31.3 Substructure of the Nucleus

The atomic nucleus is the dense core of every atom, housing protons and neutrons bound together by the strong nuclear force. Understanding nuclear substructure explains why elements have the identities they do, why isotopes exist, and why nuclear reactions release such enormous amounts of energy.

Atomic Nucleus

Structure of atomic nuclei

The nucleus contains two types of particles collectively called nucleons: protons and neutrons. Protons carry a positive charge and define which element an atom is. Neutrons are electrically neutral and contribute to the atom's mass without changing its elemental identity.

These nucleons are packed into an incredibly small space and held together by the strong nuclear force, which is powerful enough to overcome the electromagnetic repulsion that protons exert on each other. Without this force, the nucleus would fly apart since all those positive charges would repel one another.

Electrons orbit the nucleus in shells or orbitals and determine the atom's chemical properties and reactivity. But the nucleus itself controls the atom's identity and nuclear behavior.

Atomic number vs mass number

• Atomic number ($Z$): the number of protons in the nucleus. This single number defines the element. Hydrogen always has $Z = 1$, carbon always has $Z = 6$, and so on. In a neutral atom, the number of electrons equals $Z$, maintaining electrical neutrality.
• Mass number ($A$): the total number of nucleons (protons + neutrons). This gives the approximate atomic mass in atomic mass units (amu). You can find the neutron count with $A - Z$.

Isotopes are atoms of the same element (same $Z$) but with different numbers of neutrons (different $A$). For example, carbon-12, carbon-13, and carbon-14 all have 6 protons, but they have 6, 7, and 8 neutrons respectively. Isotopes share nearly identical chemical properties because they have the same electron configuration, but they differ in physical properties like mass and nuclear stability. Carbon-14, for instance, is radioactive while carbon-12 is stable.

Nuclear Properties

Nuclear density calculation and significance

Nuclear density is calculated using:

$\rho = \frac{m}{V} = \frac{m}{\frac{4}{3}\pi r^3}$

where $m$ is the nuclear mass and $r$ is the nuclear radius.

The result is roughly $2.3 \times 10^{17}$ kg/m³. To put that in perspective, water has a density of about 1000 kg/m³, so nuclear matter is on the order of $10^{14}$ times denser. A teaspoon of pure nuclear matter would weigh hundreds of millions of tons.

This extraordinary density reflects how tightly the strong force packs nucleons together. It also helps explain why nuclear reactions (fission and fusion) release so much energy compared to chemical reactions.

Properties of nuclear force

The nuclear force (also called the strong force, at the nuclear level) is the attractive force that acts between all nucleons, whether proton-proton, neutron-neutron, or proton-neutron.

Key characteristics:

• It is short-range, effective only over distances on the order of $10^{-15}$ m (about the size of a nucleus). Beyond that distance, it drops off rapidly to essentially zero.
• It is the strongest of the four fundamental forces (gravitational, electromagnetic, weak, and strong).
• It does not depend on electric charge. Neutrons feel it just as strongly as protons do.

Nuclear vs electromagnetic forces

These two forces compete inside the nucleus:

• The nuclear force attracts all nucleons to each other but only works at very short range ($\sim 10^{-15}$ m).
• The electromagnetic force causes protons to repel each other due to their like positive charges. It has infinite range and follows Coulomb's law, decreasing with distance but never fully disappearing.

At nuclear distances, the strong force is roughly 100 times stronger than the electromagnetic force. This is why the nucleus holds together: the attractive nuclear force overwhelms the repulsive electromagnetic force at close range.

However, as nuclei get larger (more protons), the electromagnetic repulsion grows because every proton repels every other proton across the entire nucleus, while the nuclear force only acts between immediate neighbors. This is why very large nuclei tend to be unstable. The energy holding the nucleus together is called the binding energy, which represents the energy you'd need to supply to completely break a nucleus apart into individual nucleons.

Nuclear Models and Stability

Two models help describe nuclear behavior:

• The liquid drop model treats the nucleus as a dense, incompressible fluid of nucleons. It's useful for understanding nuclear fission and fusion, and it captures bulk properties like density and binding energy trends.
• The nuclear shell model arranges nucleons into discrete energy levels (shells), similar to how electrons fill atomic orbitals. This model explains magic numbers (2, 8, 20, 28, 50, 82, 126), which are specific proton or neutron counts that produce especially stable nuclei.

Nuclear stability depends heavily on the ratio of protons to neutrons. Light stable nuclei tend to have roughly equal numbers of each ($Z \approx N$), while heavier stable nuclei need progressively more neutrons than protons to offset the growing electromagnetic repulsion.

When a nucleus is unstable, it undergoes radioactive decay, emitting particles or energy to reach a more stable configuration. The three main types are alpha decay (emitting a helium nucleus), beta decay (converting a neutron to a proton or vice versa), and gamma decay (emitting high-energy photons).