In words, the theorem states that
Although the spectral function G(L) is generally
unknown for an arbitrary two-phase composite, it's analytically known
or can be numerically derived for any existing mixing rule
(Stognienko et al. 1995).
Because each spectral function has to be non-negative, normalized
to unity in the interval [0,1], and obey (for isotropic systems) the
first moment equation
,
the derivation of G(L) and
the check whether these restrictions are fulfiled or not is useful to valididate any mixing rule.
Note, there are mixing rules in the literature which are not correct
with respect to the Bergman spectral representation.
E.g. the spectral function for the mixing rule by C.J.F. Böttcher
(in: "Theory of Electric Polarization", Elsevier, Amsterdam, p. 415 [1952])
which fulfiles only the normalization restriction.
In the following table some mixing rules and their corresponding spectral functions are listed.
Mixing rule | Spectral function |
---|---|
Bruggeman (D.A.G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen", Ann. Phys. (Leipzig) 24, 636-679 [1935]) | |
Maxwell Garnett (J.C.M. Garnett, "Colours in metal glasses and in metallic films", Phil. Trans. R. Soc. Lond. 203, 385-420 [1904]) | |
Looyenga (H. Looyenga, "Dielectric constants of heterogeneous mixtures", Physica 31, 401-406 [1965]) | |
PCA/CCA aggregate equivalent (Stognienko et al. 1995, Henning and Stognienko 1996) | |
Monecke (J. Monecke, "Bergman spectral representation of a simple expression for the dielectric response of a symmetric two-component composite", J. Phys.: Cond. Mat. 6, 907-912 [1994]) | |
Hollow sphere equivalent (see, e.g., C.F. Bohren and D.R. Huffman "Absorption and Scattering of Light by Small Particles", Wiley, New York, p. 149 [1983]) |