Binding Energy per Nucleon Curve (SL 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 a nucleus, there are
  • Repulsive electric forces between protons due to their positive charge
  • Attractive gravitational forces due to the mass of the nucleons

• Gravity is the weakest of the fundamental forces, so it has a negligible effect compared to electric repulsion between protons
• If these were the only forces acting, the nucleus would not hold together
• Therefore, there must be an attractive force acting between all nucleons which is stronger than the electric repulsive force
  • This is known as the strong nuclear force

• The strong nuclear force acts between particles called quarks
• Protons and neutrons are made up of quarks, so the interaction between the quarks in the nucleons 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

Properties 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 below a separation of 0.5 fm and attractive up to 3.0 fm

• The key features of the graph are:
  • The strong force is highly repulsive at separations below 0.5 fm
  • The strong force is very attractive up to a nuclear separation of 3.0 fm
  • The maximum attractive value occurs at around 1.0 fm, which is a typical value for nucleon separation
  • The equilibrium position, where the resultant force is zero, occurs at a separation of about 0.5 fm

• In comparison to other fundamental forces, the strong nuclear force has a very small range (from 0.5 to 3.0 fm)

Comparison of Electrostatic and Strong Forces

• The graph below shows how the strength of the electrostatic and strong forces between two nucleons vary with the separation between them
  • The red curve represents the strong nuclear force between nucleons
  • The blue curve represents the electrostatic repulsion between protons

At separations between 0.5 and 3.0 fm, the attraction of the strong force is far more powerful than the repulsion of the electrostatic force

• The repulsive electrostatic force between protons has a much larger range than the strong nuclear force
  • However, it only becomes significant when the proton separation is more than around 2.5 fm

• The electrostatic force is influenced by charge, whereas the strong nuclear force is not
• This means the strength of the strong nuclear force is roughly the same between all types of nucleon (i.e. proton-proton, neutron-neutron and proton-neutron)
  • This only applies for separations between 0.5 and 3.0 fm (where the electrostatic force between protons is insignificant)

• The equilibrium position for protons, where the electrostatic repulsive and strong attractive forces are equal, occurs at a separation slightly below 0.5 fm