Mass Defect and Nuclear Binding Energy – Cambridge A-Level Physics 9702
Mass Defect and Nuclear Binding Energy
Mass Defect
The mass of a nucleus is always slightly less than the sum of the masses of its constituent protons and neutrons. This difference is called the mass defect (\$\Delta m\$) and is given by
\$\Delta m = Z\,mp + N\,mn - m_{\text{nucleus}}\$
where \$Z\$ is the number of protons, \$N\$ the number of neutrons, \$mp\$ the mass of a proton, \$mn\$ the mass of a neutron and \$m_{\text{nucleus}}\$ the measured nuclear mass.
According to Einstein’s mass‑energy equivalence, the missing mass is converted into binding energy:
\$E_{\text{binding}} = \Delta m\,c^{2}\$
Thus the nucleus is held together by the energy released when the nucleons combine.
Binding Energy
The binding energy of a nucleus is the energy required to separate it into its individual protons and neutrons. It is often expressed per nucleon to compare the stability of different nuclei.
Nucleus
Mass Defect (u)
Binding Energy (MeV)
Binding Energy per Nucleon (MeV)
\$^{4}_{2}\text{He}\$
0.0304
28.3
7.1
\$^{56}_{26}\text{Fe}\$
0.5280
492.3
8.8
\$^{238}_{92}\text{U}\$
0.8620
1786
7.5
Suggested diagram: Plot of binding energy per nucleon versus mass number showing the peak near iron.
Nuclear Reactions
Nuclear Fusion
Fusion is the process in which two light nuclei combine to form a heavier nucleus, releasing energy because the final nucleus has a higher binding energy per nucleon.
Typical example (deuterium–tritium fusion):
\$^{2}{1}\text{H} + ^{3}{1}\text{H} \;\rightarrow\; ^{4}_{2}\text{He} + n + 17.6\ \text{MeV}\$
Key characteristics:
Requires extremely high temperatures (≈10⁸ K) to overcome the Coulomb barrier.
Energy is released as kinetic energy of the products, which can be converted to heat.
Occurs naturally in stars, where gravitational pressure provides the necessary conditions.
Suggested diagram: Schematic of deuterium‑tritium fusion showing the incoming nuclei, the intermediate compound nucleus, and the outgoing helium nucleus and neutron.
Nuclear Fission
Fission is the splitting of a heavy nucleus into two (or more) lighter fragments, accompanied by the release of neutrons and a large amount of energy.
Typical example (thermal neutron‑induced fission of uranium‑235):
Triggered when a neutron is captured, making the nucleus unstable.
The resulting fragments have a higher binding energy per nucleon, so the excess energy is released.
Emitted neutrons can induce further fissions, leading to a chain reaction.
Control of the chain reaction (critical mass, moderators, control rods) is essential for reactor operation.
Suggested diagram: Schematic of uranium‑235 fission showing the incoming neutron, the split into barium and krypton fragments, and the release of additional neutrons.
Comparison of Fusion and Fission
Fuel nuclei: light (e.g., isotopes of hydrogen) for fusion; heavy (e.g., \$^{235}\$U, \$^{239}\$Pu) for fission.
Energy per nucleon released: fusion releases \overline{8} Me \cdot per nucleon; fission releases \overline{0}.9 Me \cdot per nucleon, but the total energy per reaction is larger for fission because of the larger mass involved.
By‑products: fusion produces mainly helium and neutrons (low‑radioactivity); fission produces a range of radioactive fission fragments and neutrons.
Technological challenges: fusion requires sustaining extremely high temperatures and pressures; fission requires managing radioactive waste and controlling the chain reaction.
Key Points to Remember
Mass defect (\$\Delta m\$) is the difference between the summed masses of nucleons and the actual nuclear mass.
Binding energy \$E_{\text{binding}} = \Delta m\,c^{2}\$ quantifies the stability of a nucleus.
Fusion combines light nuclei, releasing energy because the product has a higher binding energy per nucleon.
Fission splits heavy nuclei, releasing energy for the same reason.
Both processes are governed by the same principle: systems evolve toward configurations of greater binding energy per nucleon.