https://doi.org/10.1351/goldbook.MT07422
Absorption @P04855@ (referred to electric dipolar absorption) for a molecular transition with its electric @T06460@ at an @A00346@ \(\theta\) with the electric vector of the light is proportional to \(\cos^{2}\theta\). For the whole sample it is proportional to the orientation factor \(K_{\theta} = \:< \cos^{2}\theta >\), averaged over all sample molecules. This average is \(1\) for a sample with all transition moments perfectly aligned along the electric vector of the light, \(1/3\) for an @I03353@ sample and \(0\) for a sample where all transition moments are perpendicular to the electric vector.
Notes:
- The directional cosines provide, especially for uniaxial samples, a simple description of exactly those orientation properties of the sample that are relevant for light absorption. With the principal coordinate system (\(x\), \(y\), \(z\)), forming angles \(\theta = \alpha,\:\beta,\:\gamma\) with the light electric vector in the \(z\) direction, all orientation effects induced by light absorption are contained in \(K_{\theta\theta} = K_{\theta}\). Since the sum of \(K_{\theta}\) for three perpendicular molecular axes is equal to \(1\), only two independent parameters are required to describe the orientation effects on light absorption.
- A related, commonly used description is based on diagonalized Saupe matrices: \[S_{\theta} = (3K_{\theta} -1)/2\] The principal (molecular) coordinate system (\(x\), \(y\), \(z\)) forming angles \(\theta = \alpha,\:\beta,\:\gamma\) with the light electric vector should be chosen such that the matrix \(K\) and the tensor \(S_{\theta}\) are diagonal.
To describe processes involving two or more photons, such as @L03641@ of a uniaxial, aligned sample, an expansion of the directional cosines to the fourth @P04792@ is required. - Order parameters (related to @WT07498@) 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. @WT07498@ are used as a @BT06999@ for an expansion of the orientation–distribution function.