Oscillator strength

An atom or a molecule can absorb light and undergo a transition from one quantum state to another.

The oscillator strength ${\displaystyle f_{12}}$ of a transition from a lower state ${\displaystyle |1\rangle }$ to an upper state ${\displaystyle |2\rangle }$ may be defined by

${\displaystyle f_{12}={\frac {2}{3}}{\frac {m_{e}}{\hbar ^{2}}}(E_{2}-E_{1})\sum _{\alpha =x,y,z}|\langle 1m_{1}|R_{\alpha }|2m_{2}\rangle |^{2},}$

where ${\displaystyle m_{e}}$ is the mass of an electron and ${\displaystyle \hbar }$ is the reduced Planck constant. The quantum states ${\displaystyle |n\rangle ,n=}$ 1,2, are assumed to have several degenerate sub-states, which are labeled by ${\displaystyle m_{n}}$. "Degenerate" means that they all have the same energy ${\displaystyle E_{n}}$. The operator ${\displaystyle R_{x}}$ is the sum of the x-coordinates ${\displaystyle r_{i,x}}$ of all ${\displaystyle N}$ electrons in the system, etc.:

${\displaystyle R_{\alpha }=\sum _{i=1}^{N}r_{i,\alpha }.}$

The oscillator strength is the same for each sub-state ${\displaystyle |nm_{n}\rangle }$.

The definition can be recast by inserting the Rydberg energy ${\displaystyle {\text{Ry}}}$ and Bohr radius ${\displaystyle a_{0}}$

${\displaystyle f_{12}={\frac {E_{2}-E_{1}}{3\,{\text{Ry}}}}{\frac {\sum _{\alpha =x,y,z}|\langle 1m_{1}|R_{\alpha }|2m_{2}\rangle |^{2}}{a_{0}^{2}}}.}$

In case the matrix elements of ${\displaystyle R_{x},R_{y},R_{z}}$ are the same, we can get rid of the sum and of the 1/3 factor

${\displaystyle f_{12}=2{\frac {m_{e}}{\hbar ^{2}}}(E_{2}-E_{1})\,|\langle 1m_{1}|R_{x}|2m_{2}\rangle |^{2}.}$

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 ${\displaystyle {\boldsymbol {p}}}$. In absence of magnetic field, the Hamiltonian can be written as ${\displaystyle H={\frac {1}{2m}}{\boldsymbol {p}}^{2}+V({\boldsymbol {r}})}$, and calculating a commutator ${\displaystyle [H,x]}$ in the basis of eigenfunctions of ${\displaystyle H}$ results in the relation between matrix elements

${\displaystyle x_{nk}=-{\frac {i\hbar /m}{E_{n}-E_{k}}}(p_{x})_{nk}.}$.

Next, calculating matrix elements of a commutator ${\displaystyle [p_{x},x]}$ in the same basis and eliminating matrix elements of ${\displaystyle x}$, we arrive at

${\displaystyle \langle n|[p_{x},x]|n\rangle ={\frac {2i\hbar }{m}}\sum _{k\neq n}{\frac {|\langle n|p_{x}|k\rangle |^{2}}{E_{n}-E_{k}}}.}$

Because ${\displaystyle [p_{x},x]=-i\hbar }$, the above expression results in a sum rule

${\displaystyle \sum _{k\neq n}f_{nk}=1,\,\,\,\,\,f_{nk}=-{\frac {2}{m}}{\frac {|\langle n|p_{x}|k\rangle |^{2}}{E_{n}-E_{k}}},}$

where ${\displaystyle f_{nk}}$ are oscillator strengths for quantum transitions between the states ${\displaystyle n}$ and ${\displaystyle k}$. This is the Thomas-Reiche-Kuhn sum rule, and the term with ${\displaystyle k=n}$ has been omitted because in confined systems such as atoms or molecules the diagonal matrix element ${\displaystyle \langle n|p_{x}|n\rangle =0}$ due to the time inversion symmetry of the Hamiltonian ${\displaystyle H}$. Excluding this term eliminates divergency because of the vanishing denominator.[5]

In crystals, the electronic energy spectrum has a band structure ${\displaystyle E_{n}({\boldsymbol {p}})}$. Near the minimum of an isotropic energy band, electron energy can be expanded in powers of ${\displaystyle {\boldsymbol {p}}}$ as ${\displaystyle E_{n}({\boldsymbol {p}})={\boldsymbol {p}}^{2}/2m^{*}}$ where ${\displaystyle m^{*}}$ is the electron effective mass. It can be shown[6] that it satisfies the equation

${\displaystyle {\frac {2}{m}}\sum _{k\neq n}{\frac {|\langle n|p_{x}|k\rangle |^{2}}{E_{k}-E_{n}}}+{\frac {m}{m^{*}}}=1.}$

Here the sum runs over all bands with ${\displaystyle k\neq n}$. Therefore, the ratio ${\displaystyle m/m^{*}}$ of the free electron mass ${\displaystyle m}$ to its effective mass ${\displaystyle m^{*}}$ 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 ${\displaystyle n}$ band into the same state.[7]

• Atomic spectral line
• Sum rule in quantum mechanics
• Electronic band structure
• Einstein coefficients