Binding nuclei…
Every square on a chart of nuclides is one combination of protons (Z) and neutrons (N). Raise each square by how much energy it carries above the most stable nucleus with the same mass number, and the stable isotopes fall to the bottom of a long, curving valley. Unstable nuclei sit up its walls and decay downhill. That is the band of stability, drawn as terrain.
Reading the valley
- Neutron-rich side (blue): β⁻ decay turns a neutron into a proton and slides the nucleus one step toward the floor. Fission fragments and carbon-14 live here.
- Proton-rich side (red): β⁺ decay or electron capture turns a proton into a neutron. PET isotopes such as fluorine-18 live here.
- Heavy end (amber): past lead, even the valley floor is uphill from lighter nuclei, so α decay sheds two protons and two neutrons at once. Uranium and thorium decay chains run down this slope.
- The bend: the floor starts on the N = Z line and drifts away from it, because extra neutrons dilute proton–proton repulsion. By lead-208 there are 126 neutrons for 82 protons.
Binding energy per nucleon and the iron peak
Switch the height to binding energy per nucleon and the terrain becomes a ridge. It climbs steeply through the light nuclei, peaks near iron and nickel at about 8.8 MeV, then slopes gently down toward uranium. Fusion of light nuclei climbs the left side; fission of heavy ones slides down the right. Both release energy, and neither can go further than the peak — which is why a star’s core ends in iron.
The model behind the chart
The heights are computed, not looked up: this page uses the semi-empirical mass formula, which treats the nucleus as a charged liquid drop. The binding energy B of a nucleus with A nucleons and Z protons is
B = 15.75A − 17.8A2/3 − 0.711Z(Z−1)/A1/3 − 23.7(A−2Z)²/A ± 11.18/√A (MeV)
The squares drawn are the nuclei the formula keeps bound against losing a single proton or neutron. Real charts list about 3,300 measured nuclides inside those drip lines. Here is how close five numbers get:
| Nucleus | Measured B/A (MeV) | Liquid drop (MeV) |
|---|---|---|
| ⁴He (Helium-4) | 7.074 | 5.710 |
| ¹²C (Carbon-12) | 7.680 | 7.468 |
| ¹⁶O (Oxygen-16) | 7.976 | 7.873 |
| ⁵⁶Fe (Iron-56) | 8.790 | 8.846 |
| ⁶²Ni (Nickel-62) | 8.795 | 8.863 |
| ²⁰⁸Pb (Lead-208) | 7.867 | 7.857 |
| ²³⁵U (Uranium-235) | 7.591 | 7.645 |
| ²³⁸U (Uranium-238) | 7.570 | 7.625 |
The formula lands within 1–2% from oxygen upward, but it badly underbinds helium-4, which is doubly magic. The yellow guide lines mark the magic numbers 2, 8, 20, 28, 50, 82, and 126, where real nuclei are extra stable and a smooth liquid drop cannot follow. Measured values are from the AME2020 atomic mass evaluation. Element-by-element data — mass, radioactivity, and discovery — lives on each profile, for example iron, lead, and uranium.
Nuclear stability questions
What is the band of stability?
Plot every known nucleus by neutron number N and proton number Z, and the stable ones fall in a narrow band. Light stable nuclei sit near N = Z. Heavier ones need extra neutrons, because neutrons add strong-force glue without adding Coulomb repulsion, so the band bends toward N ≈ 1.5 Z by lead. Nuclei off the band decay back toward it.
How do I predict whether an isotope undergoes beta-minus or beta-plus decay?
Compare it with the band. Too many neutrons (below the band on an N–Z chart with Z up) means β⁻ decay: a neutron becomes a proton, an electron, and an antineutrino. Too many protons means β⁺ decay or electron capture: a proton becomes a neutron. Very heavy nuclei, past about lead, also shed mass by α decay.
Why is iron-56 called the most stable nucleus?
Iron-56 and nickel-62 sit at the top of the binding-energy-per-nucleon curve, near 8.8 MeV. Nickel-62 is technically the most tightly bound per nucleon, while iron-56 has the lowest mass per nucleon. Either way, fusing lighter nuclei or fissioning heavier ones both move toward this peak and release energy, which is why stars stop fusing at iron.
What are magic numbers in nuclear physics?
Proton or neutron counts of 2, 8, 20, 28, 50, 82, and 126 close a nuclear shell, much like noble-gas electron counts. Nuclei with magic numbers are extra stable; doubly magic ones such as helium-4, oxygen-16, calcium-40, and lead-208 especially so. The liquid-drop model on this page has no shells, so it misses these bumps — that gap is the evidence for the nuclear shell model.
What is the semi-empirical mass formula?
It treats the nucleus as a charged liquid drop. Binding energy = a volume term (each nucleon bonds to its neighbors), minus a surface term (nucleons at the edge have fewer neighbors), minus Coulomb repulsion between protons, minus an asymmetry term (penalty for N ≠ Z), plus or minus a pairing term. Five fitted numbers reproduce binding energies to within about 1% for most nuclei.