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.
An atom or a molecule can absorb light and undergo a transition from one quantum state to another.
The oscillator strength f 12 of a transition from a lower state | 1 ⟩ to an upper state | 2 ⟩ 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 , where m e is the mass of an electron and ℏ is the reduced Planck constant. The quantum states | n ⟩ , 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 α = ∑ i = 1 N r i , α . The oscillator strength is the same for each sub-state | n m n ⟩ .
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 p . In absence of magnetic field, the Hamiltonian can be written as H = 1 2 m p 2 + V ( r ) , and calculating a commutator [ H , x ] in the basis of eigenfunctions of H results in the relation between matrix elements
x n k = − i ℏ / m E n − E k ( p x ) n k . .
Next, calculating matrix elements of a commutator [ p x , x ] in the same basis and eliminating matrix elements of x , we arrive at
⟨ n | [ p x , x ] | n ⟩ = 2 i ℏ m ∑ k ≠ n | ⟨ n | p x | k ⟩ | 2 E n − E k . Because [ p x , x ] = − i ℏ , the above expression results in a sum rule
∑ k ≠ n f n k = 1 , f n k = − 2 m | ⟨ n | p x | k ⟩ | 2 E n − E k , where f n k are oscillator strengths for quantum transitions between the states n and k . This is the Thomas-Reiche-Kuhn sum rule, and the term with k = n has been omitted because in confined systems such as atoms or molecules the diagonal matrix element ⟨ n | p x | n ⟩ = 0 due to the time inversion symmetry of the Hamiltonian H . Excluding this term eliminates divergency because of the vanishing denominator.
Sum rule and electron effective mass in crystals
In crystals, energy spectrum of electrons has a band structure E n ( p ) . Near the minimum of an isotropic energy band, electron energy can be expanded in powers of p as E n ( p ) = p 2 / 2 m ∗ where m ∗ is the electron effective mass. It can be shown that it satisfies the equation
2 m ∑ k ≠ n | ⟨ n | p x | k ⟩ | 2 E k − E n + m m ∗ = 1. Here the sum runs over all bands with k ≠ n . Therefore, the ratio m / m ∗ of the free electron mass m to its effective mass m ∗ 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 n band into the same state.
