Binding energy, fusion and fissionEdexcel International A Level Physics: Revision notes

The mass of a nucleus is always less than the total mass of its separate protons and neutrons. The difference is the mass deficit:

$\Delta m = (Zm_p + (A-Z)m_n) - m_{nucleus}$

When nucleons join to form a nucleus, energy is released and this appears as the missing mass. The binding energy is the energy needed to separate a nucleus completely into its individual nucleons. The two are linked by Einstein's mass-energy equation:

$E = c^2 m$, so binding energy $= c^2\,\Delta m$.

A large binding energy means the nucleons are held together tightly.

Nuclear masses are tiny, so they are quoted in atomic mass units. One unit, u, is one twelfth of the mass of a carbon-12 atom:

$1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg}$

To find a binding energy in joules, convert the mass deficit to kilograms, then use $E = c^2\Delta m$. To convert to MeV, divide by $1.60 \times 10^{-13}\ \text{J}$.

Worked example: a mass deficit of 0.0304 u is $0.0304 \times 1.66 \times 10^{-27} = 5.05 \times 10^{-29}$ kg. Then $E = 5.05 \times 10^{-29} \times (3.00 \times 10^8)^2 = 4.54 \times 10^{-12}$ J, which is 28.4 MeV.

The binding energy per nucleon is the binding energy divided by the nucleon number. It measures how tightly bound each nucleon is, so it shows which nuclei are most stable.

• It rises steeply from hydrogen-2 (about 1 MeV) to helium-4 (about 7 MeV).
• It reaches a maximum of about 8.8 MeV near iron-56.
• It then falls slowly, to about 7.6 MeV for uranium-235.

A nucleus that changes to give a higher binding energy per nucleon becomes more tightly bound, and the increase in total binding energy is released as energy.

Nuclear fission is the splitting of a large nucleus, such as uranium-235 after absorbing a neutron, into two medium-mass fragments. The fragments have a higher binding energy per nucleon than uranium (about 8.5 MeV compared with 7.6 MeV), so energy is released.

Nuclear fusion is the joining of two light nuclei into a heavier one, for example hydrogen-2 forming helium-4. The product has a higher binding energy per nucleon, so energy is released.

Both processes move nuclei towards iron-56 at the peak of the curve. Fusing nuclei heavier than iron, or splitting nuclei lighter than iron, would lower the binding energy per nucleon, so energy would have to be supplied.

Nuclei are all positively charged, so they repel each other electrostatically. To fuse they must get within about $10^{-15}$ m, where the short-range attractive force between nucleons takes over.

• Very high temperature gives the nuclei enough kinetic energy to overcome the repulsion. At these temperatures matter is a plasma.
• Very high density packs the nuclei close together, so collisions are frequent enough to keep the reaction going.

In stars, gravity supplies both the density and the confinement, so the Sun's core fuses hydrogen at about $1.5 \times 10^7$ K. A reactor on Earth has a much lower density, so it needs a hotter plasma, about $10^8$ K, held in place by magnetic fields because no container could survive it.