Radioactive half-life

The half-life of an isotope is the time taken for half the nuclei of that isotope in a sample to decay. It is characteristic of the isotope: changing the initial number of nuclei does not change its half-life.

After one half-life, half the original nuclei remain undecayed. During the next half-life, half of those remaining decay, so one quarter of the original nuclei remain. After a third half-life, one eighth remain. The same fraction of the remaining nuclei decays in each half-life, not the same number.

Calculating nuclei remaining

After $$n$$ whole half-lives, the number of undecayed nuclei is:

Here, $$N_0$$ is the initial number of nuclei and $$N$$ is the number remaining. The number that have decayed is $$N_0-N$$.

For example, suppose a sample initially contains $$800$$ nuclei and its half-life is $$4$$ hours. After $$12$$ hours, three half-lives have passed. The number remaining is $$800\times(\frac12)^3=100$$, so $$800-100=700$$ nuclei have decayed.

Conversely, if three half-lives have passed and each half-life is $$4$$ hours, the elapsed time is $$3\times4=12$$ hours.

Finding half-life from a table or decay curve

Choose a number of undecayed nuclei and find the time at which it occurs. Then find the time at which the number has fallen to half that value. The difference between the two times is the half-life.

For example, if a decay curve shows $$600$$ undecayed nuclei at $$2$$ hours and $$300$$ at $$6$$ hours, the half-life is $$6-2=4$$ hours. The chosen starting value need not be the initial number of nuclei.

Half-life with background radiation

A detector placed near a radioactive source records radiation from both the source and the background. To find how quickly the source’s count rate falls, the background contribution must first be removed.

Measure the background count rate separately, using the same units as the readings taken near the source. Subtract it from each measured count rate:

If the detector records counts over equal time intervals, the background count for the same interval can be subtracted from each measured count instead. The corrected counts can then be compared directly.

Half-life from corrected readings

The half-life is the time taken for the corrected source count rate to fall to half its value. The total detector reading does not have to halve over this time, because it still includes the background.

For example, suppose the background count rate is $$20$$ counts per minute. A measured rate of $$100$$ counts per minute gives a corrected source rate of $$80$$ counts per minute. If the measured rate is $$60$$ counts per minute five minutes later, the corrected source rate is $$40$$ counts per minute. The source rate has halved from $$80$$ to $$40$$, so its half-life is five minutes.

Half-life from a decay curve containing background

On a decay curve that shows total detector readings, subtract the background level from the readings before looking for a halving. Choose two corrected source count rates in a $$2:1$$ ratio, then read the time interval between them from the curve. That interval is the half-life.

Ionisation smoke alarms

An ionisation smoke alarm detects smoke by monitoring a small electric current through the air. Inside the alarm, a radioactive source emits alpha radiation into the air between two electrodes.

The alpha particles ionise air molecules: they knock electrons off the molecules, producing positive ions and free electrons. A voltage across the electrodes causes these charged particles to move, producing a small current.

When smoke enters the alarm, smoke particles absorb some alpha radiation and interfere with the ionisation of the air. Fewer charged particles are available to carry the current, so the current falls. The alarm detects this fall and sounds.

Why alpha radiation is suitable

Alpha radiation ionises air effectively, allowing the alarm to produce a measurable current. It also has low penetration, so it is largely contained within the alarm.

The isotope used has a long half-life, meaning its activity decreases slowly. The source therefore remains useful throughout the working life of the alarm.

Gamma irradiation for food and equipment

Gamma radiation can kill bacteria and other microorganisms by damaging their cells. Food can be exposed to gamma radiation to reduce the number of microorganisms it contains. Gamma radiation can also be used to sterilise equipment, killing microorganisms on it.

Gamma radiation is highly penetrating, so it can pass through packaging and reach the contents. Equipment can therefore be sterilised while sealed inside its packaging.

A gamma source with a long half-life takes a long time for its activity to fall substantially. It remains useful for an extended period and does not need frequent replacement.

Irradiation means exposure to radiation. Exposing food or equipment to gamma radiation does not make it radioactive.

Radioactive thickness control

A radioactive source and a detector can monitor the thickness of a moving sheet without touching it. The source is placed on one side of the sheet and the detector on the other. Some radiation passes through the sheet and is detected; some is absorbed by the material.

The detector measures the count rate: the number of radiation counts it records per unit time. A thicker sheet absorbs more radiation, so the detector count rate falls. A thinner sheet absorbs less, so the count rate rises.

Controlling the sheet thickness

The detector signal can be compared with the count rate expected for the required thickness. It can then control rollers that adjust the sheet:

• If the count rate is too low, the sheet is too thick. The rollers are adjusted closer together to make it thinner.
• If the count rate is too high, the sheet is too thin. The rollers are adjusted farther apart to make it thicker.

As the thickness changes, the count rate changes too, allowing the control system to keep the sheet close to the required thickness.

Choosing the radiation source

The radiation must be suitably penetrating. Enough must pass through the sheet to reach the detector, but enough must also be absorbed for a change in thickness to produce a detectable change in count rate. Beta radiation can be suitable for thin sheets; more penetrating gamma radiation can be used for thicker materials.

A source with a long half-life is used so that its activity changes relatively little over the period of use. This helps keep changes in detector count rate attributable to changes in sheet thickness rather than a rapid fall in source activity.

Gamma tracers in cancer diagnosis

A radioactive tracer is a substance containing a radioactive isotope that can be introduced into the body. The tracer is chosen so that it is taken up by particular tissues. As the isotope decays, it emits gamma radiation from those tissues.

Gamma rays can pass out of the body and be detected by a device outside it. The detected radiation is used to form an image showing where the tracer has collected. Unusual patterns of uptake can help locate abnormalities, such as cancers.

Gamma tracers: Choice of radiation and half-life

Gamma radiation is suitable because enough of it can escape the body to be detected externally. This allows the distribution of the tracer to be imaged without placing a detector inside the body.

The tracer should have a relatively short half-life so that its activity falls and the patient is not exposed to radiation for longer than necessary. However, the half-life must be long enough for the tracer to reach the target tissue and for the measurement to be completed while enough radiation is still being emitted.