Uniaxial order parameters associated with surface SFG spectra. I. Distributions with cylindrical symmetry

Order parameters provide a useful qualitative and quantitative description of the distribution of molecules in ordered materials as they are independent of the shape of the orientation distribution. For samples that exhibit uniaxial ordering with no twist preference, extracting ⟨P2⟩ from polarized IR absorption data is common and obtaining ⟨P2⟩ and ⟨P4⟩ from Raman data has also been well-described. However, such an approach is not routine in the analysis of surface sum-frequency generation (SFG) spectra. We first provide a description whereby elements of the second-order susceptibility can be expressed in terms of the order parameters ⟨P1⟩ and ⟨P3⟩. We then illustrate how experimental data would be used to extract those parameters and how a most probable orientation distribution function could be determined by subsequently maximizing the information entropy. Finally, features of this distribution function are explored.