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Radioactivity and Half-Life
Modern Physics
Radioactivity is the spontaneous decay of unstable nuclei, releasing radiation and turning into different nuclei. Its rate is measured by the half-life, and it cannot be hurried or slowed.
What radioactivity is
Some nuclei are unstable because they have too many neutrons or too much energy, and they decay by emitting radiation. In alpha decay the nucleus emits a helium nucleus, losing two protons and two neutrons. In beta decay a neutron turns into a proton, emitting an electron, or a proton turns into a neutron, emitting a positron. In gamma decay the nucleus drops to a lower energy state by emitting a high-energy photon, often after an alpha or beta decay. Each decay changes the nucleus, and the daughter nucleus may itself be radioactive, forming a decay chain that ends at a stable nucleus such as lead.
Half-life
Half-life is the time for half the undecayed nuclei in a sample to decay. It is a property of the isotope and ranges from fractions of a second to billions of years. After one half-life half the original nuclei remain, after two half-lives a quarter, after three an eighth, and so on. The activity of a sample, the number of decays per second, is measured in becquerels, where one becquerel is one decay per second. Because decay is exponential, a graph of the number of undecayed nuclei against time is a curve that falls steeply at first and then flattens out.
Where you see it
Carbon-14 dating uses the known half-life of carbon-14, about 5730 years, to date once-living material up to tens of thousands of years old. Radiotherapy uses focused gamma rays or particle beams to destroy cancer cells, and medical tracers use short-lived isotopes to image organs. Nuclear power stations use the controlled fission of uranium-235, and smoke detectors use a tiny alpha source to detect smoke particles. Background radiation is everywhere, from rocks, cosmic rays and food, and at ordinary levels it is harmless; the danger comes from dose, which depends on the type of radiation, its energy and how long the exposure lasts.
📝 Worked example: nuclei remaining after two half-lives
A sample contains N0 = 800 undecayed nuclei of an isotope with half-life T_half = 10 days. After t = 20 days, how many remain? Number of half-lives: t / T_half = 20 / 10 = 2. Use N = N0 x (1/2)^(t/T_half) = 800 x (1/2)^2. (1/2)^2 = 1/4, so N = 800 / 4 = 200. So 200 nuclei remain undecayed after 20 days. After one more half-life, 10 more days, 100 would remain.
Common mistakes
- Saying decay is predictable for one nucleus: only the fraction of a large sample that decays in a given time is predictable; each nucleus decays at random.
- Confusing half-life with total lifetime: after one half-life half the nuclei remain; after two, a quarter remain; the sample never quite reaches zero.
- Treating all radiation as the same: alpha, beta and gamma radiation differ in charge, mass, range and penetrating power.
Practice questions
- A sample of 1000 nuclei has a half-life of 5 days. How many remain after 15 days?
Show answer
After 3 half-lives, 1000 x (1/2)^3 = 1000/8 = 125 nuclei remain.
- A sample falls to one quarter of its activity in 12 hours. What is its half-life?
Show answer
t/T_half = 2, so 2 half-lives pass in 12 hours, giving T_half = 6 hours.
- Which type of radiation is the most penetrating?
Show answer
Gamma rays, because they are high-energy photons with no charge and no mass.
Frequently asked questions
Can the rate of radioactive decay be speeded up or slowed down?
No. Decay is random for individual nuclei, and temperature, pressure and chemical state do not change the half-life.
What stops each type of radiation?
Alpha particles are stopped by paper, beta particles by a few millimetres of aluminium, and gamma rays need thick lead or concrete.