9 search hits
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Electrostatic Trapping as a Self-Consistent Phenomenon in Plasmas and other Collective Systems
(2004)
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Alejandro Luque Estepa
- This thesis investigates self-consistent electrostatic structures in plasmas and related collective systems. They are coherent structures in which particles become trapped in the wave potential. These phenomena require a kinetic description i.e. a description in which the velocity distribution of the particles is taken into account. Trapping structures extend the areas in configuration space in which a plasma is unstable. The main argument and result of this work is that there exist certain kinds of perturbations of an equilibrium that can destabilize the plasma, even if linear theory predicts stability. The usual procedure in plasma theory of analyzing the stability of a plasma by means of a linearization of the equations is therefore questioned. Particle trapping is an essentially nonlinear phenomenon, still present for infinitesimally small wave amplitudes. The effect of the particle trapping is therefore not linked only with the treatment of finite amplitudes, as often assumed, but has also to be taken into account from the very beginning if one wants to arrive at generally valid predictions about stability and the associated anomalous transport. Thus it is not surprising that the problem of transport represents a not yet closed chapter in the theory of plasmas, a fact which is shown in many examples from fusion and space research, where almost collisionless plasmas are present. Particle trapping is however not confined to classic plasmas. Another result of this work is to show that the applied formalism can also be extended to other systems that present a collective behavior. Namely, a quantum extension is possible, which allow us to investigate quantum-like systems and also to draw a connection between electrostatic trapping in plasmas and envelope solitons in nonlinear optical media like e.g. optical fibers. The longitudinal dynamics of charged particle beams in accelerators and storage rings provides a further example of a collective system in which the phenomenon of particle trapping plays an essential role of the dynamics.
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Untersuchungen zur Diffusion und Reaktion von Kohlenstoff auf Nickel- und Eisenoberflächen sowie von Beryllium auf Wolfram
(2004)
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Almut Wiltner
- Die chemische Wechselwirkung und die Diffusion von dünnen C-Schichten auf Ni- und Fe-Oberflächen sowie von Be-Schichten auf polykristallinen W-Substraten werden mit der oberflächensensitiven Analysemethode Röntgenphotoelektronenspektroskopie (XPS) untersucht. Dabei erfolgt die Schichtdeponierung aus der Gasphase durch Elektronenbeschuß eines Graphitstabs bzw. von metallischem Be, das in einem BeO-Heiztiegel vorliegt. Die C-Schichten werden auf Ni(100), Ni(111) und Fe(110) deponiert. Diese Substratoberflächen repräsentieren sowohl die dichtesten als auch offene Atomanordnungen. Die Bildung der entsprechenden Ni- und Fe-Carbide verläuft in einer endothermen Reaktion. Die Carbidbildung und C-Diffusion werden vom untersuchten Substrat und der jeweiligen Oberflächenatomanordnung beeinflußt. Die bei Raumtemperatur aufgebrachten C-Schichten werden mit XPS analysiert. Im C 1s-Signal wird hauptsächlich elementarer C nachgewiesen. Dieser nicht reagierte C besteht aus dem graphitischen und dem ungeordneten Anteil. Obwohl die Carbidbildungsreaktionen endotherm sind, wird auf allen untersuchten Substratoberflächen zusätzlich ein Carbidanteil nachgewiesen. Dieser Anteil ist auf das unmittelbare Interface limitiert und wird mit der Bildung eines Oberflächencarbids erklärt. Die C-Schichten werden nachfolgend geheizt. Dabei werden zwei experimentelle Verfahren angewandt. Im Verfahren I werden die Proben in Schritten von 50 bzw. 100 K bis zu einer Temperatur von 970 K für jeweils 10 bzw. 30 Minuten geheizt. Nach Abkühlung auf Raumtemperatur werden die Systeme nach jedem Heizschritt mit XPS analysiert. Im thermischen Verhalten zeigen sich dabei verschiedene Phasen. Das anfänglich gebildete Carbid wird zunächst zersetzt (Phase I, 370-570 K), die C 1s-Signalintensität des elementaren C-Anteils nimmt entsprechend zu. Im weiteren Temperaturverlauf wird die Phase II (>370 K) identifiziert. In dieser Phase nimmt der ungeordnete C-Anteil ab, die Signalintensität des graphitischen Anteils steigt entsprechend an. Diese Reaktion wird als Umordnungsreaktion des elementaren C-Anteils bezeichnet. Phase I und II werden auf allen Substratoberflächen im gleichen Temperaturbereich beobachtet. In Abhängigkeit von Substrat und der untersuchten Oberfläche setzt die C-Diffusion (Phase III) ein. Auf Fe(110) die C 1s-Signalintensität bei 620 K ab, gefolgt von Ni(100) bei 670 K und Ni(111) bei 770 K. Der C-Verlust innerhalb dieser Phase führt zu einer Abnahme des C 1s-Signals innerhalb von 200 K. An der Oberfläche ist nach diesem Temperaturbereich eine C-Restbedeckung von weniger als einer Monolage nachweisbar. Zur Ermittlung der kinetischen Parameter (Aktivierungsenergien) für die identifizierten Phasen werden C-Schichten im Heizverfahren II untersucht. Dabei werden die jeweiligen Systeme für mehrere Stunden bei ausgewählten Temperaturen gehalten und dabei mit XPS analysiert. In diesem experimentellen Heizverfahren kann zusätzlich die endotherme Carbidbildungsreaktion, die im Verfahren I nicht eindeutig zu identifizieren ist, charakterisiert werden. Die in beiden Verfahren ermittelten Reaktionen werden sowohl vom untersuchten Substrat als auch von der vorliegenden Struktur beeinflußt. Da die gebildeten Carbide durch die Einlagerung von C-Atomen in das Metallatomgitter entstehen, geht diese Reaktion mit der C-Diffusion einher. Die Be-Schichten werden auf polykristalline W-Substrate deponiert. Das Be 1s-Signal weist hauptsächlich metallisches Be auf. Bereits bei Raumtemperatur wird sowohl im Be 1s- als auch im W 4f-Signal eine Be-W-Mischphase nachgewiesen. Diese Mischphase wird mit der Bildung einer Oberflächenlegierung erklärt. Im Be 1s-Signal wird außerdem ein oxidischer Anteil nachgewiesen. Das thermische Verhalten der Be-Schichten ist abhängig von der anfänglich deponierten Schichtdicke. Dünne Filme (bis 1.2 nm) weisen bis 970 K eine konstante Schichtdicke auf. Werden Be-Schichten mit einer anfänglichen Dicke von 1.2-3.0 nm aufgebracht, nimmt die Be-Menge ab 670 K innerhalb weniger Minuten deutlich ab. Die nach den Heizschritten bis 970 K an der Oberfläche verbleibende Be-Schichtdicke (1.0-.2 nm) ist mit den zuvor erwähnten dünnen Filmen vergleichbar. Be-Schichten mit einer anfänglichen Dicke von mehr als 3.0 nm zeigen eine wesentlich langsamere Abnahme der Schichtdicke bei vergleichbaren Temperaturen. Nach dem letzten Heizschritt bei 970 K ist die verbleibende Schichtdicke größer als 1.4 nm. Unabhängig von der Schichtdicke nimmt die Intensität des Legierungsanteils im Be 1s- und W 4f-Signal ab 670 K zu. Das stöchiometrische Verhältnis des Legierungsanteils in beiden Signalen deutet auf die Bildung von Be(2)W(1) hin. Nach Abschluß der Heizexperimente weisen Tiefenprofile auf eine Be-Diffusion, die auf die Be-W-Mischphase begrenzt ist, hin. Der Verlust des Be wird entsprechend mit der Be-Desorption erklärt.
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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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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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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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Magnetooptische Untersuchungen an Halbleitersystemen mit reduzierter Dimensionalität
(2004)
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Jens Fürst
- Diese Arbeit beschäftigt sich mit der magnetooptischen Untersuchung von niedrigdimensionalen Halbleiterstrukturen, bestehend aus III-V und IV-VI Halbleiternsystemen. Als Vertreter der III-V Halbleiter stehen InAs Quantenpunktproben zur Verfügung. Die elektronischen Eigenschaften der InAs Quantenpunkte werden mit Hilfe von Photolumineszenz- und Hanle-Effekt-Messungen analysiert. Dabei wird der Einfluss einer Vielzahl von Probenparametern wie Dotierung und Zusammensetzung der Quantenpunkte sowie des Matrixmaterials auf die Elektronenspinlebenszeit untersucht. Grundlegende Analysen werden an dem ternären IV-VI Materialsystem PbSeTe durchgeführt. Bei der Entstehung von PbSe Quantenpunkten in PbEuTe Matrix ist es nicht auszuschließen, dass es an der Grenzfläche der Quantenpunkte und des Matrixmaterials zu Durchmischungen dieser beiden Materialien kommt und sich dadurch das System PbSeTe bildet. Deshalb werden die Bandparameter dieses Mischkristalls in einer Probenserie von reinem PbSe zu reinem PbTe mit Hilfe von Magnetotransmissionsmessungen und kohärenten anti-Stokes Ramanstreuung bestimmt. Zur Untersuchung der Emission von PbSe Quantenpunkten sind diese in einen vertikalen Resonator eingebracht. Das von diesen nulldimensionalen Systemen emittierte Licht wird mit der emittierten Strahlung von Resonatoren mit zweidimensionalen und dreidimensionalen aktiven Zonen verglichen. In weiteren Experimenten werden vertikal emittierende Laserstrukturen in einem Spektralbereich nahe 8 Mikrometer bei kontinuierlicher und gepulster Anregung untersucht. Dabei zeigen sich sehr schmale Emissionslinien. In einem longitudinalen Magnetfeld konnte aufgrund der kleinen Halbwertsbreite des emittierten Lichtes eine Aufspaltung beobachtet werden.
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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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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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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.
