Radioactive Decay — Question 4

Question 4

Integrated counts recover the activity at the start of a measurement

A source has half-life 60.0 s. During a 60.0 s acquisition a detector records 3000 signal counts with constant efficiency .250 and negligible dead time or background. Each decay can contribute one count.

1. Derive the relation between integrated counts and initial activity.

2. Find the initial activity and compare it with the measured mean activity inferred from counts per time.

Question 4 - Solution

Integrated counts recover the activity at the start of a measurement

(a) Method and calculation

$$C=\varepsilon\int_0^T\mathcal A_0e^{-\lambda t}dt=\varepsilon\mathcal A_0(1-e^{-\lambda T})/\lambda.$$$$\boxed{\mathcal A_0=C\lambda/[\varepsilon(1-e^{-\lambda T})]}.$$

(b) Interpretation and further calculation

$$\boxed{\mathcal A_0=3000(\ln2/60)/(.250\times.5)=277.3\,\mathrm{Bq}}.$$$$\boxed{\overline{\mathcal A}=C/(\varepsilon T)=200\,\mathrm{Bq},\quad\overline{\mathcal A}/\mathcal A_0=1/(2\ln2)=.7213}.$$

Physical check

Dividing counts by acquisition time returns an interval average. A rapidly decaying source requires the integral to recover its activity at a specified instant.