Increasing efforts have been put into research about thermoelectric materials for the last few decades, especially recently, faced with the crucial demand for new energy and energy savings. Among the potential candidates for new generation thermoelectric materials are the intermetallic clathrates. Clathrates are cage-structured materials with guest atoms enclosed. Previous studies have shown lower thermal conductivities compared with many other bulk compounds, and it is believed that guest atom vibration modes are the reason for such thermal behaviors. Several models, including the Einstein oscillator and soft potential models, have been used to explain the guest motion. However the characterization of the anharmonic oscillating motion can be a challenge. In this work, Nuclear Magnetic Resonance (NMR), heat capacity and transport measurements have been used to study several clathrate systems, especially the well- known type-I Ba8Ga16Sn30, which has been reported to have one of the lowest thermal conductivities for bulk compounds. In this material the strong anharmonic rattling behavior was investigated and analyzed according to a double well potential model, yielding good agreement with the experimental results. Furthermore, the resistivity and heat capacity results were studied and analyzed according to the influence of the anharmonic contribution. This offered a way to connect the NMR, transport and heat capacity properties, providing an advantageous way to study strongly anharmonic systems. In further work, several related intermetallic materials were examined for their structure, motion and NMR properties. Dynamical and electrical behaviors were investigated by studying the magnetic and quadrupole NMR spin-lattice relaxation. Type-VIII Ba8Ga16Sn30 exhibits an enhanced dynamics-related term at low temperature, but no rattling response as observed for the type-I structure. Type-I Ba8In16Ge30 was compared with the type-I Ba8Ga16Sn30 because their cage structures are similar. No strong anharmonic contribution was found in the NMR T1 behavior of Ba8In16Ge30, however the T2 showed behavior characteristic of atomic motion. In all cases, the magnetic relaxation was used to characterize the electron structures, and n- type Ba8Ga16Ge30 exhibited a spin-lattice relaxation behavior which is characteristic of impurity band structures near the Fermi surface. Also, a series of Ba8CuxGe46-x clathrates were investigated and showed much more insulating like behavior. In related work, the layered BaGa4 and BaGa3Sn have shown interesting NMR spin-spin relaxation behavior that indicates atomic fluctuations. This is similar to the situation found in type-I Ba8In16Ge30. The influence of atomic motion on the NMR and also the atomic structures of these alloys is further discussed in this work.

abstract: A novel Monte Carlo rejection technique for solving the phonon and electron Boltzmann Transport Equation (BTE), including full many-particle interactions, is presented in this work. This technique has been developed to explicitly model population-dependent scattering within the full-band Cellular Monte Carlo (CMC) framework to simulate electro-thermal transport in semiconductors, while ensuring the conservation of energy and momentum for each scattering event. The scattering algorithm directly solves the many-body problem accounting for the instantaneous distribution of the phonons. The general approach presented is capable of simulating any non-equilibrium phase-space distribution of phonons using the full phonon dispersion without the need of the approximations commonly used in previous Monte Carlo simulations. In particular, anharmonic interactions require no assumptions regarding the dominant modes responsible for anharmonic decay, while Normal and Umklapp scattering are treated on the same footing. This work discusses details of the algorithmic implementation of the three particle scattering for the treatment of the anharmonic interactions between phonons, as well as treating isotope and impurity scattering within the same framework. The approach is then extended with a technique based on the multivariable Hawkes point process that has been developed to model the emission and the absorption process of phonons by electrons. The simulation code was validated by comparison with both analytical, numerical, and experimental results; in particular, simulation results show close agreement with a wide range of experimental data such as the thermal conductivity as function of the isotopic composition, the temperature and the thin-film thickness. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2018

We use an algebraic method to model the molecular collision dynamics of a collinear triatomic system. Beginning with a forced oscillator, we develop a mathematical framework upon which inelastic and reactive collisions are modeled. The model is considered algebraic because it takes advantage of the properties of a Lie algebra in the derivation of a time-evolution operator. The time-evolution operator is shown to generate both phase-space and quantum dynamics of a forced oscillator simultaneously. The model is considered semi-classical because only the molecule's internal degrees-of-freedom are quantized. The relative translation between the colliding atom and molecule in an exchange reaction (AB+C ->A+BC) contains no bound states and any possible tunneling is neglected so the relative translation is treated classically. The purpose of this dissertation is to develop a working model for the quantum dynamics of a collinear reactive collision. After a reliable model is developed we apply statistical mechanics principles by averaging collisions with molecules in a thermal bath. The initial Boltzmann distribution is of the oscillator energies. The relative velocities of the colliding particles is considered a thermal average. Results are shown of quantum transition probabilities around the transition state that are highly dynamic due to the coupling between the translational and transverse coordinate.

This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, ··· generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions Ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by Āi4(y). By investigating the small effect generated by Āi4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n Ε N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions Ψ(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| - λn) in its normalised form.

This dissertation contains research directed at investigating the behavior and properties of a class of composite materials known as phononic crystals. Two categories of phononic crystals are explicitly investigated: (I) elastic phononic crystals and (II) nano-scale phononic crystals. For elastic phononic crystals, attention is directed at two-dimensional structures. Two specific structures are evaluated (1) a two-dimensional configuration consisting of a square array of cylindrical Polyvinylchloride inclusions in air and (2) a two-dimensional configuration consisting of a square array of steel cylindrical inclusions in epoxy. For the first configuration, a theoretical model is developed to ascertain the necessary band structure and equi-frequency contour features for the realization of phase control between propagating acoustic waves. In contrasting this phononic crystal with a reference system, it is shown that phononic crystals with equifrequency contours showing non-collinear wave and group velocity vectors are ideal systems for controlling the phase between propagating acoustic waves. For the second configuration, it is demonstrated that multiple functions can be realized of a solid/solid phononic crystal. The epoxy/steel phononic crystal is shown to behave as (1) an acoustic wave collimator, (2) a defect-less wave guide, (3) a directional source for elastic waves, (4) an acoustic beam splitter, (5) a phase-control device and (6) a k-space multiplexer. To transition between macro-scale systems (elastic phononic crystals) and nano-scale systems (nano-phononic crystals), a toy model of a one-dimensional chain of masses connected with non-linear, anharmonic springs is utilized. The implementation of this model introduces critical ideas unique to nano-scale systems, particularly the concept of phonon mode lifetime. The nano-scale phononic crystal of interest is a graphene sheet with periodically spaced holes in a triangular array. It is found through equilibrium molecular dynamics simulation techniques, that phonon-boundary collision effects and coherent phononic effects (band-folding) are two competing scattering mechanisms responsible for the reduction of acoustic and optical phonon lifetimes. Conclusions drawn about the lifetime of thermal phonons in phononic crystal patterned graphene are linked with the anharmonic, one-dimensional crystal model.

Water and methanol are common solvents used in liquid chromatographic (LC) separations. It is highly desirable to model .the interactions of these solvents in order to better understand the nature of analyte solvation and its effect on retention. Therefore, structure and frequencies of complexes of these solvent molecules have been studied from a theoretical perspective as a first step in this direction. Specifically, cluster structures have been optimized at the RHF and MP2 levels in various flexible basis sets and with the counterpoise correction for basis set superposition error, and trends in the structure and binding energies of several clusters are described. Good agreement wasobtained for the water dimer with the experimental value for the binding energy of D20 using MP2 energies from 6-3 11G**/6-3 l+G** basis sets in conjunction with counterpoise optimizations and full counterpoise corrections. In this investigation harmonic frequencies have been calculated and corrected for the effects of anharmonicity by several methods, two of which are original. The first new method fits a Morse potential function to the energy computed along each normal mode. A second new method is based on fitting a quartic polynomial to energies computed along each normal mode. In cases where the quartic potential function is not very different from the harmonic well, a second order perturbation formula provides a reasonable approximation to the anharmonic vibrational frequencies. When the quartic potential is very far from the harmonic potential, a variational treatment of the vibrations is required. We find that the Morse method delivers reasonable estimates of frequencies of anharmonic motions at lower cost than multi-point potential mapping/multiple geometry optimization/Taylor series methods, and is more successful at predicting intermolecular frequencies than the anharmonic VSCF methods found in GAMESS software. Variational calculations using the quartic polynomials produce estimates of frequencies comparable to the more costly VSCF method. Both the Morse method and polynomial method are very fast computationally relative to these and other methods found in the literature.

Targeted Energy Transfer is a resonance phenomenon in coupled anharmonic oscillators. In this thesis we investigate if the concept of Targeted Energy Transfer is applicable to Bose-Einsteain condensates in optical lattices. The model used to describe Bose-Einstein condensates in optical lattices is based on the Gross-Pitaevskii equation. Targeted Energy Transfer in these systems would correspond to energy being transferred from one lattice site to another. We also try to expand the concept of Targeted Energy Transfer to a system consisting of three sites, where one of the sites are considered a perturbation to the system. We have concluded that it is possible to achieve Targeted Energy Transfer in a three-site system. The set-up of the system will in some of the cases studied lead to interesting properties, such as more energy being transferred to the acceptor site than what was initially localized on the donor site.

An expression for the dielectric susceptibility tensor of a cubic ionic crystal has been derived using the classical Liouville operator. The effect of cubic anharmonic forces is included as a perturbation on the harmonic crystal solution, and a series expansion for the dielectric susceptibility is developed. The most important terms in the series are identified and summed, yielding an expression for the complex susceptibility with an anharmonic contribution which is linearly dependent on temperature. A numerical example shows that both the real and imaginary parts of the susceptibility are continuous, finite functions of frequency.

This dissertation studies quantum decoherence in anharmonic oscillator systems to monitor and understand the way the systems evolve. It also explores methods to control the systems' evolution, and the effects of decoherence when applicable. We primarily do this by finding the time evolution of the systems using their Lie algebraic structures. We solve for a generalized Caldirola-Kanai Hamiltonian, and propose a general way to produce a desired evolution of the system. We apply the analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. This is done in a way that can be applied to any system that is made up of a Lie algebra. We also explore the evolution of an optomechanical coupled mirror-laser system while maintaining a second order coupling. This system creates anharmonic effects that can produce cat states which can be used for quantum computing. We find that the decoherence in this system causes a rotational smearing effect in the Husimi function which, with the second order term added, causes rotational smearing after a squeezing effect. Finally, we also address the dynamic evolution and decoherence of an anharmonic oscillator with infinite coupling using the Born-Markov master equation. This is done by using the Lie algebraic structure of the Born-Markov master equation's superoperators when applying a strategic mean field approximation to maintain dynamic flexibility. The system is compared to the Born-Markov master equation for the harmonic oscillator, the regular anharmonic oscillator, and the dynamic double anharmonic oscillator. Throughout, Husimi plots are provided to visualize the dynamic decoherence of these systems.

Successful enzyme catalysis depends on many noncovalent interactions between the enzyme, cofactors, and substrate that poise the system to access a productive transition state. Motions on a variety of timescales contribute to this, but some controversy exists surrounding the role of ultrafast dynamics on catalysis. Site-specific 2D IR spectroscopy using probes of vibrational dynamics provides the opportunity to explore ultrafast motions in an enzyme active site owing to the technique’s spatial and temporal resolution. In this work, a series of aryl-azide vibrational labels were assessed using a variety of 2D IR techniques for their sensitivity to solvent and energy transfer processes, and their ability to be adapted to experiments in biomacromolecules. One of these labels, 4-azido-N-phenylmaleimide, is a substrate analog for the promiscuous ene-reductase from Pyrococcus horikoshii (PhENR). The label was covalently attached in two orientations in the enzyme active site, occupying the same position as native substrates based on X-ray crystallography and molecular dynamics simulations. FTIR and 2D IR spectroscopy were used to identify close-lying conformational states based on the strong anharmonic coupling of the label, revealing that the active site itself modulates the probe’s internal vibrational coupling. More commonly used analogous aryl-nitrile labels, however, were not sensitive to such small structural and lineshape changes. This demonstrates the importance of thoughtful label design to maximize the amount of information that can be gleaned from 2D IR studies. Using the methods herein—both spectroscopic and biochemical—provides a strategy for probing ultrafast motions that could possibly be catalytically relevant.