9.1 Properties of the Nucleus

Atomic Nucleus Structure and Composition

The atomic nucleus sits at the center of every atom, packed with nearly all of the atom's mass in an incredibly small volume. Understanding its structure, composition, and the forces holding it together is the foundation for everything else in nuclear physics, from radioactive decay to energy production and stellar nucleosynthesis.

Nucleon Composition and Nuclear Force

The nucleus is made of protons (positive charge) and neutrons (no charge), collectively called nucleons. These are held together by the strong nuclear force, which operates only at very short distances (on the order of $10^{-15}$ m, roughly the size of the nucleus itself).

• The nucleus is extraordinarily small: about $10^{-15}$ m in diameter, roughly 100,000 times smaller than the atom as a whole.
• Nuclear density is on the order of $10^{17}$ kg/m³, trillions of times denser than ordinary solid matter.
• The strong nuclear force is what keeps the nucleus from flying apart. Protons all carry positive charge, so they repel each other through electrostatic (Coulomb) repulsion. The strong force is powerful enough to overcome this repulsion, but only when nucleons are extremely close together.

Nuclear Models and Subatomic Particles

Two models are commonly used to describe nuclear behavior, each capturing different aspects of how the nucleus works.

The liquid drop model treats the nucleus as a dense, incompressible fluid of nucleons. It's useful for understanding bulk nuclear properties like binding energy and fission, and it forms the basis of the semi-empirical mass formula.

The nuclear shell model is more like the electron shell model you already know from atomic physics. Nucleons fill discrete energy levels, and certain "filled shell" configurations are unusually stable. These correspond to the magic numbers (discussed below).

At a deeper level, nucleons themselves are composite particles:

• A proton is made of two up quarks and one down quark (uud).
• A neutron is made of one up quark and two down quarks (udd).
• Quarks are bound together by gluons, which mediate the strong interaction between quarks, much like photons mediate the electromagnetic force.

Nuclear Binding Energy and Stability

Nuclear binding energy is the energy you'd need to supply to completely break a nucleus apart into its individual protons and neutrons. A higher binding energy per nucleon means a more tightly bound, more stable nucleus.

The binding energy comes from the mass defect: the mass of an assembled nucleus is slightly less than the sum of its individual nucleon masses. That "missing" mass has been converted into binding energy according to Einstein's relation:

$E = mc^2$

where $m$ is the mass defect and $c$ is the speed of light.

A few key points about binding energy:

• Binding energy per nucleon peaks near iron ($^{56}_{26}\text{Fe}$). This is why iron is so abundant in stellar cores: fusion of lighter elements up to iron releases energy, and fission of heavier elements down toward iron also releases energy.
• The semi-empirical mass formula (also called the Bethe-Weizsäcker formula) models binding energy by combining several terms from the liquid drop model: volume, surface, Coulomb, asymmetry, and pairing terms.
• Nuclear stability depends on the interplay of all these factors, not just one.

Atomic Number, Mass Number, and Isotopes

Atomic Number and Element Identity

The atomic number ($Z$) is the number of protons in the nucleus. This single number defines which element you're dealing with: change $Z$ and you change the element entirely.

• A neutral atom has the same number of electrons as protons, so $Z$ also determines the electron configuration and therefore the chemical behavior.
• The periodic table is ordered by increasing $Z$.

Mass Number and Isotopes

The mass number ($A$) is the total number of nucleons (protons + neutrons):

$A = Z + N$

where $N$ is the neutron number.

Isotopes are atoms of the same element (same $Z$) but with different numbers of neutrons (different $N$, and therefore different $A$). Standard nuclide notation writes them as:

$^A_Z X$

For example, $^{12}_6\text{C}$ and $^{14}_6\text{C}$ are both carbon ($Z = 6$), but carbon-12 has 6 neutrons while carbon-14 has 8.

Because isotopes share the same electron configuration, they behave almost identically in chemical reactions. Where they differ is in their nuclear properties: stability, radioactivity, and mass. This is what makes isotopes so useful in applications.

Isotope Applications and Abundance

• Carbon-14 dating exploits the known decay rate of $^{14}\text{C}$ to determine the age of organic materials up to roughly 50,000 years old.
• Deuterium ($^{2}\text{H}$, or heavy hydrogen) has one neutron in addition to its proton. It's used as a moderator in certain nuclear reactors and is a key fuel in fusion research.
• Uranium enrichment separates $^{235}\text{U}$ (fissile) from the far more abundant $^{238}\text{U}$ for use in reactors and weapons.
• The natural abundance of different isotopes varies by element and affects the average atomic mass listed on the periodic table.

Nuclear Stability and Proton-Neutron Ratio

Neutron-to-Proton Ratio and the Band of Stability

Not every combination of protons and neutrons produces a stable nucleus. The neutron-to-proton ratio ($N/Z$) is one of the most important predictors of stability.

• For light elements ($Z < 20$), stable nuclei tend to have roughly equal numbers of protons and neutrons ($N/Z \approx 1$).
• For heavier elements, stable nuclei need proportionally more neutrons. The extra neutrons provide additional strong-force attraction without adding Coulomb repulsion, helping to hold the nucleus together. By the time you reach lead ($Z = 82$), stable isotopes have $N/Z \approx 1.5$.
• Plotting stable nuclei on a chart of $N$ vs. $Z$ produces the band of stability. Nuclei that fall outside this band are radioactive and will undergo decay to move closer to it.

Magic Numbers and Nuclear Shell Structure

Certain nuclei are exceptionally stable, and this stability correlates with specific nucleon counts called magic numbers:

$2, \; 8, \; 20, \; 28, \; 50, \; 82, \; 126$

These correspond to completely filled energy shells in the nuclear shell model, analogous to noble gas electron configurations in atomic physics.

• A nucleus with a magic number of protons or neutrons is more stable than its neighbors.
• Doubly magic nuclei have magic numbers for both protons and neutrons. Examples include $^{4}_{2}\text{He}$ (2 protons, 2 neutrons), $^{16}_{8}\text{O}$ (8 and 8), and $^{208}_{82}\text{Pb}$ (82 protons, 126 neutrons). These are among the most stable nuclei known.

Factors Affecting Nuclear Stability

Several factors work together to determine whether a nucleus is stable:

• Pairing effect: Nuclei with even numbers of both protons and neutrons (even-even) tend to be more stable than those with odd numbers. Of the roughly 250 stable nuclides, the vast majority are even-even. Only four stable odd-odd nuclei exist.
• Semi-empirical mass formula: This quantifies stability by summing contributions from volume energy (bulk attraction), surface energy (nucleons at the surface have fewer neighbors), Coulomb energy (proton-proton repulsion), asymmetry energy (penalty for $N \neq Z$), and pairing energy.
• Island of stability: Theoretical predictions suggest that certain superheavy elements near specific magic numbers could have unexpectedly long half-lives, forming an "island of stability" beyond the current periodic table.

Nuclear Charge and Mass vs. the Atom

Charge Distribution

All of the atom's positive charge is concentrated in the nucleus, which occupies roughly $10^{-15}$ of the atom's linear dimension (or about $10^{-15}$ of the atomic radius). The total nuclear charge is:

$Q = Ze$

where $e$ is the elementary charge ($1.6 \times 10^{-19}$ C).

This extreme concentration of charge creates an intense electric field near the nucleus, which is what shapes electron orbitals and drives chemical bonding. The surrounding electron cloud balances the nuclear charge, making a neutral atom in its ground state electrically neutral overall.

Mass Distribution

The nucleus contains over 99.9% of the atom's total mass, packed into that tiny volume. This gives nuclear matter its extraordinary density of about $10^{17}$ kg/m³.

While this density implies large gravitational forces at the nuclear scale, gravity is still utterly negligible compared to the strong nuclear force and electromagnetic force at these distances. The mass defect, the difference between the total mass of separated nucleons and the actual nuclear mass, directly gives the binding energy through $E = mc^2$. Precise mass measurements using mass spectrometers are one of the primary tools for probing nuclear structure.

Comparison of Nuclear and Atomic Scales

• The ratio of atomic to nuclear radius is roughly $10^{5}$, meaning if the nucleus were a marble, the atom would be a football stadium.
• Electrostatic potential energy between protons inside the nucleus is on the order of MeV, far larger than electron binding energies (eV to keV range).
• The strong force dominates at distances below about $10^{-15}$ m but drops off sharply beyond that, which is why it doesn't affect the electron cloud.
• Quantum tunneling allows nuclear reactions to occur even when particles don't have enough kinetic energy to classically overcome the Coulomb barrier. This is essential for fusion in stellar cores, where temperatures alone wouldn't be sufficient without tunneling.