43 search hits
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Instabilities in layered liquids induced by external fields
(2003)
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Günter Auernhammer
- In this thesis, we have shown that the inclusion of a nematic degree of freedom in the macroscopic hydrodynamic description of smectic-A-like liquids leads to a number of interesting results. While the director and the layer normal are coupled such that they are parallel in equilibrium, in non-equilibrium situations, the director needs not be parallel to the smectic layer normal. This is in contrast to standard smectic-A hydrodynamics. Using irreversible thermodynamics and symmetry arguments, we derived a complete set of macroscopic hydrodynamic equations for the director variables, the layer displacement, the velocity field, and the moduli of the nematic and smectic order parameters. Recent experiments find that the parallel orientation of smectic-A- like liquids is destabilized by an applied shear. After destabilization, two typical scenarios are observed in a steady state situation: i) The layers are oriented perpendicular to the vorticity direction of the flow, i.e., they lie in the plane spanned by the velocity and the gradient direction (`perpendicular' orientation). ii) Closed multi-lamellar vesicles (`onions') form. A number of experiments indicate that the onset of this reorientation is controlled by the applied shear rate. In contrast to standard smectic-A hydrodynamics where shear in the parallel orientation has no effect on the layers, this destabilizing effect comes out naturally from our extended smectic-A hydrodynamics. The argumentation goes along the following lines. The shear field exerts a torque on the director that must be balanced by the coupling to the layer normal. In the limit of small angles, balancing these torques leads, in the steady state, to a shear-induced director tilt proportional to the shear rate. The preferred thickness of a smectic layer is directly connected to the projection of the averaged molecular axes on the layer normal, or, in terms of our model, the thickness is proportional to the projection of the director on the layer normal. If the director is tilted, this projection is shorter. This decrease of the projection is equivalent to an effective dilation, because the actual layer thickness is larger than the preferred layer thickness. Similar to the case of low molecular weight smectic-A liquid crystals under a dilative strain, this effective dilation leads to an undulation instability. To investigate the stability of the parallel alignment, we performed a linear and weakly non-linear analysis of the governing equations. The initial state is the above described spatially homogeneous director tilt with the smectic layers in the parallel orientation. The linear stability analysis showed an undulation instability which sets in above a critical tilt angle (or equivalently, a critical shear rate). This critical tilt angle turned out to depend strongly on the material parameters. For a typical low molecular weight thermotropic liquid crystal, we estimated the critical tilt angle to be on the order of a few degrees. The linear stability analysis also revealed that the nematic and smectic order is modulated close to the boundaries. Since the probability for the formation of defects is larger in regions with a decreased modulus of the order parameter, these variations in the modulus of the order parameter open the way for a destabilization of the layered structure. We note that a detailed investigation of this point is beyond the scope of the present work. Finally, we could exclude an oscillatory instability for all physically reasonable regions in parameter space. The weakly non-linear analysis shows that the bifurcation is supercritical for most physically relevant regions in the parameter space. A detailed comparison to an independent approach was undertaken in a collaboration with simulation physicists from the Max-Planck- Institute for Polymer Research in Mainz. In a molecular dynamics simulation, a model layered liquid consisting of chains of four particles (AABB) was considered. The interaction potential of particles not connected by springs is attractive for like particles and repulsive for particles of a different nature. The simulation demonstrated the two main predictions of our analytic theory: The director tilts in the flow direction and, above a critical shear rate, the layers show stationary undulations with a wave vector in the vorticity direction. Besides this good qualitative agreement, a reasonable quantitative agreement for the critical shear rate was found.
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Maximum Entropy Method in Superspace Crystallography
(2003)
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Lukas Palatinus
- This thesis discusses several aspects of the combination of the Maximum Entropy Method (MEM) for the reconstructions of the electron density with the superspace approach to the description of structures of aperiodic crystals. It is shown that the MEM in superspace provides a parameter-free reconstruction of the modulation functions with sufficient accuracy. The MEM in superspace has been applied to diffraction data of several compounds. The computer program BayMEM was developed for this purpose. BayMEM allows electron densities of the ordinary 3D structures and the superspace electron densities of the aperiodic structures to be reconstructed using the same general principles. The program has been extended by adding features improving its versatility and accuracy of the results. The improvements include attaching of the set of subroutines MemSys5 to BayMEM, implementation of the method of the Generalized F-constraints and the static weighting, implementation of the G-constraints, of the Prior-derived F-constraints and of the two-channel entropy. The second major computer program EDMA is a software tool for analysis of the electron densities in arbitrary dimension. The program analyzes the MEM electron density and extracts quantitative information about the atoms according to Bader's formalism “Atoms in molecules“. Two new variants of the constraints in the MEM have been developed in order to solve the problems with artifacts in the MEM reconstructions. The two methods are the Generalized F-constraints and the Prior-derived F-constraints. The concept of the Generalized F-constraints is based in the observation, that the standard F-constraint is not sufficiently strong to constrain the histogram of the normalized residuals of the structure factors to the expected Gaussian shape. Higher moments of the distribution of the normalized residuals were therefore used as the constraint in the MEM calculations. With these constraints significantly improved histograms were obtained. The source of some artifacts in the MEM electron densities was identified to be the tendency of the MEM to estimate incorrectly those structure factors, that are not included in the experimental dataset. It is shown that the missing structure factors can successfully be replaced by the structure factors derived from the procrystal electron density. If the structure factors derived from the procrystal prior electron density (the Prior-derived F-constraints) are used as additional constraints in the MEM calculation, the result is free of sharp artifacts and the quality of the reconstruction of the electron density is comparable with the results of multipole refinements. To test the accuracy of the MEM in superspace, the method was applied to the dataset of the misfit-layer composite structure of (LaS)1.14NbS2. It has been shown, that the MEM on the model structure factors reproduces the model modulation functions with accuracy better that 10% of the pixel size of the grid, on which the electron density was sampled. The structure of the high-pressure phase III of Bi provided a prominent example illustrating the advantages of the MEM in superspace over the standard structure refinements. The MEM in superspace was applied to the diffraction data of Bi-III to extract more information about the modulation than obtained from the standard structure refinement. The modulation functions extracted from the MEM electron density revealed a block-wave-like shape of the modulation function of the Bi atom of the host structure that indicates shifts of the atom between two stable environments rather than smooth harmonic variation of the position indicated by the modulation function from the standard refinement. Secondly, the MEM modulation function of the Bi atoms in channels allowed to better understand the nature of the most prominent feature of the modulated structure - the occurrence of the pairs of Bi atoms along the channels. The incommensurately modulated structure of ammonium tetrafluoroberyllate (NH4)2BeF4, stable between 175K and 182K, was solved and refined in superspace. The known two-fold low-temperature superstructure of (NH4)2BeF4, that is stable below 175K has been described in superspace as a commensurately modulated structure. With aid of this description the close relationship between the two structures has been found. The MEM was applied to the incommensurate structure to test the appropriateness of the refined harmonic structure model. The MEM has shown that the harmonic model is very accurate. The MEM in superspace was established in this thesis as a reliable tool for the structure solutions of the modulated structure. The individual chapters together form a framework that allows to use the MEM in superspace to extract novel information from the diffraction data of both the periodic and aperiodic structures, that cannot be obtained from the structure refinements.
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The two-dimensional vibrating reed technique: a study of anisotropic pinning in high-temperature superconductors
(2004)
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Anna Karelina
- In this work the anisotropy of the pinning forces of vortices in a-b plane of high temperature-supraconductors was examined. For this purpose vibrating reed with two degrees of freedom of the oscillation was constructed. The pinning forces were examined in single crystals of YBa2Cu3O7 and Bi2Sr2CaCu2O8. Because of the d waves symmetry of the order parameter is expected four-fold anisotropy of the pinning potential. The reed consists of a sapphire fiber which is fast clamped at one end. Other end of the reed glued into a hole of the ruby disc, which is used as a sample holder. The superconducting sample was glued on top of the disc with the c-axis parallel to the sapphire fiber. The dielectric reed is covered by a thin conducting layer for driving the reed electrostatically and detecting its elongation from equilibrium by capacitance method. Thus we have the possibility to vibrate the sample in any direction of the ab plane. The magnetic field directed along the c axis creates vortices in the superconductor. Vibrating of the superconductor leads to the distortion of the flux lines and displacement of the vortices from the point-like oxygen defects, which act as pinning centers. The pinning force leads to additional restoring force which causes a typical magnetic field dependence of the resonance frequency. Such an experimental configuration has never been used before and gives the possibility to study the symmetry of the pinning potential. A mathematical model was developed to describe the effect of anisotropy of the pinning potential on the vibrating reed motion. The two simplest cases of two- and four-fold symmetry of the potential were considered in this model. The experiments with the two-dimensional vibrating reed were carried out with single crystals of YBa2Cu3O7, glued with some angle between the easy axis of the reed (the x direction) and the crystallographic axis a and b. These experiments demonstrated the presence of the two-fold symmetry of the pinning potential of the sample. The anisotropy of the pinning potential manifests itself by the appearance of coupling between main axes of the vibrating reed and by beating in the free oscillations. From the comparison of measurement and mathematical model anisotropy parameter was obtained equal to 10%. For the investigations of the fourfold symmetry of the pinning potential, the crystal was glued so that the crystallographic axes coincide with the easy axes of the reed to eliminate the effect of the two-fold symmetry of the pinning potential. The experiments with YBa2Cu3O7 show that at temperatures lower than 78K the vortices are in a nonequilibrium state. This leads to a flux creep and to a drift of the resonance frequency with time. This prevents the comparison of resonance curves in different directions of oscillations. In Bi2Sr2CaCu2O8 single crystals the vortices are in more stable state, but the measurements of the resonance curves in different directions show no indication of the four-fold symmetry. At temperatures below 60 K a strong hysteresis of the resonance frequency and the resonance-oscillation amplitude was found in YBa2Cu3O7 crystals as a function of the magnetic field. The hysteresis of the amplitude caused probably by bending of the reed because of the irreversible magnetisation.
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Convection and Magnetic Field Generation in Rotating Spherical Fluid Shells
(2004)
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Radostin D. Simitev
- The dissertation reports results from numerical and analytical studies of convection and dynamo action in rotating fluid spheres and spherical shells. This research is motivated by the geophysical problem of the origin and properties of the Earth's magnetism. Extensive numerical simulations are performed in order to advance the understanding of the basic physical components and mechanisms believed to be responsible for the generation and the variations in time of the main geomagnetic field. Questions such as linear onset and nonlinear finite-amplitude properties of rotating convection, generation and equilibration of magnetic fields in electrically conducting fluids, nonlinear feedback effects of the generated magnetic fields on convection, spatio-temporal structures of magnetic and velocity fields, oscillations and coherent processes in turbulent regimes and other questions are studied in dependence on all basic parameters of the problem, as well as for various choices of the magnetic, thermal and velocity boundary conditions and for some secondary assumptions such as a finitely-conducting inner core and various basic temperature profiles. Because of the lack of knowledge of the properties of the Earth's core and the uncertain details of the processes that take place there, this research is necessary in order to provide the tools for extrapolation to realistic models of the geodynamo. Of particular interest are various types of oscillations of dipolar fields. In contrast to quadrupolar and hemispherical dynamos dipolar dynamos have been originally considered to be non-oscillatory. But the six different types of dipolar oscillations, among which is the ``invisible'' one, reported in this dissertation alter this view. Generation of magnetic fields by convection shows a strong dependence on the Prandtl number P of the fluid. But this fact has received little attention in the past. Convection-driven dynamo action at Prandtl numbers larger than unity is studied with the goal to test the validity of the magnetostrophic approximation. The latter is found to be poorly satisfied for P < 300. Dynamos in this regime require magnetic Prandtl numbers Pm which increase with P. The same trend continues to hold for values of P less then unity and this regime thus seems to be best suited to reach the goal of minimal values of Pm. For Pm=P=0.1 a hemispherical dynamo is obtained in the case of a rotation parameter tau=10**5. A further reduction of Pm leads to a decay of magnetic field irrespective of the Rayleigh numbers used. Apart from numerical simulations and parameter studies of basic physical mechanisms, the dissertation includes an analytical study of inertial convection in rotating spheres in the limit of small Prandtl numbers and large rotation rates. Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber and on the product of P tau are derived and new results for the case of a nearly thermally insulating boundary are obtained. Limited comparisons with actually observed features of the geomagnetic field are also presented. An example are the torsional Alfven waves found in the numerical simulations of this dissertation. They are geophysically relevant as a possible cause for the observed secular variation impulses of the Earth's magnetic field. Reversals of the magnetic field polarity have also been observed in our simulations. Dynamo intermittency and interaction between dipolar and quadrupolar components are preconditions for aperiodic dipolar reversals similar to those of the Earth's main field. However, the opportunities for quantitative comparisons with geophysical observations are rather limited by the complexity of the self-consistent dynamo problem and by the computational restrictions of our numerical simulations.
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Pigment-Pigment Interactions and Protein Dynamics in Light-Harvesting Complexes: a Single-Molecule Study
(2004)
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Clemens Hofmann
- Light harvesting complexes that are involved in the first steps of photosynthesis in purple bacteria were studied with low-temperature optical single molecule spectroscopy. In one series of experiments the spectral properties of the bacteriochlorophyll a molecules within the complexes were studied in the view of model systems of molecular aggregates. It was found that the excitations in the B800 band of the light harvesting 2 complex are mainly localised on individual chromophores although evidence was found for an electronic coupling in the weak to intermediate range between individual bacteriochlorophyll a molecules. In contrast, for the B850 band of the light harvesting complex 2 as well as for the B870 band of the light harvesting complex 1 it was found that the assembly of bacteriochlorophyll molecules represents a strongly coupled system and that the excitation is coherently delocalised over a substantial part of the chromophores. By performing Monte-Carlo simulation an estimate on the amount of random and correlated energetic disorder in the site energies of the chromophores as well as on structural properties of the complexes could be given. In experiments on individual LH2-LH1-RC complexes, the energy transfer within a single photosynthetic unit was observed. In further experiments the chromophores were used as local probes to monitor conformational fluctuations of the protein residues in their binding pocket. Looking at the spectral diffusion of individual chromophores allowed to elucidate the organisation of the protein-energy landscape in tiers. In addition a clear correlation for the transition rates between those states and the energy separation of the levels involved could be uncovered. To simplify and automatise the analysis of a large number of consecutively recorded spectra a pattern recognition approach using multivariate statistical analysis proved to be a very useful tool. Apart from elucidating spectral diffusion processes detailed information about the line shape of individual chromophore absorptions could be gained from which it was inferred that the electron-phonon coupling in the B800 pigment pool is very weak. In summary, this thesis demonstrates that low-temperature single-molecule spectroscopy provides a unique method to reveal details of pigment-pigment interactions in the weak to intermediate as well as strong coupling limit that are inaccessible by other experimental methods.
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Optically induced orientational transitions in nematic liquid crystals
(2004)
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Dmitry Krimer
- I have presented in this thesis a theoretical study of some dynamical phenomena and orientational transitions induced by intense light in homeotropically oriented nematic layers. A large number of experiments has been performed in such systems and various interesting dynamical regimes have been identified. However, systematical theories capable of describing the observed phenomena have been derived for some cases only. In other cases oversimplified models exist with limited applicability. In Chapter 2 I considered the case of a circularly polarized plane light wave incident perpendicularly on the layer. I have constructed a theory that is capable of describing the observed regimes of director motion and the transitions between them in detail. The first instability is the Freedericksz transition from the homeotropic state to a small-amplitude reoriented state with uniform director precession around the layer normal. With increasing light intensity, this state destabilizes via a supercritical Hopf bifurcation and a new frequency in the time Fourier spectra of the dynamical variables appears. This regime is quasiperiodic and corresponds to a precession and nutation of the director. As the intensity increases further, this state disappears at a certain critical value where the period of nutation becomes infinite. There a strongly hysteretic transition to a state with large reorientation occurs via a homoclinic bifurcation. The homoclinic orbit involved is of the simplest type where a limit cycle collides with a saddle point having one unstable direction. The new state corresponds to a uniform precession of the director, however, with very large period and with large reorientation. I have also investigated the influence of an additional static electric field on the dynamical scenario described above. In Chapter 3 the treatment is generalized to the case of elliptically polarized light. The complete bifurcation diagram with the light intensity and the ellipticity as control parameters has been calculated in the region where rotating states exist. I have shown that for a fairly narrow region of ellipticities close to circular polarization the first periodic rotating state loses its stability in a supercritical Hopf bifurcation. I have found that with increasing light intensity at different ellipticities different sequences of transitions all finally lead to a state with large director distortion as the intensity is increased. The nature of this largely distorted state, as well as intermediate regimes vary with ellipticity. Some of the regimes that appear at lower intensities were studied previously, both experimentally and theoretically, but a complete picture up to the largely distorted regime was missing. In the theoretical treatments developed in the Chapters above, as in all other treatments, the velocity field induced by the director motion (backflow) has been neglected. In Chapter 4 I have investigated the influence of backflow on the dynamical scenario described in Chapter 2 and have shown that the backflow leads to substantial quantitative changes. It turns out that the regime of nonuniform precession shifts to higher light intensities and exists in a larger interval. I have also found unanticipated spatial oscillations of the backflow across the layer for the state with large director distortion. This is a signature of the interference pattern of the light within the layer. Actually, in the theory presented, for the first time, a light-induced dynamical phenomenon has been derived from the full nematodynamic equations. Thus, for the first time, full quantitative comparison with experiments using a transversally extended laser light could be done. Also, in all previous theoretical treatments involving plane wave incident light, it was assumed that the director distortion does not depend on the coordinates in the plane of the layer, i.e. one dealt with a one dimensional situation. In Chapter 5 I have studied the instabilities induced by a linearly polarized ordinary light wave incident at a small oblique angle allowing for spatial variations of the director in the plane of the layer and including the case of a dye-doped nematic. It was previously known that for sufficiently small angles of incidence the homeotropic state looses stability in a stationary, homogeneous pitchfork bifurcation. I have shown that the resulting stationary distorted state looses stability via a secondary Hopf bifurcation to spatially inhomogeneous state (nonzero critical wavenumber) that leads to the formation of travelling waves in the plane of the layer. The wavelength of these waves depend on the angle of incidence and the ratios of the elastic constants. It is typically several times larger than the thickness of the layer.
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Pattern Formation in Rotating Fluid Systems under the Influence of Magnetic Fields
(2004)
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Erol Kurt
- Patterns are observed in many different systems in nature. They are seen in the cloud streets, in sand ripples, in the morphology of plants and animals, on weather maps, in chemical reactions. In all these cases one deals with open, continuous dissipative systems which are driven out of equilibrium by an external stress. If this stress is larger than a certain threshold value, the symmetry of the temporally and spatially homogeneous ground state is spontaneously broken. The resulting patterns show then periodicity in space and/or in time. One of the best studied examples is the convection instability when a fluid layer is subjected to a temperature gradient. For instance, in a horizontal fluid layer heated from below and cooled from above a striped patterns of convection rolls develop. This scenario describes the famous Rayleigh- Benard convection (RBC), as a standard paradigm of pattern formation. Many concepts and mathematical tools to analyze the patterns have been developed and tested for this case. This thesis deals with two different pattern forming systems, namely a particular example of a convection instability and the case of a shear flow driven instability. In the first part of the thesis, a variation of the standard RBC is investigated. We consider the problem of convection induced by radial buoyancy in an electrically conducting fluid contained in a rotating (angular frequency, Omega) cylindrical annulus which is cooled at the inner surface and heated from outside. In addition, an azimuthal magnetic field (B) is applied for instance by an electrical current through the cylinder axis. The motivation of this study has come originally from the geophysical context. This setup is hoped to capture some important features of convection patterns in rotating stars and planets near the equatorial regions. The problem is also of considerable interest from a more general point of view in that it is concerned with formations of patterns in the presence of two competing directional effects, in this case rotation and the magnetic field. The second part of the thesis is devoted to the the pattern formation by a shear flow between two rotating and infinitely electrically conducting plates with a magnetic field perpendicular to the plates. This geometry is called the magnetic Ekman-Couette layer and has been a basic model for magnetic activities at the boundary of the Earth's liquid core or at the tachocline in the Sun below the convection zone for a few decades. To analyze the forementioned problems, various codes and computational tools had to be developed, for instance, we were able to describe complex spatio-temporal patterns by the direct simulations of the underlying hydrodynamic equations for our problems. The discussion of the physical details of the systems are postponed to the introductory sections of the corresponding parts of the thesis. In Chapter 1, a general formulation of the linear and nonlinear analysis, methods, which are applicable to both pattern forming systems in this work will be presented. The investigation of thermal convection in a plane layer which is a geometry equivalent to the cylindrical annulus will be discussed in Chapter 2. The next chapter (Chapter 3) covers both the linear and nonlinear analyses in the case of magnetic Ekman-Couette layer problem. Finally, in Chapter 4, we will present the general conclusions on both of the systems.
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Dynamics of vortices in the two-dimensional anisotropic Heisenberg model with magnetic fields.
(2003)
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Juan Pablo Zagorodny
- The subject of this work is the dynamics of a vortex in a classical 2-dimensional spin system with anisotropic exchange interaction under the combined action of magnetic fields and damping. Static as well as dynamic magnetic fields were employed (as dynamical field we used a homogeneous field which is rotating in the XY-plane). The most important goal of this work was to demonstrate that there is a coupling between the inner and translational freedom degrees of the vortex, coupling which is responsible for at least 2 phenomena that we study in detail in this Thesis: 1. the switching or flipping of the vortex polarization (for negative field frequency), and 2. the formation of stable orbits of the vortex center around the center of the system driven by the rotating field (for positive frequency). It was known to us that the polarization can change abruptly its sign under the action of a field rotating in the XY-plane, for p omega < 0 and appropriate field amplitudes. In the Chapter 4 we have investigated the possible underlying mechanisms for this phenomenon. Our main results can be summarized as follows: a) The flipping times do not depend essentially on the size of the system, provided that the lattice is large enough (radius L >~ 36 lattice constants). In other words, the switching of the vortex polarization is not much affected by the presence of boundaries. b) In our numerical simulations we observed a clear correlation between the core magnetization dynamics (the oscillations of the core spins in the out-of-plane direction) and the velocity of the vortex center in the plane of the lattice. c) A diagram of flipping events as a function of the field parameters, from extensive numerical simulations with an OP vortex in a rotating magnetic field, was presented. We found out that in the (omega, h) parameters space there is no well-defined curve which separates the regime where the flips do not occur from the regime where they do. We found intervals ("windows'') of intermittent flip and non-flip events. d) The switching of the vortex polarization can be achieved also by applying a static magnetic field with both in-plane (IP) and out-of-plane (OP) components. The IP component of the field sets the vortex into translational movement in the XY-plane, while the OP component breaks the vertical symmetry favoring one of the two possible orientations. e) The switching dynamics may be described in terms of a core model which takes into account a coupling between the vortex polarization dynamics and the motion of the vortex center. We showed that a reduced core model, which is valid near the threshold of the IP-OP vortex instability (lambda ~ lambda c), can be mapped to a generalized Thiele equation with an inertial term. f) It is plausible that the phenomenon of switching we described will not be essentially affected by the inclusion of a dipole-dipole interaction. The experimental works on nanodisks mentioned in the Introduction of this Thesis reported the observation of vortices in either of two polarization states, and the switching between them was forced by means of static fields perpendicular to the plane of the disks. Rotating magnetic fields might be used as well static fields with both IP and OP components to make this switching more favorable. In the Chapter 5 we turned to the study of the movement of the vortex in the XY plane, in the presence of the IP rotating field. Attention was directed to the existence of stable orbits, where the vortex stays inside the system in a stationary movement, forming circular limit cycles. We discussed then the failure of the conventional Thiele approach to describe this phenomenon, and this motivated us to formulate an extended collective coordinate Theory, which leads to a qualitative agreement with the results of the simulations. A diagram of the different types of trajectories, as a function of the field parameters, showed the presence of non-monotonous effects and "windows'', like in the case of the switching diagram. We are led to conclude that for some regions of the field parameters space, the system exhibits chaos -which is typical for many-body systems-, though no particular tool of the chaos theory was used to study our discrete and collective coordinate models, from this viewpoint. Our theoretical work qualitatively suggests that it would be interesting to apply in the experiments weak rotating fields like those used here, to control both the mean position of a vortex in larger magnetic dots (where the vortex center could show dynamics) and at the same time the sign of the out-of-plane core magnetization. Future directions of this work may include the use of inhomogeneous fields, particularly with a gaussian localization in a small region of the lattice or "spot'', as a model of the field of a laser beam.
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Supercurrents in Restricted Geometries and Driven by Time-Dependent Electric Fields
(2004)
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Martin Endres
- NS Contact We studied the linear response of a normal metal superconducting metal contact to a small electric field. In a preparatory section the order-parameter profile and the density of states were calculated in equilibrium. We showed that the density of states in the normal metal is unaltered if the impurity self-energies are not taken into account while the coherence in the superconductor is always affected by the presence of the normal metal. Self-consistent calculations result in an impurity-induced proximity effect in the normal metal. This proximity effect causes a spatially constant gap in the density of states of the normal metal if the normal metal is sandwiched between two superconductors. The dynamics of the NS contact is strongly dominated by the conservation law for charge and local charge neutrality which together fully determine the current in one-dimensional systems. For answering the question how this constant current is established in the non-homogeneous NS contact, the quasiclassical equations were solved including the self-consistencies for the order parameter, the impurity self-energies, and the electrochemical potential. The latter was used to deduce an internal electric field as response to the external perturbation. The internal field is of same order as the perturbation and is caused by charges which are either bound to the interface or spread over several coherence lengths. The surface charges are not due to the step in the order parameter at the interface but solely to abrupt changes of the impurity scattering. The order parameter itself can only produce continuous charge densities. The charges are indirectly calculated using Maxwell's equations. They are of higher order in the expansion parameters of Fermi-liquid theory and are hence beyond this theory. Nevertheless, their effect has to be considered to be consistent in leading order. Weak links in He3 We investigated several methods of calculating the current-phase relation of weak links in He3. In the limit of small holes the hole itself and the current through it does not affect the order parameter in the superfluid and the current can hence be calculated using the pinhole model. This leads to a periodic current-phase relation. It was shown that the pair-breaking effect of the separating wall has no significant influence on the functional dependence of the current on the phase difference. The wall mainly reduces the amplitude of the current. For orifices with radii comparable with the coherence length, self-consistent order-parameter fields were calculated. The two fixed phases of the pinhole model are then replaced by a field which allows the phase to wind up continuously. This not only breaks the periodicity, but also leads to multivalued current-phase relations. Over a wide range the current through the orifice is linear in the phase difference between the reservoirs. Although this is expected in the hydrodynamic limit, the hydrodynamic equations are not applicable as they always fail at the edges of the circular apertures. However, calculating the current quasiclassically with the phase determined via the Laplace equation gives a fairly good approximation to the fully self-consistent solution. This approximation becomes weak for larger phase differences when pair-breaking due to the current itself has to be taken into account. Remarkably, the maximal current through the aperture is sandwiched between the pinhole current and the depairing current for a homogeneous superfluid which differ only by a factor of about two at low temperature in spite of the drastic difference of the models. A quasiclassical free-energy functional was introduced and it was stressed that this choice is not unique and that a whole zoo of different functionals exists. The functional was used to investigate the change in free energy due to the wall, the orifice, and the phase difference.
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Ratchet dynamics in nonlinear Klein-Gordon systems
(2005)
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Luis Morales Molina
- In the first part of the work we have studied a directed energy transport in homogeneous nonlinear extended systems in the presence of a biharmonic force and dissipation. We have shown that the mechanism responsible for unidirectional motion of topological excitations is the coupling of their internal and translation degrees of freedom. Our results lead to a selection rule for the existence of such motion based on resonances that explains earlier symmetry analysis of this phenomenon. We also found in the framework of the collective coordinate theory an explanation to the dynamics dependence on the damping. In the second part of the work we have presented and studied a novel design for a ratchet potential for soliton excitations. The investigation was focused on the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic and asymmetric lattice of point-like inhomogeneities in the overdamped regime. In addition, we explained the underlying rectification mechanism within a collective coordinate framework, which shows that such a system behaves as a rocking ratchet for point particles.This was supported by numerical simulations. A quantitative agreement was found in an improved version of the collective coordinate approach that regards the kink width in addition to the fundamental translational degree of freedom. An explanation for the to the kink width dynamics and its role in the transport was presented. We also studied the robustness of our kink rocking ratchet in the presence of noise. For this situation it was shown that noise activates unidirectional motion in a parameter range where the motion is not observed in the noiseless case. This is subsequently corroborated by the collective variable theory. The study was also extended to the weak underdamped regime, where higher values of the mean kink velocity were found. An explanation for this new phenomenom was given.
