Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.

A taxa de relaxação magnética nuclear de uma impureza diluída em um meio metálico foi calculada como função da temperatura. Nossos cálculos são aplicados ao modelo de Anderson com degenerescência de spin, originalmente desenvolvido para descrever ligas magnéticas diluídas. Nossos cálculos são aplicados ao modelo de Anderson com degenerescência de spin, originalmente desenvolvido para descrever ligas magnéticas diluídas. Discutimos a relevância a sistemas de férmions pesados, valência flutuante e adsorção química em superfícies metálicas. As taxas de relaxação como função da temperatura exibem picos que concordam qualitativamente com resultados experimentais. No limite de T &#8594 0 as taxas de relaxação são proporcionais a temperatura, mesmo nos casos em que efeitos de muitos corpos invalidam clássica derivação da lei de Korringa. O coeficiente linear é proporcional ao quadrado da suscetibilidade magnética à temperatura zero; isto generaliza a relação derivada por Shiba no limite Kondo. / The nuclear magnetic relaxation rate for an impurity in a metallic environment has been calculated as a function of temperature. Our calculations are based on the spin-degenerate Anderson model originally developed to describe dilute magnetic alloys. The relevance to heavy férmions, Valence fluctuation, and chemisorption on metallic surfaces is discussed. The temperature dependent rates exhibit peaks in qualitative agreement with experimental results. As expected, in the limit T &#8594 0 the rates are proportional to the temperature, even for cases in which many-body effects invalidate the classical derivation of the Korringa Law. The linear coefficient is shown to be proportional to the square of the zero temperature suscetibility; this generalizes a relation derived by Shiba in the Kondo limit.

Neste trabalho, descrevemos um espectrômetro ótico para medidas de Dicroismo Circular Magnético (DCM), utilizado nas seguintes experiências: 1) Medidas de tempo de relaxação spin-rede (T1) para centros F em NaCl, NaBr, CsBr e CsCl, a temperatura de 1,8&#176K em campos magnéticos até 17000Gs. Verificamos a validade da teoria da referência (8) para explicar as diferenças observadas, no comportamento de \'T IND.1\', para halogenetos com diferentes íons alcalinos, bem como diferentes estruturas. Comprovamos que a interação hiperfina ainda continua a ser o mecanismo mais importante para esses centros. Verificamos também que, para temperaturas entre 6&#176K e l5&#176K, os valores de experimentais para T1, em KBr, concordam razoavelmente com a teoria da referência (21). Esta terapia é uma extensão daquela da referência (8). 2) Espectros de DCM para KCl: Co++ e CaF2: Co++ foram obtidos para campos magnéticos ate 56KGs e temperaturas entre 1,8&#176K e 4,2 &#176K. Os resultados obtidos mostraram ser concordantes com a hipótese dos centros Co++ ocuparem sítios intersticiais na rede de KCl. / In this work we describe a Magnetic Circular Dicroism Spectrometer wich was used in the following experiments: 1) We measured the spin-lattice relaxation time (T1) for F centers in NaCl, NaBr, CsBr and CsCl, at 1,8&#176K in magnetic fields up to 15000Gs. We verified the suitability of the theory of ref.(O8) to explain the differences observed for halides of differents alkali ions as well as for different structures. This proves that the hyperfine interaction is the most important mechanism for this kind of centers. We also verified that, for temperatures between 6&#176K and l5&#176K, T1, the T1 experimental values fits the theory of ref.(21) reasonably well ,for F centers in KBr. This theory is an extension of that of ref.(8). 2) We obtained the MCD spectra for KCl: Co++ and CaF2: Co++ in different magnetic fields up to 56KGs, and in temperature range between 1,8 &#176K and 4,2&#176K. Our results are consistent with the assumption that Co++ centers are intersticial in KCl lattice.

Realizamos medidas de tempos de relaxação de centros F em cristais de KCN, pela técnica de saturação do sinal de absorção por pulsos de microonda. Observamos um comportamento não exponencial para a recuperação da magnetização da amostra ap6so pulso. Este fato deve-se ao efeito combinado de relaxação cruzada ou difusão da magnetização pelo espectro de absorção, juntamente com a relaxação spin-rede. Medidas à 4,2 °K e 1,8 °K fornecem indícios que, ap6s algum tempo, apenas a relaxação spin-rede sobrexiste. O modelo teórico baseado na modulação da interação hiperfina e no processo direto fornece tempos menores que os obtidos experimentalmente. Isto deve-se à formação de conglomerados de centros. Também elaboramos um modelo levando em conta processo Raman, para prever tempos de relaxação spin-rede a temperaturas mais elevadas. / We measure the spin-lattice relaxation time of F centers in KCN at 1.8 and 4.2 K using the saturation recovery technique. The observed recovery is non exponential. This result is interpreted in terms of the combined effect of cross-relaxation and spin lattice relaxation. The values obtained for T1 (- sec) are found to be shorter than those calculated theoretically. This discrepancy is throught to arise from the presence of clusters. The characteristic time found from the initial part of recovery is found to be -6 sec in agreement with that found in KC for the recovery of a hole burnt in the resonance line (45). Finally the value of T1 was calculated for the Raman processes at high temperature and compared with known results.

In the past two decades polymer/clay nanocomposites (PCNs) have emerged as promising materials that exhibit remarkably improved properties when compared to conventional composites and pristine polymers. Such improvements strongly depend on the dispersion of clay nanoparticles in the polymer matrix. In spite of great efforts expended in characterizing clay dispersion, effective, simple and quantitative techniques are still needed. This work addresses this challenge by presenting new aspects of 1H solid-state NMR for quantifying clay dispersion in PCNs filled with clay containing paramagnetic ions. Employing these 1H solid-state NMR methods, some structure-processing-deformation relationships of PCNs were derived, and basic insights into nuclear relaxation and spin diffusion in PCNs were gained as well. Detailed models and analyses were described for 1H spin-lattice relaxation in the presence of paramagnetic clays in PCNs. Relaxation recovery was analytically correlated to clay dispersion in two ways: one is the initial relaxation recovery which is related to clay surface area, and the other is the spin-lattice relaxation time which is related to interparticle spacing. These two NMR observables were employed to quantitatively observe the evolution of clay morphology in poly(propylene)/clay (PP/MMT) nanocomposites upon equibiaxial stretching, as well as upon in situ uniaxial deformation. The initial relaxation recovery was independently utilized to determine the polymer-clay interfacial surface area and the degree of clay exfoliation. We demonstrated the capabilities of our models in quantitatively characterizing several materials, including poly(vinyl alcohol), nylon 6, poly(å-caprolactone) (PCL), poly(lactic acid) (PLA) and PP nanocomposites. These results were used to examine the dependence of clay morphology upon processing (strain ratio, strain rate, temperature), deformation (extension), component characteristics (polymer molecular weight, clay surface modification) and clay content. Effects of paramagnetic Fe3+ concentration and external magnetic field strength on 1H spin-lattice relaxation in PCNs were also investigated and discussed. In particular, low field separates the initial relaxation recovery into two stages: one related to clay content and the other related to the polymer-clay interfacial surface area. The low field was observed to enhance the paramagnetic contribution to the spin-lattice relaxation rate, increasing its sensitivity to clay morphology. In addition, measurements of long-distance spin diffusion coefficients for a variety of polymers and paramagnetic characteristics of organically modified clay were explored. Overall, the utility of NMR relaxometry in characterizing PCNs has been significantly expanded and successfully demonstrated in this dissertation.

Thesis (M.S.)--Mississippi State University. Department of Physics and Astronomy. / Title from title screen. Includes bibliographical references.

Realizamos medidas de tempos de relaxação de centros F em cristais de KCN, pela técnica de saturação do sinal de absorção por pulsos de microonda. Observamos um comportamento não exponencial para a recuperação da magnetização da amostra ap6so pulso. Este fato deve-se ao efeito combinado de relaxação cruzada ou difusão da magnetização pelo espectro de absorção, juntamente com a relaxação spin-rede. Medidas à 4,2 °K e 1,8 °K fornecem indícios que, ap6s algum tempo, apenas a relaxação spin-rede sobrexiste. O modelo teórico baseado na modulação da interação hiperfina e no processo direto fornece tempos menores que os obtidos experimentalmente. Isto deve-se à formação de conglomerados de centros. Também elaboramos um modelo levando em conta processo Raman, para prever tempos de relaxação spin-rede a temperaturas mais elevadas. / We measure the spin-lattice relaxation time of F centers in KCN at 1.8 and 4.2 K using the saturation recovery technique. The observed recovery is non exponential. This result is interpreted in terms of the combined effect of cross-relaxation and spin lattice relaxation. The values obtained for T1 (- sec) are found to be shorter than those calculated theoretically. This discrepancy is throught to arise from the presence of clusters. The characteristic time found from the initial part of recovery is found to be -6 sec in agreement with that found in KC for the recovery of a hole burnt in the resonance line (45). Finally the value of T1 was calculated for the Raman processes at high temperature and compared with known results.

In this work I present a 23Na/51V-NMR study of sodium vanadate as pure compound, and the influence of very small amounts of Sodium substitution by Calcium and Lithium is reported and discussed. The measurements of spin-lattice relaxation for 23Na and 51V are also presented. The sodium vanadate is found to have a double phase transition. The two transitions are close together and take place around 34 K. Above the transition temperature there is one V site in the mixed oxidation state 4.5+ and there is one Na site. A consistency for a number of un-doped and very slightly doped samples of three vanadium valences is argued, confirming a charge ordering transition at transition temperature, in good qualitative agreement with Bernert’s model and as well in quantitative and qualitative agreement with Sawa’s monoclinic structure. Below the transition temperature sodium is found to have ten sites. The number of the Na site is continuously developed with decreasing the temperature below transition temperature. This continues development of the Na sites demonstrates that the second transition is continues. The detailed spin-lattice relaxation rate for 23Na in both, the pure and the doped samples, in transition region, provided evidence for a non-symmetric and complex transition peak structure which we relate to the onset of more than one transition occurring at slightly different temperatures. This scales with the transition temperature reduction provided by lithium and calcium doping. The two transitions (i.e. dimerization and charge ordering) are intimately related. We investigated the spin-gap by means of Vanadium and Sodium spin-lattice relaxation temperature dependence well below transition temperature. The analysis of the single crystal data reveals a significant anisotropy in the nature of the gap, which is sensitive to Calcium and Lithium doping, indicating that is constrained to the ladder plane.

The spin-lattice relaxation rates are calculated for a rigid magnetic spin cluster in an elastic medium in the presence of a magnetic eld using the latticerame method. This rate is then compared with both the rate calculated using the quantum mechanical densitymatrix method and with the Glauber dynamics. These calculation results are used in the contribution of various heat baths, such as a phonon bath in various dimensions or a fermionic bath, to transition rates that enter into dynamic Monte Carlo simulations of molecular magnets and nanomagnets.

Dans cette thèse l’ancienne question philosophique “tout événement a-t-il une cause ?” sera examinée à la lumière de la mécanique quantique et de la théorie des probabilités. Aussi bien en physique qu’en philosophie des sciences la position orthodoxe maintient que le monde physique est indéterministe. Au niveau fondamental de la réalité physique – au niveau quantique – les événements se passeraient sans causes, mais par chance, par hasard ‘irréductible’. Le théorème physique le plus précis qui mène à cette conclusion est le théorème de Bell. Ici les prémisses de ce théorème seront réexaminées. Il sera rappelé que d’autres solutions au théorème que l’indéterminisme sont envisageables, dont certaines sont connues mais négligées, comme le ‘superdéterminisme’. Mais il sera argué que d’autres solutions compatibles avec le déterminisme existent, notamment en étudiant des systèmes physiques modèles. Une des conclusions générales de cette thèse est que l’interprétation du théorème de Bell et de la mécanique quantique dépend crucialement des prémisses philosophiques desquelles on part. Par exemple, au sein de la vision d’un Spinoza, le monde quantique peut bien être compris comme étant déterministe. Mais il est argué qu’aussi un déterminisme nettement moins radical que celui de Spinoza n’est pas éliminé par les expériences physiques. Si cela est vrai, le débat ‘déterminisme – indéterminisme’ n’est pas décidé au laboratoire : il reste philosophique et ouvert – contrairement à ce que l’on pense souvent. Dans la deuxième partie de cette thèse un modèle pour l’interprétation de la probabilité sera proposé. Une étude conceptuelle de la notion de probabilité indique que l’hypothèse du déterminisme aide à mieux comprendre ce que c’est qu’un ‘système probabiliste’. Il semble que le déterminisme peut répondre à certaines questions pour lesquelles l’indéterminisme n’a pas de réponses. Pour cette raison nous conclurons que la conjecture de Laplace – à savoir que la théorie des probabilités présuppose une réalité déterministe sous-jacente – garde toute sa légitimité. Dans cette thèse aussi bien les méthodes de la philosophie que de la physique seront utilisées. Il apparaît que les deux domaines sont ici solidement reliés, et qu’ils offrent un vaste potentiel de fertilisation croisée – donc bidirectionnelle. / In this thesis the ancient philosophical question whether ‘everything has a cause’ will be examined in the light of quantum mechanics and probability theory. In the physics and philosophy of science communities the orthodox position states that the physical world is indeterministic. On the deepest level of physical reality – the quantum level – things or events would have no causes but happen by chance, by irreducible hazard. Arguably the clearest and most convincing theorem that led to this conclusion is Bell’s theorem. Here the premises of this theorem will be re-evaluated, notably by investigating physical model systems. It will be recalled that other solutions to the theorem than indeterminism exist, some of which are known but neglected, such as ‘superdeterminism’. But it will be argued that also other solutions compatible with determinism exist. One general conclusion will be that the interpretation of Bell’s theorem and quantum mechanics hinges on the philosophical premises from which one starts. For instance, within a worldview à la Spinoza the quantum world may well be seen as deterministic. But it is argued that also much ‘softer’ determinism than Spinoza’s is not excluded by the existing experiments. If that is true the ‘determinism – indeterminism’ is not decided in the laboratory: it remains philosophical and open-ended – contrary to what is often believed. In the second part of the thesis a model for the interpretation of probability will be proposed. A conceptual study of the notion of probability indicates that the hypothesis of determinism is instrumental for understanding what ‘probabilistic systems’ are. It seems that determinism answers certain questions that cannot be answered by indeterminism. Therefore we believe there is room for the conjecture that probability theory cannot not do without a deterministic reality underneath probability – as Laplace claimed. Throughout the thesis the methods of philosophy and physics will be used. Both fields appear to be solidly intertwined here, and to offer a large potential for cross-fertilization – in both directions.