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Not all nuclei are stable. Some isotopes — once formed in stellar nucleosynthesis or in nuclear reactors — exist forever; others decay within microseconds. The dividing line is set by an intricate balance between the short-range, attractive strong nuclear force and the long-range, repulsive electromagnetic force between protons. This lesson introduces the N–Z stability diagram, the three principal modes of radioactive decay (α, β⁻, β⁺), and the conservation equations that govern each. It is the qualitative scaffold for the much more quantitative treatment in the nuclear-physics course.
Specification mapping. This lesson develops AQA A-Level Physics (7408) Particles and Radiation strand, sub-strand 3.2.1.2 (stability of the nucleus, the strong nuclear force as the binding mechanism, and the modes of radioactive decay including α, β⁻ and β⁺) and bridges to 3.2.1.3 (particles and antiparticles, including positrons and antineutrinos as decay products). Refer to the official AQA 7408 specification document for the authoritative wording. The lesson sits between the constituents-of-the-atom introduction (order 0) and the dedicated particles-antiparticles-and-photons treatment (order 2) because students need to know that decay produces the antiparticles whose properties will be developed next.
Synoptic links. Three principal synoptic threads run through this content. First, the quark model and the weak interaction (3.2.1.4–6, our orders 3–4) — at the quark level, β⁻ decay is d → u + e⁻ + ν̄ₑ with W⁻ emission, and β⁺ decay is u → d + e⁺ + νₑ with W⁺ emission; the qualitative picture given here is sharpened to a Feynman-diagram picture in the later lessons. Second, conservation laws (3.2.1.7, our order 6) — every decay equation must conserve charge, mass number, lepton number and (in non-weak processes) strangeness; the antineutrino in β⁻ decay is required by lepton-number conservation, not merely a convenience. Third, nuclear physics (course 6, 3.8) — half-life, activity, decay constant and decay law are developed quantitatively in the dedicated nuclear-physics course; this lesson gives the qualitative framework that course expands.
Why does the nucleus hold together at all? At first sight the situation looks impossible: the nucleus contains tightly packed positive protons that should repel each other electromagnetically. Without an additional attractive force, no nucleus heavier than hydrogen could exist.
The strong nuclear force is that additional force. Its key properties:
The force-distance graph for the strong nuclear force has a characteristic shape:
graph LR
A["r < 0.5 fm"] --> B["Strong repulsion"]
C["r ≈ 1 fm"] --> D["Strong attraction (binds nucleons)"]
E["r > 3 fm"] --> F["Force essentially zero"]
Compare with the electromagnetic force between two protons, which follows Coulomb's inverse-square law at all separations and is repulsive everywhere. At very short separations (~1 fm) the strong attractive force is roughly 100 times stronger than the electromagnetic repulsion, allowing stable bound nuclei to exist. At separations beyond ~3 fm, the strong force has dropped to essentially zero, so an external proton or alpha particle approaching a heavy nucleus only "sees" the long-range Coulomb repulsion until it gets within touching distance.
Plot all known stable and unstable nuclei on a chart with neutron number N on one axis and proton number Z on the other. The stable nuclei form a narrow band — the band of stability — and the shape of this band reveals the underlying physics.
Two competing effects:
To compensate, heavier nuclei pack in additional neutrons. Extra neutrons contribute to the strong-force binding (which is short-range, so only neighbours matter) without adding to the Coulomb repulsion (since neutrons are neutral). The result is a stability line that bends upward toward neutron-rich compositions as Z increases.
Nuclei off the stability band decay back toward it:
In alpha decay, a heavy unstable nucleus emits an alpha particle — a tightly bound cluster of 2 protons and 2 neutrons, identical to a ⁴_₂He nucleus. The parent nucleus loses Z = 2 (atomic number decreases by 2, so the element changes) and A = 4 (mass number decreases by 4).
ᴬ_Z X → ᴬ⁻⁴_Z⁻₂ Y + ⁴_₂He
²³⁸_₉₂U → ²³⁴_₉₀Th + ⁴_₂He
Uranium-238 decays to thorium-234 with a half-life of about 4.5 billion years — a timescale comparable to the age of the Earth. This decay is the first step in the natural uranium decay chain that eventually terminates at stable lead-206.
²²⁶_₈₈Ra → ²²²_₈₆Rn + ⁴_₂He
Radium-226 decays to radon-222 with a half-life of 1600 years; this is the origin of the radon gas that accumulates in basements in some regions, a notable indoor-air-quality concern.
The alpha particle is exceptionally tightly bound — its binding energy per nucleon (~7 MeV) is large for such a small system. When a heavy nucleus has enough surplus mass-energy, it can emit a pre-formed alpha cluster with a substantial release of energy, while a single-proton or single-neutron emission would be less favourable. The high binding energy of the alpha "tunnel-out" channel makes it the dominant decay mode for heavy proton-rich (and just plain heavy) nuclei.
In beta-minus decay, a neutron in the nucleus converts into a proton, emitting an electron (the β⁻ particle) and an electron antineutrino (ν̄ₑ):
n → p + e⁻ + ν̄ₑ
At the quark level: d → u + e⁻ + ν̄ₑ (mediated by the W⁻ boson). At the nucleus level:
ᴬ_Z X → ᴬ_Z₊₁ Y + ⁰_₋₁e⁻ + ν̄ₑ
The mass number A is unchanged; the proton number Z increases by 1 (so the element changes); the neutron number N decreases by 1. The net effect is to move the nucleus downward and rightward on the N–Z diagram — toward the stability band if the nucleus was neutron-rich.
¹⁴_₆C → ¹⁴_₇N + ⁰_₋₁e⁻ + ν̄ₑ
Carbon-14 has 8 neutrons and 6 protons (N − Z = 2, neutron-rich compared with stable ¹²C and ¹³C). It β⁻-decays to nitrogen-14, with a half-life of 5,730 years — the basis of radiocarbon dating.
³_₁H → ³_₂He + ⁰_₋₁e⁻ + ν̄ₑ
Tritium (hydrogen-3) decays to helium-3 with a half-life of about 12 years.
The antineutrino was originally postulated by Wolfgang Pauli in 1930 to "save" the conservation laws in beta decay. Without the antineutrino, the emitted electron would have a definite kinetic energy (determined by mass-energy conservation), but experiment showed a spectrum of electron energies — there had to be another, undetected particle carrying off the missing momentum and energy. The antineutrino is required for two further reasons: conservation of lepton number (the electron has Lₑ = +1, so an antiparticle with Lₑ = −1 must accompany it) and conservation of angular momentum (the spin-½ nature of the electron and the spin-½ nature of nucleons require an additional spin-½ particle to balance).
In beta-plus decay, a proton converts into a neutron, emitting a positron (e⁺, the antiparticle of the electron) and an electron neutrino (νₑ):
p → n + e⁺ + νₑ
At the quark level: u → d + e⁺ + νₑ (mediated by the W⁺ boson). At the nucleus level:
ᴬ_Z X → ᴬ_Z₋₁ Y + ⁰_₊₁e⁺ + νₑ
The mass number A is unchanged; the proton number Z decreases by 1; the neutron number N increases by 1. The nucleus moves upward and leftward on the N–Z diagram.
²²_₁₁Na → ²²_₁₀Ne + ⁰_₊₁e⁺ + νₑ
Sodium-22 has 11 protons and 11 neutrons (proton-rich relative to the stability band for A = 22). It β⁺-decays to neon-22 with a half-life of 2.6 years.
¹⁸_₉F → ¹⁸_₈O + ⁰_₊₁e⁺ + νₑ
Fluorine-18 is the radioisotope used in PET (Positron Emission Tomography) medical imaging. With a half-life of 110 minutes, it emits positrons that annihilate almost immediately with nearby electrons, producing pairs of 511 keV gamma rays detected by the scanner.
A free proton cannot undergo β⁺ decay because the proton (mass 938.272 MeV/c²) is lighter than the neutron (mass 939.565 MeV/c²) — the decay would violate energy conservation. β⁺ decay can occur for bound protons in nuclei only because the nuclear binding-energy difference between parent and daughter nucleus can supply the necessary mass-energy. Free neutrons, by contrast, β⁻-decay with a half-life of about 10.5 minutes — the mass difference is more than enough to supply the electron rest energy.
A close cousin of β⁺ decay is electron capture, in which an inner-shell electron is absorbed by the nucleus, converting a proton into a neutron and emitting an electron neutrino:
p + e⁻ → n + νₑ
At the quark level: u + e⁻ → d + νₑ (the W boson is exchanged between the electron and the up quark). At the nucleus level:
ᴬ_Z X + ⁰_₋₁e⁻ → ᴬ_Z₋₁ Y + νₑ
Mass number A is unchanged; Z decreases by 1. Electron capture is an alternative decay channel for proton-rich nuclei and can occur for nuclei where β⁺ decay is energetically forbidden (because the mass difference is insufficient to supply the positron rest energy of 0.511 MeV).
⁷_₄Be + e⁻ → ⁷_₃Li + νₑ
Beryllium-7 (half-life 53 days) decays exclusively by electron capture, because the mass difference between ⁷Be and ⁷Li is too small to permit β⁺ decay.
| Decay mode | Particle emitted | Quark transition | ΔZ | ΔN | ΔA | Conservation requires |
|---|---|---|---|---|---|---|
| α | ⁴_₂He nucleus | (no quark flavour change) | −2 | −2 | −4 | charge, A, momentum, energy |
| β⁻ | e⁻ + ν̄ₑ | d → u | +1 | −1 | 0 | charge, A, Lₑ |
| β⁺ | e⁺ + νₑ | u → d | −1 | +1 | 0 | charge, A, Lₑ |
| Electron capture | νₑ (no charged lepton out) | u → d | −1 | +1 | 0 | charge, A, Lₑ |
The half-life t_½ of a radioactive isotope is the time for half of any sample of nuclei to decay. Decay is a probabilistic process governed by quantum mechanics: each nucleus has a fixed probability per unit time of decaying, regardless of its age or its neighbours' decays. The half-life is a characteristic of the isotope, like its mass or charge — it does not depend on temperature, pressure or chemical environment (with negligible exceptions).
Half-lives span an astonishing range:
| Isotope | Half-life | Decay mode |
|---|---|---|
| Polonium-214 | 164 μs | α |
| Beryllium-8 | 7 × 10⁻¹⁷ s | α (decays as soon as it forms) |
| Carbon-14 | 5,730 years | β⁻ |
| Uranium-238 | 4.5 × 10⁹ years | α |
| Tellurium-128 | 2 × 10²⁴ years | β⁻ (essentially stable in practice) |
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