Nuclear binding energy

Nuclear binding energy is the energy it would take to pull an Atomic nucleus completely apart into free Protons and Neutrons. Run backwards, it is the energy released when those particles snap together. It is the largest energy budget in ordinary matter, and it is paid for out of mass.

The mass that isn't there

Weigh a helium-4 atom. Separately weigh its ingredients — two hydrogen atoms and two free neutrons — and add them up. The parts weigh more than the whole.

item	mass (u)
2 × ¹H atom	2.015650
2 × free neutron	2.017330
sum of the parts	4.032980
actual ⁴He atom	4.002603
missing mass	0.030377

Helium-4 is 0.75% lighter than the pieces it is made of. That missing mass is not lost. It left as Energy when the nucleus formed, at the exchange rate Albert Einstein wrote down: E = mc². One atomic mass unit is worth 931.494 103 72 MeV, so the missing 0.030377 u is 28.296 MeV of binding energy.

Do that arithmetic for every nucleus and you get the whole subject. The recipe, in atomic masses:

B = ( Z·m(¹H) + N·m(neutron) − m(atom) ) × 931.494 MeV/u

where Z is the proton count and N the neutron count.

Per nucleon is the number that matters

Total binding energy just grows with size — uranium has more of it than helium, the way a brick wall has more mortar than a brick. The useful figure is binding energy per nucleon, B/A. That says how tightly each particle is held, and it is what decides which reactions release energy.

nuclide	A	B/A (keV)	total B (MeV)
²H (deuterium)	2	1 112.283	2.225
⁶Li	6	5 332.331	31.994
⁴He	4	7 073.916	28.296
¹²C	12	7 680.145	92.162
²³⁸U	238	7 570.126	1 801.69
²³⁵U	235	7 590.915	1 783.87
¹⁶O	16	7 976.207	127.619
²⁰⁸Pb	208	7 867.453	1 636.43
⁵⁶Fe	56	8 790.356	492.26
⁵⁸Fe	58	8 792.253	509.95
⁶²Ni	62	8 794.556	545.26

⁶²Ni holds the record: 8 794.556 keV per nucleon, the most tightly bound nuclide known. ⁵⁸Fe and ⁵⁶Fe are close behind, separated by a few keV out of nearly nine thousand. Textbooks usually say "iron" because ⁵⁶Fe is by far the commonest of the three — 91.75% of natural iron — but the strict winner is nickel.

Why the curve has a hump

Two forces fight over the nucleus.

• The strong force binds each nucleon to its immediate neighbours only. It saturates: past a couple of nucleon-widths it stops helping. So its contribution grows roughly with the number of nucleons.
• Electric repulsion between protons has no such limit. Every proton pushes every other proton, however far apart, so the penalty grows faster than the size of the nucleus.

Small nuclei are loosely bound because too many of their nucleons sit on the surface with nothing on the other side to hold. Large nuclei are loosely bound because the protons have piled up too much repulsion. In between, around A ≈ 60, the two effects balance and binding is tightest.

Everything downhill releases energy

The hump is why both directions of nuclear energy work:

• nuclear fusion — climbing the steep left slope. Joining light nuclei moves them to higher B/A.
• nuclear fission — sliding down the gentle right slope. Splitting a heavy nucleus moves both fragments to higher B/A.

Both end nearer the peak. Nothing releases energy by fusing iron, which is why stellar cores stop there and then collapse.

Worked example: one fission

Take a common split of ²³⁵U:

²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n

nucleus	total binding energy
²³⁵U	1 783.87 MeV
¹⁴¹Ba	1 173.98 MeV
⁹²Kr	783.17 MeV
released (products − fuel)	173.28 MeV

That is one nucleus. A kilogram of ²³⁵U holds 2.56 × 10²⁴ of them, so fissioning all of it would release about 71 terajoules — what an 800-megawatt power station delivers in a full day, out of a lump you could hold in one hand.

Worked example: one star

The Sun runs the other slope. Its net reaction turns four hydrogen atoms into one helium atom:

4 ¹H → ⁴He + 2 neutrinos + light

Weigh both sides: 4 × 1.007825 u in, 4.002603 u out, leaving 0.028697 u — 26.73 MeV per helium atom made. Per unit of fuel that beats fission by about a factor of nine — 6.6 MeV per nucleon consumed against 0.73 — because hydrogen's climb up the curve is far steeper than uranium's slide down it.

What it is not

• Not chemical bond energy. Burning carbon releases a few eV per atom; nuclear reactions release millions of eV per atom. The gap is about a million to one, and it is entirely because the strong force is stronger than the electric force that holds molecules together.
• Not the same as radioactive decay energy. A nucleus can be bound and still unstable — it decays because a rearrangement is downhill, not because it is unbound.
• Not where most of a proton's mass comes from. A Proton's own mass is mostly confined strong force energy among its Quarks, a different accounting from the binding between nucleons described here.

Sources and precision

The masses above come from the 2020 Atomic Mass Evaluation (AME2020), served through the IAEA Live Chart of Nuclides. Modern mass measurements are extraordinarily good: ⁴He's mass is known to four parts in 10¹¹, which is why binding energies quoted to the keV are meaningful rather than decorative.

See also: half-life, gamma ray, Isotope, Nuclear reactor, Nucleosynthesis.