Binding Energy per Nucleon Curve (HL IB Physics)

• In order to compare nuclear stability, it is more 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, it requires more energy to pull the nucleus apart

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

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 tend to have a lower binding energy per nucleon, hence, they are generally less stable
  • This means the lightest elements have weaker electrostatic forces and are the most likely to undergo fusion

• 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

• At high values of A:
  • The general binding energy per nucleon is high and gradually decreases with A
  • This means the heaviest elements are the most unstable and likely to undergo fission

Comparing Fusion & Fission

• Fusion occurs at low values of A because:
  • Attractive nuclear forces between nucleons dominate over repulsive electrostatic forces between protons

• In fusion, the mass of the nucleus that is created is slightly less than the total mass of the original nuclei
  • The mass defect is equal to the binding energy that is released since the nucleus that is formed is more stable

• Fission occurs at high values of A because:
  • Repulsive electrostatic forces between protons begin to dominate, and these forces tend to break apart the nucleus rather than hold it together

• In fission, an unstable nucleus is converted into more stable nuclei with a smaller total mass
  • This difference in mass, the mass defect, is equal to the binding energy that is released

• Fusion releases much more energy per kg than fission
• The energy released is the difference in binding energy caused by the difference in mass between the reactant and products
  • Hence, the greater the increase in binding energy, the greater the energy released

• At small values of A (fusion region), the gradient is much steeper compared to the gradient at large values of A (fission region)
• This corresponds to a larger binding energy per nucleon being released

• 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)

The Strong Nuclear Force & Fusion

• For two nuclei to fuse, both nuclei must have high kinetic energy
  • This is because nuclei must be able to get close enough to fuse

• However, two forces acting within the nuclei make this difficult to achieve
• Electrostatic Repulsion 
  • Protons inside the nuclei are positively charged, which means that they electrostatically repel one another
• Strong Nuclear Force
  • The strong nuclear force, which binds nucleons together, acts at very short distances within nuclei
  • Therefore, nuclei must get very close together for the strong nuclear force to take effect
• It takes a great deal of energy to overcome the repulsive electrostatic forces, hence fusion can only be achieved in an extremely hot environment where particles have a great deal of kinetic energy, such as the core of a star