1.1.3.1 Paramagnetism

It is well known that most of the materials, subject to magnetic fields, exhibits either diamagnetic or paramagnetic behavior [5]. This reflects in a value of the magnetic permeability slightly different from the vacuum permeability $\mu_0$. Conversely, few materials, like Fe, Ni, and Co behaves differently and are referred to as ferromagnetic materials. In the following, we will briefly explain the paramagnetism, since it is helpful for describing ferromagnetic materials. Thus, let us consider a medium whose elementary particles possess magnetic moment. Let us suppose that no external field is applied, and that the body is in thermodynamic equilibrium. Due to the random orientation of the elementary magnets, the magnetization vector $\textbf{M}$ is zero everywhere in the medium. When an external field $\mathbf{H}_$a is applied, an equilibrium between the tendency of dipoles to align with the field and the thermal agitation establishes. This produces the magnetization of the body in the same direction and orientation as the external field. If we call $\textbf{{m}}_0$ the permanent magnetic moment of the generic dipole and $\theta$ the angle between $\textbf{{m}}_0$ and $\mathbf{H}_$a, the contribution $dM$ to the total magnetic moment of the body, given by the single dipole, is the component of $\textbf{{m}}_0$ along the field direction

$$dM=m_0 \cos \theta \quad.$$ (1.19)

Now we have to determine the distribution of the dipoles with respect to the angle $\theta$ and then to compute the average value of $m_0\cos\theta$. To this end, we can use Boltzmann statistic which gives the probability $p(E)$ for a dipole to have suitable potential energy $E$ as:

$$p\propto\exp\left(-\frac{E}{k_B T}\right) \quad,$$ (1.20)

where $k_B$ is the Boltzmann constant and $T$ is the temperature. The potential energy of a dipole subject to the field $\mathbf{H}_$a is:

$$E=-\mu_0\textbf{{m}}_0 \cdot \mathbf{H}_ \quad.$$ (1.21)

If $N$ is the number of dipoles per unit volume, the total magnetic moment $M$ per unit volume can be expressed as the following statistical average:

$$M =\frac{\int_{E_\text{min}}^{E_\text{max}} N m_0 \cos\theta p(E) dE}{\int_{E_\text{min}}^{E_\text{max}} p(E) dE} =$$

$$=\frac{\int_{E_\text{min}}^{E_\text{max}} N m_0 \cos\theta \e... ...t{a} \cos\theta}{k_B T}\right) d(-\mu_0 m_0 H_\text{a} \cos\theta) } \quad.$$ (1.22)

In Eq. (1.22) the denominator takes into account the fact that the probability density function $p(E)$ has to be normalized to unity. With the positions

$$x=\cos\theta \quad,\quad \beta=\frac{\mu_0 m_0 H_\text{a}}{k_BT} \quad,$$ (1.23)

Eq. (1.22) becomes:

$$M=M_s \left(\coth\beta - \frac{1}{\beta} \right)=M_s \mathcal{L}(\beta) \quad,$$ (1.24)

where $M_s=N m_0$ is the saturation magnetization, corresponding to the case in which all the dipoles are aligned, and $\mathcal{L}(\beta)$ is the Langevin function. Generally, in experiments on paramagnetic substances, typical temperatures and fields are such that

$$\beta=\frac{\mu_0 m_0 H_\text{a}}{k_B T} \ll 1 \quad.$$ (1.25)

Since the Langevin function can be developed in Taylor series

$$\mathcal{L}(\beta)=\frac{\beta}{3}+\mathcal{O}(\beta^2) \quad,$$ (1.26)

for small $\beta$ we can take the first order expansion and rewrite Eq. (1.24) as

$$M=\frac{\mu_0 M_s m_0}{3k_BT} H_ = \chi H_ \quad.$$ (1.27)

where the magnetic susceptibility $\chi$ is in the order of $10^{-4}$ for typical values of the parameters. One can clearly see that Eq. (1.27) explains the inverse dependance of the susceptibility on the temperature observed experimentally by Curie.