On the effective permittivity of grainy systems

High permittivity grainy materials such as ferroelectric and related perovskite ceramics and thin films, in which grain boundaries and/or microscopic cracks occupying very small fraction of the sample volume can result in strong reduction of permittivity and increase of effective phonon frequencies. It is shown how the effective permittivity of such grainy systems can be theoretically described using the Bergman representation. The grainy system is considered as a two-component mixture of close-packed but non-percolated grains and sparse but percolated grain boundaries (or cracks). The response is controlled mainly by the depolarizing factor which is given by the mean value of the spectral function, while the width of the spectral function determines validity criterion of the previously proposed generalized brick model.