Additivity of vibrational density of states in two-dimensional mesoscopic systems

We investigate the vibrational density of states (DOS) in two-dimensional (2D) composite systems with nonhomogeneous geometry. The following three objects are selected as case studies: (i) the union between a 1D and a 2D crystallite; (ii) the union of a mass fractal with a 2D crystallite; and (iii) the union of a surface fractal with a 2D crystallite. In each case, it is found that the DOS of the composite system is, within a very good approximation, equal to the sum of the DOS of the components. This indicates the absence of a long-range contribution to the DOS of 2D macroscopic systems. This quantity can therefore be directly evaluated from the simple average of the DOS of its tessellated mesoscopic elements. The calculation of the vibrational DOS of a macroscopic solid can then be reduced to a feasible computational operation.