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:

-dN_j/dt = N_j SUM(i<j) A_ji

where the A_jis 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_jis 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 E_j =  ----------
            2pi tau_j

             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.

Further Information

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updated 2/25/96