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Craig F. Bohren

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Craig F. Bohren

Abstract

Effective-medium theories yield effective dielectric functions (or, equivalently, refractive indices) of composite media. Such theories have been formulated that go beyond the Maxwell-Garnett and Bruggeman theories, which art restricted to media composed of grains much smaller than the wavelength. These extended effective-medium theories do not, however, yield effective dielectric functions that can be used for the same purposes for which we unhesitatingly use the dielectric functions of substances such as pure water and pure ice (e.g., reflection and transmission by smooth interfaces; absorption and scattering by particles). Extended dielectric functions can lead to unphysical results; for example, absorption in composite media with nonabsorbing components. Moreover, if the grains in composite media are large enough to give rise to magnetic dipole and higher-order multipole radiation, then the effective permeability of the composite medium cannot be taken to be that of free space even if the grains are nonmagnetic.

Recently, extended effective-medium theories have been applied to the problem of determining, the effective dielectric function of ice in which soot grains are embedded in order to explain a factor of 2 discrepancy between measurements of the albedo of soot-contaminated snow and calculations based on a snow albedo model. Setting aside questions about the applicability of these theories, reasonable alternative explanations for the discrepancy exist: (i) Soot is not an invariable substance; measured refractive indices of carbonaceous materials vary appreciably, especially the imaginary part (about a factor of 5). (ii) Absorption by a small soot particle depends on its shape, varying by as much as a factor of 2. (iii) Absorption by a soot particle may be enhanced by porosity; for a fixed particle volume, the enhancement is roughly proportional to the porosity. To predict exactly how much a given amount of soot reduces the visible albedo of snow requires, therefore, more detailed information about the soot than is likely to be readily obtainable.

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Craig F. Bohren
and
Louis J. Battan

Abstract

Calculations of radar backscattering by inhomogeneous precipitation particles require values of the dielectric function of two-component mixtures. Four such dielectric functions are critically examined and their relative merits are weighed. Although apparently different, two are shown to be equivalent: the effective-medium and Polder-van Santen theories. All the dielectric functions agree when the two components are dielectrically similar. All except the Maxwell-Garnet dielectric function are symmetric with respect to interchange of the components. When compared with measurements on ice-air mixtures, the effective-medium and Maxwell-Garnet dielectric functions are marginally better than the Debye function, which has previously been used in backscattering calculations. When the fraction of water is high, the effective-medium function gives calculated values of radar backscattering that are in good agreement with measurements on ice spheres coated with a mixture of ice and water. The Maxwell-Garnet theory, with ice inclusions embedded in a water matrix, is also in good agreement with these measurements, and is so over a wider range of water-volume fractions than the effective-medium theory. Although there are no compelling reasons for preferring one above the other, on the basis of the evidence presented, we would be inclined to use the Maxwell-Garnet dielectric function in radar backscattering calculations.

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Craig F. Bohren
and
Louis J. Battan

Abstract

The radar backscattering cross section of a spongy ice hailstone—a mixture of ice and liquid water—depends on its size, shape and dielectric function. There are two types of theories of the effective dielectric function of two-component mixtures: Maxwell-Garnet and Bruggeman theories. In the latter, the two components are treated symmetrically, whereas in the former they are not. We have generalized the Maxwell-Garnet expression, originally derived for spherical inclusions in an otherwise homogeneous matrix, to ellipsoidal inclusions. When this expression, with all ellipsoidal shapes equally probable, is used in calculations of radar backscattering by ice spheres coated with spongy ice, the results are in generally good agreement with measured cross sections. Agreement is better if the inclusions are ellipsoidal rather than spherical, but only slightly so. Shape considerations are less important than taking ice to be the inclusions and liquid water to be the matrix.

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Craig F. Bohren
,
Jeffrey R. Linskens
, and
Michael E. Churma

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