photon, and this is known as gamma decay. This is analogous to de-excitation of an atomic electron. Gamma decay is represented symbolically by

where the asterisk (*) on the nucleus indicates an excited state. In $\gamma$ decay, neither the atomic number nor the mass number changes, so the type of nucleus does not change.

Radioactive Decay Series

Nuclei with $Z>82$ are unstable and decay naturally. Many of these nuclei have very short lifetimes, so they are not found in nature. Notable exceptions include ${}_{90}^{232}\text{T}\text{h}$ (or Th-232) with a half-life of $1.39\times {10}^{10}$ years, and ${}_{92}^{238}\text{U}$ (or U-238) with a half-life of $7.04\times {10}^8$ years. When a heavy nucleus decays to a lighter one, the lighter daughter nucleus can become the parent nucleus for the next decay, and so on. This process can produce a long series of nuclear decays called a decay series. The series ends with a stable nucleus.

To illustrate the concept of a decay series, consider the decay of Th-232 series (Figure 10.13). The neutron number, N, is plotted on the vertical y-axis, and the atomic number, Z, is plotted on the horizontal x-axis, so Th-232 is found at the coordinates $\left(N,Z\right)=\left(142,90\right).$ Th-232 decays by $\alpha$ emission with a half-life of $1.39\times {10}^{10}$ years. Alpha decay decreases the atomic number by 2 and the mass number by 4, so we have

The neutron number for Radium-228 is 140, so it is found in the diagram at the coordinates $\left(N,Z\right)=\left(140,88\right).$ Radium-228 is also unstable and decays by ${\beta }^{-}$ emission with a half-life of 5.76 years to Actinum-228. The atomic number increases by 1, the mass number remains the same, and the neutron number decreases by 1. Notice that in the graph, $\alpha$ emission appears as a line sloping downward to the left, with both N and Z decreasing by 2. Beta emission, on the other hand, appears as a line sloping downward to the right with N decreasing by 1, and Z increasing by 1. After several additional alpha and beta decays, the series ends with the stable nucleus Pb-208.

The relative frequency of different types of radioactive decays (alpha, beta, and gamma) depends on many factors, including the strength of the forces involved and the number of ways a given reaction can occur without violating the conservation of energy and momentum. How often a radioactive decay occurs often depends on a sensitive balance of the strong and electromagnetic forces. These forces are discussed in Particle Physics and Cosmology.

Figure 10.13 In the thorium ${}_{90}^{232}\text{T}\text{h}$ decay series, alpha $(\alpha )$ decays reduce the atomic number, as indicated by the red arrows. Beta (${\beta }^{-}$) decays increase the atomic number, as indicated by the blue arrows. The series ends at the stable nucleus Pb-208.

As another example, consider the U-238 decay series shown in Figure 10.14. After numerous alpha and beta decays, the series ends with the stable nucleus Pb-206. An example of a decay whose parent nucleus no longer exists naturally is shown in Figure 10.15. It starts with Neptunium-237 and ends in the stable nucleus Bismuth-209. Neptunium is called a transuranic element because it lies beyond uranium in the periodic table. Uranium has the highest atomic number $\left(Z=92\right)$ of any element found in nature. Elements with $Z>92$ can be produced only in the laboratory. They most probably also existed in nature at the time of the formation of Earth, but because of their relatively short lifetimes, they have completely decayed. There is nothing fundamentally different between naturally occurring and artificial elements.

Figure 10.14 In the Uranium-238 decay series, alpha ($\alpha$) decays reduce the atomic number, as indicated by the red arrows. Beta (${\beta }^{-}$) decays increase the atomic number, as indicated by the blue arrows. The series ends at the stable nucleus Pb-206.

Notice that for Bi (21), the decay may proceed through either alpha or beta decay.

Figure 10.15 In the Neptunium-237 decay series, alpha ($\alpha$) decays reduce the atomic number, as indicated by the red arrows. Beta (${\beta }^{-}$) decays increase the atomic number, as indicated by the blue arrows. The series ends at the stable nucleus Bi-209.

Radioactivity in the Earth

According to geologists, if there were no heat source, Earth should have cooled to its present temperature in no more than 1 billion years. Yet, Earth is more than 4 billion years old. Why is Earth cooling so slowly? The answer is nuclear radioactivity, that is, high-energy particles produced in radioactive decays heat Earth from the inside (Figure 10.16).

Figure 10.16 Earth is heated by nuclear reactions (alpha, beta, and gamma decays). Without these reactions, Earth’s core and mantle would be much cooler than it is now.

Candidate nuclei for this heating model are ${}^{238}\text{U and}{}^{40}\text{K}$, which possess half-lives similar to or longer than the age of Earth. The energy produced by these decays (per second per cubic meter) is small, but the energy cannot escape easily, so Earth’s core is very hot. Thermal energy in Earth’s core is transferred to Earth’s surface and away from it through the processes of convection, conduction, and radiation.