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Please read the abstract in the section 00front of this document / Dissertation (MSc (Computer Science))--University of Pretoria, 2006. / Computer Science / unrestricted

Predictions of heavy quark parameters are an integral component of precision tests of the Standard Model of particle physics. Experimental measurements of electroweak processes involving heavy hadrons provide stringent tests of Cabibbo-Kobayashi-Maskawa (CKM) matrix unitarity and serve as a probe of new physics. Hadronic matrix elements parameterise the strong dynamics of these interactions and these matrix elements must be calculated nonperturbatively. Lattice quantum chromodynamics (QCD) provides the framework for nonperturbative calculations of QCD processes. Current lattices are too coarse to directly simulate b quarks. Therefore an effective theory, nonrelativistic QCD (NRQCD), is used to discretise the heavy quarks. High precision simulations are required so systematic uncertainties are removed by improving the NRQCD action. Precise simulations also require improved sea quark actions, such as the highly-improved staggered quark (HISQ) action. The renormalisation parameters of these actions cannot be feasibly determined by hand and thus automated procedures have been developed. In this dissertation I apply automated lattice pertubartion theory to a number of heavy quark calculations. I first review the fundamentals of lattice QCD and the construction of lattice NRQCD. I then motivate and discuss lattice perturbation theory in detail, focussing on the tools and techniques that I use in this dissertation. I calculate the two-loop tadpole improvement factors for improved gluons with improved light quarks. I then compute the renormalisation parameters of NRQCD. I use a mix of analytic and numerical methods to extract the one-loop radiative corrections to the higher order kinetic operators in the NRQCD action. I then employ a fully automated procedure to calculate the heavy quark energy shift at two-loops. I use this result to extract a new prediction of the mass of the b quark from lattice NRQCD simulationsby the HPQCD collaboration. I also review the calculation of the radiative corrections to the chromo-magnetic operator in the NRQCD action. This computation is the first outcome of our implementation of background field gauge for automated lattice perturbation theory. Finally, I calculate the heavy-light currents for highly-improved NRQCD heavy quarks with massless HISQ light quarks and discuss the application of these results to nonperturbative studies by the HPQCD collaboration.

Hadrons containing heavy quarks, in particular b quarks, play an important role in high energy physics. Measurements of their electroweak interactions are used to test the Standard Model and search for new physics. For the comparison of experimental results with theoretical predictions, nonperturbative calculations of hadronic matrix elements within the theory of quantum chromodymanics are required. Such calculations can be performed from first principles by formulating QCD on a Euclidean spacetime grid and computing the path integral numerically. Including b quarks in lattice QCD calculations requires special techniques as the lattice spacing in present computations usually can not be chosen fine enough to resolve their Compton wavelength. In this work, improved nonrelativistic lattice actions for heavy quarks are used to perform calculations of the bottom hadron mass spectrum and of form factors for heavy-to-light decays. In heavy-to-light decays, additional complications arise at high recoil, when the momentum of the light meson reaches a magnitude comparable to the cutoff imposed by the lattice. Discretisation errors at high recoil can be reduced by working in a frame of reference where the heavy and light mesons move in opposite directions. Using a formalism referred to as moving nonrelativistic QCD (mNRQCD), the nonrelativistic expansion for the heavy quark can be performed around a state with an arbitrary velocity. This dissertation begins with a review of the fundamentals of lattice QCD. Then, the construction of effective Lagrangians for heavy quarks in the continuum and on the lattice is discussed in detail. A highly improved lattice mNRQCD action is derived and its effectiveness is demonstrated by nonperturbative tests involving both heavy-heavy and heavy-light mesons at several frame velocities. This mNRQCD action is then used in combination with a staggered action for the light quarks to calculate hadronic matrix elements relevant for rare B decays, including B --> K* gamma and B --> K l l. A major contribution to the uncertainty of the results also comes from statistical errors. The effectiveness of random-wall sources to reduce these errors is studied. As another application of a nonrelativistic heavy quark action, the spectrum of bottomonium is calculated and masses of several bottom baryons are predicted. In these computations, the light quarks are implemented with a domain wall action.

This thesis describes the development of a Lattice Boltzmann (LB) model for a binary gas mixture. Specifically, channel flow driven by a density gradient with diffusion slip occurring at the wall is studied in depth. The first part of this thesis sets the foundation for the multi-component model used in the subsequent chapters. Commonly used single component LB methods use a non-physical equation of state, in which the relationship between pressure and density varies according to the scaling used. This is fundamentally unsuitable for extension to multi-component systems containing gases of differing molecular masses that are modelled with the ideal gas equation of state. Also, existing methods for implementing boundary conditions are unsuitable for extending to novel boundary conditions, such as diffusion slip. Therefore, a new single component LB derivation and a new method for implementing boundary conditions are developed, and validated against Poiseuille flow. However, including a physical equation of state reduces stability and time accuracy, leading to longer computational times, compared with 'incompressible' LB methods. The new method of analysing LB boundary conditions successfully explains observations from other commonly used schemes, such as the slip velocity associated with 'bounce-back'.The new model developed for multi-component gases avoids the pitfalls of some other LB models, a single computational grid is shared by all the species and the diffusivity is independent of the viscosity. The Navier-Stokes equation for the mixture and the Stefan-Maxwell diffusion equation are both recovered by the model. However, the species momentum equations are not recovered correctly and this can lead to instability. Diffusion slip, the non-zero velocity of a gas mixture at a wall parallel to a concentration gradient, is successfully modelled and validated against a simple one-dimensional model for channel flow. To increase the accuracy of the scheme a second order numerical implementation is needed. This can be achieved using a variable transformation method which does not result in an increase in computational time. Simulations were carried out on hydrogen and water diffusion through a narrow channel, with varying total pressure and concentration gradients. For a given value of the species mass flux ratio, the total pressure gradient was dependent on the species concentration gradients. These results may be applicable to fuel cells where the species mass flux ratio is determined by a chemical reaction and the species have opposing velocities. In this case the total pressure gradient is low and the cross-channel average mass flux of hydrogen is independent of the channel width. Finally, solutions for a binary Stefan tube problem were investigated, in which the boundary at one end of a channel is permeable to hydrogen but not water. The water has no total mass flux along the channel but circulates due to the slip velocity at the wall. The cross-channel average mass flux of the hydrogen along the channel increases with larger channel widths. A fuel cell using a mixture of gases, one being inert, will experience similar circulation phenomena and, importantly, the width of the pores will affect performance. This thesis essentially proves the viability of LB models to simulate multi-component gases with diffusion slip boundaries, and identifies the many areas in which improvements could be made.

Selective excitation pulsed NMRON, CW-NMRON and Thermal NMR methods have been used to study the low temperature ⁵⁴Mn nuclear spin-lattice relaxation mechanisms in magnetic insulators. The selective single and double quantum excitation sequences have been used for the first time in NMRON to obtain single and double quantum rotation patterns, Free Induction Decays, Hahn spin echoes and pulsed T₁ measurements. Two insulating magnets have been studied; MnCl₂.4H₂O and Mn(COOCH₃ )₂ .4H₂O. In the ⁵⁴Mn-MnCl₂ .4H₂O system the temperature dependence of the ⁵⁴Mn spin-lattice relaxation time at zero field was measured between 35 mK and 90 mK and it was found that the dominant relaxation process between 65 mK and 90 mK is an electronic magnon Raman process and below 65 mK a direct relaxation process dominates. Single and double quantum Free Induction Decays and Hahn spin echoes have been used to determine the magnitude and nature of the spin-spin relaxation mechanism for ⁵⁴Mn oriented in MnCl₂.4H₂O at zero applied field. NMRON was observed for the first time in the paramagnetic phase of MnCl₂.4H₂O. The resonance lines are inhomogeneously broadened and 300 kHz wide. A value of <⁵⁴AS>/h=-513.6(3) MHz has been determined for the paramagnetic phase hyperfine coupling constant, and this value has been used to determine the zero point spin deviation of the antiferromagnetic phase. The field and temperature dependence of the ⁵⁴Mn T₁ was measured for values of field above the spin flop paramagnetic phase transition and a field dependent T₁ minimum was discovered at Ba=2.64 T. For the ⁵⁴Mn-Mn(COOCH₃) .4H₂O system two ⁵⁴Mn resonances have been observed and the value of the hyper fine coupling constants for the two sites were found to be <⁵⁴AS>/h=-435 (1) MHz for the ⁵⁴Mn1 site and <⁵⁴AS>/h=-478(1) MHz for the ⁵⁴Mn2 site. The high field spin-lattice relaxation behavior has also been investigated and a T₁ minimum at Ba =2.74 T analogous to that observed in MnCl₂ .4H₂O was discovered. A Hahn echo study of the low field single quantum spin-spin relaxation processes has been performed and anomalous behavior of the spin echo amplitude revealed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Spiral waves occur in several natural phenomena, including reaction fronts in two-dimension excitable media. In this thesis we attempt to characterize the motion of the spiral tip of a rigidly rotating wave and a linearly travelling wave in the context of a lattice perturbation. This system can be reduced to its center manifold, which allows us to describe the system as ordinary differential equations. This in turn means dynamical systems methods are appropriate to describe the motion of the tip. It is in such a context that we work on spiral waves. We study perturbed rotating waves and travelling waves using standard techniques from dynamical systems theory.

The use of accelerator technology, particularly Graphics Processing Units (GPUs), for scientific computing has increased greatly over the last decade. While this technology allows larger and more complicated problems to be solved faster than before it also presents another opportunity: the real-time and interactive solution of problems. This work aims to investigate the progress that GPU technology has made towards allowing fluid-structure interaction (FSI) problems to be solved in real-time, and to facilitate user interaction with such a solver. A mesoscopic scale fluid flow solver is implemented on third generation nVidia ‘Kepler’ GPUs in two and three dimensions, and its performance studied and compared with existing literature. Following careful optimisation the solvers are found to be at least as efficient as existing work, reaching peak efficiencies of 93% compared with theoretical values. These solvers are then coupled with a novel immersed boundary method, allowing boundaries defined at arbitrary coordinates to interact with the structured fluid domain through a set of singular forces. The limiting factor of the performance of this method is found to be the integration of forces and velocities over the fluid and boundaries; the arbitrary location of boundary markers makes the memory accesses during these integrations largely random, leading to poor utilisation of the available memory bandwidth. In sample cases, the efficiency of the method is found to be as low as 2.7%, although in most scenarios this inefficiency is masked by the fact that the time taken to evolve the fluid flow dominates the overall execution time of the solver. Finally, techniques to visualise the fluid flow in-situ are implemented, and used to allow user interaction with the solvers. Initially this is achieved via keyboard and mouse to control the fluid properties and create boundaries within the fluid, and later by using an image based depth sensor to import real world geometry into the fluid. The work concludes that, for 2D problems, real-time interactive FSI solvers can be implemented on a single laptop-based GPU. In 3D the memory (both size and bandwidth) of the GPU limits the solver to relatively simple cases. Recommendations for future work to allow larger and more complicated test cases to be solved in real-time are then made to complete the work.

Nuclear magnetic resonance experiments on impurity atoms in a ferromagnetic host have shown that the measured spin-lattice relaxation time of those nuclei located in domains is strongly dependent on the degree of magnetic saturation of the host material (1, 2, 3). The relaxation time increases as the applied magnetic field is increased and reaches a constant value for a magnetically saturated specimen. Wall nuclei show a much shorter relaxation time than those in the bulk. This fact, together with the increased number of walls present in a magnetically non-saturated specimen could explain the observed field-dependent decrease of the relaxation time if an increasing fraction of wall nuclei is observed. Nuclei located in walls experience a much larger enhancement than those in domains. Therefore special techniques have to be applied to exclusively observe nuclei located in the bulk (1, 4). For this reason some uncertainty exists in the interpretation of the nuclear magnetic resonance measurements. The theory of the spin-lattice relaxation in ferromagnetic metals (5) gives an estimate for the relaxation rate observed in magnetically saturated specimens. No field dependence the relaxation time is predicted. Partly due to the uncertainty in the NMR results, this theoretical problem has received little attention so far. We therefore have employed low temperature nuclear orientation which predominantly measures bulk nuclei to investigate this problem. In most of these experiments the combined technique of nuclear orientation and nuclear magnetic resonance (NMR/ON) (6) has been applied td prepare the initial state from which the relaxation takes place. Some experiments have also been performed by an entirely non-resonant technique (7). Our experimental results on ⁶⁰Co-Fe, ⁵⁴Mn-Fe and ⁵⁴Mn-Ni clearly confirm the field dependence of the relaxation time observed in nuclear magnetic resonance experiments (8). Thus the need for a detailed theoretical study is evident. Performing an NMR/ON experiment the resonance is detected by a change in the observed y-ray intensity. Resonance lines for ⁶⁰Co-Fe, ⁵⁴Mn-Fe and ⁵⁴Mn-Ni have been recorded. We have for the first time observed that their full widths at half maximum show a strong field dependence. An explanation in terms of a local distribution in the demagnetizing field is offered. We have also measured the intensity of the resonance line as a function of the applied field. An estimate shows that this is inadequately explained in terms of the expected field dependence of the enhancement factor. The distribution of hyperfine fields has never before been studied by NMR/ON. We have employed this technique successfully to investigate an alloy of one atomic percent ⁵⁹Co-Fe which has been doped with a small amount of ⁶⁰Co. A strong, well resolved satellite line of the impurity nuclei is observed. These data are interpreted in terms of the effect of near neighbor impurity nuclei on the hyperfine field (9, 10). We have computed a theoretical curve based on parameters given in the literature (10). This provides a moderately good fit for most portions of our spectra. This pilot study demonstrates that NMR/ON is indeed a valuable tool for the investigation of hyperfine field distributions. The advantages over nuclear magnetic resonance studies are that essentially only bulk as compared to wall nuclei are studied and that the sensitivity is independent of the alloy concentration. Based partially on our own data we present a short discussion of the question whether a spin temperature is maintained by the impurity nuclei during relaxation. Finally we offer a comparison between relaxation data measured by NMR/ON and other nuclear orientation techniques (11). For ⁶⁰Co-Fe the relaxation times measured by NMR/ON are found to be almost 50% longer than those measured by techniques in which the initial condition is known. This discrepancy is generally attributed to the incomplete knowledge of the initial conditions when the NMR/ON technique is employed. We have computed theoretical relaxation curves for a number of initial conditions and find that the resulting spread in relaxation time for those curves that allow a good fit to the measured curve is larger than the difference obtained from the experiments. Thus our model indeed could explain the observed discrepancy. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The purpose of this paper is to present a study of partly ordered sets. It includes a rigorous development of relations based on the notion of a relation as a set, lattices, and theorems concerning the lattice of subgroups of a group.