Nuclear Radius Equation (AQA A Level Physics): Revision Note

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, A 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 A^{1 / 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 A^{1 / 3})

ln R = ln R₀ + ln (A^{1 / 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