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Zugl.: Aachen, Techn. Hochsch., Diss., 1982.

In this thesis, anisotropic Heisenberg magnets with arbitrary spin are investigated within the second-order Green-function theory. Three models are considered. First, the second-order Green-fuction theory for one-dimensional and two-dimensional Heisenberg ferromagnets with arbitrary spin S in a magnetic field is developed. For the determination of the introduced vertex parameters sum rules, higher-derivative sum rules, and regularity conditions are derived, and the equality of the isothermal and the longitudinal uniform static Kubo susceptibilities is required. Thermodynamic quantities, such as the specific heat, magnetic susceptibility, transverse and longitudinal correlation lengths are calculated. Empirical formulas describing the dependence of the position and height of the susceptibility maximum on the magnetic field are given. An anomal behavior of the longitudinal correlation length is observed. The appearance of two maxima in the temperature dependence of the specific heat is discussed. Further, as an example of a system with an anisotropy in the spin space, the S=1 ferromagnetic chain with easy-axis single-ion anisotropy is studied. Justified by the up-down symmetry of the model with respect to $S_i^z -> -S_i^z$, $\\langle S_i^z \\rangle=0$ is set. Two different ways of the determination of the introduced vertex parameters are presented. The transverse nearest-neighbor correlation function, spin-wave spectrum and longitudinal correlation length are analyzed. The effects of the single-ion anisotropy on the transverse and longitudinal uniform static susceptibilities as well as on the appearance of two maxima in the temperature dependence of the specific heat are examined. Finally, as examples of spatial anisotropic spin systems,layered Heisenberg ferromagnets and antiferromagnets with arbitrary spin are studied within the rotation-invariant Green-function theory. The long-range order is described by the condensation term, which is determined from the requirement that in the ordered state the static susceptibility has to diverge at the ordering wave vector. For determination of the introduced vertex parameters, the sum rule and the isotropy condition are used and also assumptions regarding the temperature dependence of some parameters are made. The main focus is put on the calculation of the specific heat, the Curie temperature, and the Néel temperature in dependence on the interlayer coupling and the spin-quantum number. Empirical formulas describing the dependence of the transition temperatures on the ratio of interlayer and intralayer couplings are given. For all three models, the results of the Green-function theory are compared to available results of exact approaches (Quantum Monte Carlo, exact diagonalization, Bethe-ansatz method) and to available experimental data.

In this thesis, anisotropic Heisenberg magnets with arbitrary spin are investigated within the second-order Green-function theory. Three models are considered. First, the second-order Green-fuction theory for one-dimensional and two-dimensional Heisenberg ferromagnets with arbitrary spin S in a magnetic field is developed. For the determination of the introduced vertex parameters sum rules, higher-derivative sum rules, and regularity conditions are derived, and the equality of the isothermal and the longitudinal uniform static Kubo susceptibilities is required. Thermodynamic quantities, such as the specific heat, magnetic susceptibility, transverse and longitudinal correlation lengths are calculated. Empirical formulas describing the dependence of the position and height of the susceptibility maximum on the magnetic field are given. An anomal behavior of the longitudinal correlation length is observed. The appearance of two maxima in the temperature dependence of the specific heat is discussed. Further, as an example of a system with an anisotropy in the spin space, the S=1 ferromagnetic chain with easy-axis single-ion anisotropy is studied. Justified by the up-down symmetry of the model with respect to $S_i^z -> -S_i^z$, $\\langle S_i^z \\rangle=0$ is set. Two different ways of the determination of the introduced vertex parameters are presented. The transverse nearest-neighbor correlation function, spin-wave spectrum and longitudinal correlation length are analyzed. The effects of the single-ion anisotropy on the transverse and longitudinal uniform static susceptibilities as well as on the appearance of two maxima in the temperature dependence of the specific heat are examined. Finally, as examples of spatial anisotropic spin systems,layered Heisenberg ferromagnets and antiferromagnets with arbitrary spin are studied within the rotation-invariant Green-function theory. The long-range order is described by the condensation term, which is determined from the requirement that in the ordered state the static susceptibility has to diverge at the ordering wave vector. For determination of the introduced vertex parameters, the sum rule and the isotropy condition are used and also assumptions regarding the temperature dependence of some parameters are made. The main focus is put on the calculation of the specific heat, the Curie temperature, and the Néel temperature in dependence on the interlayer coupling and the spin-quantum number. Empirical formulas describing the dependence of the transition temperatures on the ratio of interlayer and intralayer couplings are given. For all three models, the results of the Green-function theory are compared to available results of exact approaches (Quantum Monte Carlo, exact diagonalization, Bethe-ansatz method) and to available experimental data.

Die verlässliche Vorhersage der Wärmeentzugsleistungen als auch der Soletemperaturen in den Sonden sind wichtig für deren Auslegung und Betriebsoptimierung. Es ist ebenso wichtig für die Auslegung und Optimierung der Anlagen im versorgten Gebäude. In der vorliegenden Dissertation wurde versucht, durch eine mathematische Weiterentwicklung von Greenschen Funktionen (g-Funktionen) eine präzisere Lösung für Temperaturverteilung im Erdreich infolge des von einer oder mehreren Sonden verursachten Wärmeentzuges mit verschiedenen Randbedingungen im geologischen Untergrund zu erreichen. Hierzu wurden sechs „neue“ g-Funktionen entwickelt, die vertikal variable Wärmeentzüge einzelner Sonden und Sondenfelder, eine Asymmetrie des Wärmeentzuges der Sonde, den Einfluss einer zusätzlichen Grundwasserströmung und den realen, geschichteten Untergrund berücksichtigen. Die mathematischen Modelle des Erdreichs wurden mit Modellen für die Soleströmung und Wärmeübergabe in der Hinterfüllung der Sonde gekoppelt und anschließend auf ein praktisches Betriebsbeispiel angewendet. Die Arbeit enthält ebenfalls umfangreiche Sichtung existierender Modelle sowie deren Anwendung und vergleichende Bewertung der teilweise komplexen Modellansätze.:Formelzeichen und Abkürzungen 1 Einführung 2 Energiequellen und Aufbau der Erdwärmeübertrager 2.1 Quellen der geothermalen Energie 2.2 Aufbau der Erdwärmeübertrager 2.3 Betriebsverhalten von Erdwärmesonden 2.4 Auslegung der Sonden 3 Vorhandene Modelle 3.1 Soleströmung 3.2 Wärmeübergang in den Rohren der Sonde 3.3 Wärmeleitung in der Hinterfüllung 3.4 Erdreichmodellierung – numerisch 3.5 Erdreichmodellierung mit g-Funktionen 4 Weiterentwicklung der analytischen Modelle 5 Anwendungsbeispiele 185 5.1 Ein praktisches Beispiel 5.2 Auswirkung auf die Jahresarbeitszahl 6 Zusammenfassung Literatur A Ableitung der Bohrlochwiderstände B Ableitung der Funktionen für Randbedingungen C Eidesstattliche Erklärung / The design of the ground heat exchangers (GHE) systems demands the precise prediction of their heat output and the brine temperature. The same information is needed for design and optimization of the HVAC systems coupled to GHEs. In the thesis at hand the Green’s functions (g-Functions) have been used to develop the more accurate solutions for the temperature distribution in soil resulting from the heat extraction from one GHE or a field of GHEs. These solutions consist of six novel g-functions that take account of the vertical variation of the extracted heat flux in one GHE or field of GHEs, of the horizontal ground water flow and of the horizontal variation of the soil properties. The models for prediction of the soil temperature have been coupled with models for brine flow and heat transfer in the GHE’s grout and eventually applied to the simulation of the real world object. Additionally, the thesis contains broad review of the known models and their applications as well as the comparative analysis of the complex modelling assumptions.:Formelzeichen und Abkürzungen 1 Einführung 2 Energiequellen und Aufbau der Erdwärmeübertrager 2.1 Quellen der geothermalen Energie 2.2 Aufbau der Erdwärmeübertrager 2.3 Betriebsverhalten von Erdwärmesonden 2.4 Auslegung der Sonden 3 Vorhandene Modelle 3.1 Soleströmung 3.2 Wärmeübergang in den Rohren der Sonde 3.3 Wärmeleitung in der Hinterfüllung 3.4 Erdreichmodellierung – numerisch 3.5 Erdreichmodellierung mit g-Funktionen 4 Weiterentwicklung der analytischen Modelle 5 Anwendungsbeispiele 185 5.1 Ein praktisches Beispiel 5.2 Auswirkung auf die Jahresarbeitszahl 6 Zusammenfassung Literatur A Ableitung der Bohrlochwiderstände B Ableitung der Funktionen für Randbedingungen C Eidesstattliche Erklärung

The development of materials with tailored functionalities and with continuously shrinking linear dimensions towards (and below) the nanoscale is not only going to revolutionize state of the art fabrication technologies, but also the computational methodologies used to model the materials properties. Specifically, atomistic methodologies are becoming increasingly relevant in the field of materials science as a fundamental tool in gaining understanding on as well as for pre-designing (in silico material design) the behavior of nanoscale materials in response to external stimuli. The major long-term goal of atomistic modelling is to obtain structure-function relationships at the nanoscale, i.e. to correlate a definite response of a given physical system with its specific atomic conformation and ultimately, with its chemical composition and electronic structure. This has clearly its pendant in the development of bottom-up fabrication technologies, which also require a detailed control and fine tuning of physical and chemical properties at sub-nanometer and nanometer length scales. The current work provides an overview of different applications of atomistic approaches to the study of nanoscale materials. We illustrate how the use of first-principle based electronic structure methodologies, quantum mechanical based molecular dynamics, and appropriate methods to model the electrical and thermal response of nanoscale materials, provides a solid starting point to shed light on the way such systems can be manipulated to control their electrical, mechanical, or thermal behavior. Thus, some typical topics addressed here include the interplay between mechanical and electronic degrees of freedom in carbon based nanoscale materials with potential relevance for designing nanoscale switches, thermoelectric properties at the single-molecule level and their control via specific chemical functionalization, and electrical and spin-dependent properties in biomaterials. We will further show how phenomenological models can be efficiently applied to get a first insight in the behavior of complex nanoscale systems, for which first principle electronic structure calculations become computationally expensive. This will become especially clear in the case of biomolecular systems and organic semiconductors.

The development of materials with tailored functionalities and with continuously shrinking linear dimensions towards (and below) the nanoscale is not only going to revolutionize state of the art fabrication technologies, but also the computational methodologies used to model the materials properties. Specifically, atomistic methodologies are becoming increasingly relevant in the field of materials science as a fundamental tool in gaining understanding on as well as for pre-designing (in silico material design) the behavior of nanoscale materials in response to external stimuli. The major long-term goal of atomistic modelling is to obtain structure-function relationships at the nanoscale, i.e. to correlate a definite response of a given physical system with its specific atomic conformation and ultimately, with its chemical composition and electronic structure. This has clearly its pendant in the development of bottom-up fabrication technologies, which also require a detailed control and fine tuning of physical and chemical properties at sub-nanometer and nanometer length scales. The current work provides an overview of different applications of atomistic approaches to the study of nanoscale materials. We illustrate how the use of first-principle based electronic structure methodologies, quantum mechanical based molecular dynamics, and appropriate methods to model the electrical and thermal response of nanoscale materials, provides a solid starting point to shed light on the way such systems can be manipulated to control their electrical, mechanical, or thermal behavior. Thus, some typical topics addressed here include the interplay between mechanical and electronic degrees of freedom in carbon based nanoscale materials with potential relevance for designing nanoscale switches, thermoelectric properties at the single-molecule level and their control via specific chemical functionalization, and electrical and spin-dependent properties in biomaterials. We will further show how phenomenological models can be efficiently applied to get a first insight in the behavior of complex nanoscale systems, for which first principle electronic structure calculations become computationally expensive. This will become especially clear in the case of biomolecular systems and organic semiconductors.

Eine wichtige Invariante von Modulkurven ist die arithmetische Selbstschnittzahl der relativ dualisierenden Garbe. Auf dem minimalen regulären Modell von X(N) ist diese Selbstschnittzahl durch den gewöhnlichen Schnitt einiger ausgezeichneter vertikaler Divisoren (dem geometrischen Beitrag) und durch die Auswertung der kanonischen Greenschen Funktion an einigen Spitzen (dem analytischen Beitrag) vollständig festgelegt. Das Ziel dieser Arbeit ist es, jeden dieser Beiträge in Abhängigkeit von der Stufe N zu bestimmen und das asymptotische Verhalten der Selbstschnittzahl zu studieren, wenn die Stufe N gegen unendlich geht. / An important invariant of modular curves is the arithmetic self-intersection of the relative dualizing sheaf. On the minimal regular model of X(N) this self-intersection is completely described by the usual intersection of some distinguished vertical divisors (geometric contribution) and the evaluation of the canonical Green’s function at certain cusps (analytic contribution). The aim of this thesis is to determine each of these contributions in terms of the level N and study the asymptotic behaviour of the self-intersection as N tends to infinity.

Wellenleiter Quantenelektrodynamik (Wellenleiter QED) ist ein wichtiger Baustein in vielen zukünftigen, auf Quantenmechanik basierenden Technologien wie z.B. Quantencomputer. Ein typisches Modellsystem besteht aus einem Zwei-Niveau-System (two level system, TLS), das an einen eindimensionalen Wellenleiter gekoppelt wurde. Der Wellenleiter ist dabei durch eine Dispersionsrelation charakterisiert und kann unter anderem Bandkanten enthalten. Wir haben in der Dissertation einen neuartigen Zugang zur Wellenleiter QED präsentiert. Dieser Zugang basiert auf der Quantenfeldtheorie und ermöglicht die Berechnung Greenscher Funktionen im ein- und zwei-Anregungs Unterraum. Diese Greenschen Funktionen wurden benutzt um die Streumatrix und die spektrale Dichte in beiden Unterräumen zu berechnen. Desweiteren konnten wir mit Hilfe von Feynman-Diagrammen die physikalischen Prozesse in der Störungsreihe der Greenschen Funktionen identifizieren. Dies war besonders im zwei-Anregungs-Unterraum von Nutzen. In diesem Fall verhält sich das System nichtlinear, da das TLS nur eine Anregung absorbieren kann. Dadurch werden Effekte induziert wie photon bunching und die effiziente Anregung eines gebundenen Atom-Photon Zustandes. Es war uns möglich diese Effekte in der Störungsreihe der Greenschen Funktion wieder zu finden. Desweiteren haben wir die Greenschen Funktionen im Orts-Zeit-Raum benutzt um ein- und zwei-Photon-Wellenpakete zu propagieren. Es hat sich herausgestellt dass das Verhältnis von Pulsbreite zur spontanten Emissions-zeit sowohl das Streuverhalten als auch die maximale Anregung des TLS bestimmt. Letztendlich haben wir den Einfluss von Unordnung im Wellenleiter auf das Zerfallsverhalten des TLS untersucht. Wir haben entdeckt dass der gebundene Atom-Photon Zustand instabil wird sobald die Unordnung einen kritischen Wert erreicht. Darüberhinaus haben wir eine spezielle Klasse Feynman Diagramme identifiziert, die dem Zerfall eine nichtmarkovsche Dynamik verleihen. / Waveguide quantum electrodynamics (waveguide QED) can be considered as a building block for many prospective technologies like quantum computing. A prototypical system consists of a two-level system (TLS) coupled to a one-dimensional waveguide. The waveguide is characterized by its dispersion relation and can also feature a band edge/slow-light regime. In this thesis we have presented a new theoretical framework for waveguide QED, based on quantum field theory. The framework provides the Green''s functions of the system in the single- and two-excitation sectors for an arbitrary dispersion relation. We have calculated the scattering matrix and the spectral density in both sectors. Furthermore, we have also represented the Green''s functions in the form of Feynman diagrams, from which we can identify the underlying physical processes. A special property of the system is that it behaves nonlinear in the case of two or more photons. This is rooted in the structure of the TLS, which can at most absorb one excitation. The nonlinearity leads to two effects: photon bunching and the efficient excitation of an atom-photon bound state. We have found both effects within our framework and we were able to assign them individual terms in the perturbation series of the Green''s function. Furthermore, we have used the Green''s function in space-time domain to propagate Gaussian one- and two-photon wavepackets. Here, we have identified the ratio of the pulsewidth and the spontaneous emission time as the parameter which governs both the scattering behavior of the photons and the maximal TLS excitation. Eventually, we have investigated the effects of disorder in the waveguide on the decay properties of the TLS. We have found here that the atom-photon bound state is stable for small disorder, but breaks down at sufficiently strong disorder. Furthermore, we have identified a special class of diagrams which render the system non-Markovian even for energies far away from the band edge.

Zur vollständigen Beschreibung eines monochromatischen Wellenfeldes ist die Kenntnis über die Amplituden- und Phasenverteilung unabdingbar. Während sich die messtechnische Erfassung der Amplitudenverteilung durch lichtempfindliche Sensoren recht einfach realisieren lässt, gestaltet sich die Bestimmung der Phasenverteilung weitaus schwieriger. Die Phasenverteilung eines optischen Wellenfeldes kann nur über indirekte Verfahren gewonnen werden. Es ergibt sich ein sogenanntes phase retrieval Problem. Zur Lösung dieses Problems bieten sich verschiedene Verfahren aus dem Bereich der berührungslosen und zerstörungsfreien optische Messtechnik an. In dieser Arbeit wird ein deterministisches Verfahren zur Phasenrekonstruktion mit Hilfe Greenscher Funktionen vorgestellt. Die erste Greensche Identität dient als Grundlage zur Entwicklung einer Gleichung, welche in der Lage ist, bei der Rekonstruktion einer Phasenverteilung spezifische Randbedingungen zu berücksichtigen. Dies ermöglicht unter anderem eine genaue Charakterisierung von Phasenobjekten bzw. ihren optischen Eigenschaften, wie beispielsweise der Brechzahlverteilung. Das vorgestellte Verfahren zur Phasenrekonstruktion basiert einerseits auf schnellen Algorithmen, welche die Leistung von parallelen Prozessoren ausnutzen und andererseits auf geschickten experimentellen Aufbauten, mit welchen die notwendigen Eingangsdaten zur Lösung der Gleichung simultan gewonnen werden können. Es ergibt sich damit die Möglichkeit, die Amplituden- und Phasenverteilung eines Wellenfeldes in Echtzeit zu bestimmen und daraus folgend ein Mittel zur quantitativen Bewertungen und Analyse von dynamischen Prozessen sowohl in der Industrie als auch im Bereich der Life Sciences. / In order to describe a monochromatic wave field entirely, knowledge about the amplitude and phase distribution is elementary. While it is easy to measure the amplitude distribution of an optical wave field by the use of photosensitive detectors, the determination of the phase distribution is by far more difficult. Due to the fact, that the phase distribution can not be measured directly, a problem of phase retrieval is presented. This problem may be solved by applying a non-contacting and non-destructive optical metrology technique. In this thesis a deterministic method for phase retrieval based on Green''s functions will be introduced. Green''s first identity serves as a starting point to derive an equation for phase retrieval considering different boundary conditions. Among others, this allows an exact characterization of phase objects, or their optical properties, as for example the refractive index distribution. On the one hand, the presented phase retrieval technique is based on fast algorithms which take advantage of the performance of parallel processors. On the other hand, skilful experimental setups allow the simultaneous acquisition of the input data, which are necessary to solve the phase retrieval equation. It follows that the presented technique is able to determine the amplitude and phase distribution of a wave field in real-time. Hence this technique enables the quantitative evaluation and analysis of dynamic processes in industry as well as in the area of life sciences.