Nuclear binding energy Knowledge-Based Study

Study notes

1. The Missing Mass in Nuclei

   When the individual masses of the constituent protons and neutrons that make up an atomic nucleus are summed, the total is always greater than the experimentally measured mass of the bound nucleus itself.

   This difference in mass is known as the mass defect ($\Delta m$). For a nucleus with atomic number $Z$ and mass number $A$, containing $Z$ protons and $N = A - Z$ neutrons, the mass defect is defined as:

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

   where:

   • $m_p$ is the mass of a free proton,
   • $m_n$ is the mass of a free neutron,
   • $m_{\text{nucleus}}$ is the rest mass of the bound nucleus.

   According to Albert Einstein's special theory of relativity, mass and energy are mutually convertible through the mass-energy equivalence relation:

   $$E = mc^2$$

   The mass that disappears during the formation of a nucleus is released as energy into the surroundings. In atomic mass units ($\text{u}$), an equivalent energy conversion factor is frequently employed: $1\,\text{u} \approx 931.5\,\text{MeV}/c^2$.

   Defining Binding Energy

   Nuclear binding energy ($E_b$) is the minimum energy required to completely disassemble a nucleus into its individual, unbound protons and neutrons. Conversely, it corresponds to the quantity of energy released when unbound nucleons assemble to form the bound nucleus.

   Using the mass defect $\Delta m$, the binding energy is expressed as:

   $$E_b = \Delta m \cdot c^2 = \left[ (Z m_p + N m_n) - m_{\text{nucleus}} \right] c^2$$

   When atomic masses $m(A, Z)$ are used instead of bare nuclear masses, the electron masses cancel out to a very close approximation by using the mass of neutral hydrogen atoms ($m_H$):

   $$E_b = \left[ Z m_H + (A - Z) m_n - m(A, Z) \right] c^2$$

   Because the strong nuclear force acts attractively between all nucleons at short ranges, nucleons fall into a potential energy well upon binding, leading to a negative potential energy relative to separated nucleons.

   The Stability Curve

   While total binding energy increases generally with the total number of nucleons $A$, the true measure of relative nuclear stability is the binding energy per nucleon ($E_b / A$):

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

   A higher value of $E_b / A$ means the nucleons are more tightly bound, indicating a more stable nucleus.

   Key Features of the Binding Energy Curve

   • Rapid Rise: For very light nuclei (such as $^2\text{H}$, $^3\text{H}$), $E_b / A$ is relatively low but climbs steeply. Notable local peaks exist for tightly bound light nuclei with even numbers of nucleons, such as $^4\text{He}$, $^ {12}\text{C}$, and $^ {16}\text{O}$.
   • The Peak of Stability: The curve reaches a broad maximum around $A \approx 56\text{--}62$, peaking near iron-56 ($^{56}\text{Fe}$) and nickel-62 ($^{62}\text{Ni}$) at approximately $8.8\,\text{MeV}$ per nucleon. Nuclei in this region are the most stable in the universe.
   • Gradual Decline: For heavier elements ($A > 60$), the binding energy per nucleon steadily decreases toward approximately $7.6\,\text{MeV}$ for uranium ($^{238}\text{U}$). This occurs because the repulsive long-range electrostatic Coulomb force among protons begins to compete effectively with the short-range attractive strong nuclear force.

   Consequences for Nuclear Reactions

   The shape of the curve explains why energy can be extracted from two distinct nuclear processes:

   • Nuclear Fusion: Combining very light nuclei into heavier ones moves upward along the steep initial slope, releasing large quantities of energy.
   • Nuclear Fission: Splitting a very heavy, less tightly bound nucleus into two medium-mass fragments moves upward along the curve toward the peak of stability, also releasing energy.