Binding Energy per Nucleon Curve (HL IB Physics)

• In order to compare nuclear stability, it is useful to look at the binding energy per nucleon
• The binding energy per nucleon is defined as:

The binding energy of a nucleus divided by the number of nucleons in the nucleus

• A higher binding energy per nucleon indicates a higher stability
• In other words, more energy is required to separate the nucleons contained within a nucleus

By plotting a graph of binding energy per nucleon against nucleon number, the stability of elements can be inferred

Key Features of the Graph

• At low values of A:
  • Nuclei have lower binding energies per nucleon than at large values of A, but they tend to be stable when N = Z
  • This means light nuclei have weaker electrostatic forces and will undergo fusion
  • The gradient is much steeper compared to the gradient at large values of A
  • This means that fusion reactions release a greater binding energy than fission reactions

• At high values of A:
• Nuclei have generally higher binding energies per nucleon, but this gradually decreases with A
• This means the heaviest elements are the most unstable and will undergo fission
• The gradient is less steep compared to the gradient at low values of A
• This means that fission reactions release less binding energy than fission reactions

• Iron (A = 56) has the highest binding energy per nucleon, which makes it the most stable of all the elements

• Helium (⁴He), carbon (¹²C) and oxygen (¹⁶O) do not fit the trend
  • Helium-4 is a particularly stable nucleus hence it has a high binding energy per nucleon
  • Carbon-12 and oxygen-16 can be considered to be three and four helium nuclei, respectively, bound together

Comparing Fusion & Fission

Similarities

• In both fusion and fission, the total mass of the products is slightly less than the total mass of the reactants
• The mass defect is equivalent to the binding energy that is released
• As a result, both fusion and fission reactions release energy

Differences

• In fusion, two smaller nuclei combine into a larger nucleus
• In fission, an unstable nucleus splits into two smaller nuclei

• Fusion occurs between light nuclei (A < 56)
• Fission occurs in heavy nuclei (A > 56)

• In light nuclei, attractive nuclear forces dominate over repulsive electrostatic forces between protons, and this contributes to nuclear stability 
• In heavy nuclei, repulsive electrostatic forces between protons begin to dominate over attractive nuclear forces, and this contributes to nuclear instability

• Fusion releases much more energy per kg than fission
• Fusion requires a greater initial input of energy than fission

• In the nucleus, there are electrostatic forces between the protons due to their electric charge and gravitational forces due to their mass
• Comparatively, gravity is a very weak force and the electrostatic repulsion between protons is therefore much stronger than their gravitational attraction
• If these were the only forces, the nucleus wouldn’t hold together
• Therefore, the force that does hold the nucleus together is called the strong nuclear force
• The strong nuclear force keeps the nucleus stable since it holds quarks together
• Since protons and neutrons are made up of quarks, the strong force keeps them bound within a nucleus

Whilst the electrostatic force is a repulsive force in the nucleus, the strong nuclear force holds the nucleus together

Range of the Strong Nuclear Force

• The strength of the strong nuclear force between two nucleons varies with the separation between them
• This can be plotted on a graph which shows how the force changes with separation

The strong nuclear force is repulsive before a separation of ~ 0.5 fm and attractive up till ~ 3.0 fm

• The key features of this graph are that the strong nuclear force is:
  • Repulsive closer than around 0.5 fm (femtometres, 10^-15 m)
  • Attractive up to around 3.0 fm
  • Reaches a maximum attractive value at around 1.0 fm (the typical nuclear separation)
  • Becomes zero after 3.0 fm
• In comparison to other fundamental forces, the strong force has a very small range (only up to 3.0 fm)