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
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.
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.
To obtain an expression for the radiative lifetime in terms of the transition oscillator strength, we can use the relation between the radiative lifetime, the transition oscillator strength, and the speed of light.
The radiative lifetime (τ) of an excited state can be related to the transition oscillator strength (f) through the following equation:
τ = (4πε₀hbar^2)/(3m_e*c^3 * f)
where:
- ε₀ is the vacuum permittivity (8.854 x 10^-12 F/m)
- hbar is the reduced Planck's constant (1.055 x 10^-34 J*s)
- m_e is the electron mass (9.109 x 10^-31 kg)
- c is the speed of light (3 x 10^8 m/s).
Now, let's calculate the lifetime for a transition with a wavelength of 500 nm (or 500 x 10^-9 m) and an oscillator strength of f = 0.5.
Substituting the given values into the equation, we have:
τ = (4π * 8.854 x 10^-12 F/m * (1.055 x 10^-34 J*s)^2) / (3 * 9.109 x 10^-31 kg * (3 x 10^8 m/s)^3 * 0.5)
Simplifying the expression further, we get:
τ = 1.18 x 10^-8 seconds
Therefore, the radiative lifetime for a transition with a wavelength of 500 nm and f = 0.5 is approximately 1.18 x 10^-8 seconds.