Relaxation in systems with exponential or Gaussian distributions of activation energies

Abstract

New expressions are presented, simplified, and discussed for the small-signal-frequency response of systems involving distributions of activation energies with either exponential or Gaussian probability densities. The results involve the possibility of separate but related thermal activation of energy-storage and energy-loss processes, and apply to the response of both dielectric and conductive systems. Response with a Gaussian distribution of activation energies (GDAE) may be either symmetric or asymmetric in log frequency, and typical GDAE responses are compared with those associated with several exponential distributions of activation-energy (EDAE) models, using complex nonlinear least-squares fitting. The GDAE model does not lead to the frequently observed fractional-exponent power-law response in time or frequency as does the EDAE; thus, the GDAE cannot fit any EDAE response well which involves an appreciable range of such behavior, but it is found that, conversely, the general EDAE model can often fit a GDAE response very well over a wide frequency range. Recent (KBr)0.5(KCN)0.5 dielectric data covering a range from T=13.7 to 34.7 K are analyzed with the Cole–Cole, EDAE, and GDAE models, and the GDAE is found to yield the best overall fits. The results of the GDAE fits are analyzed in detail to illustrate the application of the GDAE model to real data. Contrary to the conclusions of an earlier analysis of the same data using an idealized, symmetric, and approximate GDAE model, we find that much of the data are better fit by a somewhat asymmetric, exact GDAE model which may involve a temperature-independent, finite-width Gaussian probability density. The present analysis suggests an alternative to the earlier results and suggestions that the width of the probability-density distribution increases with decreasing temperature and that the activation energies or barrier heights themselves depend linearly on temperature. The present data fit yield estimates of the lower limit of the temperature independent distribution of activation energies E0 and of the more or less central activation energy E1, but only set a lower limit for the value of the maximum activation energy of the distribution, E∞. There is some evidence from the fitting that there may be a glasslike transition below about 4 K, but other effects outside the GDAE model may intervene before that temperature region is reached.