Metal-nonmetal transition in metal-ammonia solutions

In this paper we present a coherent physical picture of the metal-nonmetal transition in metal-ammonia solutions in the intermediate concentration range. We propose that in Li-N${\mathrm{H}}_3$ and Na-N${\mathrm{H}}_3$ solutions the metallic propagation regime is separated from a nonmetallic regime by a microscopically inhomogeneous regime in which the concentration fluctuates locally about either of two well-defined values $M_0$ and $M_1$, $M_0>M_1$, the local concentration remaining near $M_0$ or $M_1$ over radii approximately equal to the Debye short correlation length $b$ for concentration fluctuations. Provided that the concentration-fluctuation decay length is much smaller than $b$, we can define a percolation problem in which a volume fraction $C$ of the material is occupied by metallic regions of concentration $M_0$, the remainder containing the low concentration $M_1$ of dissociated electron-cation complexes. $M_0$ and $M_1$ constitute the upper and the lower bounds of the inhomogeneous regime, respectively, while $C$ exhibits a linear dependence on $M$. This physical picture is borne out by concentration-fluctuation determinations based on chemical-potential measurements in Li and Na solutions and by small-angle x-ray and neutron scattering in Li solutions. Assuming that the phase-coherence length of the conduction electrons is shorter than $b$ and having demonstrated that tunneling corrections are negligible, we can define local electronic structure and transport properties. The limits of the inhomogeneous regime were determined from a combination of concentration-fluctuation measurements, electrical conductivity, Hall effect, and paramagnetic susceptibility data to be $M_0=9\mathrm{}$ mole percent metal (MPM) and $M_1=2\left(\frac{1}{3}\right)$ MPM, which yield the $C$ scale, $C=\frac{[M-2(\frac{1}{3}\left)\right]}{6\left(\frac{2}{3}\right)}$, for both Li-N${\mathrm{H}}_3$ at 223°K and for Na-N${\mathrm{H}}_3$ at 240°K. We have also established the consistency of our picture with the available magnetic data for Na solutions. An analysis of the electronic and the thermal transport properties was carried out in terms of an effective-medium theory, modified to account for scattering from the boundaries of the metallic clusters. For low values of the conductivity ratio (∼ ${10}^{-3\mathrm{}}$) between the nonmetallic and the metallic regions the modified effective-medium theory is valid for $C>\mathrm{}0.4\mathrm{}$. In an attempt to mimic the features of continuous percolation, we have carried out numerical simulations of the conductivity in a simple cubic lattice incorporating correlation between metallic bonds. An excellent fit of the experimental conductivity data for Li and Na with the results of the numerical simulations has been obtained over a three order of magnitude variation of the conductivity throughout the entire inhomogeneous regime. A small systematic negative deviation of the conductivity from the predictions of the effective-medium theory for $C>\mathrm{}0.4\mathrm{}$ can be properly accounted for in terms of boundary scattering corrections resulting in $b\simeq 15$ Å for Li at 223°K and $b\simeq 32$ Å for Na at 240°K. The overall agreement of the experimental Hall effect, Hall mobility, thermalconductivity, and thermoelectric-power data with the effective-medium theory is good. The proposed inhomogeneous regime in Li and Na solutions resembles a macroscopic mixed phase at a concentration inside a coexistence curve but with mixing on a microscopic scale. The concentration fluctuations in the inhomogeneous state have nothing to do with critical fluctuations; nevertheless, this state seems to be closely associated with the occurrence of a phase separation.