The shape of complex impedance plane arcs for ionically conducting glasses

Abstract

The complex resistivity (ϱ ∗) plots have been calculated for a conducting Cole-Cole dielectric, for the Fourier transform of the fractional power decay law, exp(− t b ), and for Dyre's uniform distribution law. It is shown that a depressed circular arc in the ϱ ∗ plane can never truly represent the properties of a real material, since the high frequency permittivity ϵ ∞ is not included. Nevertheless the calculated arcs are mostly not far from circular. For the Cole-Cole dielectric the shape is found to depend as much on the ratio ϵ ∞/( ϵ s − ϵ ∞) and on Namikawa's p = ( ϵ s − ϵ ∞) ϵ 02 πf m / σ 0 as on the Cole-Cole α. For the exp(− t b ) law, the ϱ ∗ arc becomes increasingly depressed as b decreases. It is shown that a more accurate extrapolation for a limited range of frequency points to the real axis at low frequencies maybe obtained by fitting an on-axis ellipse rather than a depressed circular arc.