TPDSci Problems Power law in spectral scattering

Problem: Spectral attenuation by a dispersion with a power-law PSD

A model of the attenuation coefficient spectrum of a dispersion of particles (Eq. 4 in Mie theory: Integration of particle size-dependent patterns for a power-law PSD) can be recast in the following simplified form:

c(λ) = const λ^{3 - sD} ∫_xmin^xmax F(x, m) dx

(1)

where λ is the wavelength of light in the dispersion medium, s_D is the slope of the power-law PSD of the dispersion, x_min = π D_min / λ, x_max = π D_max / λ, where D is the particle size, and F denotes the integrand, a function of the relative particle size, x, and the refractive index, m, of the particles. This equation implies an explicit power-law dependence of the attenuation coefficient spectrum, with a slope of 3 - s_D, referred to as the Ångström law (for example, Heitzenberg and Charlson 1996).

Problems

Identify sources of the wavelength dependence of the attenuation coefficient, c, implied by Eq. 1.
Under what conditions is the wavelength dependence of c expressed by the term: λ^{3 - sD} ? Does Eq. 1 represent a spectrum of c of a real dispersion under all these conditions?
What is the sign of the power law exponent, 3 - s_D, for the attenuation spectra of typical natural dispersions, such as atmospheric aerosols and suspensions in natural waters? (Hint topic).
Under what condition(s) is the wavelength dependence of c likely to change from that expressed by the term: λ^{3 - sD} ?

CITATION:
Jonasz M., Twardowski M. S. 2006. Problem: Spectral attenuation by a dispersion with a power-law PSD (www.tpdsci.com/Tpc/AtnCfSptPwLw_P.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 23-Sep-2006
Modified: 28-Sep-2006
Peer-reviewed: 13-Nov-2006

Topics in Particle and Dispersion Science