Half-life and nuclear instability

Reading decay data

The decay law has two equivalent forms, N = N₀e^−λt and successive halving, with T_½ = ln 2/λ relating the half-life to the decay constant. Here you use the equations in reverse. A detector records real data, with random scatter, and you find the half-life from it.

From a decay curve, read the half-life directly. Find the time for the activity to halve, then check it again from a different starting point, because a genuine exponential halves in the same time everywhere along its length.

For scattered data the better tool is a log graph, the twin of the capacitor practical. Taking natural logs gives ln N = ln N₀ − λt, a straight line of ln N against t with gradient −λ. Straightness is the evidence the decay is exponential, and the gradient gives λ, from which T½ = ln 2/λ follows. Symbol for symbol, this is the RP9 analysis with λt in place of t/RC.

Measuring the absorption of gamma radiation by lead

Pearson Edexcel (9PH0) sets Core Practical 15: Investigate the absorption of gamma radiation by lead. OCR A (H556) assesses it in PAG7 under statement 6.4.3(b)(ii): techniques and procedures used to investigate the absorption of α-particles, β-particles and γ-rays by appropriate materials. The analysis is the log graph above.

You need a sealed gamma source handled with tongs, a Geiger–Müller tube and counter, lead sheets, and a micrometer to measure each sheet at several points.

Count the background first, with the source in its store, for at least ten minutes. Clamp the source and tube a few centimetres apart and keep that distance fixed, or the inverse-square law changes the count rate along with the lead. Add sheets one at a time, recording the total thickness x and the count over a fixed time. A count of N is uncertain by about √N, so 10 000 counts give 1 per cent: count for several minutes, longer behind thick stacks, and repeat each count.

Subtract the background rate C_b from each rate C. The corrected rate falls as C₀e^−μx, where μ is the linear attenuation coefficient, so

Plot ln(C − C_b) against x. A straight line confirms exponential absorption, the gradient is −μ, and the half-value thickness is ln 2/μ.

Idealised readings: background 240 counts in 10.0 min, so C_b = 24.0 min⁻¹, and each count took 5.00 min.

The round 0.080 mm⁻¹ keeps the arithmetic simple, and a cobalt-60 source gives less. Its gamma photons, at 1.17 and 1.33 MeV, give lead a narrow-beam μ of about 0.067 mm⁻¹, a half-value thickness near 10 mm, and an uncollimated beam usually measures lower still, for the reason given next.

Limitations. Photons scattered in the lead can still reach the tube, so a wide beam makes μ too small; collimating the source helps. Photons crossing the sheets at an angle pass through more lead than x. The tube misses events during its dead time after each pulse, so keep the unshielded rate modest or the thinnest absorbers read low.

Safety. Sealed sources are used only under the school's local rules and risk assessment. Handle the source with tongs, point its window away from people, minimise the time it is out and maximise your distance from it. The school's radiation protection arrangements take precedence over anything here.

Half-life at work

The half-life sets the applications, because it fixes how long a source stays significantly active. Radioactive waste must be stored securely for many half-lives of its longest-lived components, which for some isotopes means centuries: after ten half-lives about a thousandth of the activity remains.

Living things hold a known fraction of carbon-14, maintained for as long as they exchange carbon with the air. After an organism dies, it no longer exchanges carbon with its environment. The remaining carbon-14 activity can therefore be used to estimate the time since death, with T½ = 5730 years. The same method with far longer-lived isotopes dates rocks.

Choosing an isotope for a job is mostly choosing a half-life. Too short and the source fades before the work is done; too long and it lingers as a hazard after the work is over. That trade-off returns with technetium-99m below, where six hours is precisely the point.

Which nuclei decay, and how

Plot every stable nuclide as a point with proton number Z across and neutron number N up, and they hug a narrow stable band, with N roughly equal to Z for light nuclei and bending neutron-rich as Z grows. Extra neutrons add strong-force glue without adding proton repulsion, and that is the reason for the bend.

Describe the drawing

Neutron number against proton number: the cyan band of stable nuclei starts along N equals Z and bends neutron-rich, while amber arrows show beta-minus decay stepping neutron-rich nuclei down towards the band, beta-plus and electron capture stepping proton-rich nuclei up, and alpha decay trimming the heaviest nuclei along the band's end.

Position on the map predicts the decay. Neutron-rich nuclei, above the band, undergo β⁻ decay, in which a neutron becomes a proton, so N falls by 1 and Z rises by 1. ⁹⁰Sr becomes ⁹⁰Y, Z climbing from 38 to 39, plus e⁻ and an antineutrino ν̅_e. Proton-rich nuclei, below the band, travel the other way by β⁺ decay or by electron capture, each turning a proton into a neutron.

The heaviest nuclei shed bulk instead. In α decay, ²²⁶Ra becomes ²²²Rn with Z falling from 88 to 86, plus ⁴He, so A drops by 4 and Z by 2. In every equation, A and Z balance across the arrow.

Decay often leaves the daughter in an excited state. Nuclei have energy levels much as atoms do, drawn as nuclear energy level diagrams, and the drop to the ground state emits a γ photon. Medicine exploits one such state deliberately. Technetium-99m is a long-lived excited state that emits gamma alone, with a six hour half-life. Injected as a tracer, its photons escape the body to a camera, and within a couple of days the activity has fallen to almost nothing.