The aim of this work is to provide a mathematical and numerical analysis of the static quasicontinuum (QC) method. The QC method is, in essence, a finite element method for atomistic material models. By restricting the set of admissible deformations to linear splines with respect to a finite element mesh, the computational complexity of atomistic material models is reduced considerably. We begin with a general review of atomistic material models and the QC method and, most importantly, a thorough discussion of the correct concept of static equilibrium. For example, it is shown that, in contrast to global energy minimization, a ‘dynamic’ selection procedure based on gradient flows models the physically correct behaviour. Next, an atomistic model with long-range Lennard–Jones type interactions is analyzed in one dimension. A rigorous demonstration is given for the existence and stability of elastic as well as fractured steady states, and it is shown that they can be approximated by a QC method if the mesh is sufficiently well adapted to the exact solution; this can be measured by the interpolation error. While the a priori error analysis is an important theoretical step for understanding the approximation properties of the QC method, it is in general unclear how to compute the QC deformation whose existence is guaranteed by the a priori analysis. An a posteriori analysis is therefore performed as well. It is shown that, if a computed QC deformation is stable and has a sufficiently small residual, then there exists a nearby exact solution and the error is estimated. This a posteriori existence idea is also analyzed in an abstract setting. Finally, extensions of the ideas to higher dimensions are investigated in detail.

Systems of ultracold atoms in optical potentials have taken a place at the forefront of research into many-body atomic systems because of the clean experimental environment they exist in and the tunability of the system parameters. In this thesis we study how light scattered from these ultracold atomic gases reveals information about the state of the atomic gas and also leads to changes in that state. We begin by investigating the angular dependence of light scattered from atoms in optical lattices at finite temperature. We demonstrate how correlations in the superfluid and Mott insulator states affect the scattering pattern, and we show that temperature affects the number of photons scattered. This effect could be used to measure the temperature of the gas, however, we show that when the lattice band structure is taken into account the efficiency of this temperature measurement is reduced. We then investigate light scattering from small optical lattices where the Bose-Hubbard Hamiltonian can be solved exactly. For small lattices, scattering a photon from the atomic system significantly perturbs the atomic system. We develop a model of the evolution of the many-body state that results from the consecutive scattering and detection of photons. This model shows that light scattering pushes the system towards eigenstates of the light scattering measurement process, in some cases leading to a superposition of atomic states. In the second half of this thesis we study light scattering that depends on the internal hyperfine spin state of the atoms, in which case the scattered light can form images of the spatial atomic spin distribution. We demonstrate how scattering spatially correlated light from the atoms can result in spin state images with enhanced spatial resolution. We also show how using spatially correlated light can lead to direct measurement of the spatial correlations of the atomic spin distribution. We then apply this theory of spin-dependent light scattering to the detection of different spin states of ultracold gases in synthetic magnetic fields. We show that it is possible to distinguish between ground states in the quantum Hall regime using light scattering. Moreover, we show how noise correlation analysis of the spin state images can be used to identify the correlations between atoms and how a variant on phase-contrast imaging can reveal the relationship between the atomic spins.

In this thesis, a joint experimental and theoretical study of the dynamics of atom- molecule collisions will be presented. The focus of this study will be conducted towards the precise, quantitative theoretical description of the collision dynamics in terms of the vectors <strong>k</strong>, <strong>k'</strong>, <strong>j</strong>, and <strong>j'</strong> (the incoming and outgoing relative momenta associated with the collision, and the initial and final rotational angular momentum of the target diatom respectively) that define the collision, and on the experimental measurement of these vector correlations. Chapter 1 is introductory, providing an overview of the field of reaction dynamics, and the experimental and theoretical methods that exist to treat the collisions of atoms and molecules. This work focusses on the collisions of the spherically symmetric rare gas atoms Ar and He with the open-shell heteronuclear diatomic radicals NO and OH. In particular, the fully quantum state-to-state resolved differential cross-sections for the collisions of NO(X) with Ar (reflecting the <strong>k</strong> - <strong>k'</strong> vector correlation), and the collisional cross-sections for the depolarisation of the rotational angular momenta of the NO(A) and OH(A) radicals (reflecting the <strong>j</strong> - <strong>j'</strong> vector correlation) have been determined experimentally and theoretically, and the results have been discussed and interpreted in terms of the mechanistic aspects of the collision dynamics, and the features of the potential energy surface that give rise to these. In Chapter 2, the atom-molecule systems that constitute the subject of this work will be introduced in detail. The close-coupled quantum mechanical and quasi-classical trajectory scattering calculations performed as part of this work will be discussed in greater detail, providing a greater insight into molecular scattering theory. The explicit calculation of the quantities of interest (most significantly the differential cross-section, and the tensor/depolarisation cross-sections) will be presented for the quasi-classical and quantum cases, offering the most transparent definitions of these quantities. Finally the mathematical description of the spatial probability distribution of a single vector, a pair of correlated vectors, and three correlated vectors is described in detail, including a discussion of the quantum mechanical nature of the vectors in question. Chapter 3 describes the experimental measurement of the differential cross-sections for the collisions of NO(X) with Ar. A hexapole was used to select uniquely those NO molecules in the |&Omega; = 0.5; j = 0.5, f> quantum state, allowing full experimental quantum state-to-state selection for the first time. A crossed molecular beam apparatus with (1+1') resonantly enhanced multi-photon ionisation detection coupled with velocity mapped ion- imaging was employed to measure the differential cross-section, and the details of the experimental set-up are provided. The accurate extraction of the true, centre of mass frame differential cross-section from the laboratory frame information yielded by the experiment is something of an involved process, and much of this Chapter will be concerned with the development of a Monte Carlo method to achieve this end. In Chapter 4, the experimental and theoretical fully quantum state-to-state resolved differential cross-sections for the collisions of NO(X) with Ar are presented, having been measured for the first time. Full resolution of the initial parity of the rotational wave- function of the NO molecule has enabled the observation of parity dependent structures within the differential cross-section, and the origin of these structures has been investi- gated, employing quasi-classical, quantum mechanical and semi-classical methods in order to elucidate the mechanism by which they arise. Chapter 5 introduces the measurement of the collisional depolarisation of the rotational angular momentum of the diatom. Rate constants for the collisional depolarisation of <strong>j</strong> were measured by monitoring the time dependence of the amplitude of Zeeman and hyperfine quantum beats in the (1+1) laser induced fluorescence decays of an ensemble of NO(A) or OH(A) radicals in the presence of a series of background pressures of a collision partner. The creation and subsequent evolution of the polarisation of <strong>j</strong> induced by the absorption of polarised laser light is described, and the magnitude of this polarisation is linked to the amplitude of the quantum beat in the laser induced fluorescence decay. The extraction of the depolarisation cross-sections from the raw experimental data is discussed, and a Monte Carlo simulation of the experiment is described to account for any additional unwanted experimental factors that may contribute to the loss of polarisation of <strong>j</strong>. A formalism is also introduced that makes use of the tensor opacities to recover spin- rotation conserving and spin-rotation changing open-shell rotational energy transfer and depolarisation cross-sections from the intrinsically closed shell quasi-classical trajectory scattering calculations. In Chapter 6, the experimentally determined collisional depolarisation cross-sections for the collisions of NO(A) with He/Ar, and of OH(A) with Ar at collision energies of 39 meV/757meV are presented along with their theoretical counterparts. The relative magnitudes of the cross-sections are rationalised in terms of the potential energy surface over which the collision takes place, and the importance of spin-rotation conserving and spin-rotation changing transitions in the depolarisation process is assessed. A detailed study of the ensemble of quasi-classical trajectories is performed to determine the character of the various atom-molecule collisions, and to identify which conditions lead to the most efficient depolarisation of <strong>j</strong>. The relative importance of the potential energy surface and the collision kinematics is also assessed at this point. The results presented in this thesis thus investigate two complementary expressions of the collision dynamics, the <strong>k</strong> - <strong>k'</strong> and <strong>j</strong> - <strong>j'</strong> vector correlations, and encompass a variety of collision partners exhibiting vastly differing collision characteristics. As such, this work serves as an illustrative overview of atom-molecule scattering dynamics, containing both experimental and theoretical reflections of the collision dynamics, and relating this information back to the fundamentals of scattering theory.

In this thesis, resonance enhanced multiphoton ionisation (REMPI) in combination with time-of-flight mass spectrometry (TOF-MS) has been used to detect nascent photofragments resulting from the UV dissociation of a variety of small molecules. The translational anisotropy and angular momentum polarisation of these photofragments has been measured and used to elucidate the underlying photodissociation dynamics. Firstly, the photodissociation of NO₂ at 320nm has been investigated and the vector correlations of the nascent NO photofragments have been measured in terms of a set of semi-classical bipolar moments. The measured angular momentum alignment is found to be consistent with an impulsive model for the dissociation, with <b>&mu;</b> and <b>&nu;</b> in the same molecular plane and both preferentially perpendicular to <b>J</b>, whilst angular momentum orientation measurements provide evidence for an additional torque due to the O-N-O bond opening during dissociation. These measurements were taken using a rotationally cooled, skimmed molecular beam and significant deviations were found between the bipolar moments measured using this source and previous measurements using a rotationally hotter source. The effect of parent molecular rotations on the measured bipolar moments has been quantified and successfully used to explain these deviations. The photodissociation of Cl₂ has been studied in the wavelength region (320-350)nm. UV absorption in this wavelength region may result in two dissociation channels, (Cl+Cl) and (Cl+Cl*), and the angular momentum polarisation of both the Cl(²P_3/2) and Cl*(²P_1/2) photofragments has been measured. This angular momentum polarisation has been reported in terms of a polarisation parameter formalism which, together with the measured translational anisotropies, has been used to determine the different potential energy surfaces contributing to the dissociation process. Translational anisotropy measurements of the Cl(²P_3/2) fragments have shown that, for the ground-state channel, dissociation results from a pure perpendicular transition to the C state, whilst alignment measurements show that non-adiabatic transitions to the A state are significant at large internuclear separations. The measured alignment parameters are found to be relatively constant for all dissociation wavelengths and are consistent with theoretical predictions. Translational anisotropy measurements of the Cl(2P_1/2) photofragments show that, for the excited-state channel, dissociation occurs following a mixed parallel and perpendicular excitation to the B and C states respectively and the interference between these two dissociation pathways has been shown to result in angular momentum orientation. The predissociation dynamics of the C ³&Pi;_g (&nu;=0) and (&nu;=1) Rydberg states of O₂ has been extensively studied. The translational anisotropy and angular momentum alignment of the O(³P) and O(¹D) photofragments resulting from this predissociation has been measured in terms of a polarisation parameter formalism, which has been extended for a two-photon dissociation process. Measurements have been taken at various fixed wavelengths within the two bands in order to investigate the differences in the predissociation dynamics of intermediate levels with different values of |&Omega;|(=0,1,2 in this case). The translational anisotropy is found to be dependent on the dissociation wavelength with the variations found to be consistent with rotational depolarisation due to the long lifetime of the excited C state. All photofragments have been found to be aligned, with the relationship between the measured O(³P) and O(¹D) alignment being found to be consistent with a diabatic model of the dissociation. In addition, all photofragments are found to display coherent orientation resulting from interference between two possible two-photon absorption pathways. The measured orientation is affected by rotational depolarisation due to the long lifetime of the excited C state; once this effect is accounted for the orientation is found to be nearly constant over all dissociation wavelengths. The origin of the coherent orientation is attributed to two-photon absorption to different spin-orbit components of the C state.

Thesis (MSc.) (Physics) --University of Limpopo, 2007 / Molecular dynamics simulations have been carried out in order to examine the mechanism of diffusion of molecules in amorphous polymer matrix. PDMS model was folded in to a periodic cell, generated by rotational isomeric state (RIS) method at a prescribed temperature and density. Molecular dynamics was used to study transport properties of cyclic PDMS oligomers (hexa-methylcyclotrisiloxane (D3), octa-methylcyclotetrasiloxane (D4) and deca-methylcyclopentasiloxane (D5) using Dreiding and COMPASS force fields. Diffusion coefficients were calculated from the Einstein relation. Only D3 penetrant reached the long time limit from which the Einstein relation is satisfied. Analysis of displacement versus time for all the penetrants in PDMS matrix indicates that the penetrant motion is characterized by relatively long periods interspersed with fairly long and small jumps. Transport of solvent molecules occurs by jumps between individual sections of free volume (cavity/hole) through temporarily open channels. / The National Research Foundation (NRF) and Eskom

Introduction An atom or ion in a solid is associated with a given lattice site. Translational motion can however occur in solids and such diffusion processes can be studied by nuclear magnetic resonance, tracer diffusion and other methods. In ionic compounds the diffusion is also a mechanism for charge transport and hence there is an associated ionic conductivity. [continued in text ...] NOTE: The Abstract contains many indistinct letters in mathematical formulas and it is impracticable to reproduce it here.

L’objectif de ce travail de recherche est d’étudier la possibilité d’utiliser la spectrométrie Raman dite "conventionnelle" pour l’analyse, voire le contrôle in situ de solutions ioniques et d'en déterminer les limites. Nous avons étudié différentes solutions salines, composées d'ions polyatomiques comme le nitrate (NO3-) donnant des raies Raman autour de 1000 cm-1, et d'ions halogénures (F-, Cl-, Br-, I-) qui ne possèdent pas de signature directe mais influencent de manière indirecte le spectre de l’eau. Les méthodologies proposées dans ce travail permettent une identification et une quantification des solutions salines et sont basées sur des analyses classiques de spectres à l'aide de pré-traitements (normalisation, correction de ligne de base). Les limites de détection des ions étudiés correspondent pour certains ions aux limites de potabilité de l’eau (nitrates et sulfates par exemple). D’autres substances restent impossibles à quantifier (nitrite, ammonium). Des méthodes statistiques ont ensuite été employées. Ainsi, l’analyse en composante principale (ACP) permet d’identifier la nature d’une solution saline, que l’ion présente une signature directe ou indirecte, et d'accéder aux arrangements moléculaires des ions avec l’eau. La régression PLS permet de quantifier les ions en solution à travers des modèles de prédiction. Cette méthode a été utilisée sur des solutions pures de nitrate et de chlorure avec des erreurs sur les prédictions en concentration 4 à 5 fois plus faibles que celles trouvées par les méthodes classiques. Ces méthodes statistiques ont également permis de descendre à des gammes de concentrations en ion Cl- beaucoup plus faibles (10-100 mM) / The objective of this research is to study the possibility of using conventional Raman spectroscopy for analysis, or in situ control of substances dissolved in water and to determine the limits of this technique. Several salt solutions were studied. Polyatomic ions such as nitrate (NO3-) give intense Raman lines around 1000 cm-1. Halide ions (F-, Cl-, Br-, I-) have no direct signature but indirectly affect the water spectrum. The methodology proposed in this work for identification and quantification of salt solutions is based on classical analysis of spectra using pre-treatment methods (normalization, baseline correction). The limits of detection of the studied ions correspond to the water potability limits. It is possible to go below the potability limits for nitrates and sulfates for example but it is impossible to reach them for some other substances (nitrite, ammonium). Then, statistical methods were used. Thus, the principal component analysis (PCA) allows to identify the nature of a salt solution (with direct or indirect signatures) and to acces to the molecular arrangements of ions with water. PLS regression quantifies the ions through predictive models. This method was used on pure solutions of nitrate and chloride with errors on the concentration predictions 4 to 5 times lower than those found by classical methods. Lower concentration ranges (10-100 mM) for the Cl- ion are reachable using these statistical methods

Cette thèse étudie l’influence des fluctuations de l’environnement au cours de l’évolution sur l’architecture fonctionnelle des protéines.L’apparition de groupes restreints d’acides aminés – les secteurs, possédant des propriétés particulières tant du point de vue structurel et évolutif que fonctionnel ne trouve en effet pas d’explication simple dans le paradigme classique de la physique des protéines. Nous avons donc choisi d’étudier le rôle de l’histoire évolutive dans la construction de cette architecture particulière et des propriétés qui en découlent.Nous avons pour cela construit un modèle de protéine fonctionnelle inspiré des modèles de réseaux élastiques, que nous avons soumis à une évolution in silico en variant au cours du temps, avec différentes fréquences, la fonction recherchée. Nous avons montré que ces fluctuations induisent une concentration semblable à celle observée dans les protéines et avons pu déterminer les paramètres clés contrôlant ce phénomène. Nous avons finalement abordé le lien entre la statistique temporelle de l’environnement et l’apparition de différents secteurs indépendants. / This thesis studies the influence of an evolutionary fluctuating environment on the functional architecture of proteins.The appearance of restricted groups of amino acids – sectors, with particular functional, evolutional and structural properties has no simple explanation in the classical paradigm of proteins physics. So we choose to study the role of evolutionary history on the construction of this particular architecture and the resulting properties.We have thus constructed a model of functional protein inspired by the elastic network models, that we have evolved in silico while temporarily varying the targeted function with various frequencies. We have shown that these fluctuations induce a form of sparsity close to that observed in proteins and has identified the key parameters of this phenomenon. We finally investigate the link between the temporal statistics of the environment and the appearance of different independent sectors.

Les résultats décrits dans cette thèse démontrent une relation complexe entre le confinement géométrique de molécules à 2D et les propriétés de chiralité. Il a été démontré qu’une chiralité multimodale peut être observée pour des systèmes non chiraux par le biais du confinement 2D. Les molécules H5T forment, sur Au(111), des domaines pour lesquels deux types de chiralité peuvent être observés. Les études sur le système H5T/Au(111) ont prouvé que des interactions simples, de type van der Waals sont susceptibles de créer des chiralités complexes, ce qui donne un caractère générique a ce type de phénomène. Une autre question importante est l’apparition, pour les molécules C-12, des domaines chiraux induits par des interactions intermoléculaires spécifiques: les liaisons hydrogène entre les groupements azobenzènes de molécules voisines. L’orientation des deux domaines chiraux est déterminée par l’orientation de l’un des deux dimères d’azobenzène stable par rapport à l’Au(111).La seconde partie de la thèse est dédiée à l’étude de l’influence du confinement 3D sur la chiralité de gouttes de cristal liquide cholestérique (CLC). Grâce à la nature photo-sensible du dopant chiral utilisé, nous avons été capables de modifier le pas choléstérique du mélange de CLC et par conséquent de déterminer l’expression de la chiralité pour un ratio rayon/pas choléstérique variable. Nous sommes parvenus à contrôler le pas choléstérique au sein de gouttes par irradiation UV et nous avons étudié les variations structurelles que l’irradiation induite. Nous avons observé que le confinement 3D d’un mélange de CLC photo-sensible a une influence positive sur l’inversion de l’hélice dans la goutte. / The first part of the thesis is devoted to studies of the self-assembled monolayers of discotic liquid crystals by the STM measurements at the liquid/solid interface. For the case of a model H5T molecule the self-assembled monolayers have evidenced both: point and organizational types of chirality, despite of the fact that neither the molecule nor the substrate was chiral. For another molecular system, C-12 - a triphenylene peripherally substituted with azobenzene moieties, self-assemblies bearing the chiral nature were also evidenced. Chirality was induced by formation of six dimers of azobenzene subunits coming from the neighboring molecules, which formed “rosettes” of clockwise or counter-clockwise rotation. For the H5T, the chirality was mediated by classical van der Waals interactions between molecules and between molecules and substrate. In the case of C-12 it was shown that the self-assembly originates from the substrate-mediated hydrogen bonding between the azobenzene moieties of neighboring molecules. The second part of the thesis presents studies of the influence of the 3D confinement on the chirality of the photoactive cholesteric liquid crystal (CLC) droplets. Due to the photo-responsive character of a chiral dopant we were able to modify the cholesteric pitch of the CLC mixture and thus to map the expression of chirality for a varying geometrical confinement parameter: radius-to-pitch. We evidenced a successful control of the droplets structure by UV irradiation and we studied the induced structural changes. In particular, the 3D confinement of a photo-responsive CLC mixture was shown to positively support the helix inversion within the cholesteric droplets.

This thesis presents a theoretical and experimental investigation of the interaction of electronically excited Rydberg hydrogen atoms with metal surfaces and the associated charge-transfer process. As a Rydberg atom approaches a metal surface, the energies of the Rydberg states are perturbed by the surface potential generated by the image charges of the Rydberg electron and core. At small atom-surface separations, the Rydberg atom may be ionised by resonant charge transfer of the Rydberg electron to the continuum of delocalised unoccupied metal states, with which the Rydberg electron is degenerate in energy. Typically, this ‘surface ionisation’ can be measured by extracting the remaining positively charged ion-cores with externally applied electric fields. By applying various levels of theory, from classical to fully time-dependent quantum calculations, this thesis explores various experimentally relevant effects on the charge-transfer process, such as the magnitude and direction of the externally applied electric field, the atom collisional velocity, the presence of local surface stray fields and electronically structured surfaces. The theoretical results give insight into the previous experimental work carried out for the xenon atom and hydrogen molecule, and point out some of the fundamental differences from the hydrogen atom system. Experiments involving Rydberg hydrogen atoms incident on an atomically flat gold surface, a rough machined aluminium surface and a single crystal copper (100) surface are presented, providing for the first time the opportunity to make a quantitative comparison of theory and experiments. The ability to control the critical distance at which charge-transfer occurs is demonstrated by using Rydberg states of varying dimensions and collisional velocities. By changing the collisional angle of the incident Rydberg beam, the effect of Rydberg trajectory is also investigated. By manipulating the polarisation of the Rydberg electron with electric fields, genuine control over the orientation of the electron density distribution in the charge-transfer process is demonstrated. This property was predicted by the theory and should be unique to the hydrogen atom due to its intrinsic symmetry. By reversing the direction of the electric field with respect to the metal surface, electrons rather than positive ions are detected, with ionisation dynamics that appear to be very different, as predicted by quantum calculations. Experiments involving the single crystal Cu(100) surface also suggests possible resonance effects from image states embedded in the projected bandgap which are shown from quantum calculations to play an important role in the surface charge transfer of electronically structured metal substrates. The experimental technique developed in this work provides some exciting future applications to study quantum confinement effects with thin films, nanoparticles and other bandgap surfaces. The ability to control the Rydberg orbital size, electronic energy, collisional velocity and orientation in the charge-transfer process will provide novel ways of probing the surface’s electronic and physical structure, as well as being a valuable feature in offering new opportunities for controlling reactive processes at metallic surfaces.