Nuclear Binding Energy
Every atomic nucleus is a tightly packed collection of protons and neutrons held together by the strong nuclear force. But here is a surprising fact: the mass of an assembled nucleus is always slightly less than the sum of the masses of its individual protons and neutrons measured separately. This 'missing' mass, called the mass defect, has been converted into energy according to Einstein's famous equation E = mc². That energy — the binding energy — is what you would need to supply to completely pull the nucleus apart into free protons and neutrons. The larger the binding energy, the more stable the nucleus. Nuclear binding energy is usually expressed in megaelectronvolts (MeV), where 1 MeV = 1.602 × 10⁻¹³ J. A single proton's rest mass is about 938.3 MeV/c², and a neutron's is about 939.6 MeV/c². When they combine into, say, a helium-4 nucleus, the resulting mass corresponds to only 3727.4 MeV/c² — roughly 28.3 MeV less than the four free nucleons combined. That 28.3 MeV is the binding energy of helium-4, which works out to about 7.07 MeV per nucleon. This 'per nucleon' figure is the key to comparing how tightly bound different nuclei are. The stability curve — a graph of binding energy per nucleon versus mass number A — is one of the most important graphs in all of modern physics. It rises steeply for light nuclei, peaks near iron-56 (around 8.8 MeV per nucleon), then slowly decreases for heavier nuclei. This shape immediately explains both nuclear fission and nuclear fusion: any process that moves a nucleus toward iron-56 releases energy. Light nuclei can fuse together (fusion) and heavy nuclei can split apart (fission), both moving up the curve toward the iron peak and releasing enormous amounts of energy in the process.
- Identify the Constituents
- Calculate the Mass Defect
- Convert to Energy Using E = mc²
- Compute Binding Energy Per Nucleon
- Reading the Stability Curve
Identify the Constituents
For a nucleus with atomic number Z (number of protons) and mass number A (total nucleons), the number of neutrons is N = A − Z. You need the rest masses of a free proton (mₚ = 1.007276 u) and a free neutron (mₙ = 1.008665 u), where 1 unified atomic mass unit u = 931.5 MeV/c².
Calculate the Mass Defect
The mass defect Δm is the difference between the total mass of free nucleons and the actual measured nuclear mass M(Z, A): Δm = Z·mₚ + N·mₙ − M(Z, A). This is always a positive number for stable nuclei. For carbon-12, with Z = 6 protons and N = 6 neutrons: Δm = 6(1.007276) + 6(1.008665) − 11.996709 = 0.098931 u.
Convert to Energy Using E = mc²
Multiply the mass defect by the conversion factor 931.5 MeV/u to get the total binding energy: BE = Δm × 931.5 MeV/u. For carbon-12: BE = 0.098931 × 931.5 ≈ 92.16 MeV. This is the total energy required to disassemble a carbon-12 nucleus into 6 separate protons and 6 separate neutrons.
Compute Binding Energy Per Nucleon
Divide the total binding energy by A to get BE/A, the figure used to compare nuclear stability: BE/A = 92.16 / 12 ≈ 7.68 MeV per nucleon. This places carbon-12 on the stability curve well below the iron-56 peak of ~8.8 MeV/nucleon, confirming that energy would be released if carbon nuclei fused into heavier nuclei approaching iron.
Reading the Stability Curve
Think of the binding-energy-per-nucleon curve like a valley with iron at the bottom. Heavy nuclei like uranium-235 (BE/A ≈ 7.6 MeV) sit on the right slope; very light nuclei like hydrogen isotopes sit on the left slope. Fission splits a heavy nucleus into two mid-size fragments, releasing the energy difference. Fusion merges two light nuclei into a heavier one, also releasing energy. Both reactions 'roll downhill' toward the iron valley floor.
The Key Equation
- BE — Binding Energy (MeV): Total energy required to completely separate the nucleus into free protons and neutrons.
- Z — Atomic number (dimensionless): Number of protons in the nucleus.
- m_p — Proton rest mass (u): Mass of a free proton: 1.007276 u.
- N — Neutron number (dimensionless): Number of neutrons; equals A − Z.
- m_n — Neutron rest mass (u): Mass of a free neutron: 1.008665 u.
- M_{\text{nucleus}} — Nuclear mass (u): Measured mass of the assembled nucleus.
- \Delta m — Mass defect (u): The 'missing' mass converted to binding energy.
Worked Example
Problem. Calculate the total binding energy and the binding energy per nucleon for helium-4 (⁴He), which has a measured nuclear mass of 4.001506 u. Use mₚ = 1.007276 u and mₙ = 1.008665 u.
Answer. Total binding energy ≈ 28.3 MeV; binding energy per nucleon ≈ 7.07 MeV/nucleon
Origins · 1 minHans Bethe and how this was discovered
Hans Bethe worked out the detailed nuclear fusion reaction chains that power stars — the proton-proton chain and the CNO cycle — earning him the 1967 Nobel Prize in Physics. His work required precise knowledge of nuclear binding energies to determine which fusion reactions release energy and by how much. His 1939 paper 'Energy Production in Stars' is a landmark of astrophysics and nuclear physics combined. Bethe lived to be 98 years old and published research papers well into his 90s. Legend has it that when colleagues asked him how he maintained such productivity, he replied, 'You should do physics. It keeps you young.'
Related Concepts
- Mass Defect — The mass defect is the direct source of binding energy; understanding Δm = Zm_p + Nm_n − M is the first step in every binding energy calculation.
- Nuclear Fission — Fission of heavy nuclei releases energy because the products have higher binding energy per nucleon, as read directly off the stability curve.
- Nuclear Fusion — Fusion of light nuclei releases energy for the same reason — products lie higher on the binding energy curve, closer to the iron-56 peak.
- Mass–Energy Equivalence (E = mc²) — Einstein's relation is the fundamental equation that converts the mass defect into the binding energy value in joules or MeV.
- Radioactive Decay — Unstable nuclei with low binding energy per nucleon undergo decay to reach more stable configurations — a process governed by the same stability principles shown on the binding energy curve.