Elastic Spectra of Two-Dimensional Disordered Lattices

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 and interacting with force constants $\alpha$, which determines the resistance to compression, and $\beta$, which determines the resistance to shear along the direction [10]. In one case, the perturbations are $\mathrm{Nq}$ randomly positioned isotopic impurities of mass $m^{\prime }$, where $N$ is the number of lattice sites. It is shown that the elastic spectrum for this and all other isotopic impurity systems is completely determined by the average mass, $\overline{m}=(1-q)m+q{m}^{\prime }$. In the other case, corresponding to certain order-disorder situations, the constants describing the interactions between $2Nq$ randomly positioned pairs of nearest neighbors are replaced by ${\alpha }^{\prime }$ and ${\beta }^{\prime }$. To first order in $q$, the resulting elastic modes are then completely determined by an average $\alpha$, $\overline{\alpha }=\alpha \left[1+\frac{q({\alpha }^{\prime }-\alpha )}{\alpha +({\alpha }^{\prime }-\alpha )\left(\frac{2}{\pi }\right) invtan{\left(\frac{\alpha }{\beta }\right)}^{\frac{1}{2}}}\right]$ 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. It is shown, however, that two different forms of the virtual-crystal approximation place bounds on the $\overline{\alpha }$ and $\overline{\beta }$, in accordance with a general theorem of Paul. A physical interpretation of the results is also presented.