The stability of a nucleus depends on how strongly its protons and neutrons are bound together by the strong nuclear force. A tightly bound nucleus requires a large amount of energy to separate into its individual nucleons, whereas a weakly bound nucleus can be broken apart more easily. By studying how much energy is needed to separate different nuclei, physicists gain valuable insight into the nature of nuclear forces and the stability of matter.

The binding energy (BE) of a nucleus is defined as the energy required to completely separate the nucleus into its individual protons and neutrons. Binding energy can be determined from the nucleus's mass through Einstein's mass-energy relationship:

One of the remarkable consequences of this relationship is that a bound nucleus has a smaller mass than the combined masses of the free protons and neutrons from which it is made. The stronger the binding between the nucleons, the greater the missing mass. This difference in mass represents energy that was released when the nucleus formed.

Imagine separating a nucleus into its individual protons and neutrons, as shown in Figure 121.1. Energy must be supplied to overcome the attractive nuclear force that holds the nucleus together. When the nucleons are finally separated and at rest, the added energy appears as an increase in the total rest mass of the system. By definition, this energy equals the binding energy of the nucleus.

The corresponding increase in mass is called the mass defect. It is the difference between the combined masses of the individual nucleons and the actual mass of the bound nucleus. For a nuclide [latex]{}^{A}X[/latex], containing [latex]Z[/latex] protons and [latex]N[/latex] neutrons, the mass defect is

Because mass and energy are equivalent, the binding energy is

Here, [latex]m_{\mathrm{tot}}[/latex] is the mass of the nucleus, [latex]m_p[/latex] is the proton mass, and [latex]m_n[/latex] is the neutron mass.

In practice, atomic masses are measured much more accurately than nuclear masses. Since tabulated atomic masses include the electrons surrounding the nucleus, it is convenient to rewrite the binding-energy equation using neutral atoms. Adding [latex]Z[/latex] electrons to both sides replaces the proton masses with the mass of hydrogen atoms, leading to the commonly used expression

In this equation, [latex]m\left({}^{A}X\right)[/latex] is the measured atomic mass of the nuclide, while [latex]m\left({}^{1}\mathrm{H}\right)[/latex] is the mass of a hydrogen atom. Using atomic masses greatly simplifies binding-energy calculations because accurate values are readily available in reference tables.

Atomic masses are commonly expressed in unified atomic mass units (u), where

This conversion factor makes it straightforward to convert a measured mass defect into the corresponding binding energy expressed in mega-electron volts (MeV).

Things Great and Small

Nuclear Decay Helps Explain Earth's Hot Interior

Radioactive decay not only allows scientists to determine the ages of rocks and fossils, but also helps explain why Earth's interior remains hot billions of years after the planet formed. This is an excellent example of how processes occurring inside atomic nuclei can influence the evolution of an entire planet.

Radiometric dating shows that the oldest surviving rocks on Earth formed approximately [latex]3.5\times10^{9}[/latex] years ago, while many independent lines of geological and astronomical evidence indicate that Earth itself is about [latex]4.5\times10^{9}[/latex] years old. Shortly after its formation, Earth was extremely hot because gravitational energy released during planetary formation was converted into thermal energy. As the planet cooled, its surface eventually became cool enough for the first solid rocks to form.

Simple cooling calculations, however, predict that Earth's interior should have cooled much more than it actually has. Seismic waves generated by earthquakes reveal that large regions of Earth's interior remain partially or completely molten. For example, shear (transverse) waves cannot travel through liquids and disappear when passing through the liquid outer core, while compressional (longitudinal) waves continue through it. These observations provide strong evidence that Earth's deep interior remains extremely hot.

A major source of this long-term heating is the radioactive decay of naturally occurring isotopes that have been present since Earth's formation. Isotopes such as [latex]{}^{238}\mathrm{U}[/latex], [latex]{}^{235}\mathrm{U}[/latex], [latex]{}^{232}\mathrm{Th}[/latex], and [latex]{}^{40}\mathrm{K}[/latex] have half-lives comparable to the age of Earth and continue to release energy today. Although each individual decay produces only a tiny amount of energy, the enormous quantity of radioactive material distributed throughout Earth's mantle and crust generates a substantial amount of heat over geological timescales.

Some radioactive isotopes also decay through long chains of intermediate daughter nuclei. For example, uranium-238 undergoes a series of radioactive decays before ultimately becoming stable lead-206. Each step in the decay chain releases additional energy, contributing further to Earth's internal heat.

The amount of power generated by radioactive decay within any cubic meter of rock is very small. However, because Earth's interior occupies an enormous volume and because heat escapes only slowly through the crust, this continuous energy production has significantly slowed the planet's cooling. Modern geophysical studies estimate that radioactive decay supplies roughly half of Earth's present internal heat, with the remainder coming primarily from residual heat left over from planetary formation.

Radioactive decay also produces helium. During alpha decay, an alpha particle is emitted from the nucleus. After slowing down, the alpha particle captures two electrons and becomes a helium atom. Much of the helium extracted from natural gas reservoirs originated through this process over hundreds of millions of years. Because helium atoms are light and move rapidly, any helium that reaches Earth's atmosphere gradually escapes into space, making radioactive decay the principal long-term source of terrestrial helium.

What can we learn by comparing the binding energies of different nuclei? One of the first patterns that emerges is that the total binding energy (BE) is approximately proportional to the number of nucleons, [latex]A[/latex], in the nucleus. For example, it takes roughly twice as much energy to separate the nucleons in [latex]{}^{24}\mathrm{Mg}[/latex] as it does for [latex]{}^{12}\mathrm{C}[/latex], because the larger nucleus contains about twice as many nucleons.

Although total binding energy is useful, it does not allow a fair comparison between nuclei of different sizes. Instead, physicists divide the binding energy by the number of nucleons and examine the binding energy per nucleon, [latex]\mathrm{BE}/A[/latex]. This quantity measures, on average, how tightly each proton or neutron is bound within the nucleus and serves as an indicator of nuclear stability.

Figure 121.3 shows that the average binding energy per nucleon is about 8 MeV for many stable nuclei. However, it is noticeably smaller for both the lightest and the heaviest nuclei. The curve reaches a maximum for nuclei with mass numbers near [latex]A\approx60[/latex], close to iron and nickel. These nuclei are the most tightly bound and are therefore among the most stable found in nature.

This behavior results from the competition between two fundamental forces. The strong nuclear force is attractive and is approximately 100 times stronger than the electrostatic (Coulomb) force between protons. However, unlike the Coulomb force, the strong nuclear force acts only over extremely short distances.

In light nuclei, each nucleon lies close enough to interact strongly with nearly all of the others. As additional nucleons are added, each one forms new attractive interactions, causing the binding energy per nucleon to increase steadily. This trend continues until nuclei reach mass numbers of approximately [latex]A\approx60[/latex].

For larger nuclei, the situation changes. Because the strong nuclear force has such a short range, newly added nucleons are too far away to interact strongly with every other nucleon. In contrast, the Coulomb force between protons acts over much longer distances, so every proton repels every other proton in the nucleus. As nuclei become larger, this increasing electrostatic repulsion gradually reduces the average binding energy per nucleon.

This competition between short-range nuclear attraction and long-range electrostatic repulsion also explains why stable heavy nuclei contain more neutrons than protons. Additional neutrons increase the attractive nuclear force without adding electrical repulsion, helping stabilize the nucleus by separating the positively charged protons.

The graph in Figure 121.3 also contains several pronounced peaks that represent nuclei with exceptionally large binding energies per nucleon. These peaks provide additional evidence about nuclear structure. Many correspond to closed-shell, or magic-number, nuclei, whose proton or neutron shells are completely filled, making them especially stable. Nuclei with even numbers of both protons and neutrons, particularly those with [latex]Z=N[/latex], also tend to be more tightly bound than neighboring nuclei.

The same trends help explain the abundance of elements throughout the universe. Hydrogen is by far the most abundant element, followed by helium, while carbon and oxygen are among the next most common. These elements are produced through nuclear fusion inside stars. Heavier elements are synthesized primarily during explosive stellar events such as supernovae and neutron-star mergers. The exceptional stability of [latex]{}^{4}\mathrm{He}[/latex], visible as one of the prominent peaks in Figure 121.3, also helps explain why alpha particles are commonly emitted during the radioactive decay of heavy nuclei.

The overall trend in binding energy per nucleon has profound consequences throughout physics. It explains why energy is released when light nuclei fuse together inside stars, why heavy nuclei release energy when they undergo fission, and why elements are produced in different astrophysical environments. These same principles also underpin many practical applications of nuclear physics, including nuclear power generation and the production of medical isotopes used for diagnostic imaging and cancer therapy.

Interactive Exploration: Nuclear Fission

Nuclear fission occurs when a heavy nucleus, such as uranium-235, absorbs a neutron and splits into two smaller nuclei. This process releases a large amount of energy, along with two or three additional neutrons. If these neutrons go on to trigger fission in neighboring nuclei, a chain reaction can develop. Depending on how many neutrons continue the process, the chain reaction may quickly die out, remain steady, or grow rapidly.

In this interactive simulation, you will investigate how chain reactions begin, why they occur only under certain conditions, and how nuclear reactors safely control the energy released during fission. You will also explore how neutron-absorbing control rods and non-fissionable materials influence the behavior of the reactor.