• 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

   

• 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 (R0A1 / 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

   

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