Quasicrystal is the term used for a solid that possesses an essentially discrete diffraction pattern without having translational symmetry. Compared to periodic crystals, this difference in structure gives quasicrystals new properties that make them interesting to study -- both from a mathematical and from a physical point of view. In this thesis we review a mathematical description of quasicrystals that aims at generalizing the well-established theory of periodic crystals. We see how this theory can be connected to the cohomology of groups and how we can use this connection to classify quasicrystals. We also review an experimental method, NIXSW (Normal Incidence X-ray Standing Waves), that is ordinarily used for surface structure determination of periodic crystals, and show how it can be used in the study of quasicrystal surfaces. Finally, we define the reduced lattice and show a way to plot lattices in MATLAB. We see that there is a connection between the diffraction pattern and the reduced lattice and we suggest a way to describe this connection.

La qualité des interfaces dans les multicouches périodiques est essentielle au développement de miroirs réfléchissant efficacement dans les domaines des rayons X et extrême ultraviolet (X-EUV). De manière générale, la structure des interfaces dépend des possibles interdiffusion et processus chimiques aux interfaces entre couches. L'idée principale de cette thèse est d'appliquer la technique des ondes stationnaires dans le domaine X à la caractérisation de matériaux, principalement mais non exclusivement aux multicouches périodiques. Cette méthode est basée sur l'interférence de deux faisceaux de rayons X cohérents. L'interférence constructive sur un plan anti-nodal amplifie le champ électrique tandis que l'interférence destructive minimise ce dernier sur un plan nodal. Cette technique des ondes stationnaires dans le domaine X permet l'excitation (photoémission, fluorescence, ...) d'endroits spécifiques dans un empilement périodique de matériaux. De cette manière, les spectres expérimentaux ainsi obtenus sont principalement les spectres caractéristiques des atomes situés sur un plan anti-nodal. Combinée avec d'autres techniques expérimentales telles que la spectroscopie d'émission X (XES) ou la spectroscopie de photoélectrons dans le domaine X (XPS), une information sélective en profondeur, avec une sensibilité sub-nanométrique, peut être obtenue. / The interfacial information of periodic multilayers can be crucial for the development of reflecting mirrors which operate in the X-ray and extreme ultraviolet (X-EUV) ranges. Such information may contain the interdiffusion and chemical process at the interfaces of the layers. The idea of this thesis is to apply the X-ray standing wave technique to the characterization of materials, mainly but not limited to the periodic multilayers. X-ray standing wave technique enables to enhance the excitation (photoemission, fluorescence etc.) of specific locations within a periodic stack. The nature of such advantage is the interference of two coherent X-ray beams. One may compare the X-ray standing waves with the mechanical standing waves. The constructive interference at the anti-nodal plane amplifies the electric field; while the destructive interference at the nodal plane minimizes the electric field. In this way, the experimental spectra obtained under standing wave field will be mostly the material located on the anti-nodal plane. Combined with other techniques such as X-ray emission spectroscopy and X-ray photoelectron spectroscopy, a depth-selective information with a sub-nanoscale sensitivity can be obtained.