Relating macroscopic electrical relaxation to microscopic movements of the ions in ionically conducting materials by theory and experiment

Abstract

Whatever the preferred representation of the frequency dependence of electrical relaxation experimental data, complex conductivity ${\ensuremath{\sigma}}^{*}(\ensuremath{\omega}),$ complex permittivity ${\ensuremath{\varepsilon}}^{*}(\ensuremath{\omega}),$ or complex electric modulus ${M}^{*}(\ensuremath{\omega}),$ there is no escape from the fact that we are dealing with measurements which are macroscopic in nature. The question of how to relate the macroscopic measurement, which contains the high-frequency dielectric constant ${\ensuremath{\varepsilon}}_{\ensuremath{\infty}},$ to the microscopic movement of the ions remains to be answered. Comparing the results of a stochastic transport theory and of the electric modulus formalism, we find that the electric modulus faithfully reproduces the shape of the dispersion of the microscopic ionic movement. However, the electric modulus relaxation time is different from the microscopic relaxation time by a known and calculable factor that is proportional to the product of the high-frequency dielectric constant and temperature. Consequently, the entire electric modulus relaxation time spectrum is shifted uniformly away from the microscopic ion relaxation time spectrum by the same frequency-independent factor, and these two relaxation time spectra have effectively the same dependence on temperature, isotope mass, etc. In contrast to electrical conductivity relaxation, nuclear spin relaxation is a microscopic probe of ionic movement, and from its result we can directly infer the microscopic dynamics of the ions. A combined study of ionic motion using electrical relaxation and nuclear spin relaxation in a crystalline ionic conductor by Le\'on et al. provides the experimental data to enable us to verify the theoretical relation between the macroscopic electric modulus spectrum and the microscopic ionic hopping relaxation spectrum.