AQA A Level Physics

Revision Notes

8.3.4 Nuclear Radius

Estimating Nuclear Radius

Closest Approach Method

  • In the Rutherford scattering experiment, alpha particles are fired at a thin gold foil
  • Some of the alpha particles are found to come straight back from the gold foil
  • This indicates that there is electrostatic repulsion between the alpha particles and the gold nucleus





  • At the point of closest approach, r, the repulsive force reduces the speed of the alpha particles to zero momentarily
  • At this point, the initial kinetic energy of an alpha particle, Ek, is equal to electric potential energy, Ep
  • The radius of the closest approach can be found be equating the initial kinetic energy to the electric potential energy

Pros & Cons of Closest Approach Method


  • Alpha scattering gives a good estimate of the upper limit for a nuclear radius
  • The mathematics behind this approach are very simple
  • The alpha particles are scattered only by the protons and not all the nucleons that make up the nucleus


  • This method does not give an accurate value for nuclear radius as it will always be an overestimate
    • This is because it measures the nearest distance the alpha particle can get to the gold nucleus, not the radius of it
  • Alpha particles are hadrons, therefore, when they get close to the nucleus they are affected by the strong nuclear force and the mathematics do not account for this
  • The gold nucleus will recoil as the alpha particle approaches
  • Alpha particles have a finite size whereas electrons can be treated as a point mass
  • It is difficult to obtain alpha particles which rebound at exactly 180°
    • In order to do this, a small collision region is required
  • The alpha particles in the beam must all have the exact same initial kinetic energy
  • The sample must be extremely thin to prevent multiple scattering

Electron Diffraction Method

  • Electrons accelerated to close to the speed of light have wave-like properties such as the ability to diffract and have a de Broglie wavelength equal to:

  • Where:
    • h = Planck's constant
    • m = mass of an electron (kg)
    • v = speed of the electrons (m s−1)


  • The diffraction pattern forms a central bright spot with dimmer concentric circles around it
  • From this pattern, a graph of intensity against diffraction angle can be used to find the diffraction angle of the first minimum
  • Using this, the size of the atomic nucleus, R, can be determined from:

  • Where:
    • θ = angle of the first minimum (degrees)
    • λ = de Broglie wavelength (m)
    • R = radius of the nucleus (m)

Pros & Cons of Electron Diffraction Method


  • Electron diffraction is much more accurate than the closest approach method
  • This method gives a direct measurement of the radius of a nucleus
  • Electrons are leptons; therefore, they will not interact with nucleons in the nucleus through the strong nuclear force as an alpha particle would


  • Electrons must be accelerated to very high speeds to minimise the de Broglie wavelength and increase resolution
    • This is because significant diffraction takes place when the electron wavelength is similar in size to the nuclear diameter
  • Electrons can be scattered by both protons and neutrons
    • If there is an excessive amount of scattering, then the first minimum of the electron diffraction can be difficult to determine

Electron Diffraction by a Nucleus

  • The graph of intensity against angle obtained through electron diffraction is as follows:

Worked Example

The graph shows how the relative intensity of the scattered electrons varies with angle due to diffraction by the oxygen-16 nuclei. The angle is measured from the original direction of the beam.

Worked Example - Electron Diffraction Intensity GraphWorked Example - Electron Diffraction Intensity Graph, downloadable AS & A Level Physics revision notes

The de Broglie wavelength λ of each electron in the beam is 3.35 × 10−15 m.

Calculate the radius of an oxygen-16 nucleus using information from the graph.

Step 1: Identify the first minimum from the graph


  • Angle of first minimum, θ = 42°

Step 2: Write out the equation relating the angle, wavelength, and nuclear radius

Step 3: Calculate the nuclear radius, R


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