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Physics

7.8 - Radioactive Decay & Half-Life

11 min

Activity and half-life

Why activity falls

Half-life from a graph

Measuring activity

A half-life calculation

note 1 of 8

Count-rate and activity

Radioactive substances emit radiation from the nuclei of their atoms.

Count-rate

The count-rate is the number of radiation counts picked up by a detector each second.
A common detector is a Geiger-Muller tube, connected to a counter that displays the number of counts.
Photographic film can also detect radiation: it becomes darker the more radiation it's exposed to.

Activity

The activity is the rate at which a source decays.
It's measured in becquerels (Bq), where 1 Bq is 1 decay (one nuclear disintegration) per second.
A larger radioactive sample has a higher activity, because it contains more radioactive nuclei that can decay.

note 2 of 8

Radioactive decay and half-life

Radioactive decay is random and spontaneous:

Random - It's impossible to predict which nucleus will decay next, or exactly when any one nucleus will decay.
Spontaneous - The rate of decay isn't affected by physical conditions like temperature, pressure or chemical bonding.

Why activity falls

Each time a nucleus decays it becomes more stable, so the number of undecayed radioactive nuclei steadily falls.
With fewer radioactive nuclei left, the activity (and count-rate) of the sample decreases over time.

Half-life

The half-life is the time taken for the number of radioactive nuclei in a sample to halve.
This is also the time taken for the activity (or count-rate) of a sample to halve.
Each isotope has its own fixed, characteristic half-life.

Three grids of circles joined by arrows marked one half-life, the purple ones labelled radioactive nuclei and trailing purple squiggles, with fewer purple and more grey circles in each successive grid. — Purple nuclei are still radioactive; grey nuclei have already decayed.

Half-life and hazard

Short half-life - Activity falls quickly: high radiation at the start, but the source soon becomes safe.
Long half-life - Activity falls slowly: smaller amounts of radiation released, but for a very long time, so nearby areas stay exposed.

note 3 of 8

Background radiation and correcting the count rate

Background radiation is the low-level ionising radiation that's around us all the time.

Natural sources

Radon gas and radioactive elements in rocks, soil, air, food and building materials.
Cosmic rays - Radiation reaching Earth from space.

Man-made sources

Fallout from nuclear weapons testing.
Medical uses (e.g. X-rays), nuclear power accidents, and radioactive waste.

Correcting for background radiation

Measure the background count rate with no source present.
Measure the count rate with the source in place.
Subtract the background count rate to get the corrected count rate from the source alone.

note 4 of 8

Finding half-life from a decay graph

A graph of activity (or count-rate, or number of radioactive nuclei) against time curves downwards, becoming less steep as time goes on.

Reading the half-life off the curve

Read the initial value off the vertical axis.
Work out half of this value.
Find the time on the horizontal axis at which the curve reaches that halved value - that time is one half-life.
Check by halving again: each successive halving should take the same length of time.

A curve falling steeply then levelling off on axes labelled activity (Bq) and time (s), starting at 800 Bq, with dashed lines marking 2 s from 800 to 400 and 400 to 200.

Subtract the background count first

Before plotting, subtract the background count from every reading.
If it isn't removed, the calculated half-life will be incorrect.

note 5 of 8

The half-life formula for activity after n half-lives

$Activity = \frac{initial activity}{2^{n}}$

Where:

$n$ = The number of half-lives that have passed

The same relationship applies to count-rate and the number of undecayed nuclei.

note 6 of 8

Worked example - Activity after n half-lives

The initial activity of a radioactive source is 2,400 Bq. Calculate the activity after 5 half-lives.

Step 1 - Formula

$Activity = \frac{initial activity}{2^{n}}$

Step 2 - Substitution and calculation

There have been 5 half-lives, so substitute $n = 5$ :

$Activity = \frac{2,400}{2^{5}} = \frac{2,400}{32} = 75 Bq$

note 7 of 8

Worked example - Time to reach a fraction of activity

The half-life of a radioactive sample is 90 minutes. How long does it take for the activity to fall to one quarter of its initial value?

Step 1 - Count the half-lives needed

full → one half → one quarter

That's two halvings, so two half-lives are needed.

Step 2 - Multiply by the half-life

Time = 2 × 90 minutes = 180 minutes (3 hours)

note 8 of 8

How your background dose varies

The radiation dose measures the potential harm to the body from radiation exposure.
It's measured in sieverts (Sv) or millisieverts (mSv), where 1,000 mSv = 1 Sv.

What affects your dose

Location - Living where more radon gas seeps out of the ground raises the dose. Living at higher altitudes also increases dose because there's less atmosphere to absorb cosmic rays.
Occupation - Radiographers and nuclear power workers spend more time near radioactive sources. Aircrew receive higher doses from spending time at high altitude.

key terms

TermDefinition

Count rateThe number of decays recorded by a detector in a given time, usually each second.

ActivityThe rate at which a source of unstable nuclei decays, measured in becquerels (Bq).

Half-lifeThe time taken for the number of radioactive nuclei in a sample to halve, or for the activity (count rate) of the sample to fall to half its original value.

Background radiationThe low-level ionising radiation that is around us all the time.

Radiation doseA measure of the risk of harm to the body from exposure to radiation.

end of notes

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