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

while spontaneous decay rate:
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}

Approach II: Judd-Ofelt theory,

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

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}

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

Here
\omega_p^2=\frac{e^2 \rho}{m\epsilon_0}

[Reference:]

Kumar G.A. et al, Journal of Luminescence vol. 99, p. 141-148 (2002)

Conservation of

Conservation of Σλ_i

Description:

For a Helbert space (wave function subspace) {φ_i} defined by the operator O whose eigenvalues are λ_i, corresponding to wavefunction  φ_i.

Simply from linear algebra, Σλ_i is actually the trace of the operator matrix O, which is of course conserved, no matter what unitary transformation of the {φ_i} is taken to form new set of basis, say {ψ_i}.

To generalize this, Σλ_iⁿ, is also conserved, because Oⁿ also defines the space.

Example:

For a 4f electron multiplet, say, Nd³⁺, ⁴I_9/2, ΣJ_z,iⁿ is conserved and of course Σμ_z,iⁿ is conserved too because they differ only by a factor of gμ_B

Just to give a few numbers,

sqrt(3Σμ_z,i²/5)=

3.6 μ_B(theoretical value)
3.4, 3.5 and 3.8 from measurement on crystal field split ⁴I_9/2 multiplet.