31.4 Nuclear Decay and Conservation Laws

Learning Objectives

By the end of this section, you will be able to:

• Define and discuss nuclear decay.
• State the conservation laws.
• Explain parent and daughter nucleus.
• Calculate the energy emitted during nuclear decay.

Alpha Decay

In alpha decay, a ${}^4\text{He}$ nucleus simply breaks away from the parent nucleus, leaving a daughter with two fewer protons and two fewer neutrons than the parent (see Figure 31.15). One example of $\alpha$ decay is shown in Figure 31.14 for ${}^{\text{238}}\text{U}$. Another nuclide that undergoes $\alpha$ decay is ${}^{\text{239}}\text{Pu}$. The decay equations for these two nuclides are

$${}^{\text{238}}\text{U}\to {}^{\text{234}}\text{Th}+{}^4\text{He}$$

and

$${}^{\text{239}}\text{Pu}\to {}^{\text{235}}\text{U}+{}^4\text{He}.$$

If you examine the periodic table of the elements, you will find that Th has $Z=\text{90}$, two fewer than U, which has $Z=\text{92}$. Similarly, in the second decay equation, we see that U has two fewer protons than Pu, which has $Z=\text{94}$. The general rule for $\alpha$ decay is best written in the format ${}_Z^A{\text{X}}_N$. If a certain nuclide is known to $\alpha$ decay (generally this information must be looked up in a table of isotopes, such as in Appendix B), its $\alpha$ decay equation is

$${}_Z^A{\text{X}}_N\to {}_{Z-2}^{A-4}{\text{Y}}_{N-2}+{}_2^4{\text{He}}_2\left(\alpha \text{decay}\right)$$

where Y is the nuclide that has two fewer protons than X, such as Th having two fewer than U. So if you were told that ${}^{\text{239}}\text{Pu}$ $\alpha$ decays and were asked to write the complete decay equation, you would first look up which element has two fewer protons (an atomic number two lower) and find that this is uranium. Then since four nucleons have broken away from the original 239, its atomic mass would be 235.

It is instructive to examine conservation laws related to $\alpha$ decay. You can see from the equation ${}_Z^A{\text{X}}_N\to {}_{Z-2}^{A-4}{\text{Y}}_{N-2}+{}_2^4{\text{He}}_2$ that total charge is conserved. Linear and angular momentum are conserved, too. Although conserved angular momentum is not of great consequence in this type of decay, conservation of linear momentum has interesting consequences. If the nucleus is at rest when it decays, its momentum is zero. In that case, the fragments must fly in opposite directions with equal-magnitude momenta so that total momentum remains zero. This results in the $\alpha$ particle carrying away most of the energy, as a bullet from a heavy rifle carries away most of the energy of the powder burned to shoot it. Total mass–energy is also conserved: the energy produced in the decay comes from conversion of a fraction of the original mass. As discussed in Atomic Physics, the general relationship is

$$E=(\Delta m)c^2.$$

Here, $E$ is the nuclear reaction energy (the reaction can be nuclear decay or any other reaction), and $\mathrm{\Delta }m$ is the difference in mass between initial and final products. When the final products have less total mass, $\mathrm{\Delta }m$ is positive, and the reaction releases energy (is exothermic). When the products have greater total mass, the reaction is endothermic ($\mathrm{\Delta }m$ is negative) and must be induced with an energy input. For $\alpha$ decay to be spontaneous, the decay products must have smaller mass than the parent.

Beta Decay

There are actually three types of beta decay. The first discovered was “ordinary” beta decay and is called ${\beta }^{-}$ decay or electron emission. The symbol ${\beta }^{-}$ represents an electron emitted in nuclear beta decay. Cobalt-60 is a nuclide that ${\beta }^{-}$ decays in the following manner:

$${}^{\text{60}}\text{Co}\to {}^{\text{60}}\text{Ni}+{\beta }^{-}+\text{neutrino.}$$

The neutrino is a particle emitted in beta decay that was unanticipated and is of fundamental importance. The neutrino was not even proposed in theory until more than 20 years after beta decay was known to involve electron emissions. Neutrinos are so difficult to detect that the first direct evidence of them was not obtained until 1953. Neutrinos are nearly massless, have no charge, and do not interact with nucleons via the strong nuclear force. Traveling approximately at the speed of light, they have little time to affect any nucleus they encounter. This is, owing to the fact that they have no charge (and they are not EM waves), they do not interact through the EM force. They do interact via the relatively weak and very short range weak nuclear force. Consequently, neutrinos escape almost any detector and penetrate almost any shielding. However, neutrinos do carry energy, angular momentum (they are fermions with half-integral spin), and linear momentum away from a beta decay. When accurate measurements of beta decay were made, it became apparent that energy, angular momentum, and linear momentum were not accounted for by the daughter nucleus and electron alone. Either a previously unsuspected particle was carrying them away, or three conservation laws were being violated. Wolfgang Pauli made a formal proposal for the existence of neutrinos in 1930. The Italian-born American physicist Enrico Fermi (1901–1954) gave neutrinos their name, meaning little neutral ones, when he developed a sophisticated theory of beta decay (see Figure 31.16). Part of Fermi’s theory was the identification of the weak nuclear force as being distinct from the strong nuclear force and in fact responsible for beta decay. Chinese-born physicist Chien-Shiung Wu, who had developed a number of processes critical to the Manhattan Project and related research, set out to investigate Fermi’s theory and some experiments whose failures had cast the theory in doubt. She first identified a number of flaws in her contemporaries’ methods and materials, and then designed an experimental method that would avoid the same errors. Wu verified Fermi’s theory and went on to establish the core principles of beta decay, which would become critical to further work in nuclear physics.

The neutrino also reveals a new conservation law. There are various families of particles, one of which is the electron family. We propose that the number of members of the electron family is constant in any process or any closed system. In our example of beta decay, there are no members of the electron family present before the decay, but after, there is an electron and a neutrino. So electrons are given an electron family number of $+1$. The neutrino in ${\beta }^{-}$ decay is an electron’s antineutrino, given the symbol ${\overline{\nu }}_e$, where $\nu$ is the Greek letter nu, and the subscript e means this neutrino is related to the electron. The bar indicates this is a particle of antimatter. (All particles have antimatter counterparts that are nearly identical except that they have the opposite charge. Antimatter is almost entirely absent on Earth, but it is found in nuclear decay and other nuclear and particle reactions as well as in outer space.) The electron’s antineutrino ${\overline{\nu }}_e$, being antimatter, has an electron family number of $\mathrm{–1}$. The total is zero, before and after the decay. The new conservation law, obeyed in all circumstances, states that the total electron family number is constant. An electron cannot be created without also creating an antimatter family member. This law is analogous to the conservation of charge in a situation where total charge is originally zero, and equal amounts of positive and negative charge must be created in a reaction to keep the total zero.

If a nuclide ${}_Z^A{\mathrm{X}}_N$ is known to ${\beta }^{-}$ decay, then its ${\beta }^{-}$ decay equation is

$${}_Z{}^A{\text{X}}_N\to {}_{Z+1}{}^A{\text{Y}}_{N-1}+{\beta }^{-}+{\stackrel{-}{\nu }}_e({\beta }^{-}\text{decay}),$$

where Y is the nuclide having one more proton than X (see Figure 31.17). So if you know that a certain nuclide ${\beta }^{-}$ decays, you can find the daughter nucleus by first looking up $Z$ for the parent and then determining which element has atomic number $Z+1$. In the example of the ${\beta }^{-}$ decay of ${}^{\text{60}}\text{Co}$ given earlier, we see that $Z=\text{27}$ for Co and $Z=\text{28}$ is Ni. It is as if one of the neutrons in the parent nucleus decays into a proton, electron, and neutrino. In fact, neutrons outside of nuclei do just that—they live only an average of a few minutes and ${\beta }^{-}$ decay in the following manner:

$$\text{n}\to \text{p}+{\beta }^{-}+{\stackrel{-}{\nu }}_e.$$

We see that charge is conserved in ${\beta }^{-}$ decay, since the total charge is $Z$ before and after the decay. For example, in ${}^{\text{60}}\text{Co}$ decay, total charge is 27 before decay, since cobalt has $Z=\text{27}$. After decay, the daughter nucleus is Ni, which has $Z=\text{28}$, and there is an electron, so that the total charge is also $\mathrm{28\; +\; (–1)}$ or 27. Angular momentum is conserved, but not obviously (you have to examine the spins and angular momenta of the final products in detail to verify this). Linear momentum is also conserved, again imparting most of the decay energy to the electron and the antineutrino, since they are of low and zero mass, respectively. Another new conservation law is obeyed here and elsewhere in nature. The total number of nucleons $A$ is conserved. In ${}^{\text{60}}\text{Co}$ decay, for example, there are 60 nucleons before and after the decay. Note that total $A$ is also conserved in $\alpha$ decay. Also note that the total number of protons changes, as does the total number of neutrons, so that total $Z$ and total $N$ are not conserved in ${\beta }^{-}$ decay, as they are in $\alpha$ decay. Energy released in ${\beta }^{-}$ decay can be calculated given the masses of the parent and products.

The second type of beta decay is less common than the first. It is ${\beta }^{+}$decay. Certain nuclides decay by the emission of a positive electron. This is antielectron or positron decay (see Figure 31.18).

The antielectron is often represented by the symbol $e^{+}$, but in beta decay it is written as ${\beta }^{+}$ to indicate the antielectron was emitted in a nuclear decay. Antielectrons are the antimatter counterpart to electrons, being nearly identical, having the same mass, spin, and so on, but having a positive charge and an electron family number of $\mathrm{–1}$. When a positron encounters an electron, there is a mutual annihilation in which all the mass of the antielectron-electron pair is converted into pure photon energy. (The reaction, $e^{+}+e^{-}\to \gamma +\gamma$, conserves electron family number as well as all other conserved quantities.) If a nuclide ${}_Z^A{\mathrm{X}}_N$ is known to ${\beta }^{+}$ decay, then its ${\beta }^{+}$decay equation is

$${}_Z^A{\text{X}}_N\to {}_{Z-1}{}^A{\text{Y}}_{N+1}+{\beta }^{+}+{\nu }_e({\beta }^{+}\text{decay}),$$

where Y is the nuclide having one less proton than X (to conserve charge) and ${\nu }_e$ is the symbol for the electron’s neutrino, which has an electron family number of $\mathrm{+1}$. Since an antimatter member of the electron family (the ${\beta }^{+}$) is created in the decay, a matter member of the family (here the ${\nu }_e$) must also be created. Given, for example, that ${}^{\text{22}}\text{Na}$ ${\beta }^{+}$ decays, you can write its full decay equation by first finding that $Z=\text{11}$ for ${}^{\text{22}}\text{Na}$, so that the daughter nuclide will have $Z=\text{10}$, the atomic number for neon. Thus the ${\beta }^{+}$ decay equation for ${}^{\text{22}}\text{Na}$ is

$${}_{\text{11}}^{\text{22}}{\text{Na}}_{\text{11}}\to {}_{\text{10}}^{\text{22}}{\text{Ne}}_{\text{12}}+{\beta }^{+}+{\nu }_e.$$

In ${\beta }^{+}$ decay, it is as if one of the protons in the parent nucleus decays into a neutron, a positron, and a neutrino. Protons do not do this outside of the nucleus, and so the decay is due to the complexities of the nuclear force. Note again that the total number of nucleons is constant in this and any other reaction. To find the energy emitted in ${\beta }^{+}$ decay, you must again count the number of electrons in the neutral atoms, since atomic masses are used. The daughter has one less electron than the parent, and one electron mass is created in the decay. Thus, in ${\beta }^{+}$ decay,

$$\mathrm{\Delta }m=m(\text{parent})-[m(\text{daughter})+{2m}_e],$$

since we use the masses of neutral atoms.

Electron capture is the third type of beta decay. Here, a nucleus captures an inner-shell electron and undergoes a nuclear reaction that has the same effect as ${\beta }^{+}$ decay. Electron capture is sometimes denoted by the letters EC. We know that electrons cannot reside in the nucleus, but this is a nuclear reaction that consumes the electron and occurs spontaneously only when the products have less mass than the parent plus the electron. If a nuclide ${}_Z^A{\mathrm{X}}_N$ is known to undergo electron capture, then its electron capture equation is

$${}_Z^A{\text{X}}_N+e^{-}\to {}_{Z-1}{}^A{\text{Y}}_{N+1}+{\nu }_e(\text{electron capture, or EC})\text{.}$$

Any nuclide that can ${\beta }^{+}$ decay can also undergo electron capture (and often does both). The same conservation laws are obeyed for EC as for ${\beta }^{+}$ decay. It is good practice to confirm these for yourself.

All forms of beta decay occur because the parent nuclide is unstable and lies outside the region of stability in the chart of nuclides. Those nuclides that have relatively more neutrons than those in the region of stability will ${\beta }^{-}$ decay to produce a daughter with fewer neutrons, producing a daughter nearer the region of stability. Similarly, those nuclides having relatively more protons than those in the region of stability will ${\beta }^{+}$ decay or undergo electron capture to produce a daughter with fewer protons, nearer the region of stability.

Gamma Decay

Gamma decay is the simplest form of nuclear decay—it is the emission of energetic photons by nuclei left in an excited state by some earlier process. Protons and neutrons in an excited nucleus are in higher orbitals, and they fall to lower levels by photon emission (analogous to electrons in excited atoms). Nuclear excited states have lifetimes typically of only about ${\text{10}}^{-\text{14}}$ s, an indication of the great strength of the forces pulling the nucleons to lower states. The $\gamma$ decay equation is simply

$${}_Z^A{\text{X}}_N^{*}\to {}_Z{}^A{\text{X}}_N+{\gamma }_1+{\gamma }_2+\cdots (\gamma \text{decay})$$

where the asterisk indicates the nucleus is in an excited state. There may be one or more $\gamma$ s emitted, depending on how the nuclide de-excites. In radioactive decay, $\gamma$ emission is common and is preceded by $\gamma$ or $\beta$ decay. For example, when ${}^{\text{60}}\text{Co}$ ${\beta }^{-}$ decays, it most often leaves the daughter nucleus in an excited state, written ${}^{\text{60}}\text{Ni*}$. Then the nickel nucleus quickly $\gamma$ decays by the emission of two penetrating $\gamma$ s:

$${}^{\text{60}}\text{Ni*}\to {}^{\text{60}}\text{Ni}+{\gamma }_1+{\gamma }_2.$$

These are called cobalt $\gamma$ rays, although they come from nickel—they are used for cancer therapy, for example. It is again constructive to verify the conservation laws for gamma decay. Finally, since $\gamma$ decay does not change the nuclide to another species, it is not prominently featured in charts of decay series, such as that in Figure 31.14.

There are other types of nuclear decay, but they occur less commonly than $\alpha$, $\beta$, and $\gamma$ decay. Spontaneous fission is the most important of the other forms of nuclear decay because of its applications in nuclear power and weapons. It is covered in the next chapter.

Test Prep for AP Courses

Section Summary

• When a parent nucleus decays, it produces a daughter nucleus following rules and conservation laws. There are three major types of nuclear decay, called alpha $\left(\alpha \right),$ beta $\left(\beta \right),$ and gamma $\left(\gamma \right)$. The $\alpha$ decay equation is

  $${}_Z^A{\mathrm{X}}_N\to {}_{Z-2}^{A-4}{\text{Y}}_{N-2}+{}_2^4{\text{He}}_2.$$

• Nuclear decay releases an amount of energy $E$ related to the mass destroyed $\mathrm{\Delta }m$ by

  $$E=(\mathrm{\Delta }m)c^2.$$

• There are three forms of beta decay. The ${\beta }^{-}$decay equation is

  $${}_Z^A{\mathrm{X}}_N\to {}_{Z+1}^A{\text{Y}}_{N-1}+{\beta }^{-}+{\overline{\nu }}_e.$$

• The ${\beta }^{+}$ decay equation is

  $${}_Z^A{\mathrm{X}}_N\to {}_{Z-1}^A{\text{Y}}_{N+1}+{\beta }^{+}+{\nu }_e.$$

• The electron capture equation is

  $${}_Z^A{\mathrm{X}}_N+e^{-}\to {}_{Z-1}^A{\text{Y}}_{N+1}+{\nu }_e.$$

• ${\beta }^{-}$ is an electron, ${\beta }^{+}$ is an antielectron or positron, ${\nu }_e$ represents an electron’s neutrino, and ${\overline{\nu }}_e$ is an electron’s antineutrino. In addition to all previously known conservation laws, two new ones arise— conservation of electron family number and conservation of the total number of nucleons. The $\gamma$ decay equation is

  $${}_Z{}^A{\text{X}}_N^{*}\to {}_Z{}^A{\text{X}}_N+{\gamma }_1+{\gamma }_2+\cdots$$

  $\gamma$ is a high-energy photon originating in a nucleus.

Conceptual Questions

Problems & Exercises

In the following eight problems, write the complete decay equation for the given nuclide in the complete ${}_Z^A{\mathrm{X}}_N$ notation. Refer to the periodic table for values of $Z$.

In the following four problems, identify the parent nuclide and write the complete decay equation in the ${}_Z^A{\mathrm{X}}_N$ notation. Refer to the periodic table for values of $Z$.