Elastic Spectra of Two-Dimensional Disordered Lattices. II. Impurities which Alter the Interactions

Abstract

The elastic vibrational spectra of perturbed square-lattice systems with nearest-neighbor central and noncentral interactions have been derived. The unperturbed system consists of masses $m$ on the lattice points interacting with force constants $\alpha$, which determines the resistance to compression, and $\beta$, which determines the resistance to shear along the direction [10]. The perturbations are $\mathrm{Nq}$ randomly positioned atoms whose mass is that of the host, but which interact with each of their nearest neighbors through spring constants ${\alpha }^{\prime }$ and ${\beta }^{\prime }$. Here $N$ is the number of lattice sites. To first order in $q$, the resulting elastic modes are then completely determined by an average: $\overline{\alpha }=\alpha \left(1+\frac{2q({\alpha }^{\prime }-\alpha )}{\alpha +\left(\frac{2}{\pi }\right)({\alpha }^{\prime }-\alpha )\left\{\right[1+\frac{\beta }{\alpha }]invtan{\left(\frac{\alpha }{\beta }\right)}^{\frac{1}{2}}-{\left(\frac{\beta }{\alpha }\right)}^{\frac{1}{2}}\}}\right),$ and a similar average, $\overline{\beta }$, which is obtained from the previous expression via the interchange of $\alpha$ and $\beta$. The appearance of $\beta$ in the expression for $\overline{\alpha }$ implies that the virtual-crystal approximation fails. The failure of the virtual-crystal approximation is related to a physical model which predicts a relationship between this failure and the lifetimes of the modes. This relationship is demonstrated for the two-dimensional binary alloy with a small amount of disorder.