Radioactivity, half-life and radiation risk

Core ideas

The radiations

Ionisation and range are linked. Each ionisation takes energy from the particle, so the most strongly ionising radiation ($\alpha$) loses its energy fastest and has the shortest range.

Identifying a radiation. Measure the background count rate. Then place the source close to a GM tube and record the count rate with no absorber, with paper, with about 3 mm of aluminium and with a few cm of lead. Correct each reading for background. A big fall when paper is added means $\alpha$. A fall to background with aluminium means $\beta$. A count that persists through aluminium and is reduced (but not removed) by lead means $\gamma$.

Decay equations and the N–Z chart

Nucleon number $A$ and proton number $Z$ are both conserved, and so are charge and lepton number.

In $\beta^-$ decay a neutron becomes a proton (a down quark becomes an up quark). In $\beta^+$ decay and electron capture a proton becomes a neutron. Gamma emission changes neither $A$ nor $Z$: the nucleus drops from an excited state after an or decay. Technetium-99m is a long-lived (metastable) excited state that emits only .

On a graph of $N$ (vertical) against $Z$ (horizontal), stable nuclei lie on a band that follows $N = Z$ for light nuclei and bends towards $N > Z$ for heavy ones, because extra neutrons add strong-force attraction without adding repulsion.

Background radiation and the inverse-square law

Background comes from radon gas (the largest share in the UK), rocks and building materials, cosmic rays, food and drink, and medical procedures. The UK average effective dose is about 2.7 mSv per year. Always subtract the background count rate before doing anything else with a count rate.

Gamma rays are barely absorbed by air, so from a point source they spread over a sphere:

In the AQA required practical you measure the corrected count rate $C$ at several distances $d$. The source sits some unknown distance $d_0$ inside its holder, so the true distance is $d + d_0$ and

A graph of $1/\sqrt{C}$ against $d$ is a straight line of gradient $1/\sqrt{k}$. Its intercept on the axis is . The systematic error is found, not ignored.

Random decay and the decay constant

Decay is random: you cannot predict which nucleus will decay next or when, but each nucleus has a constant probability of decaying per unit time. Decay is spontaneous: it is not affected by external conditions such as temperature, pressure or chemical bonding.

The decay constant $\lambda$ is the probability that a given nucleus decays per unit time (unit $\text{s}^{-1}$). For a large number of nuclei, the rate of decay is proportional to the number left:

Activity $A$ is measured in becquerels ($1\ \text{Bq} = 1$ decay per second). A count rate is always less than the activity, because radiation leaves in all directions and the detector is not 100% efficient. Because decay is random, a count of $N$ has an uncertainty of about $\sqrt{N}$, so longer counts give smaller percentage uncertainties (400 counts gives about 5%).

Exponential decay and its graphs

After $n$ half-lives, the fraction remaining is $(1/2)^n$. The same shape applies to $N$, $A$, mass and corrected count rate.

Taking logs gives $\ln A = \ln A_0 - \lambda t$. A graph of $\ln A$ against $t$ is a straight line with and intercept . If a question uses , the gradient is .

Iterative modelling

The iterative model (central on OCR B) steps forward in small time intervals:

Each step multiplies $N$ by $(1 - \lambda\Delta t)$. This matches the exponential only when $\lambda\Delta t \ll 1$. With larger steps the model decays too quickly. The same point explains the dice analogy: if a “decay” is throwing a six, the probability per throw is $1/6$, and the number left after throws is .

Choosing an isotope for a job

Dose, risk and safe handling

Absorbed dose $D = E/m$ is measured in grays ($1\ \text{Gy} = 1\ \text{J kg}^{-1}$). Effective (equivalent) dose = absorbed dose × radiation weighting factor (20 for $$, 1 for and ), in sieverts. The risk of a fatal cancer is roughly 5% per sievert. An source is harmless outside the body but most dangerous inside it. Safe handling means (minimise exposure), (use tongs, because the inverse-square law works in your favour) and (lead-lined storage, never pointing a source at anyone).

Worked examples

Exam technique

• Correct for background first. Subtract it from every count rate before halving, dividing or taking logs.
• Check units before using $A = \lambda N$. If $A$ is in Bq, $\lambda$ must be in $\text{s}^{-1}$. Convert the half-life to seconds at the start.

Common mistakes

Quick recap

• $\alpha$: strongly ionising, a few cm of air, stopped by paper. $\beta^-$: moderate, stopped by a few mm of aluminium. $\gamma$: weakly ionising, obeys $I = k/x^2$, reduced by lead.