Static Micromagnetics

The basic concept of micromagnetism is to replace the atomic magnetic moments by a continuous function of position. In a continuum theory the local direction of the magnetic moments may be described by the magnetic polarization vector 

The competitive effects of the micromagnetic energy contributions upon minimization determine the equilibrium distribution of the magnetization. 

In the approach of micromagnetism the quantum mechanical spin operators are substituted by classical vectors, S. Thus, the exchange energy in the Heisenberg model [34] 

The assumption of a constant exchange integral is justified, because experiments show only small differences when measuring along different crystallographic directions. When the angle between S and S is small Eq. (16) can be written 

The micromagnetic exchange constant (exchange stiffness) A [J/m] is proportional to the exchange integral J. For a cubic lattice having one nonequivalent atom, A is given by [36] 

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This chapter investigates how intrinsic and extrinsic properties are affected by nanostructuring. Emphasis is on model calculations and analytical approximations, as contrasted to Chapter 2 by Kashyap et al. which focuses on numerical calculations, and Ch. 4 by Schrefl et al., where emphasis is on micromagnetic models and simulations. Section 2 is devoted to static properties, whereas section 3 is concerned with magnetization dynamics. Finally, section 4 presents a number of case studies. 

Many micromagnetic investigations focus on the optimization of the magnet s nanostructure, in order to tailor its magnetic properties for specific applications. For these studies dynamic effects are less important and static energy minimization techniques can be applied. 

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