Nuclear Binding Energy: Why Nuclei Release Energy in Fission and Fusion

Mass Defect and Einstein's E=mc^2

A nucleus always weighs less than the sum of its individual protons and neutrons measured separately. This missing mass, called the mass defect, has been converted into binding energy according to Einstein's mass-energy equivalence relation E=mc^2. For example, a helium-4 nucleus (2 protons + 2 neutrons) has a mass of 4.00153 atomic mass units (amu), while 2 free protons plus 2 free neutrons have a combined mass of 4.03188 amu. The mass defect is 0.03035 amu, corresponding to 28.3 MeV of binding energy. This means you would need to supply 28.3 million electron volts of energy to tear a helium-4 nucleus apart into four separate nucleons, and conversely, 28.3 MeV of energy is released when two protons and two neutrons assemble into helium-4.

The relationship between mass and energy at the nuclear scale is astounding compared to chemical energy. The mass defect of helium-4 represents about 0.7% of the total nuclear mass, meaning 0.7% of the matter is converted to pure energy during formation. By comparison, chemical reactions (which rearrange electrons rather than nucleons) convert less than one part per billion of the reacting mass into energy. Burning one kilogram of coal releases about 32 million joules. Fusing one kilogram of deuterium-tritium fuel into helium releases about 340 trillion joules, roughly 10 million times more energy per kilogram. This million-fold energy density advantage is why nuclear energy, whether from fission or fusion, is so extraordinarily concentrated compared to chemical fuels.

The Binding Energy Curve

The binding energy per nucleon (total binding energy divided by mass number A) reveals the relative stability of different nuclei and is one of the most important graphs in nuclear physics. Starting from hydrogen-1 (zero binding energy, since a single proton has no binding), the curve rises steeply through light elements: helium-4 has 7.07 MeV per nucleon, carbon-12 has 7.68 MeV, oxygen-16 has 7.98 MeV. The curve reaches its maximum around iron-56 (8.79 MeV per nucleon) and nickel-62 (8.79 MeV per nucleon, technically the highest of any nuclide), then gradually decreases for heavier elements down to about 7.57 MeV per nucleon for uranium-238.

The shape of this curve has profound consequences. Any nuclear reaction that moves nuclei toward the peak at iron/nickel releases energy because the products are more tightly bound (have higher binding energy per nucleon) than the reactants. Light nuclei below iron can release energy by fusing together (climbing the left side of the curve upward toward the peak). Heavy nuclei above iron can release energy by splitting apart through fission (moving from the right side of the curve back toward the peak). Iron and nickel sit at the energetic minimum, representing the "ash" of nuclear burning. No energy can be extracted from iron through either fission or fusion, which is why massive stars develop iron cores that cannot generate further energy and eventually collapse in supernovae.

Several features of the curve deserve attention. Helium-4 stands out with anomalously high binding energy for its mass number, appearing as a local spike on the curve. This reflects the doubly-magic nature of helium-4 (both proton number 2 and neutron number 2 are magic numbers) and the exceptional stability of two-proton-two-neutron groups within the nucleus. This special stability is why alpha particles (helium-4 nuclei) are emitted intact during radioactive decay rather than individual nucleons. Carbon-12 and oxygen-16 also show enhanced binding relative to their neighbors for similar shell-closure reasons.

The Semi-Empirical Mass Formula

The liquid drop model of the nucleus, developed by Carl Friedrich von Weizsacker in 1935, provides a quantitative formula for nuclear binding energy based on five physically motivated terms. The volume term (proportional to A) accounts for the strong force attraction between nearest-neighbor nucleons: since each nucleon interacts only with its close neighbors (the strong force saturates), total binding scales with the number of nucleons. The surface term (proportional to A^(2/3), negative) corrects for nucleons at the nuclear surface that have fewer neighbors and thus less binding. The Coulomb term (proportional to Z^2/A^(1/3), negative) accounts for electromagnetic repulsion between all proton pairs across the full nuclear volume.

The asymmetry term (proportional to (N-Z)^2/A, negative) penalizes nuclei with unequal numbers of protons and neutrons, reflecting the Pauli exclusion principle applied to nucleons as fermions. If neutron and proton energy levels fill independently, deviating from N=Z forces nucleons into higher energy levels, reducing binding. The pairing term (positive for even-even nuclei, negative for odd-odd, zero for odd-A) accounts for the tendency of nucleons to form pairs with opposite spins, providing extra binding when both proton and neutron numbers are even. Together, these five terms reproduce measured binding energies to within about 1% across the entire chart of nuclides, a remarkable achievement for such a simple model.

The semi-empirical mass formula explains numerous nuclear phenomena. The competition between the volume term (which favors large nuclei) and the Coulomb term (which destabilizes large nuclei) explains why the binding energy curve peaks at intermediate mass numbers. The asymmetry term explains why stable heavy nuclei have more neutrons than protons. The surface term explains why very small nuclei are less tightly bound per nucleon. The Coulomb term explains why spontaneous fission becomes possible for the heaviest elements: the electrostatic repulsion energy eventually exceeds the surface energy cost of deformation, making the nucleus unstable against splitting.

Energy Release in Fission and Fusion

When uranium-235 undergoes fission, splitting into two fragments of roughly mass 95 and 140, the binding energy per nucleon increases from about 7.6 MeV (for uranium) to about 8.5 MeV (for the fragments). With 236 nucleons rearranging, the total energy released is approximately 236 x (8.5 - 7.6) = 200 MeV per fission event. This energy appears as kinetic energy of the fission fragments (about 170 MeV), kinetic energy of prompt neutrons (about 5 MeV), prompt gamma rays (about 7 MeV), and beta/gamma radiation from fission product decay (about 20 MeV). One kilogram of uranium-235 undergoing complete fission releases about 82 terajoules, equivalent to burning roughly 2,700 tonnes of coal.

Fusion of deuterium and tritium into helium-4 and a neutron releases 17.6 MeV per reaction. The binding energy per nucleon jumps from about 1.1 MeV (deuterium) and 2.8 MeV (tritium) to 7.1 MeV (helium-4), with the unbound neutron carrying away most kinetic energy. Per unit mass, fusion releases about 4 times more energy than fission because the percentage change in binding energy is much larger for light nuclei (moving from the steep left side of the curve) than for heavy nuclei (moving from the gentle right side). The proton-proton chain in the Sun, converting four hydrogen nuclei into helium-4 through a series of intermediate steps, releases 26.7 MeV total, with about 2% of the reacting mass converted to energy.

The binding energy curve also explains why iron is the end product of stellar nucleosynthesis in massive stars. Starting from hydrogen, each successive fusion stage (hydrogen to helium, helium to carbon, carbon to neon, neon to oxygen, oxygen to silicon, silicon to iron) releases less energy per nucleon as the reactions approach the curve's peak. Once the stellar core reaches iron, no further energy-releasing fusion is possible. Without a nuclear energy source to support it against gravity, the iron core collapses in seconds, triggering a core-collapse supernova. Elements heavier than iron require energy input to create and are formed during supernovae and neutron star mergers through rapid neutron capture processes powered by gravitational energy rather than nuclear binding energy.

Nuclear shell effects superimpose local variations on the smooth liquid drop binding energy trend, producing measurable deviations at the magic numbers (2, 8, 20, 28, 50, 82, 126). Nuclei with magic proton or neutron numbers are more tightly bound than the semi-empirical formula alone would predict, analogous to how noble gas atoms are more stable than their neighbors in the periodic table. The nuclear shell model, developed by Maria Goeppert Mayer and J. Hans D. Jensen, explains these magic numbers through spin-orbit coupling that splits nucleon energy levels, creating large energy gaps at specific occupancies. Measuring binding energies of exotic isotopes far from stability at modern radioactive beam facilities has revealed that shell closures can shift or disappear under extreme neutron-to-proton ratios, challenging and refining our understanding of nuclear structure at the limits of existence.