A theoretical approach has been developed to model the vibrational modes of amorphous, two-dimensional materials. The method considers that the vibrational density of states is composed primarily of states originating from embedded ring structures of medium-range order. The materials are modeled as continuous random networks comprised of a statistical distribution of symmetric, planar rings with four to eight members. The rings are treated as local structural units embedded in the material, similar to molecules within a solid. The ring potentials are approximated with a valence force model (bond-stretching and bond-angle-bending force constants) modified by a third harmonic, effective force constant coupling the rings to the surrounding network. The molecular dynamics of the rings are analyzed with the use of group theory, and the frequencies are calculated using a normal coordinate treatment. The utility of the embedded ring approach lies in its use of the material's ring statistics and ring mode frequencies, allowing determination of the structure of an amorphous material from its vibrational spectrum or prediction of vibrational spectra and density of states from structural models. The method is applied here to various graphitic carbon materials as a test case and predicts a set of force constants and Raman spectra consistent with previous data and structural models.
JR Dennison and T. E. Doyle, "An Embedded Ring Approach to the Vibrational Dynamics of Amorphous Materials," Carbon, 35(10-11) 1465-1477 (1997).