# Excited-State Lifetime and Natural Linewidth

The excited state of an atom will have an intrinsic lifetime due to radiative decay given by:
-dN_{j}/dt = N_{j} SUM(i<j) A_{ji}

where the A_{ji}s are the Einstein spontaneous emission coefficients for all of the radiative transitions originating from level j. Intergrating this equation produces:
N_{j}(t) = N_{j}(0) exp (-t/tau_{j})

where tau_{j} is defined as the radiative lifetime:
1 / SUM(i<j) A_{ji}

Strong atomic transitions have A_{ji}s of 10^{8} to 10^{9} s^{-1}, so lifetimes are 1 to 10 ns. Lifetimes can be shortened by collisions or stimulated emission.

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 E_{j} = ----------
2pi tau_{j}

or
h SUM(i<j) A_{ji}
delta E_{j} = ----------
2pi

Since E = h *nu*

delta *nu* = SUM(i<j) A_{ji}

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.

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Copyright © 1996 by Brian M. Tissue

updated 2/25/96