Oscillator strength

An atom or a molecule can absorb light and undergo a transition from one quantum state to another. The oscillator strength is a dimensionless quantity to express the strength of the transition. The oscillator strength f₁₂ of a transition from a lower state $|1 m_1\rangle$ to an upper state $|2 m_2\rangle$ may be defined by

$f_{12} = \frac{2 }{3}\frac{m_e}{\hbar^2}(E_2 - E_1) \sum_{m_2} \sum_{\alpha=x,y,z} | \langle 1 m_1 | R_\alpha | 2 m_2 \rangle |^2,$

where m_e is the mass of an electron and $\hbar$ is the reduced Planck constant. The quantum states $|n m_n\rangle, n=$ 1,2,..., are assumed to have several degenerate sub-states, which are labeled by m_n. "Degenerate" means that they all have the same energy E_n. The operator R_x is the sum of the x-coordinates r_i,x of all N electrons in the system, etc.:

$R_\alpha = \sum_{i=1}^N r_{i,\alpha}.$

The oscillator strength is the same for each sub-state $|1 m_1\rangle$.

Thomas-Reiche-Kuhn sum rule

The sum of the oscillator strength from one sub-state $|i m_i\rangle$ to all other states $|j m_j\rangle$ is equal to the number of electrons N:

∑	fij = N.
j

See also

• Atomic spectral line
• Sum rule in quantum mechanics

References

• Robert C. Hilborn, Einstein coefficients, cross sections, f values, dipole moments, and all that, Am. J. of Phys. 50, 982 (1982), arXiv:physics/0202029v1