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The current thesis is devoted to a systematic study of fluctuations and correlations in heavy-ion collisions, which might be considered as probes for the phase transition and the critical point in the phase diagram, within the Hadron-String- Dynamics (HSD) microscopic transport approach. This is a powerful tool to study nucleus-nucleus collisions and allows to completely simulate experimental collisions on an event-by-event basis. Thus, the transport model has been used to study fluctuations and correlations including the influence of experimental acceptance as well as centrality, system size and collision energy. The comparison to experimental data can separate the effects induced by a phase transition since there is no phase transition in the HSD version used here. Firstly the centrality dependence of multiplicity fluctuations has been studied. Different centrality selections have been performed in the analysis in correspondence to the experimental situation. For the fixed target experiment NA49 events with fixed numbers of the projectile participants have been studied while in the collider experiment PHENIX centrality classes of events have been defined by the multiplicity in certain phase space region. A decrease of participant number fluctuations (and thus volume fluctuations) in more central collisions for both experiments has been obtained. Another area of this work addresses to transport model calculations of multiplicity fluctuations in nucleus-nucleus collisions as a function of colliding energy and system size. This study is in full correspondence to the experimental program of the NA61 Collaboration at the SPS. Central C+C, S+S, In+In, and Pb+Pb nuclear collisions at Elab = 10, 20, 30, 40, 80, 158 AGeV have been investigated. The expected enhanced fluctuations - attributed to the critical point and phase transition - can be observed experimentally on top of a monotonic and smooth ‘hadronic background’. These findings should be helpful for the optimal choice of collision systems and collision energies for the experimental search of the QCD critical point. Other observables are fluctuations of ratios of hadrons (e.g. pions, kaons, protons, etc.) which are not so much affected by volume fluctuations. In particular HSD results for the kaon-to-pion ratio fluctuations, which has been regarded as promising observable for a long time, are presented from low SPS energies up to high energies at RHIC. In addition to the HSD calculations statistical model is also used in terms of microcanonical, canonical and grand canonical ensembles. Further a study of the system size event-by-event fluctuations causing rapidity forward-backward correlations in relativistic heavy-ion collisions is presented. The HSD simulations reveal strong forward-backward correlations and reproduce the main qualitative features of the STAR data in A+A collisions at RHIC energies. It has been shown that strong forward-backward correlations arise due to an averaging over many different events that belong to one centrality bin. An optimization of the experimental selection of centrality classes is presented, which is relevant for the program of the NA61 collaboration at CERN, the low-energy program at RHIC, as well as future experiments at FAIR.
Ein wesentliches Ziel der Physik mit schweren Ionen ist die Untersuchung der Zustände von Kernmaterie bei hohen Dichten bzw. Temperaturen. Solche Zustände lassen sich durch Kollisionen von hochenergetischen schweren Ionen in Teilchenbeschleunigern wie dem Super Proton Synchrotron SPS am Europäischen Kernforschungszentrum CERN in Genf erzeugen und untersuchen. Die vorliegende Arbeit beschäftigt sich mit der Analyse des Einflusses des in einer solchen Kollision erzeugten Mediums auf hochenergetische Teilchen, welche dieses Medium durchqueren. Hierzu werden Korrelationen zwischen Teilchen mit hohem Transversalimpuls pt als Funktion der Zentralität der Kollisionen und der Ladung der beteiligten Teilchen untersucht. Ziel ist es, hierdurch eine experimentelle Grundlage für die theoretische Beschreibung der Eigenschaften des Mediums in solchen Kollisionen bereitzustellen. ...
Kaon and pion production in centrality selected minimum bias Pb+Pb collisions at 40 and 158A GeV
(2009)
Results on charged kaon and negatively charged pion production and spectra for centrality selected Pb+Pb mininimum bias events at 40 and 158A GeV have been presented in this thesis. All analysis are based on data taken by the NA49 experiment at the accelerator Super Proton Synchrotron (SPS) at the European Organization for Nuclear Research (CERN) in Geneva, Switzerland. The kaon results are based on an analysis of the mean energy loss <dE/dx> of the charged particles traversing the detector gas of the time projection chambers (TPCs). The pion results are from an analysis of all negatively charged particles h- corrected for contributions from particle decays and secondary interactions. For the dE/dx analysis of charged kaons, main TPC tracks with a total momentum between 4 and 50 GeV have been analyzed in logarithmic momentum log(p) and transverse momentum pt bins. The resulting dE/dx spectra have been fitted by the sum of 5 Gaussians, one for each main particle type (electrons, pions, kaons, protons, deuterons). The amplitude of the Gaussian used for the kaon part of the spectra has been corrected for efficiency and acceptance and the binning has been transformed to rapidity y and transverse momentum pt bins. The multiplicity dN/dy of the single rapidity bins has been derived by summing the measured range of the transverse momentum spectra and an extrapolation to full coverage with a single exponential function fitted to the measured range. The results have been combined with the mid-rapidity measurements from the time-of-flight detectors and a double Gaussian fit to the dN/dy spectra has been used for extrapolation to rapidity outside of the acceptance of the dE/dx analysis. For the h- analysis of negatively charged pions, all negatively charged tracks have been analyzed. The background from secondary reactions, particle decays, and gamma-conversions has been corrected with the VENUS event generator. The results were also corrected for efficiency and acceptance and the pt spectra were analyzed and extrapolated where necessary to derive the mean yield per rapidity bin dN/dy. The mean multiplicity <pi-> has been derived by summing up the measured dN/dy and extrapolating the rapidity spectrum with a double Gaussian fit to 4pi coverage. The results have been discussed in detail and compared to various model calculations. Microscopical models like URQMD and HSD do not describe the full complexity of Pb+Pb collisions. Especially the production of the positively charged kaons, which carry the major part of strange quarks, cannot be consistently reproduced by the model calculations. Centrality selected minimum bias Pb+Pb collisions can be described as a mixture of a high-density region of multiply colliding nucleons (core) and practically independent nucleon-nucleon collisions (corona). This leads to a smooth evolution from peripheral to central collisions. A more detailed approach derives the ensemble volume from a percolation of elementary clusters. In the percolation model all clusters are formed from coalescing strings that are assumed to decay statistically with the volume dependence of canonical strangeness suppression. The percolation model describes the measured data for top SPS and RHIC energies. At 40A GeV, the system size dependence of the relative strangeness production starts to evolve from the saturation seen at higher energies from peripheral events onwards towards a linear dependence at SIS and AGS. This change of the dependence on system size occurs in the energy region of the observed maximum of the K+ to pi ratio for central Pb+Pb collisions. Future measurements with heavy ion beam energies around this maximum at RHIC and FAIR as well as the upgraded NA49 successor experiment NA61 will further improve our understanding of quark matter and its reflection in modern heavy ion physics and theories.
Poster presentation: The brain is autonomously active and this self-sustained neural activity is in general modulated, but not driven, by the sensory input data stream [1,2]. Traditionally one has regarded this eigendynamics as resulting from inter-modular recurrent neural activity [3]. Understanding the basic modules for cognitive computation is, in this view, the primary focus of research and the overall neural dynamics would be determined by the the topology of the intermodular pathways. Here we examine an alternative point of view, asking whether certain aspects of the neural eigendynamics have a central functional role for overall cognitive computation [4,5]. Transiently stable neural activity is regularly observed on the cognitive time-scale of 80–100 ms, with indications that neural competition [6] plays an important role in the selection of the transiently stable neural ensembles [7], also denoted winning coalitions [8]. We report on a theory approach which implements these two principles, transient-state dynamics and neural competition, in terms of an associative neural network with clique encoding [9]. A cognitive system [10] with a non-trivial internal eigendynamics has two seemingly contrasting tasks to fulfill. The internal processes need to be regular and not chaotic on one side, but sensitive to the afferent sensory stimuli on the other side. We show, that these two contrasting demands can be reconciled within our approach based on competitive transient-state dynamics, when allowing the sensory stimuli to modulate the competition for the next winning coalition. By testing the system with the bars problem, we find an emerging cognitive capability. Only based on the two basic architectural principles, neural competition and transient-state dynamics, with no explicit algorithmic encoding, the system performs on its own a non-linear independent component analysis of input data stream. The system has rudimentary biological features. All learning is local Hebbian-style, unsupervised and online. It exhibits an ever-ongoing eigendynamics and at no time is the state or the value of synaptic strengths reset or the system restarted; there is no separation between training and performance. We believe that this kind of approach – cognitive computation with autonomously active neural networks – to be an emerging field, relevant both for system neuroscience and synthetic cognitive systems.
The study of the electromagnetic structure of hadrons plays an important role in understanding the nature of matter. In particular the emission of lepton pairs out of the hot and dense collision zone in heavy-ion reactions is a promising probe to investigate in-medium properties of hadrons and in general the properties of matter under such extreme conditions. The first experimental observation of an enhanced di-electron yield in the invariant-mass region 0:3 - 0:7 GeV/c2 in p+Be collisions at 4:9 GeV/u beam energy [2] was announced by the DLS collaboration [1]. Recent results of the HADES collaboration show a moderate enhancement above n Dalitz decay contributions for 12C+12C at 1 and 2 GeV/u [3, 4] confirming the DLS results. There are several theoretical explanations of this observation, most of them focusing on possible in-medium modifications of the properties of vector mesons. At low beam energies the question whether the observed excess is related to any in-medium effects remains open because of uncertainties in the description of elementary di-electron sources. In this work the di-electron production in p+p and d+p reactions at a kinetic beam energy of 1:25 GeV/u measured by the HADES spectrometer is discussed. At Ekin = 1:25 GeV/u, i.e. below the n meson production threshold in proton-proton reactions, the delta Dalitz decay is expected to be the most abundant source above the pi 0 Dalitz decay region. The observed large difference in di-electron production in p+p and d+p collisions suggests that di-electron production in the d+p system is dominated by the n+p interaction. In order to separate delta Dalitz decays and np bremsstrahlung the di-electron yield observed in p+p and n+p reactions, both measured at the same beam energy, has been compared. The main interest here is the investigation of iso-spin effects in baryonic resonance excitations and the off-shell production of vector mesons [5]. We indeed observe a large difference in di-electron production in p+p and n+p reactions. Results of these studies will be compared to recent calculations. We will also present our experimentally defined cocktail for heavy-ion data. At much higher beam energies experimental results of the CERES [6] and NA60 [7] collaborations also show an enhancement in the invariant mass region 0:3 - 0:7 GeV/c2, in principle similar to the situation in DLS. A strong excess of lepton pairs observed by recent high energy heavy-ion dilepton experiments hint to a strong influence of baryons, however no data exist at highly compressed baryonic matter, achievable in heavy-ion collisions from 8 - 45 GeV/u beam energy. These conditions would allow to study the expected restoration of chiral symmetry by measuring in-medium modifications of hadronic properties, an experimental program which is foreseen by the future CBM experiment at FAIR. The experimental challenge is to suppress the large physical background on the one hand and to provide a clean identification of electrons on the other hand. In this work, strategies to reduce the combinatorial background in electron pair measurements with the CBM detector are discussed. The main goal is to study the feasibility of effectively reducing combinatorial background with the currently foreseen experimental setup, which does not provide electron identification in front of the magnetic field.
The multiplicity fluctuations in A+A collisions at SPS and RHIC energies are studied within the HSD transport approach. We find a dominant role of the fluctuations in the nucleon participant number for the final fluctuations. In order to extract physical fluctuations one should decrease the fluctuations in the participants number. This can be done considering very central collisions. The system size dependence of the multiplicity fluctuations in central A+A collisions at the SPS energy range – obtained in the HSD and UrQMD transport models – is presented. The results can be used as a ‘background’ for experimental measurements of fluctuations as a signal of the critical point. Event-by-event fluctuations of the K/p , K/p and p/p ratios in A+A collisions are also studied. Event-by-event fluctuations of the kaon to pion number ratio in nucleus-nucleus collisions are studied for SPS and RHIC energies. We find that the HSD model can qualitatively reproduce the measured excitation function for the K/p ratio fluctuations in central Au+Au (or Pb+Pb) collisions from low SPS up to top RHIC energies. The forward-backward correlation coefficient measured by the STAR Collaboration in Au+Au collisions at RHIC is also studied. We discuss the effects of initial collision geometry and centrality bin definition on correlations in nucleus-nucleus collisions. We argue that a study of the dependence of correlations on the centrality bin definition as well as the bin size may distinguish between these ‘trivial’ correlations and correlations arising from ‘new physics’. 5th International Workshop on Critical Point and Onset of Deconfinement - CPOD 2009, June 08 - 12 2009 Brookhaven National Laboratory, Long Island, New York, USA
In this thesis we have studied the physics of different ultracold Bose-Fermi mixtures in optical lattices, as well as spin 1=2 fermions in a harmonic trap. To study these systems we generalized dynamical mean-field theory for a mixture of fermions and bosons, as well as for an inhomogeneous environment. Generalized dynamical mean-field theory (GDMFT) is a method that describes a mixture of fermions and bosons. This method consists of Gutzwiller mean-field for the bosons, and dynamical mean-field theory for the fermions, which are coupled on-site by the Bose-Fermi density-density interaction and possibly a Feshbach term which converts a pair of up and down fermions into a molecule, i.e. a boson. We derived the self-consistency equations and showed that this method is well-controlled in the limit of high lattice coordination number z. We develop real-space dynamical mean-field theory for studying systems in an inhomogeneous environment, e.g. in a harmonic trap. The crucial difference compared to standard DMFT is that we are taking into account that different sites are not equivalent to each other and thus take into account the inhomogeneity of the system. Different sites are coupled by the real-space Dyson equation. ...
The bulk viscosity of several quark matter phases is calculated. It is found that the effect of color superconductivity is not trivial, it may suppress, or enhance the bulk viscosity depending on the critical temperature and the temperature at which the bulk viscosity is calculated. Also, is it found that the effect of neutrino-emitting Urca processes cannot be neglected in the consideration of the bulk viscosity of strange quark matter. The results for the bulk viscosity of strange quark matter are used to calculate the r-mode instability window of quark stars with several possible phases. It is shown that each possible phase has a different structure for the r-mode instability window.
In this thesis the first fully integrated Boltzmann+hydrodynamics approach to relativistic heavy ion reactions has been developed. After a short introduction that motivates the study of heavy ion reactions as the tool to get insights about the QCD phase diagram, the most important theoretical approaches to describe the system are reviewed. To model the dynamical evolution of the collective system assuming local thermal equilibrium ideal hydrodynamics seems to be a good tool. Nowadays, the development of either viscous hydrodynamic codes or hybrid approaches is favoured. For the microscopic description of the hadronic as well as the partonic stage of the evolution transport approaches have beeen successfully applied, since they generate the full phse-space dynamics of all the particles. The hadron-string transport approach that this work is based on is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach. It constitutes an effective solution of the relativistic Boltzmann equation and is restricted to binary collisions of the propagated hadrons. Therefore, the Boltzmann equation and the basic assumptions of this model are introduced. Furthermore, predictions for the charged particle multiplicities at LHC energies are made. The next step is the development of a new framework to calculate the baryon number density in a transport approach. Time evolutions of the net baryon number and the quark density have been calculated at AGS, SPS and RHIC energies and the new approach leads to reasonable results over the whole energy range. Studies of phase diagram trajectories using hydrodynamics are performed as a first move into the direction of the development of the hybrid approach. The hybrid approach that has been developed as the main part of this thesis is based on the UrQMD transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The initial energy and baryon number density distributions are not smooth and not symmetric in any direction and the initial velocity profiles are non-trivial since they are generated by the non-equilibrium transport approach. The fulll (3+1) dimensional ideal relativistic one fluid dynamics evolution is solved using the SHASTA algorithm. For the present work, three different equations of state have been used, namely a hadron gas equation of state without a QGP phase transition, a chiral EoS and a bag model EoS including a strong first order phase transition. For the freeze-out transition from hydrodynamics to the cascade calculation two different set-ups are employed. Either an in the computational frame isochronous freeze-out or an gradual freeze-out that mimics an iso-eigentime criterion. The particle vectors are generated by Monte Carlo methods according to the Cooper-Frye formula and UrQMD takes care of the final decoupling procedure of the particles. The parameter dependences of the model are investigated and the time evolution of different quantities is explored. The final pion and proton multiplicities are lower in the hybrid model calculation due to the isentropic hydrodynamic expansion while the yields for strange particles are enhanced due to the local equilibrium in the hydrodynamic evolution. The elliptic flow values at SPS energies are shown to be in line with an ideal hydrodynamic evolution if a proper initial state is used and the final freeze-out proceeds gradually. The hybrid model calculation is able to reproduce the experimentally measured integrated as well as transverse momentum dependent $v_2$ values for charged particles. The multiplicity and mean transverse mass excitation function is calculated for pions, protons and kaons in the energy range from $E_{\rm lab}=2-160A~$GeV. It is observed that the different freeze-out procedures have almost as much influence on the mean transverse mass excitation function as the equation of state. The experimentally observed step-like behaviour of the mean transverse mass excitation function is only reproduced, if a first order phase transition with a large latent heat is applied or the EoS is effectively softened due to non-equilibrium effects in the hadronic transport calculation. The HBT correlation of the negatively charged pion source created in central Pb+Pb collisions at SPS energies are investigated with the hybrid model. It has been found that the latent heat influences the emission of particles visibly and hence the HBT radii of the pion source. The final hadronic interactions after the hydrodynamic freeze-out are very important for the HBT correlation since a large amount of collisions and decays still takes place during this period.
Induced charge computation
(2009)
One of the main aspects of statistical mechanics is that the properties of a thermodynamics state point do not depend on the choice of the statistical ensemble. It breaks down for small systems e.g. single molecules. Hence, the choice of the statistical ensemble is crucial for the interpretation of single molecule experiments, where the outcome of measurements depends on which variables or control parameters, are held fixed and which ones are allowed to fluctuate. Following this principle, this thesis investigates the thermodynamics of a single polymer pulling experiments within two different statistical ensembles. The scaling of the conjugate chain ensembles, the fixed end-to-end vector (Helmholtz) and the fixed applied force (Gibbs), are studied in depth. This thesis further investigates the ensemble equivalence for different force regimes and polymer-chain contour lengths. Using coarse-grained molecular dynamic simulations, i.e. Langevin dynamics, the simulations were found to complement the theoretical predictions for the scaling of ensemble difference of Gaussian chains in different force-regimes, giving special attention to the zero force regime. After constructing Helmholtz and Gibbs conjugate ensembles for a Gaussian chain, two different data sets of thermodynamic states on the force-extension plane, i.e. force-extension curves, were generated. The ensemble difference is computed for different polymer-chain lengths by using force-extension curves. The scaling of the ensemble difference versus relative polymer-chain length under different force regimes has been derived from the simulation data and compared to theoretical predictions. The results demonstrate that the Gaussian chain in the zero force limit generates nonequivalent ensembles, regardless of its equilibrium bond length and polymer-chain contour length. Moreover, if polymers are charged in confinement, coarse-graining is problematic, owing to dielectric interfaces. Hence, the effect of dielectric interfaces must be taken into account when describing physical systems such as ionic channels or biopolymers inside nanopores. It is shown that the effect of dielectrics is crucial for the dynamics of a biopolymer or an ion inside a nanopore. In the simulations, the feasibility of an efficient and accurate computation of electrostatic interactions in the presence of an arbitrarily shaped dielectric domain is challenging. Several solutions for this problem have been previously proposed in the literature such as a density functional approach, or transforming problem at hand into an algebraic problem ( Induced Charge Computation (ICC) ) and boundary element methods. Even though the essential concept is the same, which is to replace the dielectric interface with a polarization charge density, these approaches have been analyzed and the ICC algorithm has been implemented. A new superior boundary element method has been devised utilizing the force computation via the Particle-Particle Particle-Mesh (P3M) method for periodic geometries (ICCP3M). This method has been compared to the ICC algorithm, the algebraic solutions, and to density functional approaches. Extensive numerical tests against analytically tractable geometries have confirmed the correctness and applicability of developed and implemented algorithms, demonstrating that the ICCP3M is the fastest and the most versatile algorithm. Further optimization issues are also discussed in obtaining accurate induced charge densities. The potential of mean force (PMF) of DNA modelled on a coarsed-grain level inside a nanopore is investigated with and without the inclusion of dielectric effects. Despite the simplicity of the model, the dramatic effect of dielectric inclusions is clearly seen in the observed force profile.
In this work the preparation of organic donor-acceptor thin films was studied. A chamber for organic molecular beam deposition was designed and integrated into an existing deposition system for metallic thin films. Furthermore, the deposition system was extended by a load-lock with integrated bake-out function, a chamber for the deposition of metallic contacts via stencil mask technique and a sputtering chamber. For the sublimation of the organic compounds several effusion cells were designed. The evaporation characteristic and the temperature profile within the cells was studied. Additionally, a simulation program was developed, which calculates the evaporation characteristics of different cell types. The following processes were integrated: evaporation of particles, migration on the cell walls and collisions in the gas phase. It is also possible to consider a temperature gradient within the cell. All processes can be studied separately and their relative strength can be varied. To verify the simulation results several evaporation experiments with different cell types were employed. The thickness profile of the prepared thin films was measured position-dependently. The results are in good agreement with the simulation. Furthermore, the simulation program was extended to the field of electron beam induced deposition (EBID). The second part of this work deals with the preparation and characterization of organic thin films. The focus hereby lies on the charge transfer salt (BEDT-TTF)(TCNQ), which has three known structure variants. Thin films were prepared by different methods of co-evaporation and were studied with optical microscopy, X-ray diffraction and energy dispersive X-ray spectroscopy (EDX).The formation of the monoclinic phase of (BEDT-TTF)(TCNQ) could be shown. As a last part tunnel structures were prepared as first thin film devices and measured in a He4 cryostat.
In this work we investigate phenomenological aspects of an anisotropic quark-gluon plasma. In the first part of this thesis, we formulate phenomenologicalmodels that take into account the momentumspace anisotropy of the system developed during the expansion of the fireball at early-times. By including the proper-time dependence of the parton hard momentum scale, phard(), and the plasma anisotropy parameter, Xi, the proposed models allow us to interpolate from 0+1 pre-equilibrated expansion at early-times to 0+1 ideal hydrodynamics at late times. We study dilepton production as a valuable observable to experimentally determine the isotropization time of the system as well as the degree of anisotropy developed at early-times. We generalize our interpolating models to include the rapidity dependence of phard and consider its impact on forward dileptons. Next, we discuss how to constrain the onset of hydrodynamics by demanding two requirements of the solutions to the equations of motion of viscous hydrodynamics. We show this explicitly for 0+1 dimensional 2nd-order conformal viscous hydrodynamics and find that the initial conditions are non-trivially constrained. Finally, we demonstrate how to match the initial conditions for 0+1 dimensional viscous hydrodynamics from pre-equilibrated expansion. We analyze the dependence of the entropy production on the pre-equilibrium phase and discuss limitations of the standard definitions of the non-equilibrium entropy in kinetic theory.
Chapter 1 contains the general background of our work. We briefly discuss important aspects of quantum chromodynamics (QCD) and introduce the concept of the chiral condensate as an order parameter for the chiral phase transition. Our focus is on the concept of universality and the arguments why the O(4) model should fall into the same universality class as the effective Lagrangian for the order parameter of (massless) two-flavor QCD. Chapter 2 pedagogically explains the CJT formalism and is concerned with the WKB method. In chapter 3 the CJT formalism is then applied to a simple Z(2) symmetric toy model featuring a one-minimum classical potential. As for all other models we are concerned with in this thesis, we study the behavior at nonzero temperature. This is done in 1+3 dimensions as well as in 1+0 dimensions. In the latter case we are able to compare the effective potential at its global minimum (which is minus the pressure) with our result from the WKB approximation. In chapter 4 this program is also carried out for the toy model with a double-well classical potential, which allows for spontaneous symmetry breaking and tunneling. Our major interest however is in the O(2) model with the fields treated as polar coordinates. This model can be regarded as the first step towards the O(4) model in four-dimensional polar coordinates. Although in principle independent, all subjects discussed in this thesis are directly related to questions arising from the investigation of this particular model. In chapter 5 we start from the generating functional in cartesian coordinates and carry out the transition to polar coordinates. Then we are concerned with the question under which circumstances it is allowed to use the same Feynman rules in polar coordinates as in cartesian coordinates. This question turns out to be non-trivial. On the basis of the common Feynman rules we apply the CJT formalism in chapter 6 to the polar O(2) model. The case of 1+0 dimensions was intended to be a toy model on the basis of which one could more easily explore the transition to polar coordinates. However, it turns out that we are faced with an additional complication in this case, the infrared divergence of thermal integrals. This problem requires special attention and motivates the explicit study of a massless field under topological constraints in chapter 8. In chapter 7 we investigate the cartesian O(2) model in 1+0 dimensions. We compare the effective potential at its global minimum calculated in the CJT formalism and via the WKB approximation. Appendix B reviews the derivation of standard thermal integrals in 1+0 and 1+3 dimensions and constitutes the basis for our CJT calculations and the discussion of infrared divergences. In chapter 9 we discuss the so-called path integral collapse and propose a solution of this problem. In chapter 10 we present our conclusions and an outlook. Since we were interested in organizing our work as pedagogical as possible within the narrow scope of a diploma thesis, we decided to make extensive use of appendices. Appendices A-H are intended for students who are not familiar with several important concepts we are concerned with. We will refer to them explicitly to establish the connection between our work and the general context in which it is settled.
The characterization of microscopic properties in correlated low-dimensional materials is a challenging problem due to the effects of dimensionality and the interplay between the many different lattice and electronic degrees of freedom. Competition between these factors gives rise to interesting and exotic magnetic phenomena. An understanding of how these phenomena are driven by these degrees of freedom can be used for rational design of new materials, to control and manipulate these degrees of freedom in order to obtain desired properties. In this work, we study these effects in materials with small exchange interaction between the magnetic ions such as metal-organic and inorganic dilute compounds. We overcome the dfficulties in studying these kind of materials by combining classical and quantum mechanical ab initio methods and many-body theory methods in an effective theoretical approach. To treat metal-organic compounds we elaborate a novel two-step methodology which allows one to include quantum effects while reducing the computational cost. We show that our approach is an effective procedure, leading at each step, to additional insights into the essential features of the phenomena and materials under study. Our investigation is divided into two parts, the first one concerning the exploration of the fundamental physical properties of novel Cu(II) hydroquinone-based compounds. We have studied two representatives of this family, a polymeric system Cu(II)-2,5-bis(pyrazol-1-yl)-1,4-dihydroxybenzene (CuCCP) and a coupled system Cu2S2F6N8O12 (TK91). The second part concerns the study of magnetic phenomena associated with the interplay between different energy scales and dimensionality in zero-, one- and two-dimensional compounds. In the zero-dimensional case, we have performed a comprehensive study of Cu4OCl6L4 with L=diallylcyanamide=NC-N-(CH2-CH=CH2)2 (Cu4OCl6daca4). Interpretations of the magnetic properties for this tetrameric compound have been controversial and inconsistent. From our studies, we conclude that the common models usually applied to this and other representatives in the same family of cluster systems fail to provide a consistent description of their low temperature magnetic properties and we thus postulate that in such systems it is necessary to take into account quantum fluctuations due to possible frustrated behavior. In the one-dimensional case, we studied polymeric Fe(II)-triazole compounds, which are of special relevance due to the possibility of inducing a spin transition between low and high spin state by applying a external perturbation. A long standing problem has been a satisfactory microscopic explanation of this large cooperative phenomenon. A lack of X-ray data has been one mitigating reason for the absence of microscopic studies. In this work, we present a novel approach to the understanding of the microscopic mechanism of spin crossover in such systems and show that in these kind of compounds magnetic exchange between high spin Fe(II) centers plays an important role. The correct description of the underlying physics in many materials is often hindered by the presence of anisotropies. To illustrate this difficulty, we have studied a two dimensional dilute compound K2V3O8 which exhibits an unusual spin reorientation effect when applying magnetic fields. While this effect can be understood when considering anisotropies in the system, it is not sufficient to reproduce experimental observations. Based on our studies of the electronic and magnetic properties in this system, we predict an extra exchange interaction and the presence of an additional magnetic moment at the non-magnetic V site. This sheds a new light into the controversial recent experimental data for the magnetic properties of this material.
Neutron stars are very dense objects. One teaspoon of their material would have a mass of five billion tons. Their gravitational force is so strong that if an object were to fall from just one meter high it would hit the surface of the respective neutron star at two thousand kilometers per second. In such dense bodies, different particles from the ones present in atomic nuclei, the nucleons, can exist. These particles can be hyperons, that contain non-zero strangeness, or broader resonances. There can also be different states of matter inside neutron stars, such as meson condensates and if the density is height enough to deconfine the nucleons, quark matter. As new degrees of freedom appear in the system, different aspects of matter have to be taken into account. The most important of them being the restoration of the chiral symmetry. This symmetry is spontaneously broken, which is a fact related to the presence of a condensate of scalar quark-antiquark pairs, that for this reason is called chiral condensate. This condensate is present at low densities and even in vacuum. It is important to remember at this point that the modern concept of vacuum is far away from emptiness. It is full of virtual particles that are constantly created and annihilated, being their existence allowed by the uncertainty principle. At very high temperature/density, when the composite particles are dissolved into constituents, the chiral consensate vanishes and the chiral symmetry is restored. To explain how and when chiral symmetry is restored in neutron stars we use a model called non-linear sigma model. This is an effective quantum relativistic model that was developed in order to describe systems of hadrons interacting via meson exchange. The model was constructed from symmetry relations, which allow it to be chiral invariant. The first consequence of this invariance is that there are no bare mass terms in the lagrangian density, causing all, or most of the particles masses to come from the interactions with the medium. There are still other interesting features in neutron stars that cannot be found anywhere else in nature. One of them is the high isospin asymmetry. In a normal nucleus, the amount of protons and neutrons is more or less the same. In a neutron star the amount of neutrons is much higher than the protons. The resulting extra energy (called Fermi energy) increases the energy of the system, allowing the star to support more mass against gravitational collapse. As a consequence of that in early stages of the neutron star evolution, when there are still many trapped neutrinos, the proton fraction is higher than in later stages and consequently the maximum mass that the star can support against gravity is smaller. This, between many other features, shows how the microscopic phenomena of the star can reflect into the macroscopic properties. Another important property of neutron stars is charge neutrality. It is a required assumption for stability in neutron stars, but there are others. One example is chemical equilibrium. It means that the number of particles from each kind is not conserved, but they are created and annihilated through specific reactions that happen at the same rate in both directions. Although to calculate microscopic physics of neutron stars the space-time of special relativity, the Minkowski space, can be used, this is not true for the global properties of the star. In this case general relativity has to be used. The solution of Einstein's equations simplified to static, spherical and isotropic stars correspond to the configurations in which the star is in hydrostatic equilibrium. That means that the internal pressure, coming mainly from the Fermi energy of the neutrons, balances the gravity avoiding the collapse. When rotation is included the star becomes more stable, and consequently, can be more massive. The movement also makes it non-spherical, what requires the metric of the star to also be a function of the polar coordinate. Another important feature that has to be taken into account is the dragging of the local inertial frame. It generates centrifugal forces that are not originated in interactions with other bodies, but from the non-rotation of the frame of reference within which observations are made. These modifications are introduced through the Hartle's approximation that solves the problem by applying perturbation theory. In the mean field approximation, the couplings as well as the parameters of the non-linear sigma model are calibrated to reproduce massive neutron stars. The introduction of new degrees of freedom decreases the maximum mass allowed for the neutron star, as they soften the equation of state. In practice, the only baryons present in the star besides the nucleons are the Lambda and Sigma-, in the case in which the baryon octet is included, and Lambda and Delta-,0,+,++, in the case in which the baryon decuplet is included. The leptons are included to ensure charge neutrality. We choose to proceed our calculations including the baryon octet but not the decuplet, in order to avoid uncertainties in the couplings. The couplings of the hyperons were fitted to the depth of their potentials in nuclei. In this case the chiral symmetry restoration can be observed through the behavior of the related order parameter. The symmetry begins to be restored inside neutron stars and the transition is a smooth crossover. Different stages of the neutron star cooling are reproduced taking into account trapped neutrinos, finite temperature and entropy. Finite-temperature calculations include the heat bath of hadronic quasiparticles within the grand canonical potential of the system. Different schemes are considered, with constant temperature, metric dependent temperature and constant entropy. The neutrino chemical potential is introduced by fixing the lepton number in the system, that also controls the amount of electrons and protons (for charge neutrality). The balance between these two features is delicate and influenced mainly by the baryon number conservation. Isolated stars have a fixed number of baryons, which creates a link between different stages of the cooling. The maximum masses allowed in each stage of the cooling process, the one with high entropy and trapped neutrinos, the deleptonized one with high entropy, and the cold one in beta equilibrium. The cooling process is also influenced by constraints related to the rotation of the star. When rotation is included the star becomes more stable, and consequently, can be more massive. The movement also deforms it, requiring the metric of the star to include modifications that are introduced through the use of perturbation theory. The analysis of the first stages of the neutron star, when it is called proto-neutron star, gives certain constraints on the possible rotation frequencies in the colder stages. Instability windows are calculated in which the star can be stable during certain stages but collapses into black holes during the cooling process. In the last part of the work the hadronic SU(3) model is extended to include quark degrees of freedom. A new effective potential to the order parameter for deconfinement, the Polyakov loop, makes the connection between the physics at low chemical potential and hight temperature of the QCD phase diagram with the height chemical potential and low temperature part. This is done through the introduction of a chemical potential dependency on the already temperature dependent potential. Analyzing the effect of both order parameters, the chiral condensate and the Polyakov loop, we can drawn a phase diagram for symmetric as well as for star matter. The diagram contains a crossover region as well as a first order phase transition line. The new couplings and parameters of the model are chosen mainly to fit lattice QCD, including the position of the critical point. Finally, this matter containing different degrees of freedom (depending on which phase of the diagram we are) is used to calculate hybrid star properties.
In the framework of this thesis the intense low energy ion beam transport was investigated. Especially, the beam transport in toroidal magnetic field configurations was discussed, as it may allow the accumulation of high intensive beams in the future. One of the specific tasks is to design an injection system that can be used for the proposed low energy accumulator ring. This thesis regarding beam transport investigations is related to the larger research fields, storage rings used in accelerator physics and non-neutral plasmas. The proposal of building a storage ring with longitudinal guiding magnetic fields was made. Due to natural transversal focussing in magnetic fields it is possible to accumulate very intense charged particle beams, a subject of interest within the physics community. A simulation code (TBT) was written to describe the particle motion in curved segments. Particle in Cell techniques were utilized to simulate a multi particle dynamics. This code allows the user to generate different particle distributions as input parameter. A possibility of reading an external data file was made available so that a measured distribution can be used to compare simulation results with measured ones. A second order cloud in cell method was used to calculate charge density and in turn to solve Poisson’s equation. The circular toroidal coordinate system was used. The drift motion and gyrating motion was proved to be consistent with analytical values. Further simulations were performed to study the self field effects on beam transport. The experiments with single toroidal segments find niche in the work. The experiments were performed to compare the simulation results and gain practical experience. The toroidal segment has similar dimensions (major axis R = 1:3 m, minor axis r = 0:1 m, arc angle 30°) as for a full scale ring design. The main difference lies in the magnetic field strength. The available segments can be operated at room temperature producing 0:6T on axis maximum magnetic field, while for the storage ring design this value is in the range of 5T. The preparatory experiments consisted of building and characterization of the ion source in a first step. Along with the momentum spectrometer and emittance scanner the beam properties were studied. Low mass ion beams He+ and mixed p, H2+, H3+ beams were analyzed. The proton beam consisting of a 48% H+ fraction was extracted regularly and used for further experiments. A moderate beam energy of 10 keV was chosen as operational energy for which 3.08 mA proton beam current was measured. In the second stage, beams were transported through a solenoid and the phase space distribution was measured as a function of the magnetic field for different beam energies. The phase-space as distributions measured in a first stage were simulated backward and then again forward transported through the solenoid. The simulated results were then compared with the measured distribution. The LINTRA transport program was used. The phase-space distribution was further simulated for transport experiments in a toroidal magnetic field. The experiments with a single toroidal segment give basic results necessary to compare the results between transport code (TBT) and measurements. The optical diagnostic provides measurements which can be well compared with the simulated results. A digital camera with a magnetic shield was used to record images in jpeg file format. A subroutine was written to analyze an image file to give the intensity distribution of a given image file. The integrated profile in vertical and horizontal direction was used to calculate the vertical drift and the beam size. The simulated values were in good agreement with the measured ones. The injection system needs most care. The transport program that was used to simulate the beam in the toroid was also used to design the injection system. The injection system with its special field configurations was designed to perform experiments with room temperature segments. The main point to tackle was to smoothly bring the charged particles generated outside the trap into the acceptance of the ring. The designed system consists of two sources, one representing a ring beam and the other one the injection beam. While simulations showed a clear way, how to inject the particle beam via a well positioned solenoid and in combination with a transverse electric field element causing an ExB drift into the main ring acceptance. After construction of these injection elements it will be very important to measure the robustness of such a system with respect to the beam stability- especially of the injection channel.
Perturbation theory for non-abelian gauge theories at finite temperature is plagued by infrared
divergences which are caused by magnetic soft modes ~ g2T, corresponding to gluon fields of
a 3d Yang-Mills theory. While the divergences can be regulated by a dynamically generated
magnetic mass on that scale, the gauge coupling drops out of the effective expansion parameter
requiring summation of all loop orders for the calculation of observables. Some gauge invariant
possibilities to implement such infrared-safe resummations are reviewed. We use a scheme based
on the non-linear sigma model to estimate some of the contributions ~ g6 of the soft magnetic
modes to the QCD pressure through two loops. The NLO contribution amounts to ~ 10% of the
LO, suggestive of a reasonable convergence of the series.
The so-called sign problem of lattice QCD prohibits Monte Carlo simulations at finite baryon
density by means of importance sampling. Over the last few years, methods have been developed
which are able to circumvent this problem as long as the quark chemical potential is m=T <~1.
After a brief review of these methods, their application to a first principles determination of the
QCD phase diagram for small baryon densities is summarised. The location and curvature of the
pseudo-critical line of the quark hardon transition is under control and extrapolations to physical
quark masses and the continuum are feasible in the near future. No definite conclusions can as
yet be drawn regarding the existence of a critical end point, which turns out to be extremely quark
mass and cut-off sensitive. Investigations with different methods on coarse lattices show the lightmass
chiral phase transition to weaken when a chemical potential is switched on. If persisting on
finer lattices, this would imply that there is no chiral critical point or phase transition for physical
QCD. Any critical structure would then be related to physics other than chiral symmetry breaking.
We present a lattice QCD calculation of the heavy-light decay constants fB and fBs performed with Nf = 2 maximally twisted Wilson fermions, at four values of the lattice spacing. The decay constants have been also computed in the static limit and the results are used to interpolate the observables between the charmand the infinite-mass sectors, thus obtaining the value of the decay constants at the physical b quark mass. Our preliminary results are fB = 191(14)MeV, fBs = 243(14)MeV, fBs/ fB = 1.27(5). They are in good agreement with those obtained with a novel approach, recently proposed by our Collaboration (ETMC), based on the use of suitable ratios having an exactly known static limit.