The inverse-square law for gamma radiation

What you are trying to do

Show that the intensity of gamma radiation falls with the inverse square of distance from the source, correcting for background radiation.

Apparatus

• A sealed gamma source in its lead castle, handled only by the teacher with long tongs
• Geiger-Müller tube and counter, on a rule so the tube-to-source distance can be set
• A metre rule; a stopclock if the counter lacks a timer

Variables

• Independent: the measured distance x from source to tube window
• Dependent: the corrected count rate
• Control: the same source and tube, counting long enough at each distance for good statistics

Method

1. With the source locked away, measure the background count over several minutes, more than once, and average to a background rate.
2. With the source in place, count at each of a series of distances. Decay is random, so longer counts mean smaller fractional statistical error: count for minutes, not seconds, especially at large x where the rate is low.

1. Subtract the background from every reading before any analysis.

Analysis

1. The true distance from the radiating material to the tube's sensitive volume is x plus an unknown offset: source and detector both hide their reference points inside casings.
2. So do not plot C against 1/x². Instead use C = k/(x + x₀)², rearranged: 1/√C = (x + x₀)/√k. Plotting 1/√C against x gives a straight line whatever the offset; the gradient is 1/√k and the intercept reveals x₀.
3. A straight line here is the inverse-square law, tested without knowing either hidden reference point.

A worked set of readings

Counts converted to rates; background 25 counts per minute already subtracted in the third column:

1/√C against x is a straight line: the inverse-square law. The gradient is 0.1667, so k = 1/gradient² = 36 counts per minute at one metre equivalent, and the intercept places the hidden offset at x₀ = 2 cm inside the casings.

Where the uncertainty comes from

• Counting statistics: A count of N carries an uncertainty of about √N, so the fractional error shrinks as counts grow; this is why the long counts are not optional.
• Background drift: Background varies; measuring it before and after and averaging is the standard guard.
• The hidden offset: The biggest systematic, and the 1/√C plot is its cure; plotting against 1/x² with the raw x would curve the line and wreck the test.
• Dead time: At very high rates a GM tube misses counts; keeping the closest distance modest sidesteps it.

What earns the marks

Safety