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 α, which determines the resistance to compression, and β, which determines the resistance to shear along the direction [10]. In one case, the perturbations are Nq randomly positioned isotopic impurities of mass m^', 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, m̄=(1-q)m+qm^'. 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 α^' and β^'. To first order in q, the resulting elastic modes are then completely determined by an average α, ${{α}}={α}[1+{q({{α}}^{{'}}{-}{α})}{{α}+({{α}}^{{'}}{-}{α})({2}{{π}}) invtan{({{α}}{{β}})}1/2}]$ and a similar average, ${{β}}$, which is obtained from the previous expression via the interchange of ${α}$ and ${β}$. The appearance of ${β}$ in the expression for ${{α}}$ implies that the virtual-crystal approximation fails. It is shown, however, that two different forms of the virtual-crystal approximation place bounds on the ${{α}}$ and ${{β}}$, in accordance with a general theorem of Paul. A physical interpretation of the results is also presented.