Studies of mixing formulae in the complex plane

Abstract

Remote sensing signal monitoring is considered. Mixing rules that have been introduced to explain the permittivity of mixtures by analytically continuing these to the whole complex plane are discussed. Equipermittivity curves for both the real and imaginary parts of the mixture dielectric constant (the imaginary part of the permittivity is a measure for the dielectric losses) are studied. These form a map on the complex plant, illustrating the nature of a mixing rule better than the classical way of presenting the mixture permittivity as a function of the volume fraction of a component in the mixture. These maps naturally vary for a given mixing rule as the volume fraction is changed, but, in principle, the structure is the same. The resulting illustrations show that certain mixing formulae can be conceived as a continuum that gradually changes from the Maxwell-Garnett formula through Polder-van Santen and coherent potential formulas to the so-called exponential formulas.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>