Fig. 2.11. Subdivision of a ferromagnetic grain into domains, (a) Single-domain structure with widely separated + and - poles, (b) Two-domain structure with less pole separation, (c) Four-domain state, (d) Two-domain state with closure domains. Redrawn after Dunlop and Özdemir (1997), with the permission of Cambridge University Press.
For magnetite the wall width has been calculated to be 0.28 |am or about 300 lattice spacings (Dunlop and Özdemir, 1997) and has been determined experimentally as 0.18 |am (Moskowitz et al., 1988). The corresponding wall energy is approximately 10"3 Jm"2.
The grain shown in Fig. 2.11a, where no domains occur, is referred to as a single-domain grain (SD grain) and at some critical size the grain will subdivide into two or more domains to form a multidomain grain (MD grain). Note that it is usually energetically more favorable to subdivide the grain by a wall parallel to the long axis of the grain. Another way of reducing or eliminating the magnetostatic energy is by adding closure domains to the two-domain state (Fig. 2.1 Id). There are now no surface poles and no external field and the wall energy is also reduced when compared with that of the four-domain state. However, the closure domains generate additional magnetoelastic energy as described later. The body and closure domains in Fig. 2.1 Id are strained and both are shorter than they would be in isolation.
In addition to the magnetostatic energy of a grain, arising from its shape anisotropy (i.e., it is easier to magnetize along the long axis than in other directions), there is energy from magnetocrystalline anisotropy that arises because it is easier for the domain magnetizations to lie along certain crystallographic axes than along others. For example, in a magnetite crystal the easy direction of magnetization is along the  axis and the difficult or hard direction is along the  axis. The magnetocrystalline anisotropy energy is the difference in magnetization energy between the hard and easy directions. Obviously, the contribution to the total energy will be a minimum when the various domain magnetizations all lie along easy directions.
The magnetization of a ferromagnetic or ferrimagnetic crystal is usually accompanied by a spontaneous change in the dimensions of the crystal, giving rise to magnetostriction, which is due to strain arising from magnetic interaction along the atoms forming the crystal lattice. In addition, the presence of some impurity in the crystal lattice, or the presence of dislocations, will produce internal stress that then acts as a barrier to changes in magnetization. Strain occurs when closure domains are formed as in Fig. 2.1 Id. The strain dependence of crystalline anisotropy is termed magnetoelasticity giving rise to magnetoelastic energy or magnetostrictive strain energy.
The theory of the magnetization of an assemblage of SD particles is due to Neel (1949, 1955). Although it appears to have wide applicability, it is based on the assumption that the grains are identical and that there are no grain interactions. These assumptions can obviously lead to shortcomings in various aspects of the theory, but the essential features of the behavior of magnetic grains over the geological time scale can be adequately described in terms of this simple theory. Consideration of grain interactions has led to the Preisach-Neel theory, which has been developed by Dunlop and West (1969), but discussion of the details of this theory is beyond the scope of this book (see also Dunlop and Özdemir, 1997).
Imagine a set of identical grains with uniaxial symmetry; that is, the magnetic moment of an individual grain may be oriented in either direction along its axis of symmetry but not in any other direction. The axes of the grains are randomly oriented so that a specimen may have zero magnetic moment if the magnetizations are directed so as to cancel out one another. On application of a magnetic field in any direction, the specimen acquires a magnetic moment because the individual grain magnetizations will be in whichever of the two directions along their axis of symmetry has a component in the direction of the external field. Thus, although individual magnetic grains are themselves magnetically anisotropic, a random assemblage of grains making up the specimen is magnetically isotropic.
The magnetic behavior of one of these grains depends on its orientation with respect to the applied field. When the field is parallel to the axis of the grain a rectangular hysteresis loop results as in Fig. 2.12a. The height is twice the saturation magnetization Ms, and the width is twice the microscopic coercivity B'c (= |i0Hc = 2K/M$, where K is the anisotropy constant, see (2.3.10) below). At B = +B'c and B = -B'c there are discontinuities in the magnetization. At the other extreme, when the axis of the grain is perpendicular to the applied field, there is no hysteresis (Fig. 2.12b). For B > B'c and B < -B'c, the magnetization is +MS and -Ms respectively. As B changes from -B'c to +B'C, then Ms varies linearly from -Ms to +MS. In a randomly oriented assemblage of grains, the average limiting hysteresis cycle is as shown in Fig. 2.12c. There is a remanence Mt = 0.5Ms and a Bc « 0.5B'c. Note that the microscopic coercivity B'c of a single grain should be distinguished from the bulk coercivity Bc defined by the hysteresis of bulk materials as in §2.1.4.
In an SD grain the internal magnetization energy depends only on the orientation of the magnetic moment with respect to certain axes in the grain. For magnetically uniaxial grains, the energy E is given by
where v is the volume of the grain and 0 is the angle between the magnetic moment and the axis. K is called the anisotropy constant and can arise from three factors that contribute to the magnetic anisotropy of an SD grain: shape anisotropy, magnetocrystalline anisotropy and magnetostrictive (or stress) anisotropy (see §2.3.1). The microscopic coercivity B'c is simply related to the
anisotropy constant by the relation
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