Microscopic Theory of Rayleigh Scattering

A self-consistent, microscopic theory of scattering of light by molecular aggregates is evolved from the standpoints of random-phase-modulation theory and stochastic theory. Contemporary theory is founded on the premise that the scattered light spectrum is proportional to a four-dimensional Fourier transform of the molecular density-correlation function. This premise is justified only in a continuum representation of density, but it breaks down when the motion of discrete molecules is taken into account. The Rayleigh spectrum is shown to be a manifestation of translational degrees of freedom of molecules, much as the Raman spectrum is of internal degrees. Both the central and shifted components are attributed to propagating waves representing probability densities. Theoretical spectra are in agreement with experimental data in both the kinetic and hydrodynamic regimes, and the shifted frequencies are simply related to the rms speed of typical molecules. This theory also provides a mechanism which could account for deviations of total integrated intensity from that predicted by incoherent scattering. Such deviations are simply related to the ratio of intensities of the shifted and central components of the spectrum. Success of this theory, however, is not achieved without complete departure from conventional approaches of kinetic theory. By necessity, statistical aspects of the problem are approached through the use of a set of partial differential equations for the probability densities of continuous, differentiable stochastic processes. Statistical trajectories of molecules are characterized by a single function $h\left(\tau \right)$ defined as the logarithmic derivative of the conditional expectation value of velocity. Solutions based on the asymptotic behavior of $h\left(\tau \right)$ suggest the possibility for existence of other lines at the high-frequency end of the Rayleigh spectrum.