Physics and Engineering of Radiation Detection

1.3. Radioactivity and Radioactive Decay 13

What this equation essentially implies is that the experimental determination of the
decay constantλis independent of the efficiency of the detection system, although
the counts observed in the experiment will always be less than the actual decays.
To see how the experimental values are used to determine the decay constant, let us
rewrite equations 1.3.16 and 1.3.19 as

ln(A)=−λt+ln(A 0 ) and (1.3.20)

ln(C)=−λt+ln(C 0 ). (1.3.21)

Hence if we plotCversuston a semilogarithmic graph, we should get a straight
line with a slope equal to−λ. Figure 1.3.2 depicts the result of such an experiment.
The predicted activity has also been plotted on the same graph using equation
1.3.20. The difference between the two lines depends on the efficiency, resolution,
and accuracy of the detector.

Time

ln(C or A)

ln(C)

Regression line

ln(A)

Slope = −λd

Figure 1.3.2: Experimental determination of decay constant.

Equation 1.3.13 can be used to estimate the average time a nucleus would take before
it decays. This quantity is generally referred to as the “lifetime” or “mean life” and
denoted byτorT. In this book it will be denoted by the symbolτ. The mean life
can be calculated by using

τ=

1

λd

. (1.3.22)

Another parameter, which is extensively quoted and used, is thehalf life.Itis
defined as the time required by half of the nuclei in a sample to decay. It is given
by

T 1 / 2 =0. 693 τ=

ln(2)

λd

. (1.3.23)

Since mean and half lives depend on the decay constant, therefore the experimental
procedure to determine the decay constant can be used to find these quantities as