Calculation of Molecular Orientation
When dealing with adsorbed molecules on a surface, it is typical to find a uniaxial distribution of the molecules, resulting in x- and y- coordinate terms to be interchangeable. When analyzing the second-order susceptibility tensor χ(2), the quantities χXYZ = -χYXZ must be 0 and only three independent tensor terms remain: χzzz, χzxx, and χxxz. The intensities of the s and p polarizations in the second harmonic are given by following relationships:
where γ is the polarization angle with γ = 0 corresponding to p-polarized light. The si terms depend on the experimental geometry are functions of the total internal reflection angles of the incident and second harmonic beams and the linear and nonlinear Fresnel factors respectively which relate the electric field components at the interface to incident and detected fields.
The second-order susceptibility tensor, χ(2), is the parameter which can be measured in second order experiments, but it does not explicitly provide insight to the molecular orientation of surface molecules. To determine molecular orientation, the second-order hyperpolarizability tensor β, must be calculated. For adsorbed molecules in a uniaxial distribution, the only independent hyperpolarizability tensor terms are βz’z’z’, βz’x’x’, and βx’x’z’ where ’ terms denote the molecular coordinate system as opposed to the laboratory coordinate system. β can be related to χ(2) through orientational averages. As an example, in an isotropic distribution on the surface, χ(2) elements are given by.
where Ns is the surface number density of the adsorbed molecules, θ and Ψ are orientational angles relating the molecular coordinate system to the laboratory coordinate system, and
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