Hysterisis in model spin systems.

Abstract

In this thesis a systematic study of hysteresis in model spin systems is undertaken by constructing the first statistical-mechanical theory wherein spatial fluctuations of the order parameter are incorporated. The theory is used to study the dependence of the shapes and areas of the hysteresis loops on the frequency and amplitude of the applied field and on the temperature and magnetic anisotropy. Chapter 1, the Introduction, is devoted to a survey of experiments on hysteresis in ferromagnets, ferroelectrics, and charge-density waves. The effects of demagnetisation fields, magnetic domains, and magnetic anisotropies on hysteresis loops are presented. Some numerical-simulation studies of hysteresis are reviewed. The amplitude and frequency of the external field drastically affect hysteretic behaviour: though the experimental data are not very systematic, there has been some work on the frequency dependence of hysteresis in ferrites. After a review of the experimental work, a discussion of some phenomenological theories of hysteresis is presented. At the end of the chapter, the task set before a theory of hysteresis is outlined. Chapter 2 consists of a study of hysteresis in an N-component, (?²)² model with O(N) symmetry in three dimensions. The dynamics of the non-conserved order parameter is governed by a Langevin equation. The mean-field and the N?? approximations are investigated. The shapes and areas of the hysteresis loops are studied as functions of the amplitude and frequency of the magnetic field at various temperatures in the ordered phase. There are five qualitatively distinct shapes of the hysteresis loops. A novel scaling behaviour of the area of the loop as a function of the amplitude and the frequency of the field is also found. The response of the above system to pulsed magnetic fields is investigated and related to its hysteretic behaviour. The results obtained are compared with experiments. Chapter 3 is a study of hysteresis in a two-dimensional, nearest-neighbour, ferromagnetic Ising model on a square lattice. The system evolves via the conventional Monte Carlo method, which does not conserve the order parameter (magnetisation). The qualitative behaviour of the shapes of the hysteresis loops is the same as in the O(N) model of Chapter 2. Chapter 4 deals with hysteresis in an N-component, (?²)³ model with O(N) symmetry in three dimensions with dynamics like that of Chapter 2. This chapter is divided into two parts: the first deals with magnetic hysteresis, the second with thermal hysteresis. The analysis of magnetic hysteresis shows, in addition to the shapes described in Chapter 2, wasp-waisted hysteresis loops for certain values of the parameters of the theory. This is similar to the hysteretic behaviour found in some ferroelectrics like SbSI and BaTiO?. The areas of the hysteresis loops are also computed as functions of the amplitude of the applied magnetic field. The (?²)³ theory exhibits a temperature-induced first-order phase transition. As the temperature is cycled periodically across the first-order boundary, asymmetric hysteresis loops are obtained. In the last chapter of the thesis we study the hysteretic behaviour of the uniaxial anisotropic (?²)² theory. The order parameter ? obeys Langevin dynamics. This chapter is principally an analysis of three approximations that have been used to derive expressions for the static and dynamic n-point averages of the order parameter. The three approximations used are (i) a Gaussian approximation, (ii) a self-consistent Gaussian approximation, and (iii) a modified self-consistent Gaussian approximation. We present a formal study of these approximations and exhibit hysteresis loops obtained within each of these approximations.