Nuclear Instability and Radioactive Decay

8.1 Why Most Nuclei Are Unstable

Of the approximately 3,300 nuclides known to exist in nature or created in laboratories (as of 2025), only about 279 are truly stable. Every other nucleus is on a slow - or not so slow - journey toward a more stable configuration. Understanding why some nuclei sit peacefully in matter while others spontaneously transform is one of the central problems of nuclear physics.

A nucleus is held together by the strong nuclear force, an attractive force that acts between all nucleons - protons and neutrons alike. At distances of about 1 to 3 femtometres (where 1 fm = 10⁻¹⁵ m), this force is roughly 100 times stronger than the electromagnetic repulsion between two individual nucleons at a distance of 1 fm. However, Coulomb repulsion is infinite-range and scales as Z(Z−1), meaning the total electromagnetic repulsion across all protons in a heavy nucleus can exceed the binding capacity of the strong force. The strong force has a critical limitation: beyond about 3 fm, its influence essentially vanishes.

This tension - short-range attraction versus long-range repulsion - is the root cause of nuclear instability. For a nucleus to be stable, the binding provided by the strong force must outweigh the disruptive Coulomb repulsion. The balance depends on three numbers: the proton count (Z), the neutron count (N), and the total nucleon count, or mass number ($A = Z + N$). When this balance tips too far in either direction, the nucleus decays.

The consequences of nuclear instability extend far beyond the physics laboratory. Radioactive decay provides the clock for radiometric dating, the radiation for cancer therapy, the signatures for medical imaging, and the energy that powers nuclear reactors. Understanding why nuclei decay, how they decay, and at what rate is essential for all of these applications.

8.2 The Forces Inside the Nucleus

The Strong Nuclear Force

The strong nuclear force is one of the four fundamental forces of nature. Unlike electromagnetism, which only acts between charged particles, the strong force acts equally between all nucleon pairs: proton-proton, neutron-neutron, and proton-neutron. This charge independence explains why neutrons, despite having no electric charge, play such a vital role in nuclear stability. A nucleus made only of protons would tear itself apart under Coulomb repulsion; neutrons provide extra binding through the strong force without adding to the repulsion.

At very short distances - below about 0.5 fm - the strong force actually becomes repulsive, creating a "hard core" that prevents nucleons from collapsing into one another. This repulsive core ensures that nuclei have well-defined sizes and do not implode.

Attraction Versus Repulsion

In any nucleus with more than one proton, two competing interactions determine whether the nucleus is stable. The strong nuclear force pulls all nucleons together, while the Coulomb force pushes protons apart. For light nuclei, where protons are packed closely together, the strong force easily dominates. This is why stable light nuclei tend to have roughly equal numbers of protons and neutrons.

As the proton count increases, however, the Coulomb repulsion grows dramatically - it scales roughly as Z(Z−1), meaning the total repulsive force approximately quadruples each time the atomic number doubles. To compensate, more neutrons must be added. The additional neutrons contribute binding energy through the strong force without adding to the Coulomb repulsion. Consequently, heavy stable nuclei require significantly more neutrons than protons. For example, lead-208 contains 82 protons and 126 neutrons, giving an N/Z ratio of about 1.54. Beyond bismuth (Z = 83), no amount of neutron excess can stabilise a nucleus, and all isotopes are radioactive.

8.3 The Valley of Stability

When every known stable isotope is plotted on a graph of neutron number (N) against proton number (Z), the points form a distinct curved band called the valley of stability. Isotopes lying within this band are generally stable. Those lying outside it are radioactive, and their position relative to the valley predicts how they will decay.

For light elements with Z ≤ 20, the valley follows approximately N = Z. Carbon-12 (6p, 6n), oxygen-16 (8p, 8n), and calcium-40 (20p, 20n) all sit exactly on this line and are exceptionally stable. As Z increases, the valley gradually curves toward the neutron-rich side. By Z = 82 (lead), the stable isotope ²⁰⁸Pb requires 126 neutrons - more than 1.5 neutrons for every proton. No stable isotopes exist beyond bismuth-209 (Z = 83), although some radioactive isotopes such as uranium-238 have half-lives comparable to the age of the Earth.

Predicting Decay from Position

The position of an isotope on the N-Z diagram is a remarkably good predictor of its decay mode:

8.4 What Makes a Nucleus Stable?

The N/Z Ratio

The neutron-to-proton ratio is the single most important factor governing nuclear stability. For elements up to Z = 20, the optimal ratio is approximately 1.0. As Z increases, the optimal ratio rises: to about 1.25 for elements in the Z = 21–40 range, and to roughly 1.5 for the heaviest stable elements near lead. This trend is a direct consequence of the Coulomb problem: extra neutrons provide strong-force binding without electromagnetic repulsion. However, this strategy has a limit. Beyond lead-208, the Coulomb repulsion becomes so great that no number of additional neutrons can produce a stable nucleus.

Even-Odd Effects

A striking empirical pattern emerges when stable isotopes are sorted by whether they contain even or odd numbers of protons and neutrons. Nuclei with even numbers of both (even-even nuclei) are far more common than any other type. Only four stable isotopes have odd numbers of both protons and neutrons: ¹H, ⁶Li, ¹⁰B, and ¹⁴N.

This pattern arises from pairing energy: two protons or two neutrons can occupy the same quantum state with opposite spins, lowering their combined energy. The semi-empirical mass formula incorporates this effect through the pairing term, which adds extra binding energy for even-even configurations and reduces it for odd-odd configurations.

Magic Numbers and the Shell Model

Certain specific numbers of protons or neutrons - 2, 8, 20, 28, 50, 82, and 126 - produce nuclei of exceptional stability. These magic numbers were first identified from systematic trends: elements with magic proton numbers have more stable isotopes than their neighbours, and nuclei with magic neutron numbers have more stable isotones (nuclei sharing the same N). Nuclei with both proton and neutron counts at magic values - called doubly magic - are extraordinarily stable. Examples include ⁴He (2, 2), ¹⁶O (8, 8), ⁴⁰Ca (20, 20), and ²⁰⁸Pb (82, 126).

The magic numbers are explained by the nuclear shell model, developed by Maria Goeppert Mayer and others in 1949. In this model, each nucleon moves in an average potential created by all the other nucleons, occupying discrete quantum energy levels much like electrons in an atom. When a shell is completely filled - that is, when the proton or neutron count equals a magic number - the nucleus gains exceptional stability, analogous to the chemical inertness of noble gases.

The shell model explains many observations that the simpler liquid drop model cannot: the existence of magic numbers, the spins and parities of nuclear ground states, the energies of excited states, and the enhanced stability of doubly magic nuclei. Tin (Z = 50), for instance, has ten stable isotopes - more than any other element - because 50 is a magic proton number. Its neighbours indium (Z = 49) and antimony (Z = 51) have only two stable isotopes each.

Binding Energy and the Semi-Empirical Mass Formula

The binding energy of a nucleus is the energy required to separate it into its constituent nucleons. It corresponds to the mass difference between the separated nucleons and the actual nucleus, converted to energy via $E = mc^2$. The binding energy per nucleon (BE/A) is a direct measure of stability: higher values mean each nucleon is more tightly bound.

The binding energy per nucleon curve rises rapidly for light nuclei, reaches a maximum at iron-56 (8.79 MeV per nucleon), and then gradually falls for heavier elements. This curve explains why both nuclear fusion (combining light nuclei to climb the curve) and nuclear fission (splitting heavy nuclei to move toward the peak) release energy. Iron-56 sits at the very top of this curve, making it the most stable nucleus in existence.

The semi-empirical mass formula (SEMF), also called the Weizsäcker formula, provides a quantitative framework for understanding binding energy by combining five physically motivated terms:

The SEMF reproduces experimental binding energies within 1–2 MeV for most nuclei but systematically underestimates binding near magic numbers, where the shell model must be invoked to account for the extra stability conferred by closed shells.

Using the SEMF - A Worked Example

Let's calculate the binding energy per nucleon for iron-56 (Z=26, N=30, A=56) using the SEMF.

This is within 1% of the experimental value of 8.79 MeV/nucleon, demonstrating the predictive power of the SEMF. The slight underestimate near magic numbers (N=28 is close to a magic number) is expected.

8.5 How Unstable Nuclei Decay

Alpha Decay

Alpha decay is the emission of a helium-4 nucleus: two protons and two neutrons bound together as a single particle. It is the dominant decay mode for very heavy nuclei (typically Z > 83, A > 210) because the alpha particle is itself an exceptionally stable cluster, and its emission simultaneously reduces both the proton count and the total mass.

The Q-value - the energy released - is calculated from the mass difference between the parent nucleus and the products: $Q = (m_X - m_Y - m_\alpha)c^2$. This energy appears as kinetic energy shared between the alpha particle and the recoiling daughter nucleus. Because the alpha particle is much lighter, it carries away the majority of the energy, typically 4–9 MeV.

Alpha decay is fundamentally a quantum tunnelling process. The alpha particle exists as a pre-formed cluster inside the nucleus, bouncing against the potential wall created by the strong force. Each collision with the nuclear surface offers a small probability of tunnelling through the Coulomb barrier - the electrostatic potential that would classically trap the particle inside. This tunnelling mechanism, first explained by George Gamow in 1928, was one of the earliest compelling demonstrations of quantum mechanics.

The relationship between alpha decay energy and half-life is described by the Geiger-Nuttall law:

A modest change in decay energy produces an enormous change in half-life. Polonium-212 decays with $Q_\alpha = 8.95$ MeV and a half-life of only 0.3 microseconds, whereas uranium-238 has $Q_\alpha = 4.27$ MeV and a half-life of 4.47 billion years - a difference of twenty orders of magnitude in half-life arising from a factor of only two in decay energy.

Beta-Minus Decay

Beta-minus decay occurs in neutron-rich nuclei and converts a neutron into a proton, emitting an electron (the beta particle) and an electron antineutrino:

In nuclear notation:

The Q-value is shared as kinetic energy among the electron, antineutrino, and recoiling daughter nucleus. Unlike alpha particles, which are emitted with discrete energies, beta particles display a continuous energy spectrum from near zero up to a maximum equal to the Q-value. This continuous spectrum was historically puzzling and led Wolfgang Pauli to postulate the existence of the neutrino in 1930.

Beta decay is mediated by the weak nuclear force, operating through the exchange of W and Z bosons. It is the mechanism by which neutron-rich nuclei adjust their N/Z ratio and move toward the valley of stability.

Beta-Plus Decay and Electron Capture

For proton-rich nuclei, the reverse process occurs. A proton transforms into a neutron, emitting a positron and an electron neutrino:

Because a positron has the same rest mass as an electron (0.511 MeV/c²), β⁺ decay is only energetically possible when the Q-value exceeds 1.022 MeV. For nuclei where the Q-value is positive but below this threshold, electron capture provides an alternative pathway. The nucleus captures one of its own orbital electrons - most likely from the inner K or L shell - converting a proton into a neutron and emitting only a neutrino:

After electron capture, the resulting vacancy in an inner electron shell is filled by electrons from outer shells, producing characteristic X-rays unique to the daughter element. Both β⁺ decay and electron capture reduce Z by one while keeping A constant, shifting proton-rich nuclei toward the valley of stability.

Gamma Emission

Gamma decay involves the emission of a high-energy photon from an excited nucleus. It changes neither the proton count, the neutron count, nor the mass number - it simply allows the nucleus to shed excess energy and transition to a lower energy state:

Gamma emission typically follows alpha or beta decay, as the daughter nucleus is often left in an excited state. The excess energy is released almost instantaneously, usually within 10⁻¹² seconds, as one or more gamma photons. Gamma rays are the most penetrating form of nuclear radiation.

Build an Atom - Interactive Simulation

Use this custom interactive tool to build your own atoms by adding protons, neutrons, and electrons. Observe how the element identity, stability, and charge change as you adjust the particle counts.

Summary of Decay Modes

Conservation Laws in Nuclear Decay

The neutrino's existence was postulated precisely because beta decay appears to violate energy conservation without it. Including the neutrino ensures conservation of energy, momentum, and lepton number simultaneously.

8.6 Decay Chains

Many heavy radioactive isotopes cannot reach stability through a single decay. Instead they undergo a decay chain - a sequence of successive alpha and beta decays, each producing a new radioactive daughter, until a stable isotope is finally reached.

The uranium-238 series is the most extensively studied decay chain. It consists of 14 steps - eight alpha decays and six beta-minus decays - that transform U-238 into stable lead-206. The half-lives within this single chain vary extraordinarily, from 4.47 billion years for U-238 down to 164 microseconds for polonium-214. This variation demonstrates how sensitive radioactive decay rates are to the specific nuclear structure and Q-value of each isotope.

This decay chain is the primary natural source of background radiation and is responsible for the presence of radon-222 gas in buildings, which is a significant public health concern.

8.7 Half-Life and Activity

The Mathematics of Decay

Radioactive decay is a first-order kinetic process: the rate of decay at any instant is directly proportional to the number of radioactive nuclei present:

Here N is the number of radioactive nuclei, t is time, and λ is the decay constant - a characteristic property of each isotope. Integrating this equation gives the exponential decay law:

The half-life (t₁/₂) is the time required for half the nuclei in a sample to decay. Setting N = N₀/2 in the equation above yields the fundamental relationship:

Half-life is an intrinsic property of each radioisotope. It is independent of temperature, pressure, chemical environment, and the age of the sample - a unique feature of nuclear processes that distinguishes radioactive decay from chemical reactions.

Mean Lifetime

Related to half-life is the mean lifetime (τ), which represents the average time an individual nucleus survives before decaying. The relationship is:

While half-life is more commonly used, mean lifetime appears in many theoretical derivations and is the natural time constant in the exponential decay equation.

Activity and Its Units

The activity of a sample is its rate of decay - the number of disintegrations per unit time:

The SI unit of activity is the becquerel (Bq), defined as one disintegration per second. An older unit still in common use is the curie (Ci), originally defined as the activity of one gram of radium-226 (3.70 × 10¹⁰ Bq). Activity, like the number of nuclei, decays exponentially: $A(t) = A_0 e^{-\lambda t}$. This equation is the foundation of all radioactive dating techniques.

8.8 Applications of Radioactive Decay

Nuclear Medicine

Radioactive isotopes have transformed modern medicine. The ideal diagnostic radioisotope emits gamma rays detectable outside the body and has a half-life long enough for the procedure but short enough to minimise patient dose. Technetium-99m is the most widely used diagnostic isotope, employed in approximately 80% of all nuclear medicine scans worldwide. Its 6-hour half-life, pure gamma emission, and versatile chemistry make it suitable for imaging bones, heart muscle, kidneys, and the thyroid.

Radiometric Dating

The predictable mathematics of radioactive decay provides a precise clock for measuring the age of materials. Carbon-14 dating relies on the constant production of ¹⁴C in the atmosphere by cosmic rays and its incorporation into living organisms. After death, the ¹⁴C content decreases with its 5,730-year half-life, allowing archaeologists to date organic materials up to about 50,000 years old. For geological timescales, uranium-lead dating uses the ²³⁸U → ²⁰⁶Pb chain (half-life 4.47 billion years) to date the oldest rocks and meteorites in the solar system.

PET Imaging

Positron emission tomography (PET) directly exploits beta-plus decay. A patient is injected with a glucose analog labelled with fluorine-18 (¹⁸F), which undergoes β⁺ decay with a 110-minute half-life. The emitted positron annihilates with an electron, producing two 511 keV gamma rays travelling in opposite directions. Detecting these coincident photons allows PET scanners to build 3D images of metabolic activity - particularly useful for detecting cancers, which consume more glucose than healthy tissue.

How Radiometric Dating Actually Works

Radiometric dating doesn't just measure activity - it measures parent-to-daughter isotope ratios. In uranium-lead dating, scientists measure the ratio of ²⁰⁶Pb to ²³⁸U in a mineral. Because each U-238 decay produces exactly one Pb-206 atom, this ratio increases predictably over time. The age equation is: $t = \frac{1}{\lambda} \ln\left(1 + \frac{N_{\text{daughter}}}{N_{\text{parent}}}\right)$. A critical assumption is that the system remained closed - no lead or uranium entered or left the mineral since it formed.

Why U-235, Not U-238?

Nuclear reactors use uranium-235 as fuel, not the more abundant U-238. This is because U-235 is fissile - its nucleus splits when struck by a slow (thermal) neutron, releasing energy and more neutrons to sustain a chain reaction. U-238 absorbs neutrons without fissioning, acting as a "poison" that would stop the chain reaction if concentrated. Natural uranium contains only 0.7% U-235, so enrichment (increasing the U-235 fraction to 3–5%) is required for most reactor designs.

Energy and Industry

Radioactive decay and induced fission of heavy elements power nuclear reactors, which provide roughly 10% of global electricity. On a smaller scale, americium-241 in smoke detectors uses alpha particles to ionise air, while industrial thickness gauges employ beta radiation to monitor paper, metal, and plastic production lines.

Common Misconceptions - Don't Fall For These!

Interactive Check Questions