In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule. For example, if an emissive state has a small oscillator strength, nonradiative decay will outpace radiative decay. Conversely, "bright" transitions will have large oscillator strengths. The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition.
Theory An atom or a molecule can absorb light and undergo a transition from one quantum state to another. The oscillator strength f 12 {\displaystyle f_{12}} of a transition from a lower state
| 1 ⟩ {\displaystyle |1\rangle } to an upper state | 2 ⟩ {\displaystyle |2\rangle } may be defined by
f 12 = 2 3 m e ℏ 2 ( E 2 − E 1 ) ∑ α = x , y , z | ⟨ 1 m 1 | R α | 2 m 2 ⟩ | 2 , {\displaystyle f_{12}={\frac {2}{3}}{\frac {m_{e}}{\hbar ^{2}}}(E_{2}-E_{1})\sum _{\alpha =x,y,z}|\langle 1m_{1}|R_{\alpha }|2m_{2}\rangle |^{2},}
where m e {\displaystyle m_{e}} is the mass of an electron and ℏ {\displaystyle \hbar } is the reduced Planck constant. The quantum states | n ⟩ , n = {\displaystyle |n\rangle ,n=} 1,2, are assumed to have several degenerate sub-states, which are labeled by m n {\displaystyle m_{n}} . "Degenerate" means that they all have the same energy E n {\displaystyle E_{n}} . The operator R x {\displaystyle R_{x}} is the sum of the x-coordinates r i , x {\displaystyle r_{i,x}}
of all N {\displaystyle N} electrons in the system, i.e.
R α = ∑ i = 1 N r i , α . {\displaystyle R_{\alpha }=\sum _{i=1}^{N}r_{i,\alpha }.}
The oscillator strength is the same for each sub-state | n m n ⟩ {\displaystyle |nm_{n}\rangle } . The definition can be recast by inserting the Rydberg energy Ry {\displaystyle {\text{Ry}}} and Bohr radius a 0 {\displaystyle a_{0}}
f 12 = E 2 − E 1 3 Ry ∑ α = x , y , z | ⟨ 1 m 1 | R α | 2 m 2 ⟩ | 2 a 0 2 . {\displaystyle f_{12}={\frac {E_{2}-E_{1}}{3\,{\text{Ry}}}}{\frac {\sum _{\alpha =x,y,z}|\langle 1m_{1}|R_{\alpha }|2m_{2}\rangle |^{2}}{a_{0}^{2}}}.}
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