In physics, radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. Each nucleus has a certain probability of decaying in a given time interval, but we cannot predict exactly when a particular atom will decay.
The count rate is the number of decays detected per unit time, usually expressed as counts per second (cps). It is measured with a Geiger–Müller tube or scintillation detector.
Because each nucleus decays independently, the number of decays that occur in a short time interval is a random variable. This randomness leads to fluctuations in the observed count rate:
These fluctuations are best described by the Poisson distribution, which models the probability of a given number of events occurring in a fixed interval when events happen independently at a constant average rate.
Imagine you have a fair six‑sided die. Each roll is independent, and you cannot predict the outcome of a single roll. If you roll the die 100 times, you might get 17 sixes, 18 ones, etc. The distribution of results will fluctuate around the expected value (≈16.67 sixes). Similarly, the number of radioactive decays in a short time interval fluctuates around the mean count rate.
The probability of observing k decays in a time interval t is given by:
\$\$
P(k;\lambda) = \frac{e^{-\lambda}\lambda^k}{k!},
\$\$
where \$\lambda\$ is the expected number of decays in that interval (i.e., \$\lambda = \text{count rate} \times t\$).
Key point: The standard deviation of the count is \$\sqrt{\lambda}\$, so relative fluctuations decrease as the number of counts increases.
| Time (s) | Counts Recorded |
|---|---|
| 1 | 98 |
| 2 | 105 |
| 3 | 97 |
| 4 | 102 |
Because each nucleus decays independently, the count rate fluctuates in a predictable statistical way. These fluctuations are evidence that radioactive decay is a random process, not a deterministic one.
Tip 1: Remember that the average count rate is the decay constant times the number of undecayed nuclei. Use the formula \$N(t) = N_0 e^{-\lambda t}\$ when asked to model decay curves.
Tip 2: When asked about fluctuations, mention the Poisson distribution and that the standard deviation is \$\sqrt{\lambda}\$.
Tip 3: Use the dice analogy to explain why we cannot predict the exact time of a single decay but can predict the average behaviour over many atoms.
Good luck! 🚀