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Urbach tails are the exponential band tails observed universally in impure crystals and disordered systems. Evidence has been provided that the topological origin of the Urbach tails in amorphous materials are filaments formed by short or long bonds[20]. One aspect of my work focuses on the size effects and choice of Hamiltonian with respect to the structure of the Urbach tails. The dynamical properties of filaments have been studied by performing Molecular Dynamics simulation under constant temperature. The response of filaments under external pressure has also been explored. The second portion of this dissertation is about carbon in two-dimensional sp2 phases. Carbon has shown itself to be the most flexible of atoms, crystallizing in divergent phases such as diamond and graphite, and being the constituent of the entire zoo of (locally) graphitic balls, tubes, capsules and possibly negative curvature analogs of fullerenes, the Schwartzites. In this part, we explore topological disorder in three-coordinated networks including odd-membered rings in amorphous graphene, as seen in some experimental studies. We start with the Wooten-Weaire-Winer models due to Kumar and Thorpe, and then carry out ab-initio studies of the topological disorder. The structural, electronic and vibrational characteristics are explored. We show that topological disorder qualitatively changes the electronic structure near the Fermi level. The existence of pentagonal rings also leads to substantial puckering in an accurate density functional simulation. The vibrational modes and spectra have proven to be interesting, and we present evidence that one might detect the presence of amorphous graphene from a vibrational signature. We also explore the energy landscape of amorphous graphene and report the eigenstates near the Fermi level.

Significant progress was achieved with the application of Rietveld method to characterize the crystalline, and nanocrystalline phases in portland cement concrete. However, to obtain detailed information on the amorphous phases present in concrete, it is necessary to analyze the total scattering data. The pair distribution function (PDF) method, that takes the Sine Fourier transform of the measured structure factor over a wide momentum transfer range, has been successfully utilized in the study of liquids and amorphous solids. The method provides a direct measure of the probability of finding an atom surrounding a central atom at a radial distance away. The obtained experimental characteristic distances can be used to validate the predictions by the theoretical models, such as, molecular dynamics simulations. This research focuses on obtained results of PDF analysis on silica fume, rice husk ash, fly ash, alkali-silica reaction gel, and opals.