-dNj/dt = Nj SUM(i<j) Ajiwhere 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) AjiStrong 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.
(delta E)*(delta t) approximately= h / 2piSo the natural width of an energy level is:
h delta Ej = ---------- 2pi taujor
h SUM(i<j) Aji delta Ej = ---------- 2piSince E = h nu
delta nu = SUM(i<j) Ajiwhere 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.
Copyright © 1996 by Brian M. Tissue