Xiaoshan Xu's Blog: From oscillator strength to spontaneo...

Thursday, May 21, 2009

From oscillator strength to spontaneo...

From oscillator strength to spontaneous decay rate:

Approach I

Using
quantum mechanics, one can derive the relation between the oscillator
strength calculated from the absorption coefficient and the transition
matrix element |x|². And the spontaneous decay rate is also decided by the matrix element |x|², therefore, one can find the decay rate from the absorption coefficient.

f=\frac{2m\omega}{\hbar}|x|^2

$eq=<br />f=\frac{2m\omega}{\hbar}|x|^2$

while spontaneous decay rate:
A=\frac{e\omega^3}{3\pi\hbar c^3\epsilon_0 n^2}|x|^2

$eq=A=\frac{e\omega^3}{3\pi\hbar c^3\epsilon_0 n^2}|x|^2$

Therefore,

\frac{A}{f}=\frac{\omega^2e^2}{6m\pi c^3\epsilon_0n^2}

$eq=\frac{A}{f}=\frac{\omega^2e^2}{6m\pi c^3\epsilon_0n^2}$

Approach II: Judd-Ofelt theory,

f=\frac{2m\omega}{\hbar}S

$eq=f=\frac{2m\omega}{\hbar}S$

A=\frac{\omega^3e^2n^2}{\hbar c^3\pi\epsilon_0}S

$eq=A=\frac{\omega^3e^2n^2}{\hbar c^3\pi\epsilon_0}S$

Therefore,

\frac{A}{f}=\frac{\omega^2e^2n^2}{2mc^3\pi\epsilon_0}

$eq=\frac{A}{f}=\frac{\omega^2e^2n^2}{2mc^3\pi\epsilon_0}$

One can see that, in two approaches, there is an factor of 3n^4.

I am not so sure why two are not the same, but at this moment I tend to believe the second approach because it is specialized just for this case.

Usage

In a real experiment, oscillator strength can be found using partial sum rule:
The dimensionless oscillator strength is:
f=\frac{2c}{N_e\hbar \pi \omega_p^2}\int_{E_2}^{E_1}n\alpha dE

$eq=f=\frac{2c}{N_e\hbar \pi \omega_p^2}\int_{E_2}^{E_1}n\alpha dE$