The work presented in this thesis was motivated by the large amount of experimental investigations of the phonons in quasicrystals. The generalized vibrational density of states (GVDOS) was measured for many quasicrystalline phases and in some cases at different temperatures [suck et al (1997), Dugain et al (1997)]. The progress achieved in the structure determination of approximants to some quasicrystals was a legitimate motivation for numerical investigations of lattice dynamics in these structures. Two different types of interatomic interactions were used: the spring model and the ab-initio pair potentials. The investigations explained the shape of some experimentally measured GVDOS (d-AlNiCo, o-Al13Co4 and i-ZnMgY) via the calculation of the partial vibrational densities of states. Both calculated and measured GVDOS of the d-AlNiCo phase showed an intensity excess at low energies relatively to the ideal Debye behaviour. This excess was found to be a consequence of the existence of special modes at theses energies which are called ``quasi-localized modes''. These modes seem to be characteristic of the lattice dynamics in the complex Al-TM structures. To calculate the frequency shift due to the shift of the GVDOS through low energies observed experimentally at high temperatures, a new method based on a Monte-Carlo simulation was developed. It was shown that the quasi-localized modes introduce large frequency shifts at low energies. Finally, the vibrational entropy was also investigated, and it was found that it contributes to the stabilization of the complex structures over the relatively simple structures at high temperatures. / Die Arbeit, die in dieser Dissertation präsentiert wird, wurde durch eine Vielzahl von experimentellen Beobachtungen von Phononen in Quasikristallen motiviert. Die verallgemeinerte vibrationelle Zustandsdichte (GVDOS, generalized vibrational density of states) wurde für viele quasikristalline Phasen gemessen und für einige auch bei verschiedener Temperatur [Suck et al. (1997), Dugain et al. (1997)]. Der Fortschritt, der in der Bestimmung von Näherungen für einige Quasikristalle erreicht wurde war eine legitime Motivation für numerische Untersuchungen der Gitterdynamik auf diesen Strukturen. Es wurden zwei unterschiedliche interatomare Wechselwirkungen verwendet: Das Federmodell und die ab-initio Paar Potentiale. Die Untersuchungen erklärten die Form einiger experimenteller GVDOS-Messungen (d-AlNiCo, o-Al13Co4 und i-ZnMgY) mittels der Berechnung der partiellen vibrationellen Zustandsdichte. Beide, berechnete und gemessene, GVDOS der d-AlNiCo Phase zeigten einen Intensitätsanstieg bei kleinen Energien relativ zum idealen Debye Verhalten. Dieser Anstieg stellte sich als Konsequenz der Existenz von besonderen Moden bei diesen Energien heraus, die quasi-lokalisierte Moden genannt werden. Diese Moden scheinen charakteristisch für die Gitterdynamik in den komplexen Al-TM Strukturen zu sein. Um die experimentell beobachtete Frequenzverschiebung aufgrund der Verschiebung der GVDOS durch niedrige Energien zu berechnen, wurde eine neue, auf Monte-Carlo Simulation beruhende, Methode entwickelt. Es wurde gezeigt, daß die quasi-lokalisierten Moden große Frequenzverschiebungen bei kleinen Energien hervorrufen. Letzt-lich wurde auch die vibrationelle Entropie untersucht und es stellte sich heraus, daß sie bei hohen Temperaturen dazu beiträgt die komplexen Strukturen gegenüber den relativ einfachen zu stabilisieren.

The work presented in this thesis was motivated by the large amount of experimental investigations of the phonons in quasicrystals. The generalized vibrational density of states (GVDOS) was measured for many quasicrystalline phases and in some cases at different temperatures [suck et al (1997), Dugain et al (1997)]. The progress achieved in the structure determination of approximants to some quasicrystals was a legitimate motivation for numerical investigations of lattice dynamics in these structures. Two different types of interatomic interactions were used: the spring model and the ab-initio pair potentials. The investigations explained the shape of some experimentally measured GVDOS (d-AlNiCo, o-Al13Co4 and i-ZnMgY) via the calculation of the partial vibrational densities of states. Both calculated and measured GVDOS of the d-AlNiCo phase showed an intensity excess at low energies relatively to the ideal Debye behaviour. This excess was found to be a consequence of the existence of special modes at theses energies which are called ``quasi-localized modes''. These modes seem to be characteristic of the lattice dynamics in the complex Al-TM structures. To calculate the frequency shift due to the shift of the GVDOS through low energies observed experimentally at high temperatures, a new method based on a Monte-Carlo simulation was developed. It was shown that the quasi-localized modes introduce large frequency shifts at low energies. Finally, the vibrational entropy was also investigated, and it was found that it contributes to the stabilization of the complex structures over the relatively simple structures at high temperatures. / Die Arbeit, die in dieser Dissertation präsentiert wird, wurde durch eine Vielzahl von experimentellen Beobachtungen von Phononen in Quasikristallen motiviert. Die verallgemeinerte vibrationelle Zustandsdichte (GVDOS, generalized vibrational density of states) wurde für viele quasikristalline Phasen gemessen und für einige auch bei verschiedener Temperatur [Suck et al. (1997), Dugain et al. (1997)]. Der Fortschritt, der in der Bestimmung von Näherungen für einige Quasikristalle erreicht wurde war eine legitime Motivation für numerische Untersuchungen der Gitterdynamik auf diesen Strukturen. Es wurden zwei unterschiedliche interatomare Wechselwirkungen verwendet: Das Federmodell und die ab-initio Paar Potentiale. Die Untersuchungen erklärten die Form einiger experimenteller GVDOS-Messungen (d-AlNiCo, o-Al13Co4 und i-ZnMgY) mittels der Berechnung der partiellen vibrationellen Zustandsdichte. Beide, berechnete und gemessene, GVDOS der d-AlNiCo Phase zeigten einen Intensitätsanstieg bei kleinen Energien relativ zum idealen Debye Verhalten. Dieser Anstieg stellte sich als Konsequenz der Existenz von besonderen Moden bei diesen Energien heraus, die quasi-lokalisierte Moden genannt werden. Diese Moden scheinen charakteristisch für die Gitterdynamik in den komplexen Al-TM Strukturen zu sein. Um die experimentell beobachtete Frequenzverschiebung aufgrund der Verschiebung der GVDOS durch niedrige Energien zu berechnen, wurde eine neue, auf Monte-Carlo Simulation beruhende, Methode entwickelt. Es wurde gezeigt, daß die quasi-lokalisierten Moden große Frequenzverschiebungen bei kleinen Energien hervorrufen. Letzt-lich wurde auch die vibrationelle Entropie untersucht und es stellte sich heraus, daß sie bei hohen Temperaturen dazu beiträgt die komplexen Strukturen gegenüber den relativ einfachen zu stabilisieren.

O CVM - Cluster Variation Method, ou Método Variacional de Clusters é um método para cálculos termodinâmicos, baseado na aproximação de campo médio para a energia livre. Em sua concepção original, o CVM dispõe apenas de contribuições configuracionais. A proposta do presente trabalho é neste sentido aprimorar o método com a incorporação de outras componentes à energia livre. Acreditamos que as contribuições volumétricas, tanto dilatacionais quanto vibracionais, são de grande importância, e estas contribuições à energia livre foram aqui adicionadas ao método. Outro objetivo do presente trabalho foi verificar se esta abordagem solucionaria o problema da escala de temperaturas de cálculos ab initio aliados ao CVM. Nestes cálculos, a escala de temperaturas dos diagramas de fases geralmente é duas a três vezes maior que o verificado experimentalmente ou, equivalentemente, que os resultados CVM a partir de dados experimentais. Um novo algoritmo de minimização foi deste modo proposto para levar em conta os efeitos vibracionais e a inclusão do volume como variável. O algoritmo é baseado no NIM (Natural Iteration Method), que é utilizado para a minimização no caso configuracional. Um método para os cálculos vibracionais foi elaborado a partir do modelo de Debye-Grüneisen, com considerações adicionais elásticas, adaptado para sistemas multicomponentes (isto é, ligas). O modelo é baseado naquele desenvolvido por Anderson, quando existem dados ab initio de constantes elásticas, ou Moruzzi-Janak-Schwarz caso contrário. Em cálculos ab initio é possível determinar as constantes elásticas de cada estrutura considerada. Com estes dados, a temperatura de Debye e o módulo de volume a 0K são determinados com maior precisão. Dados de primeiros princípios relativos a energias de coesão/formação em função do volume para o sistema prototípico Fe-Al CCC (ferro-alumínio cúbico de corpo centrado) foram utilizados como exemplo para a obtenção de parâmetros para o modelo. / The Cluster Variation Method (CVM), used in thermodynamical calculations, is based in the mean-field approximation to the free energy. The CVM was originally devised to treat configurational-only cases. The scope of the present work is to enhance the method's capabilities, introducing other free energy components. The volumetric contributions, either dilatacional or vibrational, are believed to be of great importance, and are therefore incorporated here in the method. Another aim is to verify whether this approach would solve the temperature range calculated with CVM using ab initio data. In this kind of calculation, the phase diagram temperature range is usually twice or three times as large as experimentally verified or, equivalently, as the ones obtained in CVM calculations using experimental data. Therefore, a new minimisation algorithm was proposed to handle with the vibrational effects and the volume as a variable. The algorithm is based on the NIM (Natural Iteration Method), which is used for the minimization in the configurational case. The Debye-Grüneisen approximation has been adapted, with elastical considerations, for multicomponent systems (i.e., alloys). The method is based on Anderson model, when ab initio elastic constants are available, or on Moruzzi-Janak-Schwarz model otherwise. In ab initio calculations it is possible to determine the elastic constants for each structure considered. Using these data, Debye temperature and bulk modulus at 0K are determined with greater accuracy. First-principles cohesion/formation energies in function of volume for the b.c.c. Fe-Al (body-centered cubic iron-aluminum) system were used as an example to derive parameters to the model.

O CVM - Cluster Variation Method, ou Método Variacional de Clusters é um método para cálculos termodinâmicos, baseado na aproximação de campo médio para a energia livre. Em sua concepção original, o CVM dispõe apenas de contribuições configuracionais. A proposta do presente trabalho é neste sentido aprimorar o método com a incorporação de outras componentes à energia livre. Acreditamos que as contribuições volumétricas, tanto dilatacionais quanto vibracionais, são de grande importância, e estas contribuições à energia livre foram aqui adicionadas ao método. Outro objetivo do presente trabalho foi verificar se esta abordagem solucionaria o problema da escala de temperaturas de cálculos ab initio aliados ao CVM. Nestes cálculos, a escala de temperaturas dos diagramas de fases geralmente é duas a três vezes maior que o verificado experimentalmente ou, equivalentemente, que os resultados CVM a partir de dados experimentais. Um novo algoritmo de minimização foi deste modo proposto para levar em conta os efeitos vibracionais e a inclusão do volume como variável. O algoritmo é baseado no NIM (Natural Iteration Method), que é utilizado para a minimização no caso configuracional. Um método para os cálculos vibracionais foi elaborado a partir do modelo de Debye-Grüneisen, com considerações adicionais elásticas, adaptado para sistemas multicomponentes (isto é, ligas). O modelo é baseado naquele desenvolvido por Anderson, quando existem dados ab initio de constantes elásticas, ou Moruzzi-Janak-Schwarz caso contrário. Em cálculos ab initio é possível determinar as constantes elásticas de cada estrutura considerada. Com estes dados, a temperatura de Debye e o módulo de volume a 0K são determinados com maior precisão. Dados de primeiros princípios relativos a energias de coesão/formação em função do volume para o sistema prototípico Fe-Al CCC (ferro-alumínio cúbico de corpo centrado) foram utilizados como exemplo para a obtenção de parâmetros para o modelo. / The Cluster Variation Method (CVM), used in thermodynamical calculations, is based in the mean-field approximation to the free energy. The CVM was originally devised to treat configurational-only cases. The scope of the present work is to enhance the method's capabilities, introducing other free energy components. The volumetric contributions, either dilatacional or vibrational, are believed to be of great importance, and are therefore incorporated here in the method. Another aim is to verify whether this approach would solve the temperature range calculated with CVM using ab initio data. In this kind of calculation, the phase diagram temperature range is usually twice or three times as large as experimentally verified or, equivalently, as the ones obtained in CVM calculations using experimental data. Therefore, a new minimisation algorithm was proposed to handle with the vibrational effects and the volume as a variable. The algorithm is based on the NIM (Natural Iteration Method), which is used for the minimization in the configurational case. The Debye-Grüneisen approximation has been adapted, with elastical considerations, for multicomponent systems (i.e., alloys). The method is based on Anderson model, when ab initio elastic constants are available, or on Moruzzi-Janak-Schwarz model otherwise. In ab initio calculations it is possible to determine the elastic constants for each structure considered. Using these data, Debye temperature and bulk modulus at 0K are determined with greater accuracy. First-principles cohesion/formation energies in function of volume for the b.c.c. Fe-Al (body-centered cubic iron-aluminum) system were used as an example to derive parameters to the model.