# Spinel Magnetic Iron Oxide Nanoparticles: Properties, Synthesis and Washing Methods
## 2 Crystal Structure and Magnetic Properties of Spinel Iron Oxide
### Magnetic Properties
#### 2.2.5 Relaxation Time  
When the necessary energy required to return the magnetization becomes lower than the ambient thermal energy, the iron oxide nanoparticles (magnetite and maghemite) show a superparamagnetic behavior. The relaxation time \(\tau\) corresponding to the inverse of the probability of tilting magnetic spins is then defined by the Neel-Brown law [11]:  
\[\tau=\tau_{0}\cdot\exp\left(\frac{E_{an}^{tot}}{k_{B}T}\right) \tag{6}\]  
where \(\tau_{0}\) is the characteristic time of relaxation of the order of \(10^{-9}\) s [7,9,14,15].  
The relaxation time increases when the total anisotropic energy increases and it decreases when the temperature increases. The observation time of the phenomenon is dependent on the characteristic measured time \(\tau_{m}\). We distinguish two cases:  
* If \(\tau>>\tau_{m}\), the magnetic moment of the particle appears as blocked and the particle adopts a ferrimagnetic behavior.
* If \(\tau<<\tau_{m}\), the magnetic moment of the particle returns around a lot of time during the measure and the average moment is zero. The particle presents a superparamagnetic behavior.  
For a given measurement time and a defined volume \(V\), the magnetic behavior will depend on the temperature. We can then define the blocking temperature \(T_{B}\) for a particle with a volume \(V\) as the temperature at which \(\tau=\tau_{m}\). This blocking temperature can be expressed, if the field is weak or zero, by the relationship:  
\[T_{B}=\frac{K_{an}^{tot}.V}{k_{B}\ln\left(\frac{\tau_{m}}{\tau_{0}}\right)} \tag{7}\]  
For the  
Finally, also note that several studies have shown that these equations (Equations (6) and (7)) are not valid only in the case of nanoparticles isolated from each other and without dipolar interactions between them. In fact, these dipolar interactions which can be written according to the Equation (9), when they present within the sample, can be responsible for a shift from the average blocking temperature to high temperature [14; 15; 16].  
\[E_{d}=\frac{\mu_{0}}{4\;\pi\;r^{3}}\;\Big{(}\overrightarrow{m_{1}}\!\!\! \cdot\!\!\overrightarrow{m_{2}}-\Big{(}3/r^{2}\Big{)}\left(\overrightarrow{m_{ 1}}\!\!\!\cdot\!\!\overrightarrow{r}\right)\left(\overrightarrow{m_{2}}\! \!\!\cdot\!\!\overrightarrow{r}\right)\Big{)} \tag{9}\]  
with \(\overrightarrow{m_{1}}\) and \(\overrightarrow{m_{2}}\) the magnetic moments carried by two nanoparticles and \(\overrightarrow{r}\) the vector representative of the distance between the two nanoparticles.  
The numeric values associated with the magnetic properties of two iron oxide magnetite and maghemite presented in Table 1 are relatively close. These are two ferrimagnetic materials but the presence of cationic gaps in the structure of maghemite causes a decrease in its saturation magnetization. With a controlled size of nanoparticles below the limit diameter D\({}_{0}\), these two oxides adopt a superparamagnetic behavior which allows the magnetization of the material to turn spontaneously, opening the way for many applications.