Physics of Magnetism

110 CHAPTER 12. PERMANENT MAGNETS

Let us consider the case that a steadily increasing magnetic field is applied in a direction

opposite to the direction in which a perfect single crystal of a magnetic compound of

hexagonal or tetragonal symmetry is magnetized (i.e., opposite to the easy magnetization

direction). One would expect all atomic moments to reverse their direction by a process

of uniform rotation only when the applied field becomes equal in size to the anisotropy

field (see Chapter 11). However, in reality, such high coercivities are

seldom encountered. Most permanent-magnet materials show the magnetization reversal

already at field strengths that are only a small fraction (10–15%) of the value of

The reason for this comparatively easy magnetization reversal is the existence of

magnetic-domain structures. Magnetic particles of sufficiently large size will generally not

be uniformly magnetized but rather be composed of magnetic domains that are mutually

separated by domain walls or Bloch walls. A schematic representation of such a wall is

given in Fig. 12.3.1. The magnetizations in adjacent domains point into opposite directions

in order to reduce the magnetostatic energy. The magnetization in the wall between two

domains gradually changes from the one preferred magnetization direction to the other. The

thickness of the wall is determined by the relative strengths of the anisotropy energy and the

exchange energy. The former tends to reduce the wall thickness, the latter tends to increase

it. This may be seen from the argument given below.

According to Eq. (4.1.2), one may obtain the exchange energy between neighboring

spins from the formula

where is the angle between the directions of the spin-angular-momentum vectors of atom
i and its neighbors j. Generally, the widths of domain walls involve many lattice spacings,
sometimes more than a hundred. For this reason, the angle between two neighboring
spins in the wall is very small, so that one may use the approximation
The variable part of the exchange energy for a row of atoms across the wall can then be