Students will understand that fluctuations in the measured count rate provide direct evidence for the random (stochastic) nature of radioactive decay.
Key Concepts
Radioactive nuclei decay spontaneously and independently.
The probability that a given nucleus decays in a short time interval \$dt\$ is proportional to \$dt\$.
The number of decays recorded in a fixed time interval follows a Poisson distribution.
Statistical fluctuations become relatively smaller as the count rate increases.
Mathematical Description
The activity \$A(t)\$ of a sample containing \$N(t)\$ undecayed nuclei is
\$\$
A(t)=\lambda N(t)
\$\$
where \$\lambda\$ is the decay constant. The number of nuclei obeys the exponential law
\$\$
N(t)=N_0 e^{-\lambda t}
\$\$
and the expected number of counts \$ \langle n \rangle \$ in a measuring interval \$\Delta t\$ is
\$\$
\langle n \rangle = A(t)\,\Delta t = \lambda N(t)\,\Delta t .
\$\$
Statistical Fluctuations
For a Poisson process the variance equals the mean:
\$\$
\sigma^2 = \langle n \rangle ,\qquad \sigma = \sqrt{\langle n \rangle } .
\$\$
Consequently the relative standard deviation is
\$\$
\frac{\sigma}{\langle n \rangle } = \frac{1}{\sqrt{\langle n \rangle }} .
\$\$
This relationship predicts larger fractional fluctuations for low count rates and smaller fractional fluctuations for high count rates.
Experimental Demonstration
Set up a Geiger‑Müller tube connected to a counter.
Place a weak radioactive source at a fixed distance from the detector.
Record the number of counts \$n_i\$ in successive equal time intervals (e.g., 10 s each) for a total of \$N\$ intervals.
Calculate the mean count \$\langle n \rangle = \frac{1}{N}\sum{i=1}^{N} ni\$ and the standard deviation \$\sigma\$.
Compare the measured \$\sigma\$ with the Poisson prediction \$\sqrt{\langle n \rangle }\$.
Sample Data Table
Interval (s)
Counts \$n_i\$
1
12
2
9
3
15
4
11
5
13
6
10
7
14
8
12
9
8
10
13
For the data above, \$\langle n \rangle = 11.7\$ and \$\sigma{\text{meas}} \approx 2.1\$, while the Poisson prediction gives \$\sigma{\text{Poisson}} = \sqrt{11.7} \approx 3.4\$. The discrepancy can be discussed in terms of detector efficiency, background radiation, and finite counting time.
Interpretation
The random nature of decay means that each nucleus has a fixed probability per unit time to decay, independent of the behaviour of other nuclei.
The observed spread of counts around the mean is not due to experimental error alone; it is an intrinsic statistical property of the decay process.
Increasing the counting time or using a stronger source reduces the relative fluctuation, illustrating the \$1/\sqrt{N}\$ dependence.
Suggested Classroom Activity
Students simulate radioactive decay using a computer program that generates random numbers. By varying the number of simulated nuclei, they can observe how the distribution of counts approaches a smooth exponential curve as the sample size grows, reinforcing the connection between randomness and macroscopic regularities.
Suggested diagram: A schematic of a Geiger‑Müller tube with a radioactive source, showing the detection of individual decay events as discrete pulses on a counter display.
Key Take‑aways
Radioactive decay is a fundamentally random process.
Count‑rate fluctuations follow a Poisson distribution, with variance equal to the mean.
Statistical analysis of count data provides quantitative evidence for the stochastic nature of decay.
Understanding these fluctuations is essential for interpreting experimental results and for applications such as radiation safety and nuclear medicine.