Nuclear binding energy is the energy required to completely disassemble an atomic nucleus into its constituent protons and neutrons. It arises from the strong nuclear force, which binds nucleons together, and is described by the semi-empirical Bethe-Weizsäcker formula: B = a_v·A − a_s·A^(2/3) − a_c·Z(Z−1)/A^(1/3) − a_a·(A−2Z)²/A + δ. The binding energy per nucleon (B/A) peaks at iron-56 (~8.79 MeV/nucleon), which is why iron is the endpoint of stellar nucleosynthesis.
This simulation plots the B/A curve across all known nuclides. You can select specific elements to highlight their position, compare lighter nuclei that release energy through fusion with heavier nuclei that release energy through fission, and explore the five terms of the SEMF formula individually.
Why is iron-56 considered the most stable nucleus?
Iron-56 has the highest binding energy per nucleon of approximately 8.79 MeV. This means that any nuclear reaction — whether fusion of lighter nuclei or fission of heavier ones — that produces iron-56 releases energy. Stars cannot gain energy by fusing iron, so iron accumulates in stellar cores until a supernova occurs.
What is the Bethe-Weizsäcker semi-empirical mass formula?
The SEMF approximates the binding energy of a nucleus using five terms: volume energy (proportional to A), surface energy (proportional to A^(2/3)), Coulomb repulsion between protons (proportional to Z²/A^(1/3)), asymmetry energy penalising unequal numbers of protons and neutrons, and a pairing term favouring even numbers of both. The coefficients were fitted to experimental nuclear masses.
How does nuclear fusion release energy?
When two light nuclei fuse, the resulting nucleus is more tightly bound per nucleon than either parent. The mass difference — called the mass defect — is converted to energy via Einstein's E = mc². For example, fusing two deuterium nuclei releases approximately 3.27 MeV of energy.
Heavy nuclei such as uranium-235 have a lower B/A (~7.6 MeV/nucleon) than mid-mass nuclei. When uranium fissions into two fragments near mass 90 and 140, the products are more tightly bound, and the mass difference appears as roughly 200 MeV of kinetic energy and radiation per fission event.
The liquid drop model treats the nucleus as an incompressible droplet of nuclear matter. It correctly predicts the volume and surface energy terms of the SEMF by analogy with surface tension in liquids, and it underpins the theory of nuclear fission by modelling the deformation of the droplet under Coulomb repulsion.
As atomic number Z increases, the Coulomb repulsion between protons — which grows as Z² — increasingly overcomes the short-range strong nuclear force. Above lead-208 (the heaviest stable nucleus), all nuclides are radioactive, and the stability peninsula ends around Z=118 (oganesson) with half-lives of milliseconds.
The valley of stability is a region on the nuclear chart (Z vs N) where nuclides are most stable. Stable nuclides have roughly equal numbers of protons and neutrons for light elements, but heavier stable nuclei have more neutrons than protons to dilute Coulomb repulsion. Nuclei outside the valley undergo beta decay to move towards it.
The mass defect is the difference between the sum of the masses of isolated protons and neutrons and the actual mass of the assembled nucleus. For helium-4, the mass defect is about 0.030 atomic mass units, equivalent to 28.3 MeV. This "missing" mass has been converted into the binding energy holding the nucleus together.
Binding energies are determined from precise measurements of atomic masses using Penning trap mass spectrometry or time-of-flight techniques. The atomic mass excess (difference from A atomic mass units, in MeV/c²) is tabulated in the Atomic Mass Evaluation (AME), most recently updated in 2020 with data for over 2,500 nuclides.
Stars burn hydrogen into helium (releasing ~6.7 MeV/nucleon gained in binding energy) and helium into carbon/oxygen in their cores. Massive stars continue fusing up to silicon, which produces iron-group elements. Each fusion stage releases less energy per nucleon than the previous, explaining why stellar evolution slows at each stage.
This simulation plots how tightly an atomic nucleus is bound together as its mass number changes, using the semi-empirical mass formula (the Bethe–Weizsäcker liquid-drop model). Binding energy per nucleon (B/A) rises steeply for light nuclei, peaks around iron-56 and nickel-62, then falls slowly for heavier elements. Drag the four SEMF coefficient sliders to see how the volume, surface, Coulomb and asymmetry terms shape the curve, and hover any point on the plot to inspect a nuclide's mass number, proton number and binding energy.
The B/A curve for the stable-valley nuclide at every mass number from hydrogen to beyond uranium, built from the five SEMF terms: volume, surface, Coulomb, asymmetry and pairing. The curve peaks around A≈56–62 (the iron/nickel region); a green-tinted fusion zone lies to its left and a red-tinted fission zone to its right, with a dashed marker on iron-56.
Drag the aV (volume), aS (surface), aC (Coulomb) and aA (asymmetry) sliders in the top bar to retune the SEMF coefficients and watch the curve and peak readout update live. Click ↺ Defaults to restore the standard values (15.6, 17.4, 0.71, 23.4 MeV), and hover anywhere over the plotted curve to read off A, Z, N and B/A for that point.
Iron-56 and nickel-62 sit within a whisker of each other at the true summit of the binding-energy curve, close to 8.79 MeV per nucleon. Once a massive star's core is mostly iron and nickel, fusing it further would absorb energy rather than release it, which is exactly what triggers core collapse and a supernova.
It evaluates the Bethe–Weizsäcker semi-empirical mass formula, B/A = aV − aS·A^(−1/3) − aC·Z(Z−1)·A^(−4/3) − aA·(A−2Z)²/A² ± δ/√A, for every mass number A. The proton number Z is chosen from a stable-valley approximation, and the pairing term δ adds or subtracts energy depending on whether Z and N are each even or odd.
The volume term grows with every added nucleon, but the negative surface term shrinks in relative importance as A increases, so binding per nucleon rises for light nuclei. Meanwhile the Coulomb term, which penalises proton–proton repulsion, grows faster than A once nuclei get large. The peak sits where the surface penalty has become small but the Coulomb penalty has not yet taken over — around A≈56–62.
aV sets the volume term, representing the attraction each nucleon feels from its immediate neighbours. aS sets the surface term, which subtracts binding energy from nucleons sitting at the nuclear surface with fewer neighbours. aC sets the Coulomb term, the electrostatic repulsion between protons. aA sets the asymmetry term, which penalises nuclei with unequal numbers of protons and neutrons.
Any reaction that moves a nucleus towards the peak increases its binding energy per nucleon, converting the difference into released energy via E=mc². Light nuclei to the left of iron-56 release energy when they fuse into heavier ones; heavy nuclei to the right of iron-56, such as uranium-235, release energy when they split into mid-mass fragments closer to the peak.
The pairing term δ gives an extra binding bonus to even-even nuclei (even numbers of both protons and neutrons), a penalty to odd-odd nuclei, and no adjustment when A is odd. It is a small correction next to the volume and surface terms, but it explains why even-even nuclides are markedly more abundant and more stable than odd-odd ones of similar mass.