# 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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