Radioactive decay

What the exam asks

Expect either a short numerical item or a decay curve. The numerical ones are built to be done in your head: a count rate falling from 800 to 100, a sample with a sixteenth left, a mass remaining after 15 days with a 5 day half-life. Count the halvings, and if you find yourself wanting to solve an exponential for the decay constant, you have missed the intended route. The graphs are either activity against time, where you read the half-life by finding how long the curve takes to fall by half starting from anywhere on it, or log activity against time, where the straight line is the whole point being made. The trap that catches most candidates is answering with the fraction remaining when the question asked for the fraction decayed, or the reverse: an eighth and seven eighths are both correct answers to something, and both will be on offer. The second trap is the total mass of a sample. The parent isotope's mass halves, but the sample does not get lighter, because the daughter nuclei are still sitting in it.

Radioactive decay is the exam's favourite piece of exponential reasoning, and it is the topic where having no calculator is a design feature rather than an obstacle. The intended arithmetic is counting halvings. Three half-lives is an eighth, and if the numbers in front of you do not resolve into a whole number of halvings, you are meant to be reading the answer off a graph instead.

Decay is random and spontaneous. You cannot say which nucleus goes next, only what fraction of a large number will have gone in a given time, and that fraction is fixed. This is why the half-life is a constant: it does not care how much you started with, how hot the sample is, what it is chemically bonded to, or how long it has already been decaying.

The rest is bookkeeping. Alpha decay takes two protons and two neutrons out. Beta-minus turns a neutron into a proton. Gamma carries off energy and changes nothing about what the nucleus is made of. Mass number and atomic number are conserved across every decay equation, and that conservation alone is usually enough to identify an unknown product without knowing any nuclear physics at all.

What to hold

• The half-life is the time for half the radioactive nuclei present to decay, and equally the time for the activity or the count rate to halve. All three halve together, because activity is proportional to the number of nuclei left.
• After n half-lives the fraction remaining is one half to the power n, so three half-lives leaves an eighth and four leaves a sixteenth, and this counting is the arithmetic the exam intends. The fraction that has decayed is one minus that, so after three half-lives seven eighths have gone.
• The half-life is a constant of the isotope. Heating, cooling, compressing or chemically bonding the sample does not change it, because decay is a nuclear process and none of those reach the nucleus.
• Decay is random. The half-life is a statistical statement about a large number of nuclei, not a schedule any individual nucleus is following.
• Activity is decays per second, measured in becquerels, and it is proportional to the number of undecayed nuclei remaining. That proportionality is exactly what makes the decay exponential rather than linear.
• Alpha decay emits a helium-4 nucleus, so the mass number falls by 4 and the atomic number falls by 2.
• Beta-minus decay turns a neutron into a proton and emits an electron, so the mass number is unchanged and the atomic number rises by 1. The electron is created in the decay and was not orbiting the atom beforehand.
• Gamma emission changes neither the mass number nor the atomic number. It is the nucleus shedding excess energy, usually just after an alpha or beta decay.
• Alpha is the most ionising and the least penetrating; gamma is the least ionising and the most penetrating. The two properties trade off, because a particle that ionises heavily gives up its energy quickly.
• In a magnetic field alpha and beta deflect in opposite directions because their charges have opposite signs, beta deflects far more because its charge to mass ratio is thousands of times larger, and gamma is undeflected because it carries no charge.
• Plotting the natural log of the activity against time gives a straight line whose gradient is minus the decay constant. The straightness itself is the evidence that the decay is exponential.
• A measured count rate includes background, and the background must be subtracted before any half-life is read off it.