Law of Radioactive Decay

In front of you is a sample of a thousand radioactive nuclei. Each one decays at a random moment: it flashes, emits an α or β particle and changes colour, while the Geiger counter clicks. There is no way to predict which nucleus decays next, yet in every half-life T almost exactly half of the nuclei decay. Compare the step graph of the experiment with the curve N = N₀·2^(−t/T), reduce the number of nuclei to ten and watch the law become “shaky”, and in the “Single nucleus” mode check that nuclei have no age.

t = 0 T

Every nucleus decays at a random moment: its lifetime is drawn from the 2^(−t/T) law, and there is no way to predict which nucleus decays next. Nuclear sizes, particle speeds and the number of nucleons are schematic, and time is sped up. The counter is idealised — it registers every decay.

Mode

Isotope

N₀ initial number of nuclei 1000

T lasts on screen 5

Sample shape

Show emitted particles

Geiger counter sound

Logarithmic N scale

Keep previous runs on the graph

Test yourself

Compare runs with N₀ = 10 and N₀ = 5000: the more nuclei, the closer the step graph follows the N₀·2^(−t/T) curve. In the “Single nucleus” mode see for yourself that a nucleus does not “age”: however long it has lived, its chance to decay during the next half-life stays 50 %.

t in half-lives T

t real time

N — left

N₀ − N — decayed

N / N₀ (experiment)

2^−t/T (theory)

T — half-life

τ = 1/λ = T/ln 2

λ = ln 2 / T

A = λN

In 1896 Henri Becquerel discovered that uranium salts emit invisible radiation on their own — it fogged a photographic plate even through black paper. Marie and Pierre Curie found new, far more active elements, polonium and radium, and called the phenomenon radioactivity. In 1902 Ernest Rutherford and Frederick Soddy realised that in radioactive decay one chemical element turns into another, and that the activity of any sample falls off according to the same law: in equal time intervals the same fraction of nuclei decays. Rutherford called the time in which half of the nuclei decay the half-life. It differs incredibly between isotopes: 164 microseconds for polonium-214 and 4.5 billion years for uranium-238 — almost the age of the Earth.

KEY FORMULAS

Law of radioactive decay

$$N = N_0 \cdot 2^{-\frac{t}{T}}$$

N₀ is the initial number of nuclei, N is the number of undecayed nuclei at time t, T is the half-life. Every half-life the number of nuclei halves

▶ Detailed Explanation

The law is statistical: it predicts the behaviour of a large number of nuclei, but not the moment at which a single nucleus decays. After a time t = nT there are N = N₀/2ⁿ nuclei left. For example, of 1000 radon-222 nuclei (T = 3.82 d), about 1000/8 = 125 remain after 11.46 d = 3T.

The fewer the nuclei, the more noticeable the random deviations from the law: they are of order √N. With N₀ = 10 the step graph “wanders” strongly around the curve; with N₀ = 5000 it almost coincides with it.

Decay law with the decay constant

$$N = N_0 e^{-\lambda t}$$

λ is the decay constant: the probability of a nucleus decaying per unit time, s⁻¹. It is the same exponential as N₀·2^(−t/T), only with base e

▶ Detailed Explanation

In a short time dt a fraction λ·dt of the remaining nuclei decays: dN = −λN·dt. The number of decays is proportional to the number of nuclei, so the solution of this equation is the exponential N = N₀e^(−λt).

Substituting t = T and N = N₀/2 gives e^(−λT) = 1/2, hence λ = ln 2 / T. That is why both forms of the law — with base 2 and with base e — are completely equivalent.

Decay constant and half-life

$$\lambda = \frac{\ln 2}{T}$$

ln 2 ≈ 0.693. The shorter the half-life, the larger the decay constant and the faster the substance decays

▶ Detailed Explanation

For radon-222 T = 3.82 d = 3.30·10⁵ s, so λ = 0.693 / 3.30·10⁵ s ≈ 2.1·10⁻⁶ s⁻¹: each radon nucleus decays in a given second with a probability of about two millionths. For uranium-238 λ ≈ 4.9·10⁻¹⁸ s⁻¹ — billions of times smaller.

The half-life is a property of the isotope: it depends neither on temperature nor on pressure, nor on the chemical compound the atom belongs to.

Activity of a radioactive sample

$$A = \lambda N = A_0 \cdot 2^{-\frac{t}{T}}$$

A is the number of decays per second, measured in becquerels: 1 Bq = 1 decay/s. The activity falls off by the same law as the number of nuclei

▶ Detailed Explanation

Activity is what a Geiger counter measures: the more undecayed nuclei, the more frequent the clicks. In the simulator the second graph shows the number of decays ΔN in each short interval Δt — that is exactly A·Δt.

Example: 1 g of radium-226 (T = 1600 yr) contains 2.7·10²¹ nuclei; its activity A = λN ≈ 3.7·10¹⁰ Bq. This amount used to be called a curie (1 Ci = 3.7·10¹⁰ Bq).

Probability of decay of a single nucleus

$$p = 1 - 2^{-\frac{\Delta t}{T}} = 1 - e^{-\lambda \Delta t}$$

p is the probability that a nucleus decays during an interval Δt. It does not depend on how long the nucleus has already lived: for Δt = T it is always 50 %

▶ Detailed Explanation

This property is called memorylessness: a nucleus has no internal “clock” bringing its decay closer. A nucleus that has lived 5T decays during the next half-life with the same probability ½ as a freshly formed one.

For small Δt ≪ T the probability is p ≈ λΔt. In the simulator the moment of decay of each nucleus is drawn exactly from this law: t = −T·log₂U, where U is a random number between 0 and 1.

Mean lifetime of a nucleus

$$\tau = \frac{1}{\lambda} = \frac{T}{\ln 2}$$

τ is the mean lifetime of a nucleus. In a time τ the number of nuclei decreases by a factor of e ≈ 2.72

▶ Detailed Explanation

Half of the nuclei decay before T, but some live 5T, 10T and longer. Because of these long-livers the mean lifetime is larger than the median: τ ≈ 1.44T.

Check it in the “Single nucleus” mode: collect a few hundred nuclei with the “+100 nuclei instantly” button — the median lifetime will be close to T and the mean close to 1.44T.

Number of decayed nuclei

$$\Delta N = N_0 - N = N_0\left(1 - 2^{-\frac{t}{T}}\right)$$

ΔN is how many nuclei have decayed in a time t. In T half of them decay, in 2T three quarters, in 3T seven eighths of the initial number

▶ Detailed Explanation

The number of decayed nuclei equals the number of daughter nuclei formed — the grey-blue nuclei in the simulator. Ages are determined from their share: for example, radiocarbon dating compares how much ¹⁴C is left in a sample with living organisms.

Problem: what fraction of cobalt-60 nuclei (T = 5.27 yr) decays in 10.54 years? That is 2T, so 1 − 1/4 = 3/4 = 75 % of the nuclei decay.

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into another nucleus with the emission of particles. In α decay the nucleus emits an α particle — a helium-4 nucleus of two protons and two neutrons — so the atomic number drops by 2 and the mass number by 4: ²²²Rn → ²¹⁸Po + ⁴He. In β⁻ decay a neutron inside the nucleus turns into a proton, the nucleus emits an electron and an antineutrino, and the atomic number grows by 1: ¹⁴C → ¹⁴N + e⁻ + ν̃. γ rays are high-energy photons that carry away the excess energy of an excited nucleus; the composition of the nucleus does not change.

The key idea of the simulator: the decay of a single nucleus is a random event. A nucleus does not “age” or “wear out”: the probability that it decays during the next interval Δt is always the same, p = 1 − 2^(−Δt/T). Within a time T any nucleus decays with probability ½, no matter how long it has already lived. In the “Single nucleus” mode a ring clock counts the half-lives, and the nucleus decays whenever chance decides. Press “+100 nuclei instantly”: the histogram of lifetimes looks like a falling exponential, and its median is close to T.

When there are many nuclei, the randomness averages out and we get the law of radioactive decay: N = N₀·2^(−t/T) = N₀·e^(−λt). Here N₀ is the initial number of nuclei, N is the number of undecayed nuclei at time t, T is the half-life, and λ = ln 2 / T ≈ 0.693/T is the decay constant. After T half of the nuclei are left, after 2T a quarter, after 3T an eighth. On the N(t) graph the yellow dots mark ½N₀, ¼N₀ and ⅛N₀. Switch on the logarithmic scale and the exponential becomes a straight line.

The activity A = λN is the number of decays per second, measured in becquerels (1 Bq = 1 decay/s). The activity is proportional to the number of remaining nuclei and falls off by the same law: A = A₀·2^(−t/T). You can see it in the second graph — the number of decays in each interval — and hear it: at first the Geiger counter crackles, and after a few half-lives it clicks less and less often. The mean lifetime of a nucleus τ = 1/λ = T/ln 2 ≈ 1.44T is longer than the half-life, because some nuclei live for a very long time.

Things to try. Run the experiment with N₀ = 1000 and compare how many nuclei are left after T, 2T and 3T with the theory. Set N₀ = 10 and repeat the run several times: the steps are different every time, and the relative spread is about 1/√N. Change the isotope: from technetium-99m (6 h) to uranium-238 (4.5 billion years) the graphs in units of T are identical — only the scale of real time changes. The decay law underlies radiocarbon dating (¹⁴C), nuclear medicine (⁹⁹ᵐTc, ¹³¹I) and the planning of radioactive waste storage.

Law of Radioactive Decay

KEY FORMULAS

Law of radioactive decay

Decay law with the decay constant

Decay constant and half-life

Activity of a radioactive sample

Probability of decay of a single nucleus

Mean lifetime of a nucleus

Number of decayed nuclei

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