Correlated diffusion in quasi-low-dimensional systems

The space- and time-dependent tracer occupancy correlation is solved for a general quasi-low-dimensional system with arbitrary concentration of hopping ions. Our theory is exact at both small and large ionic concentrations and gives reasonable results for all intermediate concentrations. Detailed numerical results for the diffusion correlation factor $f^0$, and the incoherent neutron scattering function $S(K,\omega )$, are provided for a variety of cases as the system changes gradually from three dimensional to one dimensional (1D), and vice versa. Similarly, the corresponding results associated with travel from two dimensions to 1D, and vice versa, are also recorded. Analytical expressions for $f^0$ as a function of the direction cosines of the wave vector are worked out for all limiting cases as the system approaches one, two, or three dimensionality. In particular, it is found that as one dimensionality is approached, the diffusion correlation factor approaches zero in proportion to the quantity $c^{-1\mathrm{}}{\left(\frac{J_{\perp }}{J_{\parallel }}\right)}^{\frac{1}{2}}$.