# Spinel Magnetic Iron Oxide Nanoparticles: Properties, Synthesis and Washing Methods
## 2 Crystal Structure and Magnetic Properties of Spinel Iron Oxide
### Magnetic Properties
#### 2.2.4 Evolution of Magnetic Properties in the Case of Nanoparticles  
The reduction of the size, for example from a 3D material to a 1D or a 0D material, modifies a certain number of magnetic properties. When the size of materials becomes inferior at a critical diameter \(\mathrm{D}_{\mathrm{C}}\) (Figure 5), the creation of Bloch's walls becomes energy-defeatable and these last disappear to minimize the magnetostatic energy. We then speak of \"mono-domain\". In the case of nanoparticles, the critical diameter can be calculated by the expression:  
\[\mathrm{D}_{\mathrm{C}}=4\,\gamma/\mu_{0}\,\mathrm{N}_{\mathrm{MD}}.\mathrm{ M}\mathrm{g}^{2} \tag{4}\]  
where \(\gamma\) is the energy of creation of a Bloch's wall, \(\mu_{0}\) the vacuum permeability, \(\mathrm{M}_{\mathrm{S}}\) the saturation magnetization of the material, and \(\mathrm{N}_{\mathrm{MD}}\) the demagnetizing factor for a mono-domain. For mono-domain nanoparticles of magnetite and maghemite, \(\mathrm{D}_{\mathrm{C}}\) is around 30 nm [7; 12].  
Below this critical size, the magnetic properties of magnetite and maghemite nanoparticles also vary. Considering a mono-domain nanoparticle with a diameter below at \(\mathrm{D}_{\mathrm{C}}\), its magnetic behavior will change depending on the ratio between the total anisotropy energy \(E_{an}^{tot}\) and the thermal activation energy \(\mathrm{E}_{\mathrm{therm}}\) with: \(E_{an}^{tot}=K_{an}^{tot}\cdot\mathrm{V}\) where \(K_{an}^{tot}\) is the sum of the magnetocrystalline anisotropy and the surface anisotropy and V the average volume of nanoparticles. \(\mathrm{E}_{\mathrm{therm}}=\mathrm{k}_{\mathrm{B}}\mathrm{T}\) where \(\mathrm{k}_{\mathrm{B}}\) is the Boltzmann's constant and T the temperature.  
Figure 5: Evolution of the magnetic organization of nanoparticles according to their diameters.  
We can distinguish two cases:  
* \(E_{an}^{tot}>E_{\rm therm}\). The magnetic moments of ferrimagnetic domains are blocked in the easy magnetization direction. A rotation of these magnetic moments by application of an external field then causes the mechanical rotation of the entire nanoparticle. The nanoparticle presents a ferrimagnetic compartment in the case of the magnetite.
* \(E_{an}^{tot}<E_{\rm therm}\). The thermal agitation is more important than the total magnetic anisotropy energy and the magnetic moment is free to rotate freely. This magnetic compartment is called superparamagnetism.  
Considering these two cases, it is possible to note by the ratio presented in Equation (5) that for a temperature given (for example the room temperature), the direction of the report will depend on the average volume V of nanoparticles and their sizes.  
\[E_{an}^{tot}/E_{\rm therm}=K_{an}^{tot}.V/k_{\rm B}.T \tag{5}\]  
If the diameter of the particle is small enough (<D\({}_{0}\)), the magnetocrystalline anisotropic energy becomes lower than thermal energy at room temperature. The magnetization can then overturn spontaneously and the particle presents a superparamagnetic behavior (Figure 5). For the magnetite and the maghemite, the limit diameter from which nanoparticles present a superparamagnetic behavior is around 20 nm [1,13].