Universality, the Barton Nakajima Namikawa relation, and scaling for dispersive ionic materials

Many frequency-response analyses of dispersive relaxation for homogeneous glasses, polycrystalline materials, and single crystals involving mobile ions of a single type indicate that estimates of the β₁ shape parameter of the Kohlrausch K1 fitting model are close to $1∕3$ and are virtually independent of both temperature and ionic concentration. This model, which usually yields better fits than others, including the closely related Kohlrausch K0 one, is indirectly associated with temporal-domain stretched-exponential relaxation having the same β₁ parameter value. Here it is shown that for the above conditions several different analyses all yield a value of $1∕3$ for the β₁ of the K1 model. It is therefore appropriate to fix the β₁ parameter of this model at the constant value of $1∕3$, then defined as the U model. It fits data sets exhibiting conductive-system dispersion that vary with both temperature and concentration just as well as the K1 model with β₁ free to vary, and it leads to a correspondingly universal value of the Barton-Nakajima-Namikawa parameter p of 1.65. Composite-model complex-nonlinear-least-squares fitting, including the dispersive U model, the effects of the bulk dipolar-electronic dielectric constant ε_D∞ and of electrode polarization when significant also lead to estimates of two U-model hopping parameters that yield optimum scaling of experimental data involving temperature and concentration variation.