# Spinel Magnetic Iron Oxide Nanoparticles: Properties, Synthesis and Washing Methods
## 2 Crystal Structure and Magnetic Properties of Spinel Iron Oxide
### Magnetic Properties
#### 2.2.3 Main Sources of Magnetic Anisotropy  
Magneto-Crystalline Anisotropy  
Within a crystal, the magnetization vector is not isotropic. There are energetically more favorable crystallographic directions for the alignment of magnetic spins. These directions are called axes of easy magnetization. This anisotropy of the crystallographic structure is at the origin of the magnetic anisotropy. The energy required to deflect the magnetization vector with respect to an easy magnetization direction is the magnetocrystalline anisotropy energy denoted \(\left(\mathrm{E}_{\mathrm{an}}^{magn}\right)\) and defined by:  
\[\mathrm{E}_{\mathrm{an}}^{magn}=\mathrm{K}_{1}\mathrm{V}\mathrm{sin}^{2} \theta+\mathrm{K}_{2}\mathrm{V}\mathrm{sin}^{4}\theta \tag{1}\]  
with K\({}_{1}\) and K\({}_{2}\) as the anisotropy constants of the material of first order and \(\theta\) the angle between the direction of magnetization and the easy magnetization direction.  
For the magnetite at room temperature, K\({}_{1}=-1.35\times 10^{4}\) J\(\cdot\)m\({}^{-3}\), K\({}_{2}=-0.28\times 10^{4}\) J\(\cdot\)m\({}^{-3}\) and the easy magnetic direction is the direction [111][8; 9].  
For the maghemite at room temperature, K\({}_{1}=-2.5\times 10^{4}\) J\(\cdot\)m\({}^{-3}\), K\({}_{2}\) is negligible in front of K\({}_{1}\) and the easy magnetic direction is the direction [110][10]. In the case of a mono-domain magnetic particle, the predominant term is the term of order 2 [7] and we reduce the expression of magnetocrystalline anisotropy energy to:  
\[\mathrm{E}_{\mathrm{an}}^{magn}=\mathrm{K}\mathrm{V}\mathrm{sin}^{2}\theta \tag{2}\]  
Figure 4: Magnetization cycle depending on the field applied for a massive ferro/ferrimagnetic material.  
The magnetic anisotropy of surface is linked to the break of symmetry on the surface of the particles causing the formation of gaps in the crystallographic structure and the decrease of the number of nearest neighbors for the atoms on the extreme surface. An effect now well-known as \"spin canting\" causes the magnetic spins of atoms on the surface to reorient themselves perpendicular to the latter and thus leads to a significant decrease in the saturation magnetization of the material [11]. This source of anisotropy is more important that the magnetic materials are small because the rate surface/volume increases. This source of anisotropy becomes preponderant in the case of nanoparticles. The energy associated with this anisotropy of surface is written as:  
\[\mathrm{E}_{\mathrm{an}}^{surf}=\mathrm{K}_{\mathrm{s}}\mathrm{V}\mathrm{ sin}^{2}\alpha \tag{3}\]  
The magnetic anisotropy of shape comes from the interaction between magnetic moments within the material and the demagnetizing field specific to any material. In the facts, it will tend, in the case of a material with anisotropic dimensions, to orient the field in the direction of the largest dimension (the magnetization of a cylinder with a negligible section and with an infinite height will be oriented in the direction of the height). For a sphere, this anisotropy is negligible due to a three-dimensional uniformity.