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.
[1][2] 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.
[3] 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.
[4] An atom or a molecule can absorb light and undergo a transition from one quantum state to another.
of a transition from a lower state
is the reduced Planck constant.
1,2, are assumed to have several degenerate sub-states, which are labeled by
"Degenerate" means that they all have the same energy
is the sum of the x-coordinates
The oscillator strength is the same for each sub-state
The definition can be recast by inserting the Rydberg energy
In case the matrix elements of
are the same, we can get rid of the sum and of the 1/3 factor To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum
In absence of magnetic field, the Hamiltonian can be written as
in the basis of eigenfunctions of
results in the relation between matrix elements Next, calculating matrix elements of a commutator
in the same basis and eliminating matrix elements of
, the above expression results in a sum rule where
are oscillator strengths for quantum transitions between the states
This is the Thomas-Reiche-Kuhn sum rule, and the term with
has been omitted because in confined systems such as atoms or molecules the diagonal matrix element
due to the time inversion symmetry of the Hamiltonian
Excluding this term eliminates divergency because of the vanishing denominator.
[5] In crystals, the electronic energy spectrum has a band structure
Near the minimum of an isotropic energy band, electron energy can be expanded in powers of
is the electron effective mass.
It can be shown[6] that it satisfies the equation Here the sum runs over all bands with
of the free electron mass
in a crystal can be considered as the oscillator strength for the transition of an electron from the quantum state at the bottom of the