# Magnetic Properties of Nanocomposites
## 2 A Survey of Magnetic Measurements and Typical Magnetic Parameters of Nanocomposite Systems
### Typical Magnetic Parameters and Quantities in Nanoparticle Systems
If \(H\) is first brought to zero and then increased again in the positive region, the material will pass through the negative remanence and the positive coercive field, finally reaching the positive saturation state. In a symmetric loop, the two values of saturation magnetization, remanence, and coercive field are equal in magnitude and opposite in sign. Under an alternating magnetic field, the magnetization will cyclically change, describing a closed loop. The demagnetized state (0,0) is not easily accessed; it can be obtained by submitting the sample to a cyclic magnetizing field of slowly decreasing amplitude. It should be remarked that magnetic hysteresis stems from irreversible magnetization processes leading to a loss of energy, which is measured by the loop area. The _magnetic susceptibility_\(\chi\) provides a measure of the magnetic response of a ferromagnetic or a paramagnetic material. In the former case, one can consider either the _initial susceptibility_\(\chi_{0}\) (measured on the first magnetization curve close to \(H=0\)) and the maximum susceptibility on the loop \(\chi_{\rm max}\) (which corresponds to the maximum loop slope). In paramagnetic materials, \(\chi_{0}\) is measured to get information about the number of magnetic ions per unit volume and/or the magnetic moment per ion. In granular systems, hysteresis loops appear only when the NPs are in the blocked state (i.e., when the measurement temperature is below the blocking temperature \(T_{B}\), see Section 2.1). Above the blocking temperature, NPs are in the so-called _superparamagnetic regime_, which is characterized by complete reversibility of the magnetization process. _(iv) Superparamagnetic state_. When the size of a particle composed of magnetic atoms is small enough, the energy necessary to divide itself into magnetic domains (as in macroscopic ferromagnetic bodies) is higher than the energy needed to remain in a state of uniform magnetization; therefore, the particle hosts just one magnetic domain (_single-domain particle_). The magnetic properties of an assembly of single-domain particles are usually studied within the framework of the _superparamagnetic theory_. The first assumption of this theory is to consider that all the magnetic moments of the particle rotate coherently, so that the net magnetic moment can be represented by a single classical vector,with magnitude corresponding to the sum over all aligned atomic moments. Usually, the particle's effective anisotropy \(K_{\mathit{eff}}\) is assumed to take the simplest form (i.e., a _uniaxial_ anisotropy with a single easy axis and two equivalent (\"up\" and \"down\") directions); this leads to an energy barrier to the magnetization vector, which is proportional to \(K_{\mathit{eff}}V\), where \(V\) is the NP volume. The switch of the NP magnetization between the two antiparallel directions on the same easy axis can be described by an Arrhenius-type law with time constant \(\tau=\tau_{0}e^{\frac{K_{\mathit{eff}}V}{k_{B}T}}\), where \(k_{B}\) is the Boltzmann constant and \(\tau_{0}^{-1}\) is the jump attempt frequency of the magnetization between the opposite directions of the easy axis. In NP systems, the observed magnetic behavior strongly depends on the value of the characteristic measuring time \(t_{M}\) with respect to the intrinsic system relaxation time. If \(t_{M}>>\tau\), the magnetization jumps at a very high rate during a single measurement, allowing the system to be viewed as being at thermodinamical equilibrium. The NPs are then in the _superparamagnetic regime_. In this case, the magnetization process displays no irreversibility; the coercivity of the system is equal to zero and the _M(H)_ curve takes a smooth form similar to the response of a classical paramagnetic system (_Langevin function_). On the contrary, if \(t_{M}<<\tau\), the magnetization jumps exceedingly slowly, and one can observe quasi-static properties as in an ordered, macroscopic magnetic system. Such NPs are said to be in the _blocked regime_. In this case, a hysteresis loop is measured.