Mass Defect and Binding Energy

Why mass defect and binding energy matter

• One curve explains both fusion and fission. Light nuclei climb toward iron by fusing; heavy nuclei climb toward iron by splitting. Both release the difference in binding energy as kinetic energy of products. The Sun gets 26.73 MeV per ⁴He from the proton-proton chain, and a 1 GW reactor fissions ~3.1 × 10¹⁹ ²³⁵U nuclei per second.
• Energy density six orders of magnitude above chemistry. Burning hydrogen with oxygen releases ~2.5 eV per H₂O molecule. Fusing four hydrogens to ⁴He releases 26.73 MeV — about 10⁷× more per nucleon. That ratio is why 1 kg of fissile ²³⁵U yields ~8.2 × 10¹³ J versus ~4.6 × 10⁷ J for 1 kg of gasoline.
• Predicts which nuclei are stable. The semi-empirical mass formula (Weizsäcker 1935, refined by Bethe and Bacher 1936) gives binding energy as B = a_VA − a_SA^(2/3) − a_CZ(Z−1)/A^(1/3) − a_A(N−Z)²/A + δ(A,Z) with coefficients a_V ≈ 15.75 MeV, a_S ≈ 17.80 MeV, a_C ≈ 0.711 MeV, a_A ≈ 23.7 MeV. The valley of beta stability sits where ∂B/∂Z = 0.
• Aston's packing fraction. Francis Aston's 1919 mass spectrograph (Cavendish, Nobel 1922) measured isotope masses to 1 part in 10000 and tabulated the "packing fraction" (M/A − 1) × 10⁴, the empirical predecessor of binding energy per nucleon. Eddington (1920) used these numbers to argue stars must be powered by hydrogen-to-helium fusion.
• Cockcroft-Walton 1932 verified E = Δm·c². Bombarding ⁷Li with 700 keV protons produced two ⁴He with total kinetic energy 17.3 MeV. The mass difference between ⁷Li + p and 2 × ⁴He is 0.01861 u = 17.34 MeV. Agreement to better than 1% was the first nuclear-reaction confirmation of mass-energy equivalence.
• Stellar nucleosynthesis stops at iron. Hydrogen → helium → carbon → oxygen → silicon → iron releases energy at every step because each product sits higher on the binding-energy curve. Building elements heavier than the Fe peak costs energy; supernova cores collapse precisely because silicon burning halts at ~⁵⁶Ni and no more fuel remains. Elements above iron form via the s-process and r-process neutron capture.
• Practical reactor design. 200 MeV per ²³⁵U fission times Avogadro's number / 235 g·mol⁻¹ = 8.2 × 10¹⁰ J per gram. A 1 GWe reactor at 33% thermal efficiency consumes ~3 g of ²³⁵U per second, or about 1 ton of low-enriched uranium fuel per year per GWe.

Common misconceptions

• "Iron-56 is the most stable nucleus." ⁶²Ni actually has a slightly higher binding energy per nucleon (8.7945 MeV vs ⁵⁶Fe at 8.7903 MeV). The popular Fe-56 myth comes from Fe being more cosmically abundant — a separate question driven by silicon burning kinetics, not pure thermodynamics.
• "Mass is converted to energy." No mass is destroyed in nuclear reactions when you account for the total energy of the system. The rest mass of bound products is lower than the rest mass of separated reactants because the binding photon and kinetic energies were emitted; the inertia of a hot fission fragment equals exactly the inertia of its rest-frame mass plus the relativistic mass equivalent of its kinetic energy.
• "Binding energy is potential energy stored in the nucleus." It is the opposite — binding energy is the energy that has already been radiated away during nucleus formation. To break the nucleus apart you must put it back. A free nucleon system has higher rest mass than a bound nucleus, not lower.
• "E = mc² applies only to nuclear reactions." It applies to every reaction. A burning candle loses ~10⁻⁹ kg of mass per kJ released, but the fractional change is below detection. Nuclear reactions have fractional mass changes near 0.1% to 1%, so they show the effect.
• "Fusion is clean because no radioactive products." D-T fusion releases 80% of its 17.59 MeV as a 14.07 MeV neutron that activates structural materials. ITER expects ~3 kg of activated tungsten and steel per GWy. Aneutronic alternatives (D-³He, p-¹¹B) require 5-10× higher temperatures and have not been demonstrated at breakeven.
• "Mass defect is a relativistic correction." It is the dominant effect, not a correction. The kinetic energies and photon energies released during nucleus formation are large relative to the masses involved, so the bound state's rest mass differs from the sum by 0.1-1% — easily measurable since 1919.

Derivation

Start with the rest energies of free constituents. A nucleus of mass number A with Z protons and N = A − Z neutrons has free-nucleon rest energy (Z·m_p + N·m_n)c², using m_p = 938.272 MeV/c² and m_n = 939.565 MeV/c². The bound nucleus has measured rest energy M(A,Z)·c². The difference Δm·c² = (Z·m_p + N·m_n − M(A,Z))·c² is the binding energy E_B. For ⁵⁶Fe with Z = 26, N = 30, M = 55.934936 u: free total = 26 × 1.007276 + 30 × 1.008665 = 56.448126 u; mass defect Δm = 0.513190 u = 478.00 MeV; per nucleon 478.00/56 = 8.5357 MeV. (Including atomic-electron binding gives the standard tabulated 8.7903 MeV/nucleon when M is the atomic mass.)

The shape of the binding-energy curve follows from the semi-empirical mass formula. The volume term a_V·A reflects each nucleon binding to its ~12 nearest neighbours (the strong force saturates because of its 1.5 fm range and the Pauli exclusion principle). The surface correction −a_S·A^(2/3) penalizes nucleons near the surface. Coulomb repulsion −a_C·Z(Z−1)/A^(1/3) treats the nucleus as a uniformly charged sphere. The asymmetry term −a_A·(N−Z)²/A pushes toward N = Z. The pairing term δ adds ~12·A^(−1/2) MeV when both N and Z are even, subtracts it when both are odd, and is zero for odd-A. Plotting B/A from the formula against measured values reproduces every gross feature: the rapid rise to mass ~20, the broad maximum around mass 60, and the slow decline to A = 240.

From this curve you read off two operations that liberate energy. Fusing ²H and ³H to ⁴He moves all 5 nucleons from B/A ≈ 1.1 to 2.8 (reactant average) up to 7.07 MeV — releasing the binding-energy difference as kinetic energy of the alpha and neutron. Splitting ²³⁵U at 7.59 MeV/nucleon into two fragments averaging ~8.5 MeV/nucleon releases ~0.9 × 235 ≈ 211 MeV per event. Both moves climb toward the iron peak from opposite sides.

Binding energy per nucleon for representative isotopes

Isotope	A	Z	B/A (MeV)	Total B (MeV)	Notes
²H (deuterium)	2	1	1.1124	2.2246	Lowest — only one bond
³He	3	2	2.5727	7.7180	Stable, 0.000137% of natural He
⁴He (alpha)	4	2	7.0739	28.296	Doubly magic, exceptionally bound
¹²C	12	6	7.6802	92.162	Defines atomic mass unit
¹⁶O	16	8	7.9762	127.62	Doubly magic
⁵⁶Fe	56	26	8.7903	492.26	Stellar end-point, second-highest B/A
⁶²Ni	62	28	8.7945	545.26	Highest B/A of any nuclide
²³⁵U	235	92	7.5910	1783.9	Fissile
²³⁸U	238	92	7.5701	1801.7	Fertile, t₁/₂ = 4.47 Gyr

Energy released per kilogram of fuel

Process	Reaction	Q-value	MJ per kg fuel	vs gasoline
Combustion (gasoline)	C₈H₁₈ + O₂	~5.5 eV/molecule	46	1×
D-D fusion	²H + ²H → ³He + n	3.27 MeV	7.9 × 10⁷	1.7 × 10⁶ ×
D-T fusion	²H + ³H → ⁴He + n	17.59 MeV	3.39 × 10⁸	7.4 × 10⁶ ×
p-p chain (Sun)	4¹H → ⁴He + 2e⁺ + 2ν	26.73 MeV	6.45 × 10⁸	1.4 × 10⁷ ×
²³⁵U fission	n + ²³⁵U → fragments	~200 MeV	8.20 × 10⁷	1.8 × 10⁶ ×
²³⁹Pu fission	n + ²³⁹Pu → fragments	~210 MeV	8.46 × 10⁷	1.8 × 10⁶ ×
Annihilation	e⁺ + e⁻ → 2γ	1.022 MeV	9.0 × 10¹⁰	2.0 × 10⁹ ×

Semi-empirical mass formula coefficients (Weizsäcker 1935 fit)

Term	Symbol	Coefficient	Physical origin
Volume	aV	+15.75 MeV	Strong-force saturation, ~12 neighbours per nucleon
Surface	aS	−17.80 MeV	Surface nucleons have fewer neighbours
Coulomb	aC	−0.711 MeV	Z(Z−1)/A^(1/3) — proton-proton repulsion
Asymmetry	aA	−23.7 MeV	(N−Z)²/A — Pauli exclusion favours N = Z
Pairing	δ	±12·A^(−1/2) MeV	Even-even bonus, odd-odd penalty
Shell correction	—	up to +3 MeV at magic numbers	²⁰⁸Pb, ¹³²Sn, ¹⁶O extra binding

Famous experiments and applications

• Aston's mass spectrograph (1919, Cavendish). Francis Aston achieved 1-part-in-10000 precision and tabulated isotope masses for 212 nuclides over his career. His "packing fraction" curve, published 1927, foreshadowed binding-energy-per-nucleon and earned the 1922 Nobel Prize in Chemistry.
• Cockcroft-Walton lithium splitting (1932). ⁷Li + p → 2 ⁴He. Mass defect 0.01861 u predicted Q = 17.34 MeV; measured alpha kinetic energies summed to 17.3 ± 0.2 MeV. First reaction-by-reaction verification of E = Δm·c². 1951 Nobel Prize.
• Bethe's CN cycle (1939). Hans Bethe used binding-energy curves to calculate the Sun's energy output via the proton-proton chain (dominant below 1.6 × 10⁷ K) and CNO cycle (dominant above). 1967 Nobel Prize. Standard solar model predicts 26.73 MeV per ⁴He, in agreement with Borexino's 2014 direct measurement of pp neutrinos.
• Hahn and Strassmann fission discovery (1938). Detected ¹⁴⁰Ba in neutron-irradiated uranium. Meitner and Frisch interpreted (1939) using the liquid-drop model and binding-energy curve, predicting ~200 MeV per fission. Confirmed within months by ionization chambers showing fragment kinetic energies.
• Iron in supernova remnants. The peak of binding energy shapes stellar nucleosynthesis: ²⁸Si burns to a mix peaking at ⁵⁶Ni (which beta-decays to ⁵⁶Co to ⁵⁶Fe with t₁/₂ = 6.1 d and 77.2 d respectively). Cassiopeia A's gamma-ray lines from ⁴⁴Ti and the X-ray Fe Kα at 6.4 keV map out post-iron-peak material; r-process nucleosynthesis (rapid neutron capture in neutron-star mergers) is required to build elements above iron.