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:

$R=R_0{A}^{\frac{1}{3}}$

• Where:

  • $R$ = nuclear radius (m)

  • $A$ = nucleon / mass number

  • $R_0$ = constant of proportionality = 1.05 fm

• Plotting a graph of $R$ against $A^{\frac{1}{3}}$ gives a straight line through the origin with the gradient equal to $R_0$

• It is also possible to plot a logarithmic graph of the relationship which can be derived as follows:

$\ln \left(R\right)=\ln \left(R_0{A}^{\frac{1}{3}}\right)$

$\ln \left(R\right)=\ln \left(R_0\right)+\ln \left(A^{\frac{1}{3}}\right)$

$\ln \left(R\right)=\ln \left(R_0\right)+\frac{1}{3}\ln \left(A\right)$

• Therefore, a graph of $\ln R$ against $\ln A$ yields a straight line

• Comparing this to the straight-line equation: $y=mx+c$

  • $y=\ln \left(R\right)$

  • $x=\ln \left(A\right)$

  • Gradient $m=\frac{1}{3}$

  • y-intercept, $c=\ln \left(R_0\right)$