117 Substructure of the Nucleus

What is inside the atomic nucleus? Why are some nuclei stable for billions of years while others decay in fractions of a second? Why do some radioactive nuclei emit alpha particles, others beta particles, and others gamma rays? And why do nuclear reactions release energies that are millions of times larger than those involved in ordinary chemical reactions? Answering these questions has led physicists to discover the fundamental structure of matter and the forces that bind atomic nuclei together. These discoveries not only transformed physics but also laid the foundation for modern nuclear medicine, medical imaging, radiation therapy, and many other technologies used throughout healthcare.

The Building Blocks of the Nucleus

Earlier in this text we identified the proton as the positively charged particle found inside the nucleus. We now know that every nucleus contains two types of particles:

• Protons, which carry a positive electric charge.
• Neutrons, which have no electric charge.

Together, protons and neutrons are called nucleons because they are the particles that make up atomic nuclei. The neutron has nearly the same mass and intrinsic spin as the proton but carries zero electric charge ([latex]q=0[/latex]). Although the proton and neutron differ greatly in electrical properties, their masses differ by less than one percent. Both particles are far more massive than an electron:

The masses of these particles are summarized in Table 117.1. On atomic and nuclear scales, kilograms are inconveniently small units of mass. Instead, physicists commonly use the unified atomic mass unit (u), defined as

This unit is defined so that a neutral carbon-12 atom has a mass of exactly 12 u. Nuclear masses are also commonly expressed in units of [latex]\text{MeV}/c^2[/latex]. These units are especially convenient because Einstein's mass-energy relationship directly converts mass into energy:

For example, if the entire rest mass of a proton were converted into energy, the result would be

A particularly useful conversion factor is

This relationship appears frequently throughout nuclear physics because it allows mass differences to be converted directly into the energies released during radioactive decay and nuclear reactions.

Describing Atomic Nuclei

Every atomic nucleus is completely defined by the numbers of protons and neutrons it contains. A particular combination of these particles is called a nuclide. The complete notation for a nuclide is

where:

• [latex]Z[/latex] is the atomic number, equal to the number of protons.
• [latex]N[/latex] is the number of neutrons.
• [latex]A[/latex] is the mass number, equal to the total number of nucleons.
• [latex]\mathrm{X}[/latex] is the chemical symbol of the element.

The mass number is therefore given by

Because every element has a unique atomic number, specifying both the element symbol and [latex]Z[/latex] is somewhat redundant. For example, calcium always has [latex]Z=20[/latex], uranium always has [latex]Z=92[/latex], and oxygen always has [latex]Z=8[/latex]. Since protons and neutrons each have masses close to 1 u, the mass of an atom is approximately equal to its mass number expressed in atomic mass units. For example, an oxygen nucleus containing eight protons and eight neutrons has

so its mass is close to 16 u.

Nuclides and Isotopes

Consider a few examples. The nucleus of ordinary hydrogen contains a single proton and no neutrons, so it is written

An alpha particle is simply the nucleus of a helium atom containing two protons and two neutrons:

Hydrogen also exists in two less common forms. Deuterium contains one proton and one neutron:

while tritium, a radioactive isotope of hydrogen, contains one proton and two neutrons:

These three forms of hydrogen have nearly identical chemical properties because chemistry depends primarily on the electrons surrounding the nucleus. Their nuclear properties, however, differ substantially because they contain different numbers of neutrons. Nuclei that contain the same number of protons but different numbers of neutrons are called isotopes of the same element. Because the element symbol uniquely determines the atomic number, physicists usually write nuclides in the simplified notation

Thus, the hydrogen isotopes become

and the alpha particle is simply written as

Similarly, uranium-238 is written as

From the periodic table we know uranium has [latex]Z=92[/latex], so the number of neutrons is

Nuclear Size and Density

Experiments show that atomic nuclei behave much like tiny, densely packed spheres of nucleons. Although protons and neutrons move rapidly inside the nucleus, they remain confined to an extremely small volume by the strong nuclear force. Nucleons can be separated only by supplying enormous amounts of energy, and they strongly resist being compressed into an even smaller space. Measurements of thousands of nuclei reveal that the radius of a nucleus is well approximated by

where

and A is the mass number. (One femtometer, or 1 fm, equals [latex]10^{-15}\ \text{m}[/latex].) Because the radius increases as [latex]A^{1/3}[/latex], the nuclear volume is proportional to the number of nucleons:

This relationship indicates that nuclei have nearly the same density regardless of their size. As additional protons and neutrons are added, the nucleus grows just enough to accommodate them while maintaining almost constant density.

Nuclear Forces and Stability

One of the most important questions in nuclear physics is: What holds the nucleus together? Every nucleus contains positively charged protons, and according to Coulomb's law, these protons strongly repel one another. Because the protons are separated by only a few femtometers, the electric repulsion between them is enormous. If the electromagnetic force were the only force present, atomic nuclei could not exist. The stability of the nucleus is possible because nature includes two additional fundamental interactions: the strong nuclear force and the weak nuclear force. Together, these forces govern the structure, stability, and radioactive decay of atomic nuclei. The strong nuclear force acts only over extremely short distances—roughly a few femtometers—but within that range it is extraordinarily powerful. It attracts neighboring nucleons so strongly that it easily overcomes the electrical repulsion between protons. At very short distances the strong force becomes repulsive, preventing nucleons from collapsing into one another. As a result, nuclei behave somewhat like tightly packed spheres of hard particles that strongly resist both being pulled apart and being compressed further. Unlike gravity and the electromagnetic force, which act over long distances, the strong nuclear force rapidly decreases to essentially zero once nucleons are separated by more than a few femtometers. This short range explains why it acts only inside atomic nuclei. The tremendous strength of the nuclear forces is also responsible for the large energies released during radioactive decay and nuclear reactions. As the nucleus rearranges into a more stable configuration, the nuclear forces perform work. Since work is given by

even though the distance over which the force acts is extremely small, the force itself is so large that the resulting energy release can be millions of times greater than that of a typical chemical reaction. The existence of different types of radioactive decay demonstrates that more than one nuclear force is involved. In this simplified model, the strong nuclear force is associated with alpha decay, while the weak nuclear force is responsible for beta decay.

The Chart of the Nuclides

The known stable and radioactive nuclei can be organized on a diagram called the chart of the nuclides. Rather than arranging elements by their chemical properties as in the periodic table, this chart organizes nuclei according to their numbers of protons and neutrons. Each nucleus occupies a unique position based on:

• [latex]Z[/latex], the number of protons (atomic number).
• [latex]N[/latex], the number of neutrons.

Examining the chart reveals clear patterns in nuclear stability, radioactive decay, and the relative abundance of isotopes.

For light nuclei, stable isotopes generally have nearly equal numbers of neutrons and protons, so stable nuclei lie close to the line

Detailed measurements also show that nuclei are often especially stable when both the numbers of protons and neutrons are even. Pairing of nucleons increases the overall binding of the nucleus. As nuclei become heavier, however, stable nuclei contain progressively more neutrons than protons. The additional neutrons help separate the positively charged protons, reducing the long-range Coulomb repulsion while still contributing to the attractive strong nuclear force. Radioactive nuclei located outside the region of stability tend to decay toward more stable combinations of neutrons and protons. As a result, radioactive decay generally moves nuclei closer to the stable band shown in the chart. Another striking feature of the chart is that certain numbers of protons or neutrons produce especially stable nuclei. These values are called magic numbers. Their existence suggests that nucleons occupy discrete energy shells inside the nucleus, much like electrons occupy shells around the nucleus in atoms. This idea forms the basis of the nuclear shell model, one of the most successful models of nuclear structure. The shell model explains many observed nuclear properties, including energy levels, radioactive decay patterns, and the enhanced stability of nuclei with completely filled shells. Since the 1940s, physicists have synthesized increasingly heavy elements beyond uranium, known as transuranic elements. Today, elements with atomic numbers as high as [latex]Z=118[/latex] have been created. The nuclear shell model predicts that an island of stability may exist among even heavier nuclei, where certain combinations of protons and neutrons could produce relatively long-lived superheavy elements.