Analysis of Lattice Vibrations in Two Dimensions
Analysis of Lattice Vibrations in Two Dimensions
At first glance, the hexagonal and square lattices appear to have the same symmetry as their unit cells (a simple hexagon or square), but closer examination reveals more rotational and reflectional symmetries, not to mention a countably infinite number of translations. Of course, a real crystal lattice deviates from perfect symmetry as it interacts with photons in energy transfer that results in the creation of discrete vibrational states.
Simple restoring force models of linear crystals can be extended to planar crystals by considering angles as well as edges. Resolving the eigenvalues and eigenvectors of a Hessian matrix can help to understand how superpositions of vibrational modes relate to possibly observable spectra. Only some of the computable values will be observed in the laboratory. The best predictions of observable transitions also consider the symmetry of the vibrational modes, as determined by their equivalency class.