# Excited-State Lifetime and Natural Linewidth

### Excited-State Lifetimes of Atoms

The excited state of an atom will have an intrinsic lifetime due to radiative decay given by:
`-dNj/dt = Nj SUM(i<j) Aji`
where the Ajis are the Einstein spontaneous emission coefficients for all of the radiative transitions originating from level j. Intergrating this equation produces:
`Nj(t) = Nj(0) exp (-t/tauj)`
where tauj is defined as the radiative lifetime:
`1 / SUM(i<j) Aji`
Strong atomic transitions have Ajis of 108 to 109 s-1, so lifetimes are 1 to 10 ns. Lifetimes can be shortened by collisions or stimulated emission.

### Natural Linewidth of Atomic Transitions

The natural linewidth (the intrinsic linewidth in the absence of external influences) of an energy level is determined by the lifetime due to the Heisenberg uncertainty principle:
`(delta E)*(delta t) approximately= h / 2pi`
So the natural width of an energy level is:
```               h
delta Ej =  ----------
2pi tauj```
or
```             h SUM(i<j) Aji
delta Ej =  ----------
2pi```
Since E = h nu
`delta nu = SUM(i<j) Aji`
where delta nu is the linewidth in frequency units of a transition between an excited state and the ground state. Since the ground state has an essentially infinite lifetime, the transition linewidth is governed by the width of the excited state.

The lineshape of a transition with only natural broadening is a Lorentzian.

### Further Information

`updated 2/25/96`