8.1.7 Required Practical: Inverse Square-Law for Gamma Radiation

Aim of the Experiment

The aim of this experiment is to verify the inverse square law for gamma radiation of a known gamma-emitting source

Variables

• Independent variable = the distance between the source and detector, x (m)
• Dependent variable = the count rate / activity of the source, C
• Control variables
  • The time interval of each measurement
  • The same thickness of aluminium foil
  • The same gamma source

Equipment List

• Resolution of equipment:
  • Metre ruler = 1 mm
  • Stopwatch = 0.01 s

Method

Set up for inverse-square law investigation

1. Measure the background radiation using a Geiger Muller tube without the gamma source in the room, take several readings and find an average
2. Next, put the gamma source at a set starting distance (e.g. 5 cm) from the GM tube and measure the number of counts in 60 seconds
3. Record 3 measurements for each distance and take an average
4. Repeat this for several distances going up in 5 cm intervals

• A suitable table of results might look like this:

Analysing the Results

• According to the inverse square law, the intensity, I, of the γ radiation from a point source depends on the distance, x, from the source

• Intensity is proportional to the corrected count rate, C, so

• Comparing this to the equation of a straight line, y = mx
  • y = C (counts min^–1)
  • x = 1/x² (m^–2)
  • Gradient = constant, k

1. Square each of the distances and subtract the background radiation from each count rate reading
2. Plot a graph of the corrected count rate per minute against 1/x²
3. If it is a straight line graph through the origin, this shows they are directly proportional, and the inverse square relationship is confirmed

A straight-line graph verifies the inverse square relationship. The closer the points are to the line, the better the experiment has demonstrated the relationship

Evaluating the Experiment

Systematic errors:

• The Geiger counter may suffer from an issue called “dead time”
  • This is when multiple counts happen simultaneously within ~100 μs and the counter only registers one
  • This is a more common problem in older detectors, so using a more modern Geiger counter should reduce this problem

• The source may not be a pure gamma emitter
  • To prevent any alpha or beta radiation being measured, the Geiger-Muller tube should be shielded with a sheet of 2–3 mm aluminium

Random errors:

• Radioactive decay is random, so repeat readings are vital in this experiment
• Measure the count over the longest time span possible
  • A larger count helps reduce the statistical percentage uncertainty inherent in smaller readings
  • This is because the percentage error is proportional to the inverse-square root of the count

Safety Considerations

• For the gamma source:
  • Reduce the exposure time by keeping it in a lead-lined box when not in use
  • Handle with long tongs
  • Do not point the source at anyone and keep a large distance (as activity reduces by an inverse square law)

• Safety clothing such as a lab coat, gloves and goggles must be worn

Step 1: Determine a mean value of background radiation

Step 2: Calculate C (corrected average count rate) and C^–1/2 

Step 3: Plot a graph of C^–1/2 against x and draw a line of best fit

• • The graph shows C^–1/2 is directly proportional to x, therefore, the data follows an inverse square law

Step 4: Determine the uncertainties in the readings

Step 5: Plot the error bars and draw a line of worst fit

Step 6: Calculate the uncertainty in the gradient