No description available.

<p> Working within the framework of the linearized Boltzmann equation for the conduction electrons the existing theoretical treatments of ultrasonic attenuation in metals are extended to include realistic descriptions of the electronic structure and electron-lattice interaction. A variational solution of the Boltzmann equation which allows the inclusion of phonon drag effects is derived. An anisotropic scattering time solution is also presented. Both of these solutions are applied to calculation of the attenuation coefficient in pure metals and dilute alloys. </p> <p> The theory of the effects of electron-electron collisions on the ultrasonic attenuation in metals is also examined. </p> / Thesis / Doctor of Philosophy (PhD)

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A new discrete velocity scheme for solving the Boltzmann equation has been implemented for homogeneous relaxation and one-dimensional problems. Directly solving the Boltzmann equation is computationally expensive because in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. To best solve the collision integral, collisions between each point in velocity space with all other points in velocity space must be considered, but this is very expensive. Motivated by the Direct Simulation Monte Carlo (DSMC) method, the computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space. A collision partner selection algorithm was implemented to favor collision partners that contribute more to the collision integral. The new scheme has a built in flexibility, where the resolution in approximating the collision integral can be adjusted by changing how many collision partners are sampled. The computational cost associated with evaluation of the collision integral is compared to the corresponding statistical error. Having a fixed set of velocities can artificially limit the collision outcomes by restricting post collision velocities to those that satisfy the conservation equations and lie precisely on the grid. A new velocity interpolation algorithm enables us to map velocities that do not lie on the grid to nearby grid points while preserving mass, momentum, and energy. This allows for arbitrary post-collision velocities that lie between grid points or completely outside of the velocity space to be projected back onto the nearby grid points. The present scheme is applied to homogeneous relaxation of the non-equilibrium Bobylev Krook-Wu distribution, and the numerical results agree well with the analytic solution. After verifying the proposed method for spatially homogeneous relaxation problems, the scheme was then used to solve a 1D traveling shock. The jump conditions across the shock match the Rankine-Hugoniot jump conditions. The internal shock wave structure was then compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 6. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Although using a relatively coarse approximation for the collision integral is computationally efficient, statistical noise pollutes the solution. The effects of using coarse and fine approximations for the collision integral are examined and it is found that by coarsely evaluating the collision integral, the computational time can be reduced by nearly two orders of magnitude while retaining relatively smooth macroscopic properties. / text

In this study, we consider electrostatics contributed from the molecules in the ionic solution. It plays a significant role in determining the binding affinity of molecules and drugs. We develop the overall framework of computing electrostatic properties for three-dimensional molecular structures, including potential, energy, and forces. These properties are derived from Poisson-Boltzmann equation, a partial differential equation that describes the electrostatic behavior of molecules in ionic solutions. In order to compute these properties, we derived new boundary integral equations and designed a boundary element algorithm based on the linear time fast multipole method for solving the linearized Poisson-Boltzmann equation. Meanwhile, a higher-order parametric formulation called algebraic spline model is used for accurate approximation of the unknown solution of the linearized Poisson-Boltzmann equation. Based on algebraic spline model, we represent the normal derivative of electrostatic potential by surrounding electrostatic potential. This representation guarantees the consistent relation between electrostatic potential and its normal derivative. In addition, accurate numerical solution and fast computation for electrostatic energy and forces are also discussed. In addition, we described our hierarchical modeling and parameter optimization of molecular structures. Based on this technique, we can control the scalability of molecular models for electrostatic computation. The numerical test and experimental results show that the proposed techniques offer an efficient and accurate solution for solving the electrostatic problem of molecules. / text

Cette thèse avait comme but le développement, l'analyse et l'application de schémas numériques pour la simulation de modèles cinétiques basés sur l'équation de Vlasov, notamment de schémas basés sur le splitting de Strang et une méthode d'interpolation essentiellement non oscillatoire (WENO). Les schémas sont testés sur des cas test de plus en plus compliqués, et finalement sur un modèle Boltzmann-Schrödinger-Poisson qui décrit les états transitoires d'un transistor à l'echelle nanométrique.

In this thesis, two methods to model quantum well lasers will be examined. The first model is based on well-known techniques to determine some of the spectral and dynamical properties of the laser. For the spectral properties, an expression for TE and TM modal amplitude gain is derived. For the dynamical properties, the rate equations are shown. The spectral and dynamical properties can be examined separately for specific operating characteristics or used in conjunction with each other for a complete description of the laser. Examples will be shown to demonstrate some of the analysis and results that can be obtained. The second model used is based on Wigner functions and the quantum Boltzmann equation. It is derived from general non-equilibrium Greens functions with the application of the Kadanoff-Baym ansatz. This model is less phenomenological than the previous model and does not require the separation of physical processes such as the former spectral and dynamical properties. It therefore has improved predictive power for the performance of novel laser designs. To the Author's knowledge, this is the first time such a model has been formulated. The quantum Boltzmann equations will be derived and some calculations will be performed for a simplified system in order to illustrate some calculation techniques as well as results that can be obtained.

An experimental investigation of the electronic transport properties of bismuth telluride nanocomposite materials is presented. The primary transport measurements are electrical conductivity, Seebeck coefficient and Hall effect. An experimental apparatus for measuring Hall effect and electrical conductivity was designed, constructed and tested. Seebeck coefficient measurements were performed on a commercial instrument. The Hall effect and Seebeck coefficient measurements are two of the most important tools for characterizing thermoelectric materials and are widely used in the semiconductor industry for determining carrier types, carrier concentration and mobility. Further, these transport parameters are used to determine the thermal to electrical conversion efficiency of a thermoelectric material. The Boltzmann transport equation was used to analyze the Seebeck coefficient, carrier mobility and electrical conductivity as a function of carrier concentration for eleven samples. The relationship between the electronic transport and material/composite composition is discussed.

Após a excitação de uma amostra semicondutora por um pulso ultracurto, os fotoporadores interagem com a rede excitando modos longitudinais ópticos. Essa interação provoca variações no índice de refração do material, produzindo modulações na resposta óptica do meio (efeito eletro-óptico). Por outro lado, esta dinâmica origina polarizações dependentes do tempo o que gera emissão de radiação terahertz. Experimentos recentes (pump-probe) observaram modulações do campo através de medidas da refletividade resolvidas no tempo. A refletividade e o campo estão relacionados segundo o efeito eletro-óptico. Também se resolve temporalmente o campo irradiado pela amostra, através de antenas que operam na faixa de terahertz. Tanto as medidas eletro-ópticas quanto de emissão terahertz fornecem informações sobre a interação dinâmica do plasma com a rede após a excitação óptica. Nesse trabalho simulamos a interação dinâmica de plasma e fônons em n-GaAs e Telúrio (\"bulk\") após estes serem excitados por um pulso ultracurto. Utilizamos equações hidrodinâmicas para descrever transporte de cargas e uma equação fenomenológica de oscilador harmônico forçado, para descrever oscilações longitudinais ópticas da rede. Complementando nossa descrição temos a equação de Poisson, com a qual calculamos o campo gerado pelo plasma e pela polarização da rede semicondutora. Essas equações constituem um sistema de seis equações diferencias (quatro parciais) acopladas. Para resolvê-las utilizamos o método das diferenças finitas. Do cálculo numérico obtemos a evolução temporal do campo elétrico no interior do material. Com esse campo determinamos as freqüências de oscilação do sistema e calculamos o campo irradiado. Nossos resultados apresentam acordo qualitativo com os experimentos / Above-band-gap optical excitation of semiconductors generates highly non-equilibrium photocarriers which interact with phonons thus exciting vibrational modes in the system. This interaction induces refractive-index changes via the electro-optic effect. Moreover it gives rise to electromagnetic radiation at characteristic frequencies (terahertz). Both effects have been measured by time-resolved ultra fast spectroscopy. Recent pump-probe experiments have found strong modulations of the internal electric field through electro-optic measurements. The emitted electromagnetic radiation has also been detected by a terahertz dipole antenna. Both electro-optic and terahertz emission measurements provide information about the coupled dynamics of photocarriers and phonons. In this work we simulate the dynamics of plasmon-phonon coupled modes in n-GaAs and Tellurium (bulk) following ultrafast laser excitation. The time evolution of the photocarrier densities and currents is described semi classically in terms of the moments of the Boltzmann equation. Phonon effects are accounted for by considering a phenomenological driven-harmonic-oscillator equation, which is coupled to the electron-hole plasma via Poisson\'s equation. These equations constitute a coupled set of differential equations. We use finite differencing to solve these equations. From the numerical results for the evolution of internal fields we can calculate both the characteristic frequencies of system and its terahertz radiation spectrum. Our results are consistent with recent experimental data

This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory. The smallest positive real eigenvalue of the problem contains valuable information about the status of the fission chain reaction in the nuclear reactor (i.e. the criticality of the reactor), and thus plays an important role in the design and safety of nuclear power stations. Because of the practical importance, efficient numerical methods to solve the criticality problem are needed, and these are the focus of this thesis. In the theory we consider the time-independent neutron transport equation in the monoenergetic homogeneous case with isotropic scattering and vacuum boundary conditions. This is an unsymmetric integro-differential equation in 5 independent variables, modelling transport, scattering, and fission, where the dependent variable is the neutron angular flux. We show that, before discretisation, the nonsymmetric eigenproblem for the angular flux is equivalent to a related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator(in space only). Furthermore, we prove the existence of a simple smallest positive real eigenvalue with a corresponding eigenfunction that is strictly positive in the interior of the reactor. We discuss approaches to discretise the problem and present discretisations that preserve the underlying symmetry in the finite dimensional form. The thesis then describes methods for computing the criticality in nuclear reactors, i.e. the smallest positive real eigenvalue, which are applicable for quite general geometries and physics. In engineering practice the criticality problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inneriterations. This leads to inexact iterative methods for criticality computations, for which there appears to be no rigorous convergence theory. The fact that, under appropriate assumptions, the integro-differential eigenvalue problem possesses an underlying symmetry (in a space of reduced dimension) allows us to perform a systematic convergence analysis for inexact inverse iteration and related methods. In particular, this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. The theory is illustrated with numerical examples on several test problems of physical relevance, using GMRES as the inner solver. We also illustrate the use of Monte Carlo methods for the solution of neutron transport source problems as well as for the criticality problem. Links between the steps in the Monte Carlo process and the underlying mathematics are emphasised and numerical examples are given. Finally, we introduce an iterative scheme (the so-called “method of perturbation”) that is based on computing the difference between the solution of the problem of interest and the known solution of a base problem. This situation is very common in the design stages for nuclear reactors when different materials are tested, or the material properties change due to the burn-up of fissile material. We explore the relation ofthe method of perturbation to some variants of inverse iteration, which allows us to give convergence results for the method of perturbation. The theory shows that the method is guaranteed to converge if the perturbations are not too large and the inner problems are solved with sufficiently small tolerances. This helps to explain the divergence of the method of perturbation in some situations which we give numerical examples of. We also identify situations, and present examples, in which the method of perturbation achieves the same convergence rate as standard shifted inverse iteration. Throughout the thesis further numerical results are provided to support the theory.