Oscillator strength


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.

The oscillator strength   of a transition from a lower state   to an upper state   may be defined by


where   is the mass of an electron and   is the reduced Planck constant. The quantum states   1,2, are assumed to have several degenerate sub-states, which are labeled by  . "Degenerate" means that they all have the same energy  . The operator   is the sum of the x-coordinates   of all   electrons in the system, i.e.


The oscillator strength is the same for each sub-state  .

The definition can be recast by inserting the Rydberg energy   and Bohr radius  


In case the matrix elements of   are the same, we can get rid of the sum and of the 1/3 factor


Thomas–Reiche–Kuhn sum rule


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  , and calculating a commutator   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  , we arrive at


Because  , the above expression results in a sum rule


where   are oscillator strengths for quantum transitions between the states   and  . 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]

Sum rule and electron effective mass in crystals


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   as   where   is the electron effective mass. It can be shown[6] that it satisfies the equation


Here the sum runs over all bands with  . Therefore, the ratio   of the free electron mass   to its effective 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   band into the same state.[7]

See also



  1. ^ W. Demtröder (2003). Laser Spectroscopy: Basic Concepts and Instrumentation. Springer. p. 31. ISBN 978-3-540-65225-0. Retrieved 26 July 2013.
  2. ^ James W. Robinson (1996). Atomic Spectroscopy. MARCEL DEKKER Incorporated. pp. 26–. ISBN 978-0-8247-9742-3. Retrieved 26 July 2013.
  3. ^ Westermayr, Julia; Marquetand, Philipp (2021-08-25). "Machine Learning for Electronically Excited States of Molecules". Chemical Reviews. 121 (16): 9873–9926. doi:10.1021/acs.chemrev.0c00749. ISSN 0009-2665. PMC 8391943. PMID 33211478.
  4. ^ Hilborn, Robert C. (1982). "Einstein coefficients, cross sections, f values, dipole moments, and all that". American Journal of Physics. 50 (11): 982–986. arXiv:physics/0202029. Bibcode:1982AmJPh..50..982H. doi:10.1119/1.12937. ISSN 0002-9505. S2CID 119050355.
  5. ^ Edward Uhler Condon; G. H. Shortley (1951). The Theory of Atomic Spectra. Cambridge University Press. p. 108. ISBN 978-0-521-09209-8. Retrieved 26 July 2013.
  6. ^ Luttinger, J. M.; Kohn, W. (1955). "Motion of Electrons and Holes in Perturbed Periodic Fields". Physical Review. 97 (4): 869. Bibcode:1955PhRv...97..869L. doi:10.1103/PhysRev.97.869.
  7. ^ Sommerfeld, A.; Bethe, H. (1933). "Elektronentheorie der Metalle". Aufbau Der Zusammenhängenden Materie. Berlin: Springer. p. 333. doi:10.1007/978-3-642-91116-3_3. ISBN 978-3-642-89260-8.