3 The shape magnetic anisotropy

Magnetostatic are other sources of the total magnetic anisotropy called the macroscopic shape anisotropy. This concept is clear in the case of homogeneous magnetization in a ellipsoid, where the demagnetization tensor can be introduced in such way that the demagnetization field can be defined as:

$\displaystyle \mathbf{H_{D}}=-\mathcal{D} \mathbf{M}$ (11)

where $ \mathcal{D}$ is the demagnetization tensor, $ \mathbf{H_{D}}$ is the demagnetization field and $ \mathbf{M}$ is the magnetization of the system. Thus the density magnetostatic energy can be described as:

$\displaystyle E_{M}=2\pi\mathbf{M}\mathcal{D}\mathbf{M}$ (12)

If the semiaxes a, b, and c of the ellipsoid represent the axes of the coordination system the $ \mathcal{D}$ is a diagonal tensor. An arbitrary direction of the magnetization with respect to the semiaxes can be characterized by the direction cosine $ \alpha_{a}$, $ \alpha_{b}$, and $ \alpha_{c}$. The tensor is given by:

$\displaystyle \mathcal{D}= \left( \begin{array}{ccc} \mathcal{D}_{a} &0 & 0 \ 0 & \mathcal{D}_{b} & 0 \ 0 & 0 & \mathcal{D}_{c} \ \end{array} \right)$ (13)

and the magnetostatic energy density can be written as:

$\displaystyle \mathcal{E}_{M}=2\pi Ms(\mathcal{D}_{a}\alpha^{2}_{a}+\mathcal{D}_{b}\alpha^{2}_{b}+ \mathcal{D}_{c}\alpha^{2}_{c}) .$ (14)

For thin films, magnetostatic interaction leads to an additional anisotropy favoring the in-plane anisotropy. In the case of elongated nanoparticles, magnetostatic interactions produce an additional easy axis parallel to the long dimensions.

Rocio Yanes