Physical half-life

A radioactive nucleus can decay, changing into a different nucleus. It is not possible to predict when an individual nucleus will decay, but the behaviour of a large population of nuclei follows a predictable pattern.

The physical half-life, $$T_P$$, of a radionuclide is the time taken for the number of its undecayed nuclei to fall to half its starting value through radioactive decay. It is also the time taken for its activity—the number of decays per unit time—to fall to half its starting value. Activity falls as the number of undecayed nuclei falls.

Successive physical half-lives

Each half-life halves the number of nuclei remaining, not the original number. After one half-life, one-half remains; after two, one-quarter remains; and after three, one-eighth remains. The decrease is exponential: the same fraction of the remaining nuclei decays in each successive half-life.

For decay alone, the number of undecayed nuclei and the activity follow the same relationship:

Here, $$N_0$$ and $$A_0$$ are the initial number of undecayed nuclei and initial activity; $$N$$ and $$A$$ are their values after time $$t$$. The relationship describes the predictable behaviour of a large population, even though individual decays are random.

The physical half-life is characteristic of the radionuclide. It describes radioactive decay and does not depend on biological removal of the radionuclide from the body.

Effective half-life

A radionuclide in the body can be reduced in two ways: the body can remove it through biological processes, and its nuclei can undergo radioactive decay. Both processes reduce the amount of that radionuclide remaining in the body.

The biological half-life, $$T_B$$, is the time taken for the amount in the body to fall to half through biological removal alone. The physical half-life, $$T_P$$, is the time taken for the amount to fall to half through radioactive decay alone.

The effective half-life, $$T_E$$, is the time taken for the amount or activity of the radionuclide in the body to fall to half when both processes act together.

Relationship between the half-lives

When biological removal and radioactive decay each occur at a constant exponential rate, the half-lives are related by:

The reciprocals of the half-lives add, rather than the half-lives themselves. All three half-lives must be expressed in the same unit of time.

Because both processes reduce the amount present, the combined reduction is faster than either process acting alone. The effective half-life is therefore shorter than both the physical half-life and the biological half-life.