The first and most important energy contribution comes from the exchange interaction, which was introduced above as a purely quantum mechanical effect responsible for the alignment of atomic magnetic moments in a ferromagnetic body. The assumption T « Tc means that the energy of temperature fluctuations is negligible compared with the exchange energy, so that adjacent magnetic moments are (almost) parallel. Hence, the magnitude of the magnetization M of the body (mag netic moment per unit volume) can be considered as constant |M| = Ms; this constant is called the saturation magnetization of a ferromagnetic material. For low temperatures, only the magnetization direction can be varied inside a body under an additional condition that the distance where the magnetization direction varies considerably is much larger than the lattice constant (or mean interatomic distance for amorphous ferromagnets) of the material. The latter circumstance allows us to introduce a unit vector m = M/Ms along the magnetization direction; its spatial distribution fully describes the magnetization structure of a ferromagnetic body.
Let us now write down the phenomenological expression for the exchange energy using simple general arguments (Landau and Lifshitz, 1975a). First, we recall that in ferromagnetic materials the exchange interaction ''prefers'' the parallel alignment of magnetic moments. This means that the corresponding energy is minimal for the magnetization configuration where all magnetic moments of a body are parallel to each other - the so-called homogeneous magnetization state. We set the exchange energy of such a state to zero, thus using it as a reference point. We also point out that the exchange interaction energy does not change when the magnetization configuration is rotated as a whole with respect to a ferromagnetic body.
We hope that it is clear from the consideration above that the exchange energy density (exchange energy per unit volume) eexch can depend only on the spatial variation of the magnetization, thus being a function of its spatial derivatives 8Mi/dxk, (i, k = 1,2,3), where M; denotes Cartesian components of the magnetization and Xk = x, y, z. Moreover, eexch can be a function only of a product of even numbers of such derivatives, because M; itself and hence - SM;/dXk -changes sign due to the time inversion operation t! —t (this is because M @ [r x v] = [r x dr/dt], see Eq. 1.32) and the energy does not. The simplest expression which satisfies this condition and another condition mentioned above -that the exchange energy is invariant with respect to the rotation of the magnetization configuration as a whole - is
Here, aik are the components of a (symmetrical) tensor of exchange coefficients which means that these components form a symmetrical 3 x 3 matrix. In the simplest case of a crystal with the cubic symmetry = aâik (da is a Kronecker symbol: = 1 if i = k and zero otherwise) and Eq. (1.49) takes the form
where we have introduced a new exchange constant A = aMs2 using the unit magnetization vector m defined above. The total exchange energy of a ferromagnetic body can be evaluated, as usual, as an integral of the corresponding density Eq. (1.49) (or Eq. 1.50) over the body volume:
dm dz dV
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