Mass-energy and binding energy

Einstein showed that mass and energy are equivalent: E = mc². Mass can turn into energy and energy into mass. Because c² is huge, a tiny mass is a vast energy: 1 g is worth 9 × 10¹³ J.

In nuclear units: 1 u = 931.5 MeV/c². So a mass in u times 931.5 gives energy in MeV.

Mass defect: a nucleus always weighs less than the protons and neutrons that make it, taken separately. The missing mass is Δm.

Where did it go? When the nucleons came together, energy was released. To pull the nucleus apart again, you must put that energy back. That is the binding energy:

E_b = Δm c² = Δm (in u) × 931.5 MeV

Example: helium-4. Two hydrogen atoms (2 × 1.007825 u) plus two neutrons (2 × 1.008665 u) make 4.032980 u. A helium-4 atom is 4.002603 u. Δm = 0.030377 u, so E_b ≈ 0.030377 × 931.5 ≈ 28.3 MeV, or about 7.1 MeV per nucleon.

Use atomic masses and the mass of a hydrogen atom (not a bare proton): the electrons then cancel out on both sides.

Binding energy per nucleon, E_b/A, measures how tightly a nucleus is held:

• It is about 8 MeV for most nuclei, which shows the nuclear force is short-ranged: each nucleon feels only its neighbours.
• It is highest, about 8.8 MeV, near iron (A ≈ 56): the most stable nuclei.
• It is lower for very light nuclei and for very heavy ones.

This curve explains nuclear energy. Joining light nuclei (fusion) or splitting heavy ones (fission) both move towards iron, to more tightly bound nuclei, and release the difference.