The

Abstract

In this thesis, we have investigated the lowest-energy members of the quantized intrinsically localized modes of vibration (ILMs) of the monatomic ß Fermi-Pasta-Ulam Hamiltonian in three-dimensions. We analytically find the excitation of different center of mass momenta. Using the Ladder Approximation, we find that the ILMs occur preferentially for centre of mass momenta at which the van-Hove singularities in the two-phonon density of states coalesce. When the ILMs first form they split off from the top of the two-phonon continuum. The ILMs can be categorized as having a spin of either S=2 or S=0 and have other internal quantum numbers. Moreover, the S=0 ILMs form for lower values of the interaction than the S=2 ILMs. We also focus on the temperature dependence of the ILMs. At zero temperature, the ILMs can form in three-dimensions, but only if the interaction exceeds a minimum value. As the temperature is raised, the magnitude of the minimal interaction required to stabilize the ILM is reduced. This is in a qualitative agreement with the experiments of Manley {\it et al.}, which only found the ILMs of NaI at elevated temperatures. We have also examined the ILM many-body wave functions and find that the relative coordinate part of the wave functions has symmetries associated with internal quantum numbers. According to our numerical results, the localization length increases with decreasing values of the strength of interaction. The results are presented in D. Kanbur and P. S. Riseborough, Phil. Mag. Letts, 94, 424-432 (2014) and D. Kanbur and P. S. Riseborough, Phys. Rev. B, 90, 134301 (2014). This work was supported by the US Department of Energy, Office of Basic Energy Science, Materials Science and Engineering through the award DEFG02-84ER45872.