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Abstract
We introduce the thesis with a discussion of fundemental magnetism, the motivation for our studies and previous theories in the field. Chapter 2 develops the athermal equation of motion for magnetisation first given by Landau and Lifshitz in 1935 and later improved by Gilbert in 1955. We discuss the analysis of the equations given by Kikuchi in 1956 and the points raised by Mallinson in 1986. Simple numerical integration schemes are considered and single spin solutions are presented. Chapter 3 is devoted to the development of our finite element model of magnetisation dynamics. We give a variational formulation of the Gilbert equation with the effective magnetic field taken to be the functional derivative of the systems free energy as de-rived in chapter 1. Our initial method follows essentially from a paper by Yang and Fredkin in 1998 with some minor modifications. The magneto-static calculation is a three dimensional generalization of the method given by Ridley et al. in 1999. Technical aspects such as sparse matrix storage and the solution of systems of linear algebraic equations are discussed in brief. Some basic examples are given in chapter 4 followed by an analysis of magnetisation reversal in individual cobalt grains. The important differences between grains of longitudinal and perpendicular orientation are established. Chapter 5 describes an attempt to model an open magnetic region using periodic boundary conditions and chapter 6 is concerned with the use of a time-stepping method which naturally conserves the magnitude of magnetisation due to the int rinsic quadratic invariance of the numerical scheme; improved numerical stability is established. The self consistency condition given by Albequerque et al. in 2001 is then used to show that the numerical error of our method may be bounded arbitrarily. Chapter 7 is devoted to the consideration of finite temperature magnetisation dynamics. We use the simple energy-barrier model to highlight the importance of thermal fluctuations, giving motivation for a more rigorous analysis. We then derive the stochastic Langevin-Gilbert equation and present single-spin solutions illustrating the superparamagnetic transition of cobalt grains as well as the temperature dependence of coercivity. Our finite element model is then applied to the finite temperature case. In chapter 8 we present some applications of our model. The effects of physical microstructure on magnetisation reversal are investigated as well as the role of energy dissipation and thermal fluctuations. We conclude with an evaluation of both our computational model a nd the results we have obtained. Further model development is discussed together with some as yet unexplored applications.
