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Open Access
Hybrid plasmonic devices for sensing and thermal imaging
(2015) Tittl, Andreas; Giessen, Harald (Prof. Dr.)
Plasmonics is an emerging field in nanooptics, which focuses on the optical properties of resonant subwavelength metal nanoparticles. Historically, such geometries commonly employed noble metal nanoparticles to achieve a variety of effects ranging from nanofocusing of light to negative refraction. Building on these concepts, this thesis investigates hybrid nanoplasmonic devices, which combine passive noble metal nanostructures with chemically reactive or actively tunable materials to obtain novel functionalities. Utilizing various complex plasmonic geometries, this work pursues two complementary threads of research, covering the technological scale from fundamental science to device applications. On the one hand, it utilizes chemically synthesized hybrid plasmonic "smart dust" nanoprobes to detect progressively lower reagent concentrations. Starting from silica shell-isolated gold nanoparticles, which are used to map the catalytic reactions in adjacent extended palladium thin films, DNA-assembled bimetallic plasmonic nanosensors are investigated to resolve changes in sub-5nm Pd nanocrystals on the single antenna level, pushing the lower limit of chemical detection volume. On the other hand, it studies plasmonic perfect absorber structures, optical elements designed to absorb all radiation of a certain wavelength, which have shown promise for a variety of technological applications. Here, the focus is on both developing a theoretical model for the optical behavior of plasmonic perfect absorber structures, especially at large incident angles, as well as on the experimental realization of efficient gas sensors and active mid-infrared imaging devices.
Open Access
Spin interactions in graphene-single molecule magnets hybrid materials
(2015) Cervetti, Christian; Bogani, Lapo (Dr.)
This thesis work employs magnetic molecules, belonging to the family of single molecule magnets (SMMs), as model systems to study the relaxation of mesoscopic spins on graphene, and their interaction with graphene charge carriers in real devices. Their large uniaxial anisotropy makes SMMs behave like giant spins, with relaxation times of years at low temperatures. Their spin dynamics combines a classical and a quantum relaxation mechanism, that can be selectively switched on and off by either applying an external magnetic Field or by varying the temperature. The work is organized as follows. The First part presents a thorough structural characterization of the SMMs-graphene hybrid materials via multiple techniques, including atomic force microscopy, x-ray photoelectrons spectroscopy, mass spectrometry, Raman spectroscopy, and electronic transport measurements on graphene-based Feld-effect transistors. The analysis of the dynamical arrangement of molecular adsorbates on graphene reveals new opportunities to control the supramolecular surface arrangement. A comprehensive study of the magnetization dynamics of SMMs on graphene is carried out by means of ac-susceptibility techniques in a broad temperature range (T = 4K - 13 mK). The details of the complex spin-graphene interaction are unraveled in the framework of a newly developed theoretical model that accounts for all the possible fundamental contributions and the two-dimensional nature of graphene. The focus of the second part is the design, fabrication and characterization of graphene-based spintronic devices. Diffrent strategies for the injection of spin-polarized carriers in graphene are implemented and tested down to very low temperatures (T = 300 mK). To conclude the first spin-transport and spin-relaxation measurements in SMMs-graphene devices are presented.
Open Access
Proximity effects and Josephson currents in ferromagnet : spin-triplet superconductors junctions
(2015) Terrade, Damien; Metzner, Walter (Prof. Dr.)
Spin-triplet superconductivity, first attached to the description of He3, is now generally considered to also occur in heavy-fermions compounds and in perovskite ruthenium oxide Sr2RuO4. The latter material is especially interesting since many experiments show strong evidences for a unitary chiral spin-triplet state. Moreover, the recent fabrication of thin heterostructures made of ferromagnetic SrRuO3 on the top of Sr2RuO4 strongly encourages new theoretical studies on the interplay between spin-triplet superconductor and ferromagnet in similar fashion to spin-singlet superconductors. Using an extended tight-binding Hamiltonian to model the superconductor, we discuss in this thesis the specific proximity effects of such interface by solving self-consistently the Bogoliubov-De Gennes equations on two- and three-dimensional lattices in the ballistic limit. We obtain the spatial profile of the superconducting order parameters at the interface as well as the spin-polarisation and the current across the Josephson junctions. In contrast to heterostructures made of spin-singlet superconductor, we show that the physical properties at the interface are not only controlled by the strength of the magnetization inside the ferromagnet but also by its orientation due to the existence of a finite pair spin projection of the spin-triplet Cooper pairs. We analyse in the first part the spin-polarisation and the Gibbs free energy at the three-dimensional ferromagnet-chiral spin-triplet superconductor interface. Then, the second part of the thesis is dedicated to the study of the Josephson junctions made of a chiral spin-triplet superconductor and a ferromagnetic barrier. More precisely, we analyse the existence of 0-Pi state transitions in two- and three-dimensional junctions with respect to the strength and the orientation of the magnetization. Finally, we study the proximity effects at the interface of helical spin-triplet superconductors. They differ from the chiral superconductor by the direction of the pair spin polarisation of the Cooper pairs and by the properties of the edge states, present at the boundaries, which can sustain dissipationless spin-current.
Open Access
Role of disorder and interactions on the surface of topological superconductors
(2015) Queiroz, Raquel; Metzner, Walter (Prof. Dr.)
In this work we study the surface properties of topological systems, with a special focus on topological superconductors without inversion symmetry. These materials provide a rich playground for multiple topological phenomena, showing boundary modes with linear and (or) flat dispersion arising from complex nodal structures. A remarkable characteristic of topological phases is their robustness to local perturbations. In the present work, we explore the extent to which this robustness can be generalized to gapless topological phases. We numerically test the robustness of topological boundary modes against local disorder and explore the contrast between different disorder strengths and distributions. Additionally, we study the interplay between topology and electron-electron interactions at the surface of nodal superconductors, where the infinitely degenerate flat bands are susceptible to spontaneous symmetry breaking. Finally, we briefly look into possible symmetry preserving interactions that can lead to the destruction of the boundary modes.
Open Access
Far-infrared spectroscopy of lanthanide-based molecular magnetic materials
(2015) Haas, Sabrina; Dressel, Martin (Prof. Dr.)
This thesis demonstrates the applicability of far-infrared spectroscopy for the study of the crystal-field splitting of lanthanides in single-molecular magnetic materials. The far-infrared studies of three different kinds of single-molecular-magnetic materials, a single-ion magnet, a single-chain magnet and an exchange-coupled cluster, yielded a deeper understanding of the crystal-field splitting of the lanthanides in these materials. In addition, our results offered the opportunity to gain a deeper insight into the relaxation processes of these materials.
Open Access
High-order methods for computational astrophysics
(2015) Núñez-de la Rosa, Jonatan; Munz, Claus-Dieter (Prof. Dr.)
In computational fluid dynamics, high-order numerical methods have gained quite popularity in the last years due to the need of high fidelity predictions in the simulations. High-order methods are suitable for unsteady flow problems and long-term simulations because they are more efficient when obtaining higher accuracy than low-order methods, and because of their outstanding dissipation and dispersion properties. In the present work, the development and application of three high-order numerical methods, namely, the conservative finite difference (FD) method, the finite volume (FV) method, and the discontinuous Galerkin spectral element method (DGSEM), is presented. These methods are used here for solving three equations systems arising in computational astrophysics on flat spacetimes, specifically, the ideal magnetohydrodynamics (MHD), relativistic hydrodynamics (SRHD) and relativistic magnetohydrodynamics (SRMHD). Our computational framework has been subject to the standard testbench in computational astrophysics. Numerical results of problems having smooth flows, and problems with shock-dominated flows, are also reported. Finite volume methods are numerical methods based on the weak solution of conservation laws in integral form. Unlike finite volume methods, where cell averages of the solution are evolved in time, in the conservative finite difference schemes only the solution at specific nodal points are considered. This difference offers a high efficiency of finite difference over finite volume methods in two and three dimensional high-order calculations because of the form of the utilized stencils in the reconstruction step. Recently, a lot of effort has been put into the development of efficient high-order accurate reconstruction procedures on structured and unstructured meshes. The most widely used procedure to achieve high-order spatial accuracy in finite volume and conservative finite difference methods is the WENO reconstruction. The basic idea of the WENO schemes is based on an adaptive reconstruction procedure to obtain a higher-order approximation on smooth regions while the scheme remains non-oscillatory near discontinuities. For this reason, the WENO formulation is particularly effective when solving conservation laws containing discontinuities. In this work, the FD and FV methods are extended to very high-order accuracy on regular Cartesian meshes by making use of the arbitrary high-order reconstruction WENO operator. The time discretization is carried out with a strong stability-preserving Runge-Kutta (SSPRK) method. The MHD, SRHD and SRMHD equations are then solved with these two methods for problems having strong shock configurations. The discontinuous Galerkin (DG) methods combine the ideas of the finite element (FE) and the finite volume methods. From the FE methods, the solution and test functions in the variational formulation of the conservation law are locally represented by polynomials, allowing to be discontinuous at element faces. In order to stabilize the scheme, from the FV methods are borrowed the ideas of using Riemann solvers, which permit to connect a given element with its direct neighboring ones. One special case in the family of DG methods is the DGSEM. In these methods, the domain is decomposed into quadrilateral/hexahedral elements, and the solution and the fluxes are represented by tensor-product basis functions (high-order Lagrangian interpolants). The integrals are approximated by quadrature, and the nodal points, where the solution is computed, are the Gauss-Legendre quadrature points. With these choices, the DG operator has a dimension-by-dimension splitting form, which yields more efficiency due to less operations and less memory consumption. In this work, the DGSEM has been also extended to the equations of computational astrophysics on flat spacetimes, but restricted only to the MHD and SRHD equations. Because discontinuous solutions form part of the nature of the hyperbolic conservation laws, shock capturing strategies have to be devised, especially for the discontinuous Galerkin method. For the DGSEM, a hybrid DG/FV shock capturing approach is used as the main building block for stabilization of the solution when shocks take place. The hybrid DGSEM/FV is constructed in such a way that, in regions of smooth flows, the DGSEM method is employed, and those parts of the flow having shocks, the DGSEM elements are interpreted as quadrilateral/hexahedral subdomains. In each of these subdomains, the nodal DG solution values are used to build a new local domain composed now of finite volume subcells, which are evolved with a robust finite volume method with third order WENO reconstruction. This new numerical framework for computational astrophysics based on the hybridization of high-order methods brings very promising results.
Open Access
Realizations of PT-symmetric Bose-Einstein condensates with time-dependent Hermitian potentials
(2015) Kreibich, Manuel; Main, Jörg (Prof. Dr.)
A PT-symmetric Bose-Einstein condensate can be theoretically described using a complex optical potential, however, the experimental realization of such an optical potential describing the coherent in- and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum mechanical realization of a PT-symmetric system, where the PT-symmetric currents of a two-well system are implemented by coupling possibly time-dependently varied additional wells to the system, which act as particle reservoirs and thus form a Hermitian system. We map the time-dependence of the amplitudes of a frozen Gaussian variational ansatz to a Schrödinger equation with a Hamiltonian matrix. This relates the parameters of a realistic external potential to the matrix elements of a matrix model. On one side, we can use the matrix model as a computationally cheap model to obtain results, on the other hand we can then map the time-dependence of the matrix elements back to the parameters of the potential, which would serve as an input for an experimental setup. In terms of these simple matrix model we derive conditions under which two wells of the Hermitian multi-well system behave exactly as the two wells of the PT-symmetric system. It turns out that the matrix elements of the Hermitian system must be time-dependent to fulfill the conditions for a sufficient time. These results are applied to calculate the time-dependencies of the matrix elements, first, by means of a four-well system, then consequently building up until we arrive at a system with a large number of wells, which can be analyzed and interpreted in terms of optical lattices. As a second method we use a full time-dependent Gaussian variational ansatz, where every variational parameter of the Gaussian functions is chosen to be time-dependent. This should give more accurate results in that this ansatz can describe significant more degrees of freedom. We derive conditions analogous to that of the matrix models that must be fulfilled such that PT-symmetry can be realized. This method is applied and results are obtained for the important case of an adiabatic current ramp which is proposed as a realistic experimental method to create a PT-symmetric ground state. Finally, the results to both methods are compared and in particular the approximations that lead to the simple matrix models are justified.
Open Access
Strukturelle und spektroskopische Eigenschaften epitaktischer FeMn/Co Exchange-Bias-Systeme
(2015) Schmidt, Mathias; Goering, Eberhard (PD Dr.)
Das Thema dieser Arbeit bestand in der Präparation und Charakterisierung des Exchange-Bias Systems FeMn/Co. Hierbei wurden mittels Molekularstrahlepitaxie zwei sich durch ihre kristalline Orientierung unterscheidende FeMn/Co-Probensysteme auf (100)-orientiertem Magnesiumoxid hergestellt. Zur Erzeugung einer flachen Schichtstruktur mit einer ausgeprägten kristallinen Ordnung war der Einsatz zweier Pufferschichten (Pt,Cu) notwendig. Bei Substrattemperaturen oberhalb von 900 K eine (100)-orientierte Schicht erschaffen, bei niedrigeren Temperaturen um 670 K entstand eine (111)-Orientierung. Untersuchungen der Kristallstruktur ergaben ein epitaktisches Wachstum des (100)-orientierten Probensystems (HTPt) mit großen Kristallitstrukturen, während für das (111)-orientierte System (NTPt) eine vierfach entartete Untergitterstruktur mit kleineren Kristalliten entstand. Es entstand ein wohldefinierter und reproduzierbarer Herstellungsprozess, bei dem sämtliche Schichtparameter gezielt verändert werden konnten. Anschließend wurden magnetometrische Untersuchungen der Probensysteme durchgeführt. Mittels SQUID-Magnetometrie wurde die Temperaturabhängigkeit dieser beiden Parameter untersucht. Es stellte sich eine stärkere Temperaturabhängigkeit des NTPt-Probensystems heraus, die der kleineren Kristallitgröße und der höheren Aktivierung von Pinnningzentren in den Korngrenzen zugeschrieben werden konnte. Dann wurde die magnetische Anisotropie der Probensysteme untersucht, dies geschah mittels eines MOKE-Systems. Es zeigte sich eine stärkere magnetokristalline Anisotropie des HTPt-Systems verglichen mit dem NTPt-System, die mit der ausgeprägteren kristallinen Ordnung in den magnetischen Schichten korreliert. Für eine ausführlichere Charakterisierung wurde auf die FORC (First Order Reversal Curves)-Methode zurückgegriffen. Dieses Verfahren erbrachte den Nachweis der asymmetrischen Magnetisierungsumkehr nicht nur parallel und antiparallel zur Feldkühlrichtung sondern auch für identische Projektionen auf die Feldkühlrichtung. Dieses Verhalten lässt auf eine nicht parallele Anordnung der leichten Richtungen von Ferromagnet und Antiferromagnet schließen. Neben der Asymmetrie der Magnetisierungsumkehr konnte auch das unterschiedliche Ummagnetisierungsverhalten beider Probensysteme analysiert werden. Die Auftrennung der irreversiblen und reversiblen Magnetisierungsbeiträge mittels FORC erbrachte für das HTPt-System irreversible Anteile über den gesamten Winkelbereich, während für das NTPt-System über nahezu den gesamten Winkelbereich reversible und somit rotationsbasierte Mechanismen identifiziert werden konnten. Zuletzt wurden die Probensysteme Röntgenabsorptionsmessungen unter Ausnutzung des Röntgenzirkulardichroismus (XMCD) unterzogen. Einerseits wurde ein Vergleich beider Probensysteme erstellt, andererseits auch Veränderungen der magnetischen Eigenschaften durch gezielte Manipulationen der antiferromagnetischen Struktur untersucht. Diese bestanden in Veränderungen der Dicke der Cu-Pufferschicht sowie in der Änderungen der Zusammensetzung des Antiferromagneten. Die Ergebnisse der Absorptionsmessungen wurden mit Hilfe der Summenregeln analysiert, um die Beiträge von magnetischem Spin- und Bahnmoment zu separieren und quantitativ zu untersuchen. Sowohl Eisen als auch Mangan zeigen ein XMCD-Differenzsignal von unkompensierten rotierbaren magnetischen Momenten. Die Magnetisierung findet sich nahe der Grenzfläche, in den tiefensensitiveren TFY-Messungen konnten keine unkompensierten Momente nachgewiesen werden. Verglichen mit den für Volumenproben reiner Elemente erhaltenen Ergebnissen wurde eine Zunahme des Bahndrehmomentes an beiden Kanten festgestellt. Die Menge an unkompensierten rotierbaren Spinmomenten nimmt bei Reduktion der magnetokristallinen Anisotropie des Antiferromagneten zu, dies ist besonders an der Eisenkante der Fall, auch wenn der Effekt ebenfalls an der Mangankante feststellbar ist. Durch eine auf der Intensität des Absorptionssignals basierende Abschätzung wurde die effektive Dicke der rotierbaren Schicht ermittelt, sie beträgt je nach Probensystem bis zu drei Monolagen für vorliegenden Exchange-Bias. Zuletzt wurde aus den gemessenen Spektren das sogenannte „Branching Ratio“ ermittelt, das Rückschlüsse auf den Erwartungswert der Spin-Bahn-Kopplung in den untersuchten Probensystemen zulässt. Hierbei ergibt sich für Mangan generell ein höherer Wert der Spin-Bahn-Kopplung verglichen mit den Messungen an der Eisenkante. Die Summe der beobachteten Effekte legt zur Erklärung der nahe der Grenzfläche im Antiferromagneten stattfindenden Abläufe ein Wechselspiel der globalen magnetokristallinen Anisotropie der Probensysteme mit einer lokal erhöhten Anisotropie an Stellen mit gestörter Kristallsymmetrie wie Fehlstellen oder Korngrenzen nahe. Letztere führt zum Ausbildung von gepinnten magnetischen Momenten, die durch den Feldkühlprozess eine unidirektionale Ausrichtung erhalten und den Exchange-Bias verursachen.
Open Access
Ion beam lithographic and multilayer fresnel zone plates for soft and hard X-rays: nanofabrication and characterization
(2015) Keskinbora, Kahraman; Schütz, Gisela (Prof. Dr.)
X-ray microscopy has become an important analytical characterization method for a plethora of applications in materials science, physics, chemistry and biology, thanks to the emergence of modern synchrotron radiation facilities. These facilities enable high brilliance, energy tunable, variable polarization X-rays which gives access to mass density, elemental, chemical, electronic and magnetic properties of materials. In the soft X-ray energies nearly all elements can be probed by spectromicroscopic methods. Another important property of synchrotron radiation is the time structure in the ns to ps range, which can be utilized for sophisticated time resolution studies. These opportunities can be combined with high spatial resolution which is determined by the focusing method and the optic. Focusing of X-rays has historically been a difficult task due to strong absorption and weak phase shift of X-rays within matter. The required phase shift of X-rays, which depends on the real part of the complex refractive index, differs from 1 (the vacuum refractive index) only on the order of 10^-2 to 10^-6 and conventional lenses do not work. One very successful X-ray optic is the Fresnel Zone Plate (FZP), a diffractive optic that act as a lens under certain conditions and can focus X-rays to nanometer sized spots. The resolution of the FZP depends on the width of the outermost zone and is highly correlated with the smallest feature that can be fabricated. Conventionally, the e-beam lithography (EBL) is used for production FZPs which could resolve up to 10 nm structures with serious limitations. One difficulty of EBL is its ever increasing complexity for many-step fabrication of smaller features or intricate geometries. Therefore, EBL is mostly constrained to planar, binary geometries with moderate efficiencies strongly decreasing with energy and not effective for hard X-rays. Special 3D geometries in the form of kinoform lenses can theoretically have 100 % focusing efficiencies. Attempts to approximate these geometries via EBL increased the number of process steps even further. The smallest FZP feature size even for low aspect ratios achievable via EBL is fundamentally limited due to the proximity effect which is the interaction and spread of electrons within the resist material. We addressed these issues by focusing our research on alternative FZP fabrication techniques as high-speed ion beam lithography (IBL), and gray scale ion lithography to realize efficient kinoforms. Another approach towards full-material multilayer FZPs with infinite aspect ratio was based on atomic layer deposition (ALD) with subsequent ion beam slicing. Each of these three methods targets specific challenges faced by the e-beam lithography based FZP fabrication techniques. All the fabricated FZPs were tested for their resolution and efficiency performances at a state of the art scanning transmission X-ray microscope at BESSY for soft X-rays and/or at optical test stations at ESRF and PETRA III for hard X-rays. Using IBL the rapid preparation of a 110 nm thick Au FZP with 50 µm diameter and 50 nm ∆r in less than 13 minutes is demonstrated. Employed for X-ray microscopy, the FZP clearly resolved 28.5 nm features with a cut-off of 24.3 nm at ~1120 eV. Additional process improvements were made towards smaller zones with higher zone quality. They allowed the preparation of a FZP with 30 nm outermost half-period remarkably, in about 8 min. This FZP was shown to clearly resolve 21 nm features on a multilayer test object with large room for improvement. This high through-put FZP production route is of special interest not only concerning the low cost and easy availability. A large array of these optical components is attractive, for experiments such as one-shot ultra-high brilliance FEL investigations due to the radiation damage or for instance for coded-aperture arrays for high-angle resolving X-ray astronomy. Towards fabrication of kinoforms for high efficiency X-ray focusing, we have performed various materials optimization studies in order to achieve a high surface quality optic. After various trials the materials were finally optimized and the fabricated lenses achieved more than 14 % absolute diffraction efficiency that is almost 90 % compared to the theoretical prediction. This confirms how closely we were able to replicate the ideal three dimensional surface relief structure for the first time. It was possible to carry out imaging with these lenses with half-pitch resolutions down to 60 nm. The kinoform lenses were tested at the soft X-ray range where a significant absorption is present in materials. These results also potentially pave the way for very high efficiency hard X-ray focusing which can in principle be utilized in laboratory based X-ray sources, X-ray astronomy and the new rising field of X-ray ptychography. To fabricate high resolution ML-FZPs, Al2O3/Ta2O5multilayers, deposited on a smooth glass optical fiber via atomic layer deposition using non-dedicated instruments were carefully cut-out, sliced and polished to a high quality surface finish using focused ion beams. Following the transfer of the slice to a TEM grid as holder the slices were polished to a high surface finish quality, also via a focused ion beam. Fabricated ML-FZPs were synchrotron tested using an in-house constructed 2-axis tilt stage specially designed for aligning ML-FZP with respect to the X-ray optical axis. The results showed that it was possible to resolve 21 nm features in direct imaging at 1200 eV and sub-30 nm focusing at 8 keV. This is the highest demonstrated resolving power for a multilayer type FZP, to date to the best of our knowledge. Results exhibit the potential for high-resolution hard X-ray focusing where this type of optics are especially efficient. For ultra-high resolution hard and soft X-ray imaging, with potentially achievable ∆r of a few nm is well below what can be achieved through any lithography method available today.
Open Access
B-Spline Ansatz für die Berechnung des Motional Stark Effekts
(2015) Göttler, Thorsten; Wunner, Günter (Prof. Dr.)
Diese Arbeit beschäftigt sich mit bewegten Atomen in starken Magnetfeldern, welche z.B. auf der Oberfläche von Neutronensternen auftreten. Bei der Entstehung dieser Sterne schrumpfen diese auf Radien um 10 km zusammen, was zur Folge hat, dass die Magnetfeldlinien sich durch die näherungsweise Erhaltung des magnetischen Flusses ebenfalls zusammenziehen und sich Magnetfelder im Bereich von 10⁵ - 10⁹ T einstellen. Messungen der Spektren von Neutronensternen haben gezeigt, dass Absorptionslinien vorhanden sind, welche durch Anwesenheit einer dünnen Atmosphäre, bestehend aus Elementen von Wasserstoff bis Eisen, erklärt werden können. Um diese Vermutung zu überprüfen, müssen Energien und Wellenfunktionen, um im Endeffekt Opazitäten zu erhalten, in starken Magnetfeldern berechnet werden. Mittlerweile können für beliebige Elemente Energien und Oszillatorstärken, sowie Wirkungsquerschnitte aus Photoionisation, berechnet werden. Um aus diesen Daten Opazitäten zu erhalten müssen Linienbreiten berechnet werden, wobei Doppler- und Druckverbreiterung einfach zu handhaben sind. Der wichtigste Effekt ist die Rückwirkung der Bewegung der Atome im starken Magnetfeld auf ihre interne Struktur, was zu enormen Veränderungen der Energien und Wellenfunktionen führen kann. Im ersten Teil dieser Arbeit soll dieser sogenannte Motional Stark Effekt für Wasserstoff berechnet werden. Dafür leiten wir einen Hamiltonoperator her, welcher für unsere numerischen Rechnungen geeignet ist. Hierzu wird zuerst der unbewegte Fall besprochen und dessen Lösung präsentiert, um anschließend auftretende Änderungen für den bewegten Fall zu diskutieren. Anschließend formulieren wir einen Ansatz, welcher genaue Ergebnisse auch für niedrige Magnetfeldstärken erlaubt. Dieser besteht aus einer 2D B-Spline Basis, definiert auf finiten Elementen. Um eine numerisch auswertbare Form der Schrödingergleichung zu erhalten, müssen effektive Potentiale berechnet werden. Dies und eine genaue Besprechung des verwendeten Algorithmus, mit all seinen Herausforderungen und wie diese gemeistert wurden, ist der Inhalt von Kapitel 3. Da keine Symmetrie in den Ansatz eingeht, ist die Flexibilität enorm, was wir anhand der Ergebnisse in Kapitel 4 zeigen wollen. Es werden Energien für verschiedene Magnetfeldstärken vorgestellt und verglichen, sowie verschiedene Eigenschaften wie Geschwindigkeit und effektive Masse diskutiert. Da im Energiespektrum viele vermiedene Kreuzungen auftreten, sollen diese speziell betrachtet werden. Wir beschreiben insbesondere das Verhalten der Wellenfunktionen an diesen vermiedenen Kreuzungen. Im zweiten Teil wird für neutrales Helium ein Hamiltonoperator in Jacobi-Koordinaten hergeleitet. Dazu orientieren wir uns an vorherigen Arbeiten, welche quasi ein Rezept für eine gewisse Art der Koordinatentransformation vorgibt, um einen separierten Hamiltonoperator zu erhalten. Dieser wird anhand des Beispiels mit harmonischer Wechselwirkung untersucht, und es werden erste Ergebnisse beschrieben. Anschließend wird ein Vorschlag erarbeitet, wie die Coulombwechselwirkung mitberücksichtigt werden kann. Dieser basiert auf der Entwicklung der Wellenfunktion in der Landaubasis, welche schon früher erfolgreich angewandt wurde. Da die Gleichungen in Jacobi-Koordinaten anders aussehen als gewohnt, müssen Eigenschaften wie Symmetrie bei Teilchenaustausch und Bedeutung der Potentiale erläutert werden.

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