What Is a Half-Life? A Constant Proportion Rather Than a Constant Amount

Radioactive decay is a property of individual nuclei, and each has a fixed probability of decaying in any given interval, entirely independent of its age and of what its neighbours are doing. A nucleus that has existed for a billion years is exactly as likely to decay in the next second as one formed moments ago, which means nothing accumulates or wears out. When an enormous number of such nuclei are present, the behaviour of the population becomes extremely regular even though each individual event is random, and the result is exponential decay: the number decaying per second is proportional to the number remaining, so as the population shrinks the decay rate falls in step. That produces a curve that halves in equal successive intervals, which is what the half-life measures. Half-lives span an extraordinary range, from fractions of a second for some isotopes to billions of years for uranium and thorium, and over twenty orders of magnitude in between.

Several consequences follow directly and routinely surprise people:

• •Two half-lives leave a quarter rather than nothing, three leave an eighth, and ten leave about a thousandth, which is the rule of thumb behind treating ten half-lives as practically complete
• •A long half-life means low activity, since the same number of atoms decaying over a longer period produces fewer events per second, so a very long-lived isotope is weakly radioactive
• •A short half-life means high activity and rapid disappearance, which is why medical tracers are chosen with half-lives of hours and must be produced close to where they are used
• •The danger from a radioactive substance depends on its half-life, the type of radiation, the energy and whether it is inside the body, rather than on half-life alone
• •Biological half-life is a separate quantity describing how fast the body excretes a substance, and the effective half-life combines both
• •The same exponential mathematics describes drug elimination, capacitor discharge and any process where the rate depends on the amount present

Half-lives make several technologies possible. Dating methods rely on choosing an isotope whose half-life is comparable to the age being measured, which is why carbon fourteen dates thousands of years and uranium dates billions and neither works well outside its range. Medical imaging uses short-lived isotopes so the patient's exposure ends quickly, with technetium ninety-nine at about six hours being the workhorse of nuclear medicine and produced on site from a generator containing its longer-lived parent. Radiotherapy uses sources selected for the right combination of energy and lifetime. Smoke detectors contain a small quantity of an isotope with a long half-life, which is why they work for decades. Radioisotope thermoelectric generators power spacecraft beyond practical solar range using plutonium with an eighty-eight year half-life, which is long enough for a mission and short enough to produce useful heat. Nuclear waste management is largely a half-life problem, since the difficulty is not the intensely radioactive short-lived material, which decays within decades, but the long-lived components requiring isolation for periods far longer than any human institution has lasted.

Determining a half-life is straightforward for intermediate values and awkward at both extremes. For an isotope lasting minutes to years, a detector counts decays from a sample over time and the curve is fitted directly, which is the textbook method. For very short half-lives, down to fractions of a second, the sample must be produced and measured in the same apparatus, since it is gone before it can be moved, and timing resolution becomes the limiting factor. For very long half-lives the opposite problem arises: a sample of uranium decays so slowly that the amount lost during any experiment is unmeasurable, so the half-life is calculated instead from the activity, by counting how many decays occur per second in a precisely known number of atoms and working backwards, since the two are related directly. That method has been pushed to extremes, with some isotopes having measured half-lives of ten to the nineteenth years or longer, which exceeds the age of the universe by many orders of magnitude and is established by counting a handful of events in an enormous shielded sample over years.

Each nucleus has a fixed probability of decaying regardless of its age, so a fixed proportion of the remaining population disappears in each interval, producing exponential decay rather than a steady countdown. Two half-lives leave a quarter and ten leave about a thousandth. A long half-life means weak activity and a short one means intense activity that ends quickly, which is why medical tracers last hours and why nuclear waste disposal is a problem about long-lived components rather than intense ones.