Theoretical Investigations on the Light Scattering of Spheres. XIII. The ``Wavelength Exponent'' of Differential Turbidity Spectra

Abstract

The exact theory of the ``wavelength exponent'' of the turbidity is developed for Rayleigh scattering and ``Debye scattering'' and, particularly, for Mie scattering by nonabsorbing spheres. The basic exponent, n, defined by $\mathrm{\{\alpha [\partial }\ln {\mathrm{(\tau /c)}}_0\text{/∂α]+1}}$ is computed from Mie turbidity data, reported previously, for α=0.40 (0.04) 0.68, 0.80 (0.2) 25.0 and m=1.05 (0.05) 1.20. For m=1.25 and 1.30, the smallest α values and the Δα intervals are the same, but the upper limiting α values are 14.0 and 11.6, respectively. In addition, α values<0.40 are considered for m=1.05. The actual exponent to be expected in a given dispersed system n₀ is found to depend on three additional factors: (1) The rate of change of the turbidity with the relative refractive index of the spheres [this factor is evaluated for α=0.4, 1.0 (1.0) 7.0 and m=1.05 (0.05) 1.30 and approximating equations are given for purposes of interpolation]. (2) The dispersion of the relative refractive index of the spheres. (3) That of the refractive index of the medium. The advantages and limitations of an application of the wavelength exponent to particle size determinations are discussed and labor saving approximating n(α) relations, suitable for first approximation results on particle size, are given. The effect of heterodispersion upon the wave length exponent is briefly discussed.