Die Arbeit beschäftigt sich mit der Rolle der Atomcluster in den ikosaedrischen Quasikristallen. Untersucht werden dabei Eigenschaften der Zustandsdichte, der Leitfähigkeit und des Stabilisierungsverhaltens.

In this Dissertation thesis the spectral theory of Schrödinger operators modeling quasicrystals in dimension one ist investigated. We allow for a large class of measures as potentials covering also point interactions. The main results can be stated as follows: If the potential can be very well approximated by periodic potentials, then the correspondig Schrödinger operator does not have any eigenvalues. If the potential is aperiodic and satisfies a certain finite local complexity condition, the absolutely continuous spectrum is absent. We also prove Cantor spectra of zero Lebesgue measure for a large class of (a randomized version of) the operator.

Die Arbeit beschäftigt sich mit der Rolle der Atomcluster in den ikosaedrischen Quasikristallen. Untersucht werden dabei Eigenschaften der Zustandsdichte, der Leitfähigkeit und des Stabilisierungsverhaltens.

In this Dissertation thesis the spectral theory of Schrödinger operators modeling quasicrystals in dimension one ist investigated. We allow for a large class of measures as potentials covering also point interactions. The main results can be stated as follows: If the potential can be very well approximated by periodic potentials, then the correspondig Schrödinger operator does not have any eigenvalues. If the potential is aperiodic and satisfies a certain finite local complexity condition, the absolutely continuous spectrum is absent. We also prove Cantor spectra of zero Lebesgue measure for a large class of (a randomized version of) the operator.

In this thesis, the effect of mechanical deformation on structure, thermal stability and hardness of a single-phase spray-deposited quasicrystalline alloy with composition Al62.5Cu25Fe12.5 has been investigated in detail. The purpose of the investigation was to study the effect of mechanical milling at different milling speeds (which approximately scale with the milling intensity) on mechanically-induced phase transformations during milling and on the phase evolution during subsequent heating. The results of the milling experiments indicate that, irrespective of the milling speeds used, mechanical milling of Al62.5Cu25Fe12.5 quasicrystals leads to the formation of a disordered CsCl-type ß phase with grain size of about 10 – 20 nm. The analysis of the kinetics of the QC–to–ß phase transformation reveals that the milling intensity has a considerable effect on the characteristics of the transformation. The increase of the milling speed considerably shortens the incubation time needed to start the QC–to–ß phase transformation. Also, the overall transformation is much faster for milling at high speeds. The QC–to–ß phase transformation starts when the grain size of the quasicrystals is reduced to about 10 nm irrespective of the milling speed used and clearly indicates that a critical grain size of the quasicrystals for initiating the transformation exists. On the other hand, no critical value of lattice strain was found for the QC–to–ß transformation. This indicates that the phase transformation is controlled by the local length scale (i.e. the grain size) and by the corresponding grain boundaries rather than by the energy stored in the lattice. Energetic considerations obtained through a simple model based on the mass and velocity of the milling balls reveal that the energy needed for the QC–to–ß transformation increases with increasing the milling speed, that is, the energetic efficiency of the process decreases with increasing the milling intensity. This indicates that part the extra energy supplied during milling at high intensities is not used to induce the phase transformation but it is dissipated by heat. During heating, the milled powder displays a multi-step thermal behavior characterized by the grain growth of the disordered ß phase at low temperatures, followed, at higher temperatures, by its transformation into the original icosahedral quasicrystalline phase. The transformation is gradual and the quasicrystals and the disordered ß phase coexist over a temperature interval of more than 250 K. The phase transformations occurring during milling and subsequent annealing have a remarkable effect on the hardness, which can be tuned within a wide range of values (7–9.6 GPa) as a function of the volume fraction of the different phases. This suggests that a composite material with optimized mechanical properties can be produced by an appropriate thermo-mechanical treatment. The quasicrystals milled at a very low speed show a transition between Hall-Petch to inverse Hall-Petch behavior at a grain size of about 40 nm, which represents the critical value for grain size softening of the present Al62.5Cu25Fe12.5 quasicrystals. This behavior may be attributed to the complexity of the quasicrystalline structure and to its peculiar deformation mechanism at room temperature (i.e. shear banding), where meta-dislocation-assisted deformation is almost absent. In order to analyze the effectiveness of the Al62.5Cu25Fe12.5 quasicrystals as reinforcing agent in metal matrix composites, Al-based composites were synthesized by hot extrusion of elemental Al blended with different amounts of Al62.5Cu25Fe12.5 quasicrystalline particles. The work was focused on two specific aspects: evaluation of the mechanical properties through room temperature compression tests and modeling of the resulting properties. The addition of the quasicrystalline reinforcement is very effective for improving the room temperature mechanical properties of pure Al. The compressive strength increases from 155 MPa for pure Al to 330 and 407 MPa for the composites with 20 and 40 vol.% of reinforcement, respectively, reaching an ultimate strain of 55 % and 20 % before fracture occurs. These results indicate that the addition of the QC reinforcement leads to composite materials with compressive strengths exceeding that of pure Al by a factor of 2 – 2.5, while retaining appreciable plastic deformation. The mechanical properties of the composites have been modeled by taking into account the combined effect of load bearing, dislocation strengthening and matrix ligament size effects. The calculations are in very good agreement with the experimental results and reveal that the reduction of the matrix ligament size, which results in a similar strengthening effect as that observed for grain refinement, is the main strengthening mechanism in the current composites. Finally, the interfacial reaction between the Al matrix and the QC reinforcement has been used to further enhance the strength of the composites through the formation of a new microstructure consisting of the Al matrix reinforced with Al7Cu2Fe w-phase particles. The optimization of the structure-property relationship was done through the systematic variation of the processing temperature during consolidation. The mechanical behavior of these transformation-strengthened composites is remarkably improved compared to the parent material. The yield strength of the composites significantly increases as the Al + QC -> ω transformation progresses from 195 MPa for the sample reinforced only with QC particles to 400 MPa for the material where the Al + QC -> ω reaction is complete. These results clearly demonstrate that powder metallurgy, i.e. powder synthesis by ball milling followed by consolidation into bulk specimens, is an attractive processing route for the production of novel and innovative lightweight composites characterized by high strength combined with considerable plastic deformation. In addition, these findings indicate that the mechanical behavior of Al-based composites reinforced with Al62.5Cu25Fe12.5 quasicrystalline particles can be tuned within a wide range of strength and plasticity depending on the volume fraction of the reinforcement as well as on the extent of the interfacial reaction between Al matrix and QC reinforcing particles.

Understanding the connection of the atomic structure and the physical properties of materials remains one of the elementary questions of condensed-matter physics. One research line in this quest started with the discovery of quasicrystals by Shechtman et al. in 1982. It soon became clear that these materials with their 5-, 8-, 10- or 12-fold rotational symmetries, which are forbidden according to classical crystallography, can be described in terms of mathematical models for nonperiodic tilings of a plane proposed by Penrose and Ammann in the 1970s. Due to the missing translational symmetry of quasicrystals, till today only finite, relatively small systems or periodic approximants have been investigated by means of numerical calculations and theoretical results have mainly been obtained for one-dimensional systems. In this thesis we study d-dimensional quasiperiodic models, so-called labyrinth tilings, with separable Hamiltonians in the tight-binding approach. This method paves the way to study higher-dimensional, quantum mechanical solutions, which can be directly derived from the one-dimensional results. This allows the investigation of very large systems in two and three dimensions with up to 10^10 sites. In particular, we contemplate the class of metallic-mean sequences. Based on this model we focus on the electronic properties of quasicrystals with a special interest on the connection of the spectral and dynamical properties of the Hamiltonian. Hence, we investigate the characteristics of the eigenstates and wave functions and compare these with the wave-packet dynamics in the labyrinth tilings by numerical calculations and by a renormalization group approach in connection with perturbation theory. It turns out that many properties show a qualitatively similar behavior in different dimensions or are even independent of the dimension as e.g. the scaling behavior of the participation numbers and the mean square displacement of a wave packet. Further, we show that the structure of the labyrinth tilings and their transport properties are connected and obtain that certain moments of the spectral dimensions are related to the wave-packet dynamics. Besides this also the photonic properties are studied for one-dimensional quasiperiodic multilayer systems for oblique incidence of light, and we show that the characteristics of the transmission bands are related to the quasiperiodic structure. / Eine der elementaren Fragen der Physik kondensierter Materie beschäftigt sich mit dem Zusammenhang zwischen der atomaren Struktur und den physikalischen Eigenschaften von Materialien. Eine Forschungslinie in diesem Kontext begann mit der Entdeckung der Quasikristalle durch Shechtman et al. 1982. Es stellte sich bald heraus, dass diese Materialien mit ihren laut der klassischen Kristallographie verbotenen 5-, 8-, 10- oder 12-zähligen Rotationssymmetrien durch mathematische Modelle für die aperiodische Pflasterung der Ebene beschrieben werden können, die durch Penrose und Ammann in den 1970er Jahren vorgeschlagen wurden. Aufgrund der fehlenden Translationssymmetrie in Quasikristallen sind bis heute nur endliche, relativ kleine Systeme oder periodische Approximanten durch numerische Berechnungen untersucht worden und theoretische Ergebnisse wurden hauptsächlich für eindimensionale Systeme gewonnen. In dieser Arbeit werden d-dimensionale quasiperiodische Modelle, sogenannte Labyrinth-Pflasterungen, mit separablem Hamilton-Operator im Modell starker Bindung betrachtet. Diese Methode erlaubt es, quantenmechanische Lösungen in höheren Dimensionen direkt aus den eindimensionalen Ergebnissen abzuleiten und ermöglicht somit die Untersuchung von sehr großen Systemen in zwei und drei Dimensionen mit bis zu 10^10 Gitterpunkten. Insbesondere betrachten wir dabei quasiperiodische Folgen mit metallischem Schnitt. Basierend auf diesem Modell befassen wir uns im Speziellen mit den elektronischen Eigenschaften der Quasikristalle im Hinblick auf die Verbindung der spektralen und dynamischen Eigenschaften des Hamilton-Operators. Hierfür untersuchen wir die Eigenschaften der Eigenzustände und Wellenfunktionen und vergleichen diese mit der Dynamik von Wellenpaketen in den Labyrinth-Pflasterungen basierend auf numerischen Berechnungen und einem Renormierungsgruppen-Ansatz in Verbindung mit Störungstheorie. Dabei stellt sich heraus, dass viele Eigenschaften wie etwa das Skalenverhalten der Partizipationszahlen und der mittleren quadratischen Abweichung eines Wellenpakets für verschiedene Dimensionen ein qualitativ gleiches Verhalten zeigen oder sogar unabhängig von der Dimension sind. Zudem zeigen wir, dass die Struktur der Labyrinth-Pflasterungen und deren Transporteigenschaften sowie bestimmte Momente der spektralen Dimensionen und die Dynamik der Wellenpakete in Beziehung zueinander stehen. Darüber hinaus werden auch die photonischen Eigenschaften für eindimensionale quasiperiodische Mehrschichtsysteme für beliebige Einfallswinkel untersucht und der Verlauf der Transmissionsbänder mit der quasiperiodischen Struktur in Zusammenhang gebracht.

Understanding the connection of the atomic structure and the physical properties of materials remains one of the elementary questions of condensed-matter physics. One research line in this quest started with the discovery of quasicrystals by Shechtman et al. in 1982. It soon became clear that these materials with their 5-, 8-, 10- or 12-fold rotational symmetries, which are forbidden according to classical crystallography, can be described in terms of mathematical models for nonperiodic tilings of a plane proposed by Penrose and Ammann in the 1970s. Due to the missing translational symmetry of quasicrystals, till today only finite, relatively small systems or periodic approximants have been investigated by means of numerical calculations and theoretical results have mainly been obtained for one-dimensional systems. In this thesis we study d-dimensional quasiperiodic models, so-called labyrinth tilings, with separable Hamiltonians in the tight-binding approach. This method paves the way to study higher-dimensional, quantum mechanical solutions, which can be directly derived from the one-dimensional results. This allows the investigation of very large systems in two and three dimensions with up to 10^10 sites. In particular, we contemplate the class of metallic-mean sequences. Based on this model we focus on the electronic properties of quasicrystals with a special interest on the connection of the spectral and dynamical properties of the Hamiltonian. Hence, we investigate the characteristics of the eigenstates and wave functions and compare these with the wave-packet dynamics in the labyrinth tilings by numerical calculations and by a renormalization group approach in connection with perturbation theory. It turns out that many properties show a qualitatively similar behavior in different dimensions or are even independent of the dimension as e.g. the scaling behavior of the participation numbers and the mean square displacement of a wave packet. Further, we show that the structure of the labyrinth tilings and their transport properties are connected and obtain that certain moments of the spectral dimensions are related to the wave-packet dynamics. Besides this also the photonic properties are studied for one-dimensional quasiperiodic multilayer systems for oblique incidence of light, and we show that the characteristics of the transmission bands are related to the quasiperiodic structure. / Eine der elementaren Fragen der Physik kondensierter Materie beschäftigt sich mit dem Zusammenhang zwischen der atomaren Struktur und den physikalischen Eigenschaften von Materialien. Eine Forschungslinie in diesem Kontext begann mit der Entdeckung der Quasikristalle durch Shechtman et al. 1982. Es stellte sich bald heraus, dass diese Materialien mit ihren laut der klassischen Kristallographie verbotenen 5-, 8-, 10- oder 12-zähligen Rotationssymmetrien durch mathematische Modelle für die aperiodische Pflasterung der Ebene beschrieben werden können, die durch Penrose und Ammann in den 1970er Jahren vorgeschlagen wurden. Aufgrund der fehlenden Translationssymmetrie in Quasikristallen sind bis heute nur endliche, relativ kleine Systeme oder periodische Approximanten durch numerische Berechnungen untersucht worden und theoretische Ergebnisse wurden hauptsächlich für eindimensionale Systeme gewonnen. In dieser Arbeit werden d-dimensionale quasiperiodische Modelle, sogenannte Labyrinth-Pflasterungen, mit separablem Hamilton-Operator im Modell starker Bindung betrachtet. Diese Methode erlaubt es, quantenmechanische Lösungen in höheren Dimensionen direkt aus den eindimensionalen Ergebnissen abzuleiten und ermöglicht somit die Untersuchung von sehr großen Systemen in zwei und drei Dimensionen mit bis zu 10^10 Gitterpunkten. Insbesondere betrachten wir dabei quasiperiodische Folgen mit metallischem Schnitt. Basierend auf diesem Modell befassen wir uns im Speziellen mit den elektronischen Eigenschaften der Quasikristalle im Hinblick auf die Verbindung der spektralen und dynamischen Eigenschaften des Hamilton-Operators. Hierfür untersuchen wir die Eigenschaften der Eigenzustände und Wellenfunktionen und vergleichen diese mit der Dynamik von Wellenpaketen in den Labyrinth-Pflasterungen basierend auf numerischen Berechnungen und einem Renormierungsgruppen-Ansatz in Verbindung mit Störungstheorie. Dabei stellt sich heraus, dass viele Eigenschaften wie etwa das Skalenverhalten der Partizipationszahlen und der mittleren quadratischen Abweichung eines Wellenpakets für verschiedene Dimensionen ein qualitativ gleiches Verhalten zeigen oder sogar unabhängig von der Dimension sind. Zudem zeigen wir, dass die Struktur der Labyrinth-Pflasterungen und deren Transporteigenschaften sowie bestimmte Momente der spektralen Dimensionen und die Dynamik der Wellenpakete in Beziehung zueinander stehen. Darüber hinaus werden auch die photonischen Eigenschaften für eindimensionale quasiperiodische Mehrschichtsysteme für beliebige Einfallswinkel untersucht und der Verlauf der Transmissionsbänder mit der quasiperiodischen Struktur in Zusammenhang gebracht.

In this thesis, the effect of mechanical deformation on structure, thermal stability and hardness of a single-phase spray-deposited quasicrystalline alloy with composition Al62.5Cu25Fe12.5 has been investigated in detail. The purpose of the investigation was to study the effect of mechanical milling at different milling speeds (which approximately scale with the milling intensity) on mechanically-induced phase transformations during milling and on the phase evolution during subsequent heating. The results of the milling experiments indicate that, irrespective of the milling speeds used, mechanical milling of Al62.5Cu25Fe12.5 quasicrystals leads to the formation of a disordered CsCl-type ß phase with grain size of about 10 – 20 nm. The analysis of the kinetics of the QC–to–ß phase transformation reveals that the milling intensity has a considerable effect on the characteristics of the transformation. The increase of the milling speed considerably shortens the incubation time needed to start the QC–to–ß phase transformation. Also, the overall transformation is much faster for milling at high speeds. The QC–to–ß phase transformation starts when the grain size of the quasicrystals is reduced to about 10 nm irrespective of the milling speed used and clearly indicates that a critical grain size of the quasicrystals for initiating the transformation exists. On the other hand, no critical value of lattice strain was found for the QC–to–ß transformation. This indicates that the phase transformation is controlled by the local length scale (i.e. the grain size) and by the corresponding grain boundaries rather than by the energy stored in the lattice. Energetic considerations obtained through a simple model based on the mass and velocity of the milling balls reveal that the energy needed for the QC–to–ß transformation increases with increasing the milling speed, that is, the energetic efficiency of the process decreases with increasing the milling intensity. This indicates that part the extra energy supplied during milling at high intensities is not used to induce the phase transformation but it is dissipated by heat. During heating, the milled powder displays a multi-step thermal behavior characterized by the grain growth of the disordered ß phase at low temperatures, followed, at higher temperatures, by its transformation into the original icosahedral quasicrystalline phase. The transformation is gradual and the quasicrystals and the disordered ß phase coexist over a temperature interval of more than 250 K. The phase transformations occurring during milling and subsequent annealing have a remarkable effect on the hardness, which can be tuned within a wide range of values (7–9.6 GPa) as a function of the volume fraction of the different phases. This suggests that a composite material with optimized mechanical properties can be produced by an appropriate thermo-mechanical treatment. The quasicrystals milled at a very low speed show a transition between Hall-Petch to inverse Hall-Petch behavior at a grain size of about 40 nm, which represents the critical value for grain size softening of the present Al62.5Cu25Fe12.5 quasicrystals. This behavior may be attributed to the complexity of the quasicrystalline structure and to its peculiar deformation mechanism at room temperature (i.e. shear banding), where meta-dislocation-assisted deformation is almost absent. In order to analyze the effectiveness of the Al62.5Cu25Fe12.5 quasicrystals as reinforcing agent in metal matrix composites, Al-based composites were synthesized by hot extrusion of elemental Al blended with different amounts of Al62.5Cu25Fe12.5 quasicrystalline particles. The work was focused on two specific aspects: evaluation of the mechanical properties through room temperature compression tests and modeling of the resulting properties. The addition of the quasicrystalline reinforcement is very effective for improving the room temperature mechanical properties of pure Al. The compressive strength increases from 155 MPa for pure Al to 330 and 407 MPa for the composites with 20 and 40 vol.% of reinforcement, respectively, reaching an ultimate strain of 55 % and 20 % before fracture occurs. These results indicate that the addition of the QC reinforcement leads to composite materials with compressive strengths exceeding that of pure Al by a factor of 2 – 2.5, while retaining appreciable plastic deformation. The mechanical properties of the composites have been modeled by taking into account the combined effect of load bearing, dislocation strengthening and matrix ligament size effects. The calculations are in very good agreement with the experimental results and reveal that the reduction of the matrix ligament size, which results in a similar strengthening effect as that observed for grain refinement, is the main strengthening mechanism in the current composites. Finally, the interfacial reaction between the Al matrix and the QC reinforcement has been used to further enhance the strength of the composites through the formation of a new microstructure consisting of the Al matrix reinforced with Al7Cu2Fe w-phase particles. The optimization of the structure-property relationship was done through the systematic variation of the processing temperature during consolidation. The mechanical behavior of these transformation-strengthened composites is remarkably improved compared to the parent material. The yield strength of the composites significantly increases as the Al + QC -> ω transformation progresses from 195 MPa for the sample reinforced only with QC particles to 400 MPa for the material where the Al + QC -> ω reaction is complete. These results clearly demonstrate that powder metallurgy, i.e. powder synthesis by ball milling followed by consolidation into bulk specimens, is an attractive processing route for the production of novel and innovative lightweight composites characterized by high strength combined with considerable plastic deformation. In addition, these findings indicate that the mechanical behavior of Al-based composites reinforced with Al62.5Cu25Fe12.5 quasicrystalline particles can be tuned within a wide range of strength and plasticity depending on the volume fraction of the reinforcement as well as on the extent of the interfacial reaction between Al matrix and QC reinforcing particles.