Physics · Atomic and quantum physics · Grade 11

Radioactivity, half-life

How can we tell how old an ancient wooden object is?

The situation

Archaeologists found an ancient wooden bowl at a dig. No date is written on it — how can its age be determined? A living tree takes in carbon from the air, and a tiny part of it is radioactive carbon-14. After the tree is cut down, C-14 is no longer replenished and slowly decays. Nobody can say in advance when a single atom will decay — so why do billions of decaying atoms work as precisely as a “clock”?

Topic

Radioactivity, half-life. Radiocarbon dating: how old is an ancient find

Key ideas

Half-life
Activity A = λN
Radiocarbon dating

Formulas

N (t) = N_{0} e^{- λ t} = N_{0} \cdot 2^{- t / T_{1 / 2}}

The law of radioactive decay: $N$ is the number of nuclei that have not yet decayed.

T_{1 / 2} = \frac{\ln 2}{λ} \approx \frac{0.693}{λ}

Half-life is the time in which half of the nuclei decay; $λ$ is the decay constant.

A = λ N

Activity is the number of decays in 1 s, measured in becquerels (1 Bq = 1 decay/s).

t = \frac{T_{1 / 2}}{\ln 2} \ln \frac{N_{0}}{N}

Radiocarbon age: $N / N_{0}$ is the fraction of C-14 remaining in the sample.

P = λ Δ t

The probability that a single nucleus decays in a short time $Δ t$ — it does not depend on the “age” of the nucleus.

Theory

α, β and γ decay

An unstable nucleus turns into another nucleus by emitting an α particle (a helium nucleus — stopped by a sheet of paper), a β particle (an electron — stopped by thin aluminium) or a γ quantum (a photon — lead is needed). α decay occurs in heavy nuclei: the α particle tunnels out through the Coulomb barrier (G. Gamow, 1928 — the “Schrödinger and quantum tunnelling” experiment), so nuclei with slightly higher energy decay much faster. The types of β decay are β⁻ (neutron → proton + electron + antineutrino), β⁺ (proton → neutron + positron + neutrino) and electron capture. The energy of β particles is distributed continuously from 0 to a maximum — to save conservation of energy, W. Pauli proposed in 1930 an invisible particle, the neutrino, and E. Fermi created the theory of β decay in 1934. Among nuclei with the same A (isobars), the one with the smallest mass is β-stable. In γ emission an excited nucleus drops to a lower level; a long-lived excited state is an isomer (Ba-137m, T½ ≈ 2.55 min). Instead of a γ quantum, the energy may be given to a shell electron — internal conversion; the transitions obey selection rules for angular momentum and parity. If a nucleus in a crystal emits and absorbs a γ quantum without recoil (the Mössbauer effect, 1958), the line is extremely narrow — with it Pound and Rebka measured the gravitational redshift in 1960. In any small interval $Δ t$ every nucleus decays with the same probability $λ Δ t$ and has no “memory”: the moment a single nucleus decays is random.

In many nuclei — an exact law

In every $Δ t$ the same fraction of the nuclei present decays, so $N (t)$ decreases exponentially: it halves every $T_{1 / 2}$ . The activity $A = λ N$ falls in the same way — the clicking of the counter becomes sparser. The unit of activity is the becquerel (1 Bq = 1 decay/s); the old unit is the curie: 1 Ci = 3.7·10¹⁰ Bq (roughly the activity of 1 g of radium-226). If the sample is small, the graph “trembles”: the random deviation is about $\sqrt{N}$ . Compare samples of 64 and 4096 nuclei.

Displacement rules and radioactive series

The displacement rules (F. Soddy and K. Fajans, 1913): in α decay the nucleus moves two places to the left in the periodic table ( $Z - 2$ , $A - 4$ ), in β⁻ decay one place to the right ( $Z + 1$ , $A$ unchanged), and in γ emission its place does not change. A heavy nucleus does not become stable in a single decay: from uranium-238 to lead-206 there are 8 α and 6 β⁻ decays — this is a radioactive series. Because A changes only by 4, there are four series: thorium (A = 4n), neptunium (4n + 1, almost none left in nature), uranium–radium (4n + 2) and actinium (4n + 3). In successive decays, the numbers of daughter nuclei are found from the Bateman equations: if the parent nucleus lives much longer, then after several T½ of the daughters the activities of all members become equal — secular equilibrium. Radon in the air of a closed room and its short-lived descendants (Po-218, Pb-214, Bi-214, Po-214) reach this equilibrium in ≈ 3 hours — the graph in the “Radioactive series” mode.

The Geiger–Müller counter and measuring activity

A particle ionises the gas inside the tube, an avalanche of electrons forms in the strong field, and a short current pulse is produced — you hear a “click”. The counter registers only a fraction of the decays (10 % in “Sample” mode), but the counting rate is proportional to the activity, so $T_{1 / 2}$ can be measured with it too. The number of pulses obeys the Poisson law: the statistical error of $N$ pulses is $\sqrt{N}$ (±5 % for 400 pulses). The $T_{1 / 2}$ of a long-lived isotope cannot be measured by waiting. Instead, the background $R_{b}$ is measured separately, and the activity is found from the net counting rate and the counter efficiency $ε$ : $A = (R - R_{b}) / ε$ ; if the number of nuclei $N = (m / M) N_{A}$ is known, $T_{1 / 2} = \ln 2 \cdot N / A$ . In “Activity” mode do this with K-40 (KCl, the substitute for table salt) and Cs-137. More about detectors is in the “Detecting particles” experiment.

Radiocarbon dating

Cosmic rays produce C-14 from nitrogen in the atmosphere. In a living organism the ratio of C-14 to C-12 is almost constant (≈ 1.2·10⁻¹²): 1 g of carbon decays about 0.23 times per second. After the organism dies, C-14 is no longer renewed and decreases with $T_{1 / 2} = 5730$ years. If 25 % remains in a sample, two half-lives have passed — about 11 460 years. Model: the nuclei are enlarged, time is scaled (1 s = $T_{1 / 2} / 8$ ), and background radiation is not taken into account; in real dating, changes in the amount of C-14 in the atmosphere are corrected using a calibration curve.

Artificial radioactivity and background radiation

Radioactive nuclei that do not exist in nature can be made in nuclear reactions — artificial radioactivity (I. and F. Joliot-Curie, 1934: they irradiated aluminium with α particles and obtained phosphorus-30). Today I-131, Co-60 and Cs-137 are made in reactors, and many medical isotopes in accelerators. Heavy nuclei can also fission spontaneously and even emit a whole “cluster” — in the “Nuclear reactions” experiment. Background radiation is everywhere: cosmic rays, uranium, thorium and K-40 in soil and building materials, radon in the air (the largest share), and K-40 and C-14 in our own bodies. Cosmogenic radionuclides (C-14, H-3, Be-7) form in the atmosphere under the action of cosmic rays, and technogenic ones (Cs-137, Sr-90, I-131) arise from nuclear tests, accidents (Chernobyl, 1986) and medicine. Worldwide, the average annual dose from natural background is ≈ 2.4 mSv. Radiation ecology studies the movement of radionuclides through soil, water, plants and food.

Did you know?

Radioactivity was discovered in 1896 by the French physicist Henri Becquerel: uranium salts darkened a photographic plate wrapped in black paper. In 1903 he shared the Nobel Prize in Physics with Marie and Pierre Curie.
The radiocarbon method was developed by the American chemist Willard Libby in the late 1940s, and he received the 1960 Nobel Prize in Chemistry for it. The method dates organic remains up to about 50 000 years old.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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