Difficulty: Intermediate | Prerequisites: Basic atomic structure, mass-energy equivalence (E = mc²), electrostatics
Big Picture
Nuclear physics sits at the boundary between the familiar world of atoms and the subatomic domain. This chapter introduces the structure of the atomic nucleus, the forces holding it together, and the ways unstable nuclei transform through radioactive decay. You should already be comfortable with atomic number, mass number, and the idea that mass and energy are interchangeable via E = mc². Everything here builds toward understanding why certain isotopes are stable, why others are not, and what happens when they change.
Nuclei are held together by the strong nuclear force, and how tightly they are bound is measured by binding energy per nucleon. Nuclei with mass number around 60 are the most stable. Unstable nuclei shed energy through alpha decay (ejecting two protons and two neutrons), beta decay (converting a neutron to a proton or vice versa), or gamma decay (emitting a high-energy photon from an excited state).
Atomic mass unit (u)
1 u = 1.6605 × 10⁻²⁷ kg. By mass-energy equivalence, 1 u corresponds to 931.49 MeV. Think of it as the standard measuring stick for nuclear masses.
Proton mass
1.0073 u. The positively charged nucleon.
Neutron mass
1.0087 u. Slightly heavier than a proton, and electrically neutral.
Isotope
Atoms with the same number of protons (same element) but different numbers of neutrons. In simple terms, same element, different weight.
Binding energy (B)
The energy required to completely disassemble a nucleus into its individual protons and neutrons. In simple terms, the "glue energy" holding the nucleus together. Higher binding energy means a more tightly bound, more stable nucleus.
Binding energy per nucleon (B/A)
Binding energy divided by the total number of nucleons (protons + neutrons). This is the key measure of nuclear stability. Think of it as how strongly each particle is held, on average.
Strong nuclear force
The fundamental force that binds nucleons together. It is attractive, short-range (only works when nucleons are very close together), and stronger than the electrostatic repulsion between protons. It does not act on electrons.
Alpha decay (α)
A heavy nucleus ejects an alpha particle (2 protons + 2 neutrons, i.e. a helium-4 nucleus) to become more stable. Mass number drops by 4, atomic number drops by 2.
Beta-minus decay (β⁻)
A neutron inside the nucleus converts into a proton, emitting an electron (e⁻). The mass number stays the same, but the atomic number increases by 1. The element changes.
Beta-plus decay (β⁺)
A proton inside the nucleus converts into a neutron, emitting a positron (e⁺, the antimatter counterpart of an electron). Mass number stays the same, atomic number decreases by 1.
Positron
A positive electron (e⁺). The antimatter partner of the ordinary electron, emitted during β⁺ decay.
Gamma decay (γ)
An excited nucleus drops to a lower energy state by emitting a high-energy photon (a gamma ray, typically around 1 MeV). No change in mass number or atomic number. Think of it as the nucleus relaxing after an alpha or beta event.
Line of stability (valley of stability)
The band on a chart of neutron number vs. proton number where stable nuclei cluster. For light elements (Z ≤ 16), stable isotopes sit near N = Z. For heavier elements, stable nuclei require progressively more neutrons than protons.
Nuclear fusion
Combining light nuclei (A < 60) into a heavier one, releasing energy because the product has higher binding energy per nucleon.
Nuclear fission
Splitting a heavy nucleus (A > 60) into lighter fragments, releasing energy for the same reason.
The nucleus contains protons (Z) and neutrons (N). The total number of nucleons is the mass number, A = Z + N.
The majority of known isotopes are radioactive. Only a narrow band of neutron-to-proton combinations produces stable nuclei.
Line of stability:
For light elements (Z ≤ 16), the most stable isotopes have roughly equal numbers of protons and neutrons (N ≈ Z).
For heavier elements, stable nuclei need progressively more neutrons than protons (N > Z) to offset the growing electrostatic repulsion among protons.
Stable nuclei cluster close to the line of stability on an N vs. Z chart. Unstable nuclei sit on either side of this band.
Lead (Z = 82) is the heaviest element with a stable isotope. Beyond lead, every isotope is radioactive.
What it is: The total energy needed to pull a nucleus apart into separate protons and neutrons. A bound nucleus has less mass than the sum of its free nucleons; the "missing" mass is the binding energy, converted via E = mc².
Formula: B = 931.49 MeV × (Z × m_p + N × m_n − m_nucleus), where m_p is the proton mass, m_n the neutron mass, and m_nucleus the measured mass of the intact nucleus, all in atomic mass units.
Binding energy per nucleon (B/A):
This is the real stability metric. Higher B/A means a more tightly bound nucleus.
B/A peaks at roughly 8.8 MeV per nucleon, near A ≈ 60 (iron-nickel region).
Consequences for stability:
For A > 60: nuclei become more stable if mass decreases (splitting apart, i.e. fission, releases energy).
For A < 60: nuclei become more stable if mass increases (combining, i.e. fusion, releases energy).
This is why fusion powers stars (combining light nuclei) and fission powers reactors (splitting heavy nuclei).
An attractive force between nucleons (protons and neutrons). It does not act on electrons.
Extremely short-range: only effective when nucleons are very close together (roughly 1 femtometre, 10⁻¹⁵ m).
At close range, it is stronger than the electrostatic (Coulomb) repulsion between protons, which is what allows nuclei to exist at all.
Behaviour at different distances:
Far apart: the strong force is negligible; protons repel each other electrically.
Close together (a few femtometres): nucleons attract each other via the strong force.
Too close: the strong force becomes repulsive, preventing nucleons from collapsing into each other. At the equilibrium distance, the net force is zero, and the nucleons are said to be "touching."
A proton-neutron pair, a neutron-neutron pair, and a proton-proton pair all experience the same strong-force attraction. However, the proton-proton pair also feels the electrostatic repulsive force on top of it.
A heavy, unstable nucleus ejects an alpha particle (⁴He: 2 protons + 2 neutrons) to reduce its mass and move closer to the line of stability.
Notation: ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + ⁴₂He + energy
Mass number A decreases by 4, atomic number Z decreases by 2. The element changes.
The parent nucleus must have a greater mass than the combined mass of the daughter nucleus plus the alpha particle. The mass difference is released as kinetic energy.
Alpha decay is common in very heavy nuclei (e.g. uranium, thorium, radium) where the Coulomb repulsion among many protons destabilises the nucleus.
Beta-minus decay (β⁻)
A neutron converts into a proton inside the nucleus, emitting an electron (e⁻) and energy.
Notation: ᴬ_Z X → ᴬ_(Z+1) Y + e⁻ + energy
A stays the same, Z increases by 1. The element changes (one step higher on the periodic table).
Occurs when the nucleus has too many neutrons relative to protons (sits above the line of stability).
Condition: the mass of the parent nucleus must be greater than the mass of the daughter nucleus (M_X > M_Y).
Beta-plus decay (β⁺)
A proton converts into a neutron inside the nucleus, emitting a positron (e⁺, a positive electron) and energy.
Notation: ᴬ_Z X → ᴬ_(Z−1) Y + e⁺ + energy
A stays the same, Z decreases by 1. The element changes (one step lower on the periodic table).
Occurs when the nucleus has too many protons relative to neutrons (sits below the line of stability).
Key point for both: beta decay changes the element but not the mass number. The nucleus is adjusting its neutron-to-proton ratio to get closer to the line of stability.
A nucleus in an excited energy state drops to a lower state (or ground state) by emitting a gamma-ray photon, with energy typically around 1 MeV.
Gamma decay does not change A or Z. The nucleus remains the same isotope of the same element.
Gamma decay usually follows an alpha or beta decay that has left the daughter nucleus in an excited state.
It only occurs when nuclei are in an excited state; a nucleus already in its ground state does not undergo gamma decay.
Example: ⁹⁹Mo undergoes beta decay to produce ⁹⁹Tc in an excited state (⁹⁹Tc*), which then emits a gamma ray to reach the ground state of ⁹⁹Tc.
Mass-energy equivalence: 1 u = 931.49 MeV/c²
Proton mass: m_p = 1.0073 u
Neutron mass: m_n = 1.0087 u
Binding energy: B = 931.49 MeV × (Z × m_p + N × m_n − m_nucleus)
Binding energy per nucleon: B/A (where A = Z + N)
Alpha decay: ᴬ_Z X → ᴬ⁻⁴_(Z−2) Y + ⁴₂He + energy
Beta-minus: ᴬ_Z X → ᴬ_(Z+1) Y + e⁻ + energy
Beta-plus: ᴬ_Z X → ᴬ_(Z−1) Y + e⁺ + energy
Gamma: X* → X + γ (no change in A or Z)
Alpha, beta, and gamma radiation are used in medicine (e.g. ⁹⁹Tc gamma scans for imaging), in smoke detectors (americium-241, an alpha emitter), and in carbon dating (measuring β⁻ decay of carbon-14). Nuclear fission powers commercial reactors, while fusion is the energy source of the Sun and the focus of experimental reactors on Earth.
Students often think binding energy is the energy stored inside the nucleus. It is the opposite: it is the energy you would need to supply to take the nucleus apart. A higher binding energy means the nucleus is harder to break up, not that it contains more stored energy.
Students often confuse "heavier nucleus = more stable." Stability depends on binding energy per nucleon (B/A), not on total mass. Iron-56 has a higher B/A than uranium-238, even though uranium is far heavier.
Students sometimes think beta decay involves an electron that was already orbiting the atom. The electron (or positron) in beta decay is created at the moment of the decay, inside the nucleus, when a neutron converts to a proton (or vice versa).
Students often assume gamma decay changes the identity of the element. It does not. Gamma decay only releases excess energy from an excited nucleus; A and Z remain unchanged.
⚠️ Be able to calculate binding energy given nuclear masses. The formula B = 931.49 × (Z m_p + N m_n − m_nucleus) is a standard exam calculation.
⚠️ Know which direction on the B/A curve leads to fusion and which leads to fission. Below A ≈ 60, fusion releases energy; above A ≈ 60, fission releases energy.
⚠️ Write balanced decay equations for alpha, β⁻, and β⁺ decay, correctly tracking changes in A and Z.
⚠️ Understand that gamma decay changes neither A nor Z, and only occurs from excited states.
⚠️ Know the properties of the strong force: short-range, attractive (repulsive at very short range), acts on nucleons but not electrons, stronger than electrostatic force at nuclear distances.
True or false: The strong nuclear force acts on electrons as well as nucleons.
False. It acts only on nucleons (protons and neutrons).
Fill in the blank: 1 atomic mass unit corresponds to ______ MeV of energy.
931.49 MeV.
True or false: In beta-minus decay, the mass number A increases by 1.
False. A stays the same; it is the atomic number Z that increases by 1.
True or false: Gamma decay always follows alpha or beta decay.
Essentially true in practice. Gamma emission occurs when a nucleus is left in an excited state, which is typically the result of a preceding alpha or beta decay.
Fill in the blank: Nuclei with A > 60 release energy through ______ ; nuclei with A < 60 release energy through ______.
Fission; fusion.
Q: A nucleus of radium-226 (Z = 88) undergoes alpha decay. What are the mass number and atomic number of the daughter nucleus?
A: A = 226 − 4 = 222, Z = 88 − 2 = 86. The daughter is radon-222.
Q: Explain why binding energy per nucleon, rather than total binding energy, is the better measure of nuclear stability.
A: Total binding energy increases with the number of nucleons simply because there are more particles. B/A normalises for size, giving the average energy holding each nucleon in place. A higher B/A means each nucleon is more tightly bound, making the nucleus harder to disrupt.
Q: Carbon-12 has a nuclear mass of 12.000 u. Calculate its binding energy. (Z = 6, N = 6.)
A: Sum of free nucleons = 6(1.0073) + 6(1.0087) = 6.0438 + 6.0522 = 12.0960 u. Mass defect = 12.0960 − 12.000 = 0.0960 u. B = 0.0960 × 931.49 = 89.4 MeV.
Q: A nucleus undergoes β⁻ decay. Does the daughter element sit higher or lower on the periodic table than the parent?
A: Higher. Z increases by 1, so the daughter is one element up on the periodic table.
Q: Why does gamma decay not change the element?
A: Gamma decay emits a photon, which has no charge and no baryon number. Neither Z nor A changes, so the nucleus remains the same isotope.
This material connects directly to Chapter 30's second half on half-lives, decay rates, and activity (covered in the companion study notes). The concept of binding energy per nucleon underpins all of nuclear energy, from stellar nucleosynthesis (how stars forge elements) to the design of fission reactors and the pursuit of fusion power. The strong force introduced here is one of the four fundamental forces and reappears in particle physics, where it is described more precisely by quantum chromodynamics (QCD).
Nuclear physics, atomic nucleus, nucleon, proton, neutron, isotope, radioactive, radioactivity, binding energy, mass defect, mass-energy equivalence, E = mc², strong nuclear force, strong interaction, alpha decay, alpha particle, helium-4, beta decay, beta-minus, beta-plus, positron, electron emission, gamma decay, gamma ray, photon emission, line of stability, valley of stability, N vs Z chart, nuclear fusion, nuclear fission, B/A curve, binding energy per nucleon, iron-56, lead-208, decay equation, transmutation, nuclear stability, Coulomb repulsion, short-range force, Chapter 30, General Physics, University of Washington