Why nobody can predict one atom, but everybody can predict a million: radioactive decay

How to use this simulation

1. Start in “Nuclei decaying” with the default half-life of 3 s. The grid holds 240 nuclei, all blue. Each tick, every blue nucleus rolls the dice once, and the unlucky ones flash and turn into a faint ring.
2. Watch the amber line on the graph. It drops quickly at first, because there are many nuclei to decay, and slows down as the sample thins out. The grey dashed curve is the formula. The amber line wiggles around it, since decay is random.
3. Find the green markers. After 3 s (one half-life) the count is about 120, after 6 s about 60, after 9 s about 30. Each half-life halves whatever is left.
4. Read the activity A = λN in the top-right of the graph. It falls in step with the count, because fewer nuclei means fewer decays per second.
5. Move the half-life slider to a larger value. The run restarts and the curve stretches out: λ is smaller, so decay is slower. Try 6 s and see how long it takes to reach 25% of the nuclei.
6. Switch the view to “Penetration”. Alpha particles die at the paper, beta particles at the aluminium, and gamma rays sail through both. Drag the lead thickness slider and watch the share of gamma rays that survive fall, but never quite reach zero.

Random for one atom, predictable for many

Take a single carbon-14 atom. Can you say when it will decay? No. Not you, not a physicist, not the best computer. It could be the next second or in ten thousand years. The nucleus does not “age” the way a fruit does: an old unstable nucleus is exactly as likely to go off in the next second as a brand new one.

That sounds like the end of prediction, but it is the start of it. If every nucleus has the same small chance of decaying in each second, then a big sample behaves like a huge crowd tossing coins. You cannot say what any one coin will do, but you can say very well that about half of them will show heads. In our grid of 240 nuclei the same thing happens: the exact nucleus that goes next is random, but the total follows a smooth curve.

This is the heart of radioactive decay, and it is also why we talk about the half-life instead of the lifetime of an atom. The lifetime of one atom is unknowable. The time for half of a big sample to go is a firm, measurable number.

Half-life: the time to lose half

The half-life T½ is the time it takes for half of the undecayed nuclei in a sample to decay. After one half-life half remain. After two, half of that half: a quarter. After three, an eighth. The fraction left is always (½) raised to the number of half-lives.

The important thing is that this does not depend on how many nuclei you started with. With the default half-life of 3 s, a sample of 240 drops to 120 after 3 s, and a sample of 120 drops to half of that, in the same 3 s. A half-life is a property of the isotope, like its melting point is for a metal.

Half-lives run from absurdly short to absurdly long. Some nuclei made in accelerators last a tiny fraction of a second. Iodine-131, used in medicine, has a half-life of about eight days. Carbon-14 has 5,730 years. Uranium-238 has about 4.5 billion years, roughly the age of the Earth.

The decay law and the decay constant

Half-lives are friendly for round numbers, but the full law works for any time. The chance that a given nucleus decays each second is called the decay constant λ. Because the number of decays per second is proportional to the number of nuclei still there, the count obeys an exponential law, the same shape as a cooling cup of tea or a capacitor discharging.

The decay constant and the half-life are two names for the same speed: λ = ln 2 ÷ T½. With T½ = 3 s, λ = 0.693 ÷ 3 = 0.231 s⁻¹. That means each nucleus has about a 0.231 chance per second of decaying, which is why the early part of the graph is steep.

There is a third number, the mean life 1 ÷ λ = 4.33 s. It is the average time a nucleus survives, and it is longer than the half-life by a factor 1 ÷ ln 2 ≈ 1.44. After one mean life the count has fallen to 36.8% of its start, the same 1/e rule you meet in capacitor charging.

Activity: how fast the sample is “going off”

Activity A is the number of decays per second in the whole sample. It is measured in becquerel (Bq), where 1 Bq is one decay per second. A Geiger counter clicks at a rate that follows the activity.

Since each of the N remaining nuclei has a chance λ per second, the activity is A = λN. At the start of the default run that is 0.231 × 240 = 55.5 Bq. Because N falls exponentially, A falls with exactly the same curve and the same half-life. In the simulation, the activity readout halves each time the count halves.

Why the amber line wiggles

The amber line does not hug the dashed curve exactly. That is not a bug, it is the physics. The number of decays in any interval is random, and the size of the random scatter is about √N for a count of N. The relative scatter is therefore 1 ÷ √N.

With all 240 nuclei the relative scatter is about 6.5%. After three half-lives only 30 remain and the scatter grows to about 18.3%. The smaller the sample, the rougher the curve.

This is why real experiments count for a long time or use many atoms. A lab count of 1,200 has a scatter of about 2.9%. A count of 300 has 5.8%. To get the scatter down to 0.5% you need 40,000 counts. Keep this in mind whenever an exam asks about the accuracy of a half-life measurement.

Worked example 1: a hospital isotope

A hospital receives a source with half-life 8 days and activity 400 MBq. What is its activity after 24 days, and when does it fall to 10% of the start?

Worked example 2: measuring a half-life from counts

A student records a source with a Geiger counter. At t = 0 the counter reads 1,220 counts per minute. 30 minutes later it reads 320 counts per minute. With the source removed, the background is 20 counts per minute. Find the half-life.

Worked example 3: dating a bone with carbon-14

Living things take in carbon-14 from the air, and stop when they die. The carbon-14 then decays with a half-life of 5,730 years. A bone has 30% of the carbon-14 of a living bone. How old is it?

The number of half-lives is n = ln(100 ÷ 30) ÷ ln 2 = 1.204 ÷ 0.693 = 1.737. So the age is 1.737 × 5,730 = about 9,953 years.

Using the decay constant instead: λ = ln 2 ÷ T½ = 1.21 × 10⁻⁴ per year, and t = ln(N₀/N) ÷ λ gives the same age. Both routes must agree, which is a good self-check in an exam.

Alpha, beta and gamma: what they are

Unstable nuclei release three kinds of radiation, and they behave very differently. The chemistry side of this, how the nucleus changes and how to balance the nuclear equation, is covered in the Radioactivity animation in the Chemistry section. This page keeps to the physics of how far each kind travels.

• Alpha (α): a helium nucleus, two protons and two neutrons. Heavy, slow, and with a charge of +2. It loses energy extremely fast, so it travels only a few centimetres in air.
• Beta (β): a fast electron (or positron) from inside the nucleus. Light, fast, and charge −1. It travels around a metre in air.
• Gamma (γ): high-energy electromagnetic radiation. No mass, no charge. It is never stopped completely, only weakened.

Penetration and ionising power

Switch the simulation to “Penetration”. A sheet of paper stops every alpha particle. A few millimetres of aluminium stops beta particles but lets gamma through. A thick block of lead or concrete is needed to cut gamma rays down.

Stopping power and ionising power go hand in hand. Ionising means knocking electrons off atoms. Every time a particle ionises an atom, it loses some energy. Alpha is heavy and charged, so it ionises the most and runs out of energy first. Gamma ionises the least, so it carries on the furthest. The rough textbook ratio of ionising power is α : β : γ ≈ 10,000 : 100 : 1, and the bars in the simulation use a log scale so all three fit.

This explains a surprising safety fact. An alpha source outside the body is nearly harmless, because dead skin stops it. Swallowed or breathed in, the same source sits next to living cells and deposits all its energy there, which makes it the most dangerous of the three. Gamma is the problem when the source is outside and you cannot easily shield it.

Shielding gamma rays: half-value layers

Gamma rays do not have a fixed range. Each layer of material removes a fixed fraction, so the intensity falls exponentially with thickness, the same pattern as the decay law. The thickness that cuts the intensity to half is the half-value layer. For lead and a typical gamma ray from cobalt-60 it is about 1.2 cm.

With the default lead thickness of 3 cm that is 2.5 half-value layers, so 17.7% of the gamma rays get through. In the simulation, notice that a few gamma rays still cross the lead, even when it is thick. You can always reduce gamma, never block it completely.

Worked example 4: how much lead is enough?

Using the half-value layer of 1.2 cm, find how much lead lets only 1% of a gamma beam through. Check what 3.6 cm does.

Try this in the simulation

Set the half-life to 6 s. The decay constant drops to 0.116 s⁻¹, and the time to reach 25% of the nuclei (60 of them) becomes 6 × 2 = 12 s. Doubling the half-life doubles every time on the graph, but the shape stays the same.

Then let a run restart a few times. Each run is a little different, but every one follows the same dashed curve. This is the difference between a single random event and the law of large numbers.

Common mistakes

These cost marks every year.

• Thinking that two half-lives means everything has gone. After two half-lives a quarter remains.
• Forgetting to subtract the background count before finding the half-life.
• Saying the half-life changes with the amount or the temperature. It does not.
• Using the mean life 1/λ instead of the half-life. They differ by the factor 1.44.
• Saying lead “stops” gamma rays. It reduces them. Alpha is stopped, gamma is attenuated.
• Mixing the units of λ and t. If λ is per second, t must be in seconds.

Revision list

Say these aloud before the exam.

• Decay is random for one nucleus and predictable for many.
• N = N₀e^(−λt) = N₀(½)^(t/T½).
• λ = ln 2 ÷ T½ = 0.693 ÷ T½, and mean life = 1/λ.
• Activity A = λN, in becquerel (decays per second); it halves every half-life too.
• Half-life does not depend on the sample size or on conditions.
• α stopped by paper, β by a few mm of aluminium, γ only reduced by thick lead.
• Ionising power: α highest, γ lowest. Range is the opposite order.

Frequently asked questions