Question

Obtain an expression for the radiative lifetime in terms of the transition oscillator strength. Calculate the lifetime for a transition with wavelength 500 nm and f = 0.5

Answer 1

The relationship between the oscillator strength (also called the absorption f-number) of a radiative transition and the radiative lifetime depends upon the degeneracies of the upper and lower states, which are called gl and gu, respectively. The degeneracies, equal to 2J +1, are integers that are only 1 in the case of angular-momentum and spin-free states (J = 0) .

The relationship between f and the Einstein spontaneous emission coefficient, Aul, is
f = m*c^3*(gu/gl)*Aul/[8*pi^2*e^2*nu^2]

where "nu" is the transition frequency, e is the electron charge, and m is the electron mass.

The Einstein coefficient Aul is related to the mean radiative lifetime, tau, by
1/tau = Aul

Strictly speaking, the right side should be a sum of Aul coefficients for all possible lower levels, since a given excited electronic state can decay via transitions at different frequencies.

ref: Caltech Ph.D. thesis, William L. Shackleford, 1964, Part I.
"Measurement of gf-values for Singly Ionized Chromium Using the Reflected Shock region of a Shock Tube".

Good thing I kept a copy.

Answer 2

gl = lower state degeneracy
gu = upper state degeneracy
I had the definitions inverted in my answer, when I wrote "respectively".
The formula is correct as written.