Mathematics of the Polarized-Fluorescence Experiment

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 I_PA 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 I_ij 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 I_ij is proportional to 〈x_i²x_j²〉_Av, where x_i and x_j 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 I_ij's, and the problem of measuring biaxial orientation is discussed. The I_ij'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.