Nuclear binding energy and mass defect Knowledge-Based Study

Study notes

1. What is Mass Defect?

   When precise measurements are made of an atomic nucleus, its actual mass is always slightly less than the sum of the individual masses of its constituent protons and neutrons (nucleons). This difference in mass is known as the mass defect ($\Delta m$).

   For a nucleus containing $Z$ protons and $N$ neutrons with an observed nuclear mass of $m_{\text{nucleus}}$, the mass defect is calculated as:

   $$\Delta m = (Z \cdot m_p + N \cdot m_n) - m_{\text{nucleus}}$$

   Here, $m_p$ represents the mass of a free proton and $m_n$ represents the mass of a free neutron. The missing mass does not simply vanish; rather, it is converted into energy during the assembly of the nucleus from free nucleons.

   Einstein's Mass-Energy Equivalence

   According to Albert Einstein's special theory of relativity, mass and energy are interconvertible, described by the fundamental relationship:

   $$E = \Delta m \cdot c^2$$

   In nuclear physics, this released energy is designated as the nuclear binding energy ($E_b$). It can be understood in two equivalent ways:

   • The energy released when unbound protons and neutrons combine to form a stable nucleus.
   • The minimum energy required to completely separate a bound nucleus into its individual, non-interacting protons and neutrons.

   In practice, atomic and nuclear masses are frequently expressed in unified atomic mass units ($\text{u}$). One atomic mass unit corresponds to an energy equivalent of approximately 931.5 MeV:

   $$E_b = \Delta m \text{ (in u)} \times 931.5 \text{ MeV/u}$$

   Measuring Relative Nuclear Stability

   Total binding energy increases generally with the total number of nucleons (mass number, $A$). However, total binding energy alone does not indicate how tightly bound or stable a specific nucleus is. To compare the relative stability of different isotopes, scientists calculate the binding energy per nucleon:

   $$\text{Binding Energy per Nucleon} = \frac{E_b}{A}$$

   A higher value of binding energy per nucleon indicates a more tightly bound and energetically stable nucleus.

   The Curve of Binding Energy

   • Light Nuclei: Very light nuclei, such as hydrogen isotopes, possess relatively low binding energy per nucleon. When light nuclei combine to form heavier nuclei through nuclear fusion, binding energy per nucleon increases, releasing tremendous amounts of energy.
   • Peak of Stability: The curve reaches its maximum around mass numbers $A \approx 56\text{--}62$, with iron-56 ($^{56}\text{Fe}$) and nickel-62 ($^{62}\text{Ni}$) being among the most stable nuclei in the universe.
   • Heavy Nuclei: Very heavy nuclei, such as uranium-235 ($^{235}\text{U}$), have lower binding energy per nucleon due to increasing electrostatic repulsion between protons. When heavy nuclei split into mid-sized fragments through nuclear fission, the products have higher binding energy per nucleon, thereby releasing energy.