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Heat is transfered in crystalline semiconductor materials via lattice vibrations. Lattice vibrations are treated with a wave-particle duality just like photons are quantum mechanical representations of electro-magnetic waves. The quanta of energy of these lattice waves are called phonons. The Boltzmann Transport Equation (BTE) has proved to be a powerful tool in modeling the phonon heat conduction in crystalline solids. The BTE tracks the phonon number density function as it evolves according to the drift of all phonons and to the phonon-phonon interactions (or collisions). Unlike Fourier's law which is limited to describing diffusive energy transport, the BTE can accurately predict energy transport in both ballistic (virtually no collisions) and diffuse regimes. Motivated by the need to understand thermal transport in irradiated Uranium Dioxide at the mesoscale, this work investigates phonon transport in UO2 using Monte Carlo simulation. The simulation scheme aims to solve the Boltzmann transport equation for phonons within a relaxation time approximation. In this approximation the Boltzmann transport equation is simplified by assigning time scales to each scattering mechanism associated with phonon interactions. The Monte Carlo method is first benchmarked by comparing to similar models for silicon. Unlike most previous works on solving this equation by Monte Carlo method, the momentum and energy conservation laws for phonon-phonon interactions in UO2 are treated exactly; in doing so, the magnitude of possible wave vectors and frequency space are all discretized and a numerical routine is then implemented which considers all possible phonon-phonon interactions and chooses those interactions which obey the conservation laws. The simulation scheme accounts for the acoustic and optical branches of the dispersion relationships of UO2. The six lowest energy branches in the  direction are tracked within the Monte Carlo. Because of their predicted low group velocities, the three remaining, high-energy branches are simply treated as a reservoir of phonons at constant energy in K-space. These phonons contribute to the thermal conductivity only by scattering with the six lower energy branches and not by their group velocities. Using periodic boundary conditions, this work presents results illustrating the diffusion limit of phonon transport in UO2 single crystals, and computes the thermal conductivity of the material in the diffusion limit based on the detailed phonon dynamics. The temperature effect on conductivity is predicted and the results are compared with experimental data available in the literature.
Boltzmann, Carlo, Monte, Phonon, Thermal, Transport
Date of Defense
November 7, 2011.
A Thesis submitted to the Department of ScientiﬁC Conmputing in partial fulfillment of the requirements for the degree of Master of Science.
Includes bibliographical references.
Anter El-Azab, Professor Directing Thesis; Tomasz Plewa, Committee Member; Xiaoqiang Wang, Committee Member.
Florida State University
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