Mathematics of the Polarized-Fluorescence Experiment

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Abstract
The polarized‐fluorescence experiment for measurement of preferred orientation is analyzed mathematically. The intensity is dependent upon P and A, unit vectors describing the positions of the polarizer and analyzer, and upon the orientation distribution N(α, δ) of the chromophoric group of the polymer. In the case of biaxial or uniaxial symmetry, the intensity IPA for any combination of P and A vectors is expressed as linear combination of nine fundamental intensities characteristic of a given sample. These intensities form a 3×3 matrix, each entry Iij being the intensity obtained with P and A parallel to the ith and jth principal axes of the sample. Of the nine components of I, only five are independent for a biaxially oriented sample; two for uniaxial orientation. Each Iij is proportional to 〈xi2xj2〉Av, where xi and xj are the components of the chromophore vector along the corresponding principal axes. Calculated matrices are shown for several model distributions. Nishijima's P function is related to the Iij's, and the problem of measuring biaxial orientation is discussed. The Iij's are also shown to be directly related to the second‐and fourth‐order coefficients in the spherical harmonic expansion of N(α, δ). These coefficients can be measured directly by a few simple experiments involving spinning the sample between crossed or parallel polarizers. Thus, the experimental error in their determination can be minimized.