[...]... as soft magnetism, permanent magnetism, magnetic recording, oxides, spin electronics, magnetic alloys, micromagnetic simulations, first-principle calculations, and theory of phase transitions There is considerable overlap among some of these subfields, but there are also big gaps, and one aim of this book is to bridge these gaps As the title suggests, emphasis is on simple models of magnetism Toy models. .. turned magnetism into a field of intense research The modeling of magnetic phenomena and materials is a crucial aspect of this research To provide an introduction to magnetism and to magnetic modeling, we start with some well-known and extremely simple models In fact, some aspects of the models are simplistic rather than simple, and this introductory chapter may also be called, What is wrong with the simplest... may also be called, What is wrong with the simplest models of magnetism? However, in spite of their very limited applicability, these models are not useless First, they hint at typical problems encountered in magnetism and provide a basis and motivation for the models in the main chapters of the book Second, even the simplest models describe a piece of reality if used in an adequate context 1.1 Field... an asterisk It is often intriguing to apply simple models to complicated magnetic, thereby xvi Preface investigating both the model and the system If an unreasonable model assumption causes a simple model to fail, then it also leads to the failure of complicated models and numerical calculations Note that the focus of the book is on specific models, not on the methodology of scientific models In some cases,... character of magnetism, corresponding to the absence of magnetic monopoles, and (e) currents as one source of magnetic fields Note that the Earth has magnetic poles that change sign every few 100,000 years At present, the South Pole is near the geographical North Pole, so that the vectors of the magnetic field H point towards the Arctic 2 Introduction: The simplest models of magnetism the hard or soft character... preparation and writing of this work Lincoln, November 2006 1 Introduction: The simplest models of magnetism During the last 100 years, magnetism has made a giant step forward By the second half of the nineteenth century, Maxwell’s equations had established the relation between different electromagnetic fields, and scientists and engineers were aware of the dipolar character of magnetostatic forces and... velocity of light In fact, the spin is of relativistic origin and can be considered as a kind of a magnetic analog of the rest energy me c2 in the energy expression E = me c2 + 1 me v 2 + O(1/c4 )) As the rest energy, the spin remains 2 nonzero for v = 0, while both the kinetic energy 1 me v 2 and the orbital moment (1.5) 2 6 Introduction: The simplest models of magnetism (a) (b) mL ms Fig 1.5 Origin of. .. (1.11), and the thermal average of the magnetization of the i-th spin is obtained by summation over all spin configurations s = 1 Z exp µ −Eµ kB T (1.12) In spite of the simple character of (1.11–12), the solution of the Ising model is a complicated problem The main reason is the large number of involved spin configurations, µ = 1, 2, 3, 2N , where N is the number of spins In Chapter 5 we will see... corresponding to an average magnetization of 0.09 mT 3 Direction of circulation on a hysteresis loop Is the circulation on a typical ferromagnetic hysteresis loop clockwise or anticlockwise? 4 Paramagnetic moment of single spin Calculate the moment of a single spin in a magnetic field of 0.5 T at 4.2 K and at 300 K 5 Susceptibility of iron The susceptibility of a soft-magnetic iron piece is equal to χ =... intra-atomic exchange, which is of the order of 1 eV, so that atomic moments at Tc remain close to their zero-temperature values and typical magnetization processes in solids are caused by magnetization rotations (a) (b) Fig 2.1 Basic types of magnetic order: (a) ferromagnetism and (b) antiferromagnetism Each arrow represents one atomic spin 16 Models of exchange In the simplest case, it is sufficient .