Studies in nuclear reactor dynamics : I. The accuracy of point kinetics. II. The effect of delayed neutrons on the spectrum of the group-diffusion operator

This thesis is a theoretical work on the space-time dynamic behavior of a nuclear reactor without feedback. Diffusion theory with G-energy groups is used.

In the first part the accuracy of the point kinetics (lumped-parameter description) model is examined. The fundamental approximation of this model is the splitting of the neutron density into a product of a known function of space and an unknown function of time; then the properties of the system can be averaged in space through the use of appropriate weighting functions; as a result a set of ordinary differential equations is obtained for the description of time behavior. It is clear that changes of the shape of the neutron-density distribution due to space-dependent perturbations are neglected. This results to an error in the eigenvalues and it is to this error that bounds are derived. This is done by using the method of weighted residuals to reduce the original eigenvalue problem to that of a real asymmetric matrix. Then Gershgorin-type theorems .are used to find discs in the complex plane in which the eigenvalues are contained. The radii of the discs depend on the perturbation in a simple manner.

In the second part the effect of delayed neutrons on the eigenvalues of the group-diffusion operator is examined. The delayed neutrons cause a shifting of the prompt-neutron eigenvalue s and the appearance of the delayed eigenvalues. Using a simple perturbation method this shifting is calculated and the delayed eigenvalues are predicted with good accuracy.

Bifurcation in a reaction diffusion system

The bifurcation and nonlinear stability properties of the Meinhardt-Gierer model for biochemical pattern formation are studied. Analyses are carried out in parameter ranges where the linearized system about a trivial solution loses stability through one to three eigenfunctions, yielding both time independent and periodic final states. Solution branches are obtained that exhibit secondary bifurcation and imperfection sensitivity and that appear, disappear, or detach themselves from other branches.

Diffusion in glassy polymers

Fluid diffusion in glassy polymers proceeds in ways that are not explained by the standard diffusion model. Although the reasons for the anomalous effects are not known, much of the observed behavior is attributed to the long times that polymers below their glass transition temperature take to adjust to changes in their condition. The slow internal relaxations of the polymer chains ensure that the material properties are history-dependent, and also allow both local inhomogeneities and differential swelling to occur. Two models are developed in this thesis with the intent of accounting for these effects in the diffusion process.

In Part I, a model is developed to account for both the history dependence of the glassy polymer, and the dual sorption which occurs when gas molecules are immobilized by the local heterogeneities. A preliminary study of a special case of this model is conducted, showing the existence of travelling wave solutions and using perturbation techniques to investigate the effect of generalized diffusion mechanisms on their form. An integral averaging method is used to estimate the penetrant front position.

In Part II, a model is developed for particle diffusion along with displacements in isotropic viscoelastic materials. The nonlinear dependence of the materials on the fluid concentration is taken into account, while pure displacements are assumed to remain in the range of linear viscoelasticity. A fairly general model is obtained for three-dimensional irrotational movements, with the development of the model being based on the assumptions of irreversible thermodynamics. With the help of some dimensional analysis, this model is simplified to a version which is proposed to be studied for Case II behavior.

Quantum error correction with biased noise

Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security.

At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level.

In this thesis, we look at how additional information about the structure of noise, or "noise bias," can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations.

In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction.

In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.

I. Reactively sputtered Ti-Si-N thin films for diffusion barrier applications. II. Oxidation, diffusion and crystallization of an amorphous Zr_(60)Al_(15)Ni_(25) alloy.

Films of Ti-Si-N obtained by reactively sputtering a TiSi_2, a Ti_5Si_3, or a Ti_3Si target are either amorphous or nanocrystalline in structure. The atomic density of some films exceeds 10^23 at./cm^3. The room-temperature resistivity of the films increases with the Si and the N content. A thermal treatment in vacuum at 700 °C for 1 hour decreases the resistivity of the Ti-rich films deposited from the Ti_5Si_3 or the Ti_3Si target, but increases that of the Si-rich films deposited from the TiSi_2 target when the nitrogen content exceeds about 30 at. %.

Ti_(34)Si_(23)N_(43) deposited from the Ti_5Si_3 target is an excellent diffusion barrier between Si and Cu. This film is a mixture of nanocrystalline TiN and amorphous SiN_x. Resistivity measurement from 80 K to 1073 K reveals that this film is electrically semiconductor-like as-deposited, and that it becomes metal-like after an hour annealing at 1000 °C in vacuum. A film of about 100 nm thick, with a resistivity of 660 µΩcm, maintains the stability of Si n+p shallow junction diodes with a 400 nm Cu overlayer up to 850 °C upon 30 min vacuum annealing. When used between Si and Al, the maximum temperature of stability is 550 °C for 30 min. This film can be etched in a CF_4/O_2 plasma.

The amorphous ternary metallic alloy Zr_(60)Al_(15)Ni_(25) was oxidized in dry oxygen in the temperature range 310 °C to 410 °C. Rutherford backscattering and cross-sectional transmission electron microscopy studies suggest that during this treatment an amorphous layer of zirconium-aluminum-oxide is formed at the surface. Nickel is depleted from the oxide and enriched in the amorphous alloy below the oxide/alloy interface. The oxide layer thickness grows parabolically with the annealing duration, with a transport constant of 2.8x10^(-5) m^2/s x exp(-1.7 eV/kT). The oxidation rate is most likely controlled by the Ni diffusion in the amorphous alloy.

At later stages of the oxidation process, precipitates of nanocrystalline ZrO_2 appear in the oxide near the interface. Finally, two intermetallic phases nucleate and grow simultaneously in the alloy, one at the interface and one within the alloy.

Energy inequalities and error estimates for axisymmetric torsion of thin elastic shells of revolution

The problem motivating this investigation is that of pure axisymmetric torsion of an elastic shell of revolution. The analysis is carried out within the framework of the three-dimensional linear theory of elastic equilibrium for homogeneous, isotropic solids. The objective is the rigorous estimation of errors involved in the use of approximations based on thin shell theory.

The underlying boundary value problem is one of Neumann type for a second order elliptic operator. A systematic procedure for constructing pointwise estimates for the solution and its first derivatives is given for a general class of second-order elliptic boundary-value problems which includes the torsion problem as a special case.

The method used here rests on the construction of “energy inequalities” and on the subsequent deduction of pointwise estimates from the energy inequalities. This method removes certain drawbacks characteristic of pointwise estimates derived in some investigations of related areas.

Special interest is directed towards thin shells of constant thickness. The method enables us to estimate the error involved in a stress analysis in which the exact solution is replaced by an approximate one, and thus provides us with a means of assessing the quality of approximate solutions for axisymmetric torsion of thin shells.

Finally, the results of the present study are applied to the stress analysis of a circular cylindrical shell, and the quality of stress estimates derived here and those from a previous related publication are discussed.