Nuclear Binding Energy

Definition

The nuclear binding energy E_B of a nucleus is the energy that must be supplied to separate it completely into its constituent free nucleons — Z protons and N neutrons — with no residual interaction between them. By energy conservation, it is also the energy liberated when those free nucleons are brought together to form the nucleus.

The key insight is that a bound nucleus has less mass than the sum of its free parts. This missing mass is the mass defect:

Δm = Z·m_p + N·m_n − M_nucleus

and the binding energy is that mass defect expressed in energy units through Einstein's mass–energy equivalence:

E_B = Δm · c²

where:

• E_B — binding energy, usually quoted in mega-electron-volts (MeV; 1 MeV = 1.602 × 10⁻¹³ J).
• Δm — mass defect, in kilograms or atomic mass units (u).
• c — speed of light, 2.998 × 10⁸ m/s.
• Z — proton number; N — neutron number; A = Z + N — mass number.
• m_p ≈ 938.272 MeV/c² (proton), m_n ≈ 939.565 MeV/c² (neutron), M_nucleus — measured nuclear mass.

A practical conversion factor makes the arithmetic quick: one atomic mass unit corresponds to 931.494 MeV/c². So a mass defect of 0.1 u means a binding energy of about 93 MeV.

Binding energy per nucleon — the master curve

The single most important quantity in nuclear energetics is not the total binding energy but the binding energy per nucleon, E_B/A. Plotting E_B/A against mass number A produces the famous curve that governs all of nuclear energy:

• It rises steeply for light nuclei, from about 1.1 MeV/nucleon in deuterium to 7.07 MeV in helium-4.
• It peaks broadly near A ≈ 56–62, at roughly 8.8 MeV/nucleon — iron-56 (8.79 MeV) and nickel-62 (8.795 MeV) sit at the top.
• It declines slowly for heavy nuclei, down to about 7.6 MeV/nucleon in uranium-238.

The peak is the reason the universe releases nuclear energy in two opposite ways. Anything below iron on the left can release energy by fusion (climbing toward the peak); anything below iron on the right can release energy by fission (also climbing toward the peak). Iron sits at the bottom of the energy valley — the ash of both processes.

How it works — step by step

1. Weigh the parts. Take Z free protons and N free neutrons and add up their masses. This is what the nucleus would weigh if the nucleons did not interact.
2. Weigh the whole. Measure the actual nuclear mass with a mass spectrometer. It comes out smaller.
3. Find the deficit. The difference Δm is the mass defect — literally mass that has been converted into binding energy and radiated away when the nucleus formed.
4. Convert to energy. Multiply by c² (or by 931.494 MeV/u) to get E_B.
5. Normalize. Divide by A to get E_B/A and place the nucleus on the master curve. Its height tells you whether fusion or fission can extract more energy.

Worked example — helium-4

Helium-4 (the alpha particle) is the classic case, and its fusion powers the Sun. It has Z = 2 protons and N = 2 neutrons.

Quantity	Value
2 × proton mass	2 × 1.007276 u = 2.014552 u
2 × neutron mass	2 × 1.008665 u = 2.017330 u
Sum of free nucleons	4.031882 u
Actual He-4 nuclear mass	4.001506 u
Mass defect Δm	0.030376 u
Binding energy E_B = Δm × 931.494	≈ 28.3 MeV
Binding energy per nucleon	28.3 / 4 ≈ 7.07 MeV

That 28.3 MeV is enormous compared with chemical bonds (a few eV) — a factor of roughly a million. The unusually high binding of He-4, thanks to its doubly-magic closed shells, is why it appears as a discrete "alpha particle" in radioactive decay and why the Sun's proton–proton chain funnels hydrogen straight into helium.

The semi-empirical mass formula

You can predict binding energies without solving the full quantum many-body problem using the semi-empirical mass formula (the Bethe–Weizsäcker formula, 1935), built on the liquid-drop model of the nucleus:

E_B = a_V·A − a_S·A^(2/3) − a_C·Z(Z−1)/A^(1/3) − a_A·(A−2Z)²/A + δ(A,Z)

Each term encodes a competing physical effect:

Term	Coefficient (MeV)	Physical meaning
Volume: +a_V·A	a_V ≈ 15.8	Strong force binds each nucleon to its neighbours; scales with volume ∝ A.
Surface: −a_S·A^(2/3)	a_S ≈ 18.3	Nucleons at the surface have fewer neighbours — a "surface tension" penalty ∝ A^(2/3).
Coulomb: −a_C·Z(Z−1)/A^(1/3)	a_C ≈ 0.714	Every proton pair repels electrically; grows with Z², hurts heavy nuclei.
Asymmetry: −a_A·(A−2Z)²/A	a_A ≈ 23.2	Pauli exclusion penalizes lopsided N ≠ Z; favours N ≈ Z (and slightly N > Z for heavy nuclei).
Pairing: +δ	±a_P·A^(−1/2), a_P ≈ 12	Even numbers of protons and of neutrons pair up and bind more tightly.

Here A^(2/3) is the surface-area scaling of a sphere of volume ∝ A, and Z(Z−1)/2 counts the number of repelling proton pairs. Despite being only five terms with a handful of fitted constants, the formula reproduces measured binding energies across the whole chart of nuclides to within about 1%. It correctly predicts the shape of the E_B/A curve: the volume term wants big nuclei, but the surface term hurts small ones and the Coulomb term hurts big ones, so binding per nucleon maximizes in the middle, right around iron.

Strong force versus Coulomb repulsion

A nucleus is a paradox at first glance: dozens of positively charged protons crammed into a sphere a few femtometres across, all repelling one another. What keeps them together is the residual strong nuclear force (the leftover of the color force between quarks). Two features are decisive:

• Strength. At nucleon separations of about 1 fm, the strong force is roughly 100 times stronger than the electromagnetic repulsion between two protons.
• Range. The strong force is short-ranged — effectively 1–2 fm — because it is mediated by massive pions. Each nucleon feels attraction only from its immediate neighbours, so the volume (a_V·A) term saturates.

The Coulomb force, by contrast, is long-ranged: every proton repels every other proton across the whole nucleus, so its energy grows like Z². In light nuclei the strong force wins easily. As Z climbs, the Z² Coulomb penalty eventually overwhelms the surface-limited strong binding — this is why there is no stable nucleus beyond bismuth-209 (Z = 83) and why the heaviest nuclei undergo alpha decay and spontaneous fission.

Binding energy across the chart of nuclides

Nucleus	A	Total E_B (MeV)	E_B/A (MeV)
Deuterium (²H)	2	2.22	1.11
Helium-4 (⁴He)	4	28.3	7.07
Carbon-12 (¹²C)	12	92.2	7.68
Oxygen-16 (¹⁶O)	16	127.6	7.98
Iron-56 (⁵⁶Fe)	56	492.3	8.79
Nickel-62 (⁶²Ni)	62	545.3	8.795
Silver-107 (¹⁰⁷Ag)	107	915	8.55
Uranium-235 (²³⁵U)	235	1784	7.59
Uranium-238 (²³⁸U)	238	1802	7.57

Read the E_B/A column top to bottom: it climbs, peaks at iron/nickel, then gently falls. Every rung of nuclear energy — from the Sun to a reactor to a bomb — is a slide down this column toward the peak.

Why fusion and fission both release energy

Fusion (left side of the curve). Fusing four protons into one helium-4 nucleus (via the Sun's proton–proton chain) converts about 0.7% of the mass into energy — roughly 26.7 MeV per helium nucleus produced. The products are far more tightly bound than the reactants, and the mass difference emerges as photons, neutrinos, and kinetic energy.

Fission (right side of the curve). When uranium-235 absorbs a neutron and splits into two mid-mass fragments (say barium and krypton), the fragments sit higher on the curve at about 8.5 MeV/nucleon versus 7.6 MeV/nucleon for the parent. The gain of nearly 0.9 MeV across ~236 nucleons yields about 200 MeV per fission, mostly as fragment kinetic energy. Per kilogram of U-235 that is roughly 82 TJ — about a million times the energy of burning the same mass of chemical fuel.

Both are the same principle: rearrange nucleons so more of them end up near the top of the binding-energy curve, and the surplus binding energy is released as the mass defect Δm·c².

A brief history

• 1905 — Einstein publishes E = mc², making mass a reservoir of energy.
• 1920 — Aston's precise mass measurements reveal that helium weighs less than four hydrogens; Eddington immediately proposes this "packing fraction" as the Sun's power source.
• 1935 — Weizsäcker (and Bethe, 1936) formulate the semi-empirical mass formula from Gamow's liquid-drop picture.
• 1938–39 — Hahn and Strassmann discover fission; Meitner and Frisch explain it energetically using the binding-energy curve and c².
• 1939 — Bethe works out the CNO and proton–proton fusion cycles powering stars, tying binding energy to stellar nucleosynthesis (Nobel Prize, 1967).

Common misconceptions

• "The mass defect means mass is destroyed." Nothing is destroyed — mass and energy are two accountings of the same conserved quantity. The "missing" mass was carried away as binding energy (released as photons and kinetic energy) when the nucleus formed.
• "Binding energy is a force." It is an energy — the depth of the potential well. The force is the residual strong interaction; binding energy is the work needed to escape it.
• "Iron-56 is exactly the peak." Iron-56 has the highest binding energy per nucleon among common nuclei, but nickel-62 edges it out slightly (8.795 vs 8.790 MeV). Iron-56 wins the related "lowest mass per nucleon" contest and dominates stellar ash, hence the folklore.
• "Fusion always releases energy." Only up to iron. Fusing anything heavier than iron costs energy — which is why elements past iron are forged in supernovae, not in ordinary stellar burning.
• "Heavier nuclei are more tightly bound because they have more binding energy." Total E_B does grow with A, but binding per nucleon is what matters for stability, and that peaks at iron and then declines.
• "You use atomic masses directly." Careful bookkeeping is needed: tabulated atomic masses include electron masses and electronic binding energy. Use nuclear masses, or apply the electron corrections consistently.