To describe processes involving two or more photons, such as luminescence of a uniaxial, aligned sample, an expansion of the directional cosines to the fourth power is required.","3":"Order parameters (related to Wigner matrices) are an alternative to the directional cosine-based description of molecular alignment. Order-parameter methods also work well for non-uniaxial samples and provide a seemingly more complex, but in other ways convenient, description of molecular orientation distributions. Wigner matrices are used as a basis set for an expansion of the orientation–distribution function."},"links":[{"title":"luminescence","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/L03641"},{"title":"power","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/P04792"},{"title":"Wigner matrices","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/WT07498"},{"title":"basis set","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/BT06999"},{"title":"probability","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/P04855"},{"title":"transition (dipole) moment","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/T06460"},{"title":"angle","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/A00346"},{"title":"isotropic","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/I03353"}],"math":[{"alttext":"θ","latex":"\\theta"},{"alttext":"cos 2θ","latex":"\\cos^{2}\\theta"},{"meaning":"Orientation factor","type":"equation","alttext":"Kθ = < cos^2 θ >","latex":"K_{\\theta} = \\:< \\cos^{2}\\theta >"},{"type":"numeric value","alttext":"1","latex":"1"},{"alttext":"1\/3","latex":"1\/3"},{"alttext":"0","latex":"0"},{"alttext":"x","latex":"x"},{"alttext":"y","latex":"y"},{"alttext":"z","latex":"z"},{"alttext":"θ = α, β, γ","latex":"\\theta = \\alpha,\\:\\beta,\\:\\gamma"},{"alttext":"z","latex":"z"},{"meaning":"Orientation factor","type":"math","alttext":"Kθθ = Kθ","latex":"K_{\\theta\\theta} = K_{\\theta}"},{"meaning":"Orientation factor","type":"quantity","alttext":"Kθ","latex":"K_{\\theta}"},{"type":"numeric value","alttext":"1","latex":"1"},{"alttext":"S θ = ( 3 K θ- 1 ) \/ 2","latex":"S_{\\theta} = (3K_{\\theta} -1)\/2"},{"alttext":"x","latex":"x"},{"alttext":"y","latex":"y"},{"alttext":"z","latex":"z"},{"alttext":"θ = α, β, γ","latex":"\\theta = \\alpha,\\:\\beta,\\:\\gamma"},{"alttext":"K","latex":"K"},{"alttext":"S θ","latex":"S_{\\theta}"}],"sources":["PAC, 2007, 79, 293. 'Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)' on page 371 (https:\/\/doi.org\/10.1351\/pac200779030293)"]}],"links":{"html":"https:\/\/goldbook.iupac.org\/terms\/view\/MT07422\/html","json":"https:\/\/goldbook.iupac.org\/terms\/view\/MT07422\/json","xml":"https:\/\/goldbook.iupac.org\/terms\/view\/MT07422\/xml","plain":"https:\/\/goldbook.iupac.org\/terms\/view\/MT07422\/plain","pdf":"https:\/\/goldbook.iupac.org\/terms\/view\/MT07422\/pdf"},"citeas":"IUPAC. Compendium of Chemical Terminology, 2nd ed. (the \"Gold Book\"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https:\/\/doi.org\/10.1351\/goldbook.","license":"Licensed under Creative Commons Attribution-NoDerivatives (CC BY-NC-ND) 4.0 International (https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/)","collection_reuse":"For parties interested in reusing the entire IUPAC Gold Book please contact IUPAC here https:\/\/www.cognitoforms.com\/IUPAC1\/ContactInformationForm","disclaimer":"The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using.","accessed":"2023-10-03T09:46:11+00:00"}}