## Decay constant, radioactive

Glossary Definition for 16-19

#### Description

Radioactivity is a random process; it is impossible to predict exactly when a particular nucleus will decay.

However, it is possible to determine the probability that a nucleus will decay in a given time.

For a particular decay mechanism, the radioactive decay constant for a nuclide is defined as the probability per unit time that a given nucleus of that nuclide will decay by that mechanism. The radioactive decay constant is usually represented by the symbol *λ*. The definition may be expressed by the equation

*P* = *λ* Δ*t*

where *P* is the probability of a given unstable nucleus decaying in the time interval Δ*t* which must be much smaller than the half-life of the nuclide.

If there are initially *N* nuclei in a sample, the average change in that number, Δ*N*, resulting from decays after time Δ*t* is

Δ*N* = -*λ* *N* Δ*t*

#### Discussion

Any sample of a radioactive nuclide whose activity is practically measurable will contain a number of nuclei, *N*, in excess of ~ 10^{20} . The proportion of these nuclei that decay in time Δ*t* will get closer and closer to the expected proportion, *λ* Δ*t*, the larger the value of *N*, so this proportion can usually be considered exact in practice.

The decay constant depends only on the particular radioactive nuclide and decay mechanism involved. It does not depend on the number of nuclei present or on any external conditions (such as temperature).

In most radioactive samples, there will be more than one way of decaying, either due to different processes within one nuclide or due to there being a mixture of nuclides. In these circumstances, each type of decay process must be considered independently. It is not possible to combine decay constants in a simple way.

#### SI unit

inverse seconds, s^{-1}

#### Expressed in SI base units

s^{-1}

#### Other commonly used unit(s)

minutes^{-1}, hours^{-1}, years^{-1}

#### Mathematical expressions

*P*= λ Δ*t*
where - In calculus notation, the instantaneous rate of decay of a nuclide, i.e. its activity,
*A*, is given by *t*_{1/2}= ln 2λ
where

*P*is the probability of a given unstable nucleus decaying in the time interval Δ

*t*which must be much smaller than the half-life of the nuclide.

*A*= -d

*N*d

*t*= λ

*N*where

*N*is the number of radioactive nuclei at that instant.

*t*

_{1/2}is the half-life of the nuclide

#### Related entries

- Activity of a radioactive source
- Half-life, radioactive

#### In context

Decay constants have a huge range of values, particularly for nuclei that emit *α*-particles. For example, the most common isotope of uranium, 238U, has a decay constant of 1.546 × 10^{–10} yr^{–1} corresponding to a half-life of 4.5 billion years, whereas 212Po has * λ * = 2.28 × 10

^{6}s

^{–1}, corresponding to a half-life of 304 ns.