For a light scattering instrument composed of many detectors placed at various angles, all the detectors need to respond the same way. Usually, detectors will have slightly different quantum efficiency, different gains, and are looking at different geometrical scattering volumes. In this case, a normalization of the detectors is absolutely needed. To normalize the detectors, a measurement of a pure solvent is made first. Then an isotropic scatterer is added to the solvent. Since isotropic scatterers scatter the same intensity at any angle, the detector efficiency and gain can be normalized with this procedure. It is convenient to normalize all the detectors to the 90° angle detector.
where I(90) is the scattering intensity measured for the Rayleigh scatterer by the 90° angle detector.
The most common equation to measure the weight-average molecular weight, M, is the Zimm equation (the right-hand side of the Zimm equation is provided incorrectly in some texts, as noted by Hiemenz and Lodge):
where
and
with
and the scattering vector for vertically polarized light is
with n the refractive index of the solvent, λ the wavelength of the light source, N the Avogadro constant, c the solution concentration, and dn/dc the change in the refractive index of the solution with change in concentration. The intensity of the analyte measured at an angle is I(θ). In these equations, the subscript A is for analyte (the solution) and T is for the toluene with the Rayleigh ratio of toluene, R being 1.35×10 cm for a HeNe laser. As described above, the radius of gyration, R, and the second virial coefficient, A, are also calculated from this equation. The refractive index increment dn/dc characterizes the change of the refractive index n with the concentration c and can be measured with a differential refractometer.
A Zimm plot is built from a double extrapolation to zero angle and zero concentration from many angles and many concentration measurements. In its simplest form, the Zimm equation is reduced to:
for measurements made at low angle and infinite dilution since P(0) = 1.
There are typically several analyses developed to analyze the scattering of particles in solution to derive the above-named physical characteristics of particles. A simple static light scattering experiment entails the average intensity of the sample that is corrected for the scattering of the solvent will yield the Rayleigh ratio, R as a function of the angle or the wave vector q as follows: