Cambridge A-Level Physics 9702 – Mass Defect and Nuclear Binding Energy
Mass Defect and Nuclear Binding Energy
Learning Objective
Sketch the variation of binding energy per nucleon with nucleon number and explain the main features of the curve.
Key Concepts
Mass defect (\$\Delta m\$): the difference between the sum of the masses of the individual nucleons and the actual mass of the nucleus.
Binding energy (\$E_b\$): the energy required to separate a nucleus into its constituent protons and neutrons. It is related to the mass defect by Einstein’s relation \$E= \Delta m c^{2}\$.
Binding energy per nucleon (\$E_b/A\$): the average energy that binds each nucleon in the nucleus. It provides a measure of nuclear stability.
Calculating Mass Defect and Binding Energy
For a nucleus with \$Z\$ protons and \$N\$ neutrons (\$A = Z+N\$):
\$\Delta m = \bigl[ Z mp + N mn \bigr] - m_{\text{nucleus}}\$
where \$mp = 1.007276\,\text{u}\$ and \$mn = 1.008665\,\text{u}\$ are the atomic masses of a proton and a neutron respectively, and \$m_{\text{nucleus}}\$ is the measured atomic mass of the nuclide (including the electron masses).
The binding energy is then:
\$E_b = \Delta m \, c^{2} = \Delta m \times 931.5\ \text{MeV}\$
The binding energy per nucleon is simply \$E_b/A\$.
Typical \cdot alues (Illustrative Table)
Nucleus
Mass Number \$A\$
Mass Defect \$\Delta m\$ (u)
Binding Energy \$E_b\$ (MeV)
Binding Energy per Nucleon \$E_b/A\$ (MeV)
\$^{2}\text{H}\$ (deuterium)
2
0.0022
2.05
1.02
\$^{4}\text{He}\$ (alpha particle)
4
0.0304
28.3
7.07
\$^{56}\text{Fe}\$
56
0.4920
492.3
8.79
\$^{238}\text{U}\$
238
1.7840
1664.0
7.00
Variation of Binding Energy per Nucleon with Nucleon Number
The graph of \$E_b/A\$ against \$A\$ has a characteristic shape:
For very light nuclei (\$A \lesssim 20\$) the binding energy per nucleon rises rapidly as nucleons are added.
It reaches a broad maximum around \$A \approx 56\$ (iron‑56 and nickel‑62), where \$E_b/A \approx 8.8\ \text{MeV}\$.
Beyond the maximum the curve slowly declines for heavier nuclei, reaching about \$7.0\ \text{MeV}\$ for actinides such as uranium.
Suggested diagram: Sketch of binding energy per nucleon (\$E_b/A\$) versus nucleon number (\$A\$). The curve should rise sharply for \$A<20\$, peak near \$A=56\$, and then fall gradually for \$A>56\$.
Physical Interpretation of the Curve
Short‑range nuclear force: Each nucleon interacts attractively only with its nearest neighbours. Adding nucleons to a small nucleus greatly increases the number of attractive pairs, raising \$E_b/A\$.
Surface effects: In light nuclei a large fraction of nucleons are on the surface and experience fewer attractive neighbours, lowering \$E_b/A\$.
Electrostatic repulsion: In heavy nuclei the growing number of protons introduces increasing Coulomb repulsion, which offsets the additional nuclear attraction and causes \$E_b/A\$ to fall.
Stability: Nuclei near the peak (iron‑group) are the most tightly bound and therefore the most stable. Nuclei lighter than the peak can release energy by fusion, while those heavier can release energy by fission.
Implications for Energy Production
Because energy release corresponds to moving to a configuration with higher \$E_b/A\$, the following processes are energetically favourable:
Fusion: Light nuclei (e.g., \$^{2}\$H, \$^{3}\$He, \$^{4}\$He) combine to form nuclei nearer the peak, releasing energy.
Fission: Heavy nuclei (e.g., \$^{235}\$U, \$^{239}\$Pu) split into fragments with \$A\$ around 100–150, which have higher \$E_b/A\$ than the original heavy nucleus, releasing energy.
Summary
The binding energy per nucleon curve encapsulates the balance between the short‑range attractive nuclear force and the long‑range Coulomb repulsion. Its peak near \$A\approx56\$ explains why both nuclear fusion (for light elements) and fission (for heavy elements) can be harnessed as powerful energy sources.