Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed

X-Ray Diffraction, Calorimetric and Dielectric Relaxation Study of the Amorphous and Smectic States of a Main Chain Liquid Crystalline Polymer

Los polímeros cristales líquidos (LCP) son sistemas complejos que forman mesofases que presentan orden orientacional y polímeros amorfos. Con frecuencia, el estado amorfo isotrópico no puede ser estudiado debido a la rápida formación de mesofases. En este trabajo se ha sintetizado y estudiado un nuevo LCP: poli(trietilenglicol metil p, p '-bibenzoato), PTEMeB. Este polímero presenta una formación de mesofase bastante lenta haciendo posible estudiar de forma independiente tanto los estados amorfo y de cristal líquidos. La estructura y las transiciones de fase del PTEMeB han sido investigados por calorimetría (DSC), con MAXS / WAXS con temperatura variable que emplean radiación de sincrotrón y con difracción de rayos X. Estos estudios han mostrado la existencia de dos transiciones vítreas, relacionadas con las fases amorfa y cristal líquido. Se ha realizado un estudio de relajación dieléctrica en amplios intervalos de temperatura y presión. Se ha encontrado que la transición vítrea dinámica de la fase amorfa es más lenta que la del cristal líquido. El estudio de la relajación ? nos ha permitido seguir la formación isoterma de la mesofase a presión atmosférica. Además, con el estudio el comportamiento dinámico a alta presión se ha encontrado que se produce la formación rápida de la mesofase inducida por cambios bruscos de presión. Liquid crystalline polymers (LCPs) are complex systems that include features of both orientationally ordered mesophases and amorphous polymers. Frequently, the isotropic amorphous state cannot be studied due to the rapid mesophase formation. Here, a new main chain LCP, poly(triethyleneglycol methyl p,p'-bibenzoate), PTEMeB, has been synthesized. It shows a rather slow mesophase formation making possible to study independently both the amorphous and the liquid crystalline states. The structure and phase transitions of PTEMeB have been investigated by calorimetry, variable-temperature MAXS/WAXS employing synchrotron radiation, and X-ray diffraction in oriented fibers. These experiments have pointed out the presence of two glass transitions, related to the amorphous or to the liquid crystal phases. Additionally, the mesophase seems to be a coexistence of orthogonal and tilted smectic phases. A dielectric relaxation study of PTEMeB over broad ranges of temperature and pressure has been performed. The dynamic glass transition turns out to be slower for the amorphous state than for the liquid crystal. Monitoring of the α relaxation has allowed us to follow the isothermal mesophase formation at atmospheric pressure. Additionally, the dynamical behavior at high pressures has pointed out the fast formation of the mesophase induced by sudden pressure changes.

Ab initio Methodology for Calculating Elastic Constant and Sound Velocity for Hydrogen and Deuterium in Molecular Solid Phase

Hydrogen isotopes play a critical role both in inertial and magnetic confinement Nuclear Fusion. Since the preferent fuel needed for this technology is a mixture of deuterium and tritium. The study of these isotopes particularly at very low temperatures carries a technological interest in other applications. The present line promotes a deep study on the structural configuration that hydrogen and deuterium adopt at cryogenic temperatures and at high pressures. Typical conditions occurring in present Inertial Fusion target designs. Our approach is aims to determine the crystal structure characteristics, phase transitions and other parameters strongly correlated to variations of temperature and pressure. With this results is possible calculated the elastic constant and sound velocity for hydrogen and deuterium in molecular solid phase.

Quantum molecular dynamics and ab initio studies of the crystal structure of hydrogen and deuterium

Hydrogen isotopes play a critical role both in inertial and magnetic confinemen Nuclear Fusion. Since the preferent fuel needed for this technology is a mixture of deuterium and tritium. The study of these isotopes particularly at very low temperatures carries a technological interest in other applications. The present line promotes a deep study on the structural configuration that hydrogen and deuterium adopt at cryogenic temperatures and at high pressures. Typical conditions occurring in present Inertial Fusion target designs. Our approach is aims to determine the crystal structure characteristics, phase transitions and other parameters strongly correlated to variations of temperature and pressure.

Modulation instability-induced fading in phase-sensitive optical time-domain reflectometry

Phase-sensitive optical time-domain reflectometry (?OTDR) is a simple and effective tool allowing the distributed monitoring of vibrations along single-mode fibers. We show in this Letter that modulation instability (MI) can induce a position-dependent signal fading in long-range ?OTDR over conventional optical fibers. This fading leads to a complete masking of the interference signal recorded at certain positions and therefore to a sensitivity loss at these positions. We illustrate this effect both theoretically and experimentally. While this effect is detrimental in the context of distributed vibration analysis using ?OTDR, we also believe that the technique provides a clear and insightful way to evidence the Fermi?Pasta?Ulam recurrence associated with the MI process.

Study of helical induced anisotropy in amorphous ribbons for sensor applications

Amorphous samples with helical induced anisotropy show magnetization processes that can be controlled by applying a longitudinal magnetic field simultaneously with an alternating current flowing through the sample. By varying the current amplitude and the phase difference between current and applied field, a wide range of coercivity and susceptibility values can be achieved. This work shows that the apparent coercive field and the susceptibility can be controlled in amorphous ribbons with helical anisotropy. These characteristics make these samples very suitable for their application as sensor cores, magnetic amplifiers, variable reluctance transformer cores, etc

Phase transition in the countdown problem

We present a combinatorial decision problem, inspired by the celebrated quiz show called Countdown, that involves the computation of a given target number T from a set of k randomly chosen integers along with a set of arithmetic operations. We find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter, hence separating the algorithm's parameter space in different phases. We also find that the system is maximally efficient close to the critical point. We derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit.

Open-atmosphere structural depth profiling of multilayer samples of photovoltaic interest using laser-induced plasma spectrometry

The present work aims to assess Laser-Induced Plasma Spectrometry (LIPS) as a tool for the characterization of photovoltaic materials. Despite being a well-established technique with applications to many scientific and industrial fields, so far LIPS is little known to the photovoltaic scientific community. The technique allows the rapid characterization of layered samples without sample preparation, in open atmosphere and in real time. In this paper, we assess LIPS ability for the determination of elements that are difficult to analyze by other broadly used techniques, or for producing analytical information from very low-concentration elements. The results of the LIPS characterization of two different samples are presented: 1) a 90 nm, Al-doped ZnO layer deposited on a Si substrate by RF sputtering and 2) a Te-doped GaInP layer grown on GaAs by Metalorganic Vapor Phase Epitaxy. For both cases, the depth profile of the constituent and dopant elements is reported along with details of the experimental setup and the optimization of key parameters. It is remarkable that the longest time of analysis was ∼10 s, what, in conjunction with the other characteristics mentioned, makes of LIPS an appealing technique for rapid screening or quality control whether at the lab or at the production line.

A review of load models for the prediction of footfall-induced vibration in structures

Crowd induced dynamic loading in large structures, such as gymnasiums or stadium, is usually modelled as a series of harmonic loads which are defined in terms of their Fourier coefficients. Different values of these coefficients that were obtained from full scale measurements can be found in codes. Recently, an alternative has been proposed, based on random generation of load time histories that take into account phase lag among individuals inside the crowd. This paper presents the testing done on a structure designed to be a gymnasium. Two series of dynamic test were performed on the gym slab. For the first test an electrodynamic shaker was placed at several locations and during the second one people located inside a marked area bounced and jumped guided by different metronome rates. A finite element model (FEM) is presented and a comparison of numerically predicted and experimentally observed vibration modes and frequencies has been used to assess its validity. The second group of measurements will be compared with predictions made using the FEM model and three alternatives for crowd induced load modelling.

Noise-induced attractor annihilation in the delayed feedback logistic map

We study dynamics of the bistable logistic map with delayed feedback, under the influence of white Gaussian noise and periodic modulation applied to the variable. This system may serve as a model to describe population dynamics under finite resources in noisy environment with seasonal fluctuations. While a very small amount of noise has no effect on the global structure of the coexisting attractors in phase space, an intermediate noise totally eliminates one of the attractors. Slow periodic modulation enhances the attractor annihilation.

Vortex-induced vibrations on bridges : the interaction mechanism in complex sections and a new proposed semi-empirical model

El presente trabajo de investigación se ocupa del estudio de las vibraciones verticales inducidas por vórtices (VIV) en aquellos puentes que, por sus características geométricas y propiedades dinámicas, muestran cierta sensibilidad este tipo de fenómeno aeroelástico. El objeto principal es el análisis del mecanismo de interacción viento-estructura sobre secciones no fuseladas de geometría simple, con objeto de realizar una adecuada caracterización del problema y poder abordar posteriormente el análisis de otras secciones de geometría más compleja, representativas de los principales elementos estructurales de los puentes, como arcos, tableros, torres y pilas. Este aspecto es fundamental durante la fase de diseño del puente, donde deberán tenerse en cuenta también una serie de detalles que pueden influir significativamente su sensibilidad ante problemas aerodinámicos, como la morfología y dimensiones principales de la sección transversal del tablero, la disposición de barreras de seguridad y barreras cortaviento, o las riostras que unen diferentes elementos estructurales. La configuración de dos elementos en tándem o la construcción de un puente en las inmediaciones de otro existente son otros aspectos a considerar respecto a la sensibilidad frente a efectos aeroelásticos. El estudio se ha llevado a cabo principalmente mediante la implementación de simulaciones numéricas que reproducen la interacción entre la corriente de aire y secciones representativas de modelos estructurales, a partir de un código CFD basado en el método de las partículas de vórtices (VPM), siguiendo por tanto un esquema Lagrangiano. Los resultados han sido validados con datos experimentales existentes, valores procedentes de ensayos en túnel de viento y registros reales a partir de diferentes casos de estudio: Alconétar (2006), Niterói (1980), Trans- Tokyo Bay (1995) y Volgogrado (2010). Finalmente, se propone un modelo semi-empírico para la estimación del rango de velocidades críticas y amplitudes de oscilación basado en la utilización de las derivadas de flameo de Scanlan, y la densidad espectral de las fuerzas aerodinámicas en el dominio de la frecuencia. The present research work concerns the study of vertical vortex-induced vibrations (VIV) in bridges which show certain sensitivity to this type of aeroelastic phenomenon. It focuses on the analysis of the wind-structure interaction mechanism on bluff sections, with the objective of making a good characterisation of the problem and subsequently addressing the analysis of sections with a complex geometry, which are representative of the bridge structural elements, such as arches, decks, towers and piers. This issue is of relative importance during the bridge design phase, since minor details of the aforementioned elements can significantly influence its sensitivity to aerodynamic problems. The shape and main dimensions of the deck cross section, the addition of safety barriers and windshields, the presence of braces to enhance the structure mechanical properties, the utilisation of cross sections in tandem arrangement, or the erection of a new bridge in the vicinity of another existing one are some of the aspects to be considered regarding the sensitivity to the aeroelastic effects. The study has been carried out mainly through the implementation of numerical simulations that reproduces the interaction between the airflow and the representative cross section of a structural bridge model, by the use of a CFD code based on the vortex particle method (VPM), thus following a Lagrangian scheme. The results have been validated with existing experimental data, values from wind tunnel tests and full scale observations from the different case studies: Alconétar (2006), Niterói (1980), Trans-Tokyo Bay (1995) and Volgograd (2010). Finally, a new semi-empirical model is proposed for the estimation of the critical wind velocity ranges and oscillation amplitudes based on the use of the Scanlan’s flutter derivatives and the power spectral density of aerodynamic force time history in the frequency domain.

Atomistic modeling of Ge damage accumulation, amorphization and solid phase epitaxial regrowth

En los últimos años, el Ge ha ganado de nuevo atención con la finalidad de ser integrado en el seno de las existentes tecnologías de microelectrónica. Aunque no se le considera como un canddato capaz de reemplazar completamente al Si en el futuro próximo, probalemente servirá como un excelente complemento para aumentar las propiedades eléctricas en dispositivos futuros, especialmente debido a su alta movilidad de portadores. Esta integración requiere de un avance significativo del estado del arte en los procesos de fabricado. Técnicas de simulación, como los algoritmos de Monte Carlo cinético (KMC), proporcionan un ambiente atractivo para llevar a cabo investigación y desarrollo en este campo, especialmente en términos de costes en tiempo y financiación. En este estudio se han usado, por primera vez, técnicas de KMC con el fin entender el procesado “front-end” de Ge en su fabricación, específicamente la acumulación de dañado y amorfización producidas por implantación iónica y el crecimiento epitaxial en fase sólida (SPER) de las capas amorfizadas. Primero, simulaciones de aproximación de clisiones binarias (BCA) son usadas para calcular el dañado causado por cada ión. La evolución de este dañado en el tiempo se simula usando KMC sin red, o de objetos (OKMC) en el que sólamente se consideran los defectos. El SPER se simula a través de una aproximación KMC de red (LKMC), siendo capaz de seguir la evolución de los átomos de la red que forman la intercara amorfo/cristalina. Con el modelo de amorfización desarrollado a lo largo de este trabajo, implementado en un simulador multi-material, se pueden simular todos estos procesos. Ha sido posible entender la acumulación de dañado, desde la generación de defectos puntuales hasta la formación completa de capas amorfas. Esta acumulación ocurre en tres regímenes bien diferenciados, empezando con un ritmo lento de formación de regiones de dañado, seguido por una rápida relajación local de ciertas áreas en la fase amorfa donde ambas fases, amorfa y cristalina, coexisten, para terminar en la amorfización completa de capas extensas, donde satura el ritmo de acumulación. Dicha transición ocurre cuando la concentración de dañado supera cierto valor límite, el cual es independiente de las condiciones de implantación. Cuando se implantan los iones a temperaturas relativamente altas, el recocido dinámico cura el dañado previamente introducido y se establece una competición entre la generación de dañado y su disolución. Estos efectos se vuelven especialmente importantes para iones ligeros, como el B, el cual crea dañado más diluido, pequeño y distribuido de manera diferente que el causado por la implantación de iones más pesados, como el Ge. Esta descripción reproduce satisfactoriamente la cantidad de dañado y la extensión de las capas amorfas causadas por implantación iónica reportadas en la bibliografía. La velocidad de recristalización de la muestra previamente amorfizada depende fuertemente de la orientación del sustrato. El modelo LKMC presentado ha sido capaz de explicar estas diferencias entre orientaciones a través de un simple modelo, dominado por una única energía de activación y diferentes prefactores en las frecuencias de SPER dependiendo de las configuraciones de vecinos de los átomos que recristalizan. La formación de maclas aparece como una consecuencia de esta descripción, y es predominante en sustratos crecidos en la orientación (111)Ge. Este modelo es capaz de reproducir resultados experimentales para diferentes orientaciones, temperaturas y tiempos de evolución de la intercara amorfo/cristalina reportados por diferentes autores. Las parametrizaciones preliminares realizadas de los tensores de activación de tensiones son también capaces de proveer una buena correlación entre las simulaciones y los resultados experimentales de velocidad de SPER a diferentes temperaturas bajo una presión hidrostática aplicada. Los estudios presentados en esta tesis han ayudado a alcanzar un mejor entendimiento de los mecanismos de producción de dañado, su evolución, amorfización y SPER para Ge, además de servir como una útil herramienta para continuar el trabajo en este campo. In the recent years, Ge has regained attention to be integrated into existing microelectronic technologies. Even though it is not thought to be a feasible full replacement to Si in the near future, it will likely serve as an excellent complement to enhance electrical properties in future devices, specially due to its high carrier mobilities. This integration requires a significant upgrade of the state-of-the-art of regular manufacturing processes. Simulation techniques, such as kinetic Monte Carlo (KMC) algorithms, provide an appealing environment to research and innovation in the field, specially in terms of time and funding costs. In the present study, KMC techniques are used, for the first time, to understand Ge front-end processing, specifically damage accumulation and amorphization produced by ion implantation and Solid Phase Epitaxial Regrowth (SPER) of the amorphized layers. First, Binary Collision Approximation (BCA) simulations are used to calculate the damage caused by every ion. The evolution of this damage over time is simulated using non-lattice, or Object, KMC (OKMC) in which only defects are considered. SPER is simulated through a Lattice KMC (LKMC) approach, being able to follow the evolution of the lattice atoms forming the amorphous/crystalline interface. With the amorphization model developed in this work, implemented into a multi-material process simulator, all these processes can be simulated. It has been possible to understand damage accumulation, from point defect generation up to full amorphous layers formation. This accumulation occurs in three differentiated regimes, starting at a slow formation rate of the damage regions, followed by a fast local relaxation of areas into the amorphous phase where both crystalline and amorphous phases coexist, ending in full amorphization of extended layers, where the accumulation rate saturates. This transition occurs when the damage concentration overcomes a certain threshold value, which is independent of the implantation conditions. When implanting ions at relatively high temperatures, dynamic annealing takes place, healing the previously induced damage and establishing a competition between damage generation and its dissolution. These effects become specially important for light ions, as B, for which the created damage is more diluted, smaller and differently distributed than that caused by implanting heavier ions, as Ge. This description successfully reproduces damage quantity and extension of amorphous layers caused by means of ion implantation reported in the literature. Recrystallization velocity of the previously amorphized sample strongly depends on the substrate orientation. The presented LKMC model has been able to explain these differences between orientations through a simple model, dominated by one only activation energy and different prefactors for the SPER rates depending on the neighboring configuration of the recrystallizing atoms. Twin defects formation appears as a consequence of this description, and are predominant for (111)Ge oriented grown substrates. This model is able to reproduce experimental results for different orientations, temperatures and times of evolution of the amorphous/crystalline interface reported by different authors. Preliminary parameterizations for the activation strain tensors are able to also provide a good match between simulations and reported experimental results for SPER velocities at different temperatures under the appliance of hydrostatic pressure. The studies presented in this thesis have helped to achieve a greater understanding of damage generation, evolution, amorphization and SPER mechanisms in Ge, and also provide a useful tool to continue research in this field.