# 3. Domain Theory (248 pages)

## 3.1 The Purpose of Domain Theory

Domain theory is based on the classical paper:

[22] L.D. Landau, E. Lifshitz: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8, 153-169 (1935)

## 3.2 Energetics of a Ferromagnet

Fig. 3.11 One contribution to the energetics of a ferromagnet is magnetostrictive self energy. It may be due to angular defects around domain wall junctions as they are compiled here for cubic materials

## 3.3 The Origin of Domains

Fig. 3.28 Sandpiles on arbitrarily shaped boards can be used to demonstrate stray-field-free domain patterns either in thin film elements or in prismatic bodies with a preferred plane of magnetization. The constant gradient vector down the slope of a sandpile represents - after a rotation by 90 degrees - the unit magnetization vector in the domains. Smooth ridges can be interpreted as representing continuous domain walls

## 3.4 Phase Theory of Domains in Large Samples

Fig. 3.40 Field map indicating the equilibrium magnetization phases in the environment of the [111]-direction of iron. For a given absolute value of the field (h = 0.788) the lowest-energy magnetization direction is computed as a function of the field direction near [111]. In addition to states derived from the easy [100] axes an anomalous phase connected to the hard axis [111] is found for a narrow angular field range. The outlined phase boundaries indicate possible multi-domain states

## 3.5 Small-Particle Switching

Fig. 3.52 A continuous vortex nucleation in a low-anisotropy cube-shaped particle calculated by numerical micromagnetics. This second-order phase transition represents one of the possible transition types discussed in this chapter

## 3.6 Domain Walls

Fig. 3.82 A classical Bloch wall in a soft magnetic material is strongly modified when meeting a surface. The contour lines represent constant values of the x-component of the magnetization, which would be zero in classical Bloch walls. Here it outlines the extended surface vortex ("cap") of such walls

## 3.7 Theoretical Analysis of Characteristic Domains

Fig. 3.139 The "effective closure depth" of a multiaxially branched pattern is a measure of the degree of branching. The graph demonstrates that branched domain structures with a small closure depth can be realized more efficiently - that is with less wall energy - using 3D structures (bold continuous line) rather than 2D patterns (isolated calculation points)

## 3.8 Résumé

This is our conclusion when considering the role of domain theory:

Domain theory and micromagnetic analysis should in general not be used to replace experimental observation by theoretical prediction. Rather, theory should help in the analysis and interpretation of observations, providing guidelines for the construction of three-dimensional models based on surface observations. If more than one such model exist, domain theory can decide which model has the lower energy. Every interpretation of experimental observations should be tested for plausibility with the help of domain theory. But if the need for experimental confirmation of the theory is felt, it is wise to focus on details such as certain continuously variable critical angles or lengths in a confirmed model rather than on the model as a whole.

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