AQA A Level Physics

Revision Notes

8.3.6 Nuclear Radius Equation

Nuclear Radius Values

  • The radius of some nuclei are shown in the table below:



  • In general, nuclear radii are of the order 10–15 m or 1 fm
  • The nuclear radius, R, varies with nucleon number, as follows:




  • The key features of this graph are:
    • The graph starts with a steep gradient at the origin
    • Then the gradient gradually decreases to almost horizontal
  • This means that
    • As more nucleons are added to a nucleus, the nucleus gets bigger
    • However, the number of nucleons A is not proportional to its size r

Radius v Nucleon Number

  • The radius of nuclei depends on the nucleon number, A of the atom
  • This makes sense because as more nucleons are added to a nucleus, more space is occupied by the nucleus, hence giving it a larger radius
  • The exact relationship between the radius and nucleon number can be determined from experimental data
  • By doing this, physicists were able to deduce the following relationship:

  • Where:
    • R = nuclear radius (m)
    • A = nucleon / mass number
    • R0 = constant of proportionality = 1.05 fm


  • Plotting a graph of R against A1 / 3 gives a straight line through the origin with the gradient equal to R0
  • It is also possible to plot a logarithmic graph of the relationship which can be derived as follows:

ln R = ln (R0 A1 / 3)

ln R = ln R0 + ln (A1 / 3)

ln R = ln R0 + 1/3 ln A

  • Therefore, a graph of ln R against ln A yields a straight line
  • Comparing this to the straight-line equation: y = mx + c
    • y = lnR
    • x = lnA
    • m (the gradient) = 1/3
    • c (y-intercept) = ln R0

Worked Example

Verify the experimental relationship between R and A using the data from the table above and estimate a value of R0.

Step 1: Add a column to the table to determine the values for A1 / 3

Step 2: Plot a graph of R against A1 / 3 and draw a line of best fit

Step 3: Calculate the gradient


R0 = 1.12 fm


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