# Exponential decay#Mean lifetime

The term life ( often also mean lifetime ) denotes in physics, the average "lifetime" of the members of an ensemble of identical objects that are in the same condition and are isolated from each other. If the object is not a state of smaller energy available, and no power is supplied to him, so it is stable and its lifetime is infinite. However, the objects can spontaneously into a state of smaller energy pass ( " decay " ), form their respective lifetimes, a frequency distribution, the arithmetic mean is the life. Typically, one speaks of life associated with unstable particles, radioactive nuclei as well as in other systems, and atoms in an excited state. In the life sciences, the term life expectancy has the same meaning.

## Lifetime and decay probability

The probability density that a member of the ensemble decays, usually follows an exponential distribution:

Is the decay constant. It is also called decay probability, but a probability per unit time, and is usually given in units of 1/sec. If more than one decay channels are available, the entire (total ) decay constant the sum of the corresponding individual (partial) decomposition constants.

The life is the inverse of the decay constant:

It is therefore the time after which the number of particles is decreased to a fraction 1 / e ≈ 0.368.

For elementary particles you get an overview of the different decay channels and decay probabilities in the book published by the Particle Data Group Review of Particle Physics or in the short version, the Particle Physics Booklet.

## Partial life

If several decay channels exist, can be formally specified to each of the partial decay constants of the inverse of a " partial life "; this happens sometimes for reasons of clarity. However, the partial lifetime is a fictional, non-observable size: You'd be the life of the system, if the decay channel in question would be the only possible one. Regardless of which channels the decay is observed, shows the life of the decay is always equal to the total decay constants.

## Half-life

Sometimes - especially in the field of radioactivity - the half-life will take the life of uses, ie the time after which half of the ensemble is still available. The half-life is calculated from the life and the decay constant with the help of

So that it amounts to about 69 % of the life. In case of several decay channels are occasionally for the sake of clarity - as in life - also called fictitious partial half-lives.

Half-lives and decay modes of radionuclides are given eg in the Karlsruhe Nuclide Chart. Branching ratios and other data can be found in the comprehensive book Table of Isotopes.

## Connection with the quantum theory

Due to the Heisenberg uncertainty principle allows the following relationship between the uncertainty of any observables and their development over time, see:

This gives a connection between the energy uncertainty or decay width of a transition or decay and its lifetime:

For determining lifetimes of excited states, it may be easier for example the energy of to measure the emitted photons and to obtain from the width of the energy distribution over the lifetime of this formula.