Experimental Observation
The optical absorption spectrum of a solid is most straightforwardly calculated from the electronic band structure using Fermi's Golden Rule where the relevant matrix element to be evaluated is the dipole operator where is the vector potential and is the momentum operator. The density of states which appears in the Fermi's Golden Rule expression is then the joint density of states, which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy. The optical absorption is then essentially the product of the dipole operator matrix element (also known as the oscillator strength) and the JDOS.
The divergences in the two- and one-dimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like graphite (quasi-2D) and Bechgaard salts (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities. Van Hove singularities play a significant role in understanding optical intensities in single-walled nanotubes (SWNTs) which are also quasi-1D systems. The Dirac point in graphene is a Van-Hove singularity that is can be seen directly as a peak in electrical resistance, when the graphene is charge-neutral.
Read more about this topic: Van Hove Singularity
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