Radioactive decay

Radioactive decay is the random, spontaneous process in which an unstable nucleus gives out radiation (an alpha particle, a beta particle, gamma rays or a neutron) and becomes more stable. You cannot predict when one nucleus will decay, but a large sample decays at a steady, predictable rate. That rate is measured by its activity, in becquerels, and by its half-life.

What is radioactive decay?

Most nuclei are stable and stay the same forever. Some have the wrong balance of protons and neutrons, are too large, or have too much energy. These unstable nuclei decay: they emit radiation and turn into a different nucleus, called the daughter nucleus, or into a lower energy state of the same one. Isotopes that do this are called radioisotopes or radionuclides.

Types of radioactive decay

Nuclear equations, penetration and ionising power for each type are in alpha, beta and gamma decay.

Radioactive decay is random

Radioactive decay is random: every nucleus of a given isotope has the same chance of decaying in the next second, but there is no way to tell which one will go next or when. It is also spontaneous: heating, cooling, pressure or chemical reactions do not change the rate.

It is like rolling a large number of dice and removing every six. You cannot say which dice will show a six, but you can be sure that about a sixth of them will each roll. This is why a count rate jumps up and down from one reading to the next, and why scientists take several readings and average them.

Activity and count rate

The activity of a source tells you how many of its nuclei decay per second. It is measured in becquerels (Bq), where one becquerel means one decay every second. A detector such as a Geiger-Müller tube cannot catch every decay, so what it measures is the count rate: the number of decays it records each second (or minute).

Background radiation

A detector picks up some radiation even with no source nearby. This background radiation comes from natural sources such as rocks (including radon gas) and cosmic rays from space, and from man-made sources such as the fallout from nuclear weapons testing and nuclear accidents. Before working with a count rate, measure the background with the source removed and subtract it from every reading.

Half-life

The half-life of a radioactive isotope is how long it takes for half of the unstable nuclei in a sample to decay, or for the count rate or activity to drop to half of its starting value. Each isotope has its own half-life, from fractions of a second to billions of years, and it does not change as the sample gets older.

Example 1: correcting for background (GCSE)

A source gives 250 counts per minute. The background count is 30 counts per minute. Two hours later the reading is 85 counts per minute. Find the half-life.

• Corrected count rates: 250 − 30 = 220 at the start, and 85 − 30 = 55 after two hours.
• 220 → 110 → 55 is two halvings, so two half-lives have passed in two hours.
• The half-life is 1 hour.

Here x is the time in hours and y is the corrected count rate in counts per minute. To read a half-life from a decay curve, pick any value on the y axis, find where the curve falls to half of it, and read the time between the two. It is the same wherever you start. Without the background correction you would compare 250 and 85, which is not a whole number of halvings, and get the wrong answer.

Try a half-life sum from the Nuclear Energy course, then press Mark.

The radioactive decay equation (A-level)

At A-level, randomness is turned into equations. The decay constant, λ, is the probability that a given nucleus decays per unit time. In a sample of N nuclei, the number decaying per second is proportional to N:

The activity is the number of decays per second, so:

Solving the first equation gives exponential decay, for the number of nuclei and for the activity:

After one half-life N = ½N₀, which gives the link between half-life and decay constant:

These are all in AQA A-level Physics section 3.8.1.3, along with finding the half-life from decay curves and log graphs. Make sure λ and t are in matching units: λ in s⁻¹ needs t in seconds.

Example 2: a medical tracer

Technetium-99m gives out the gamma rays used in many hospital scans, and its half-life is 6.0 hours. A dose is prepared with an activity of 600 MBq. Work out the decay constant, how many technetium nuclei the dose contains, and the activity 15 hours after it was prepared.

Check: 15 hours is 2.5 half-lives, and 600 × 0.5^2.5 = 106 MBq. After 24 hours, exactly four half-lives, only 600 ÷ 2⁴ = 37.5 MBq is left. The short half-life is the point: the dose gives out enough gamma rays for the scan and then fades quickly.

Example 3: radiocarbon dating

A piece of old wood has 40% of the carbon-14 activity of living wood. Carbon-14 has a half-life of 5,730 years. How old is it?

That gives t = 7,570 years. A useful check: 40% is between 50% (one half-life, 5,730 years) and 25% (two half-lives, 11,460 years), so the answer must be between those.

Example 4: using the molar mass

Find the activity of 1.0 μg of iodine-131 (molar mass 131 g mol⁻¹, half-life 8.02 days).

• Number of nuclei: N = (1.0 × 10⁻⁶ ÷ 131) × 6.02 × 10²³ = 4.60 × 10¹⁵.
• Decay constant: λ = ln 2 ÷ (8.02 × 24 × 3600) = 1.00 × 10⁻⁶ s⁻¹.
• Activity: A = λN = 4.60 × 10⁹ Bq, from just a millionth of a gram.

Example 5: the half-life from a log graph

Taking logs of A = A₀e^(−λt) gives ln A = ln A₀ − λt, a straight line with gradient −λ. If ln A falls from 10.0 to 7.2 in 4.0 hours, the gradient is −2.8 ÷ 4.0 = −0.70 h⁻¹, so λ = 0.70 h⁻¹ and the half-life is ln 2 ÷ 0.70 = 0.99 hours. A straight line also confirms that the decay is exponential.

Check your understanding.

Common mistakes

• Forgetting background radiation. Subtract it from every count rate before you look for halvings.
• Thinking a sample is gone after two half-lives. Half goes in the first half-life, half of what is left in the next: a quarter remains.
• Mixing units. If λ is in s⁻¹, t must be in seconds and A comes out in Bq.
• Confusing activity with count rate. Activity counts every decay; count rate is only what the detector records.
• Saying temperature changes the rate. Radioactive decay is not affected by physical or chemical conditions.
• Dropping the minus sign in N = N₀e^(−λt), which would make the sample grow.

For quick answers you can check, use the Half-life Calculator, which finds the amount left, the half-life or the time from any two of them, and converts between half-life and decay constant. The half-life calculator guide has more worked examples.

Practise with a short timed paper

Radioactive decay comes up in GCSE Physics and in the second year of A-level Physics. The daily GCSE Physics Higher and A-level Physics: Year 2 papers are short timed papers of 8 questions, with a new one every day.

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Physics · Atoms and particlesHalf-life calculator: amount left, half-life and timeFree half-life calculator: find the amount left after any time, the half-life from your data, or the time to decay, with every step and a decay curve.