Interpretation of light-scattering spectra in terms of particle displacements

Quasielastic light-scattering spectroscopy is regularly used to examine the dynamics of dilute solutions of diffusing mesoscopic probe particles in fluids. For probes in a simple liquid, the light-scattering spectrum is a simple exponential; the field correlation function g(1)(q,tau) of the scattering particles is related to their mean-square displacements X2 identical with [(delta x(tau))2] during tau via g(1)(q,tau) = exp(-1/2 q2X2). However, demonstrations of this expression refer only to identical Brownian particles in simple liquids and show that if the form is correct then it is also true for all tau that g(1)(q,tau) = exp(-gamma tau), a pure exponential in tau. In general, g(1)(q,tau) is not a single exponential in time. A correct general form for g(1)(q,tau) in terms of the X(2n), replacing the incorrect exp(-1/2 q2X2), is obtained. A simple experimental diagnostic determining when the field correlation function gives the mean-square displacement is identified, namely, g(1)(q,tau) only reveals X2 if g(1)(q,tau) is a single exponential in tau. Contrariwise, if g(1)(q,tau) is not a single exponential, then g(1)(q,tau) depends not only on X2 but on all higher moments X(2n). Corrections to the crude approximation g(1)(q,tau) = exp(-1/2 q2X2) closely resemble the higher spectral cumulants from a cumulant expansion of g(1)(q,tau).