Magnetic properties

Magnetization

Any magnetic material will produce an auxiliary field¹ $\mathbf{H}$-field both in the space around it and within its own volume $V_m$. The significance of this auxiliary field relies on the relationship between$\mathbf{H}$ and its magnetization $\mathbf{M}$ that depends on the type of material (see also magnetic manipulation). Depending on their magnetic characteristics, materials can be classified as [2], [4]:

• Diamagnetic (e.g., $\mathrm{Cu}$, $\mathrm{C}$, $\mathrm{H}{\phantom{A}}_2$, water…) materials that have a weak, negative susceptibility, and are lightly repelled by a magnetic field;

• Paramagnetic (e.g., $\mathrm{Al}$, $\mathrm{Na}$, $\mathrm{O}{\phantom{A}}_2$…) materials that have a small, positive susceptibility, and are slightly attracted by a magnetic field;

• Ferromagnetic (e.g., $\mathrm{Fe}$, $\mathrm{Ni}$, $\mathrm{Co}$…) materials that have a large, positive susceptibility and exhibit a strong attraction to magnetic fields.

  

For diamagnetic and paramagnetic materials, a relatively large applied field is required to produce changes in magnetization; and there is no remanent magnetization if the applied field is removed. For ferromagnetic materials, magnetization curves have a typical hysteresis loop form, as represented in Figure 1, and there is no one-to-one relation between $\mathbf{H}$ and $\mathbf{M}$. Inside diamagnets and paramagnets the $\mathbf{H}$-$\mathbf{M}$ relationship is commonly linear given by: $$\mathbf{M}=\chi \mathbf{H}$$ with $\chi$ the magnetic susceptibility.
Inside a magnetized sample, $\mathbf{M}$ gives also rise to a reversed field ${\mathbf{H}}_d$. This field is called the demagnetization-field (or the stray-field) and is expressed as: $${\mathbf{H}}_d=-\mathbf{N}\mathbf{M}$$ where $\mathbf{N}$ is the demagnetization-factor matrix which has values between 0 and 1 depending on the shape of the sample. The internal field ${\mathbf{H}}_i$ is a function of the applied field $\mathbf{H}$, as well as a demagnetizing field produced by the magnetization distribution of the sample itself: $${\mathbf{H}}_i=\mathbf{H}+{\mathbf{H}}_d$$

Magnetic field

The relation between the magnetic field $\mathbf{B}$, the auxiliary field $\mathbf{H}$ and the magnetization $\mathbf{M}$ of the medium can be then expressed as: $$\mathbf{B}=\mu \mathbf{H}={\mu }_0(1+\chi )\mathbf{H}$$

Furthermore, according to their coercivity² $H_c$, ferromagnetic materials can be divided in two groups: soft and hard magnets (see Figure 1). Hard magnets, or permanent magnets, provide high coercivity, lower permeability and high hysteresis loop. While soft magnetic materials are easier to demagnetize after the saturation state, the $H_c$ value is low and $\chi$ value is high. When the size of ferromagnetic particles are small enough (e.g. ≲100 µm) a superparamagnetic behavior occurs, and their magnetization appears to be in average zero when no field is applied (i.e. $H_c\approx 0$). Commonly, materials with a steep magnetization curve, high saturation magnetization $M_s$ and low coercivity $H_c$ are commonly desired to be able to work with minimal fields and gradients [3].

Initially, a ferromagnetic or superparamagnetic material consists of magnetic domains with random orientation of their magnetic moment $\mathbf{m}$, as represented in the inset in Figure 1. Applied magnetic field leads to reorientation of magnetic moments towards its direction. When all the moments are aligned with magnetic field, the magnetization saturation state $\Vert \mathbf{M}\Vert =M_s$ is reached. For hard magnetic materials the magnetization magnitude is usually independent of the external magnetic field $\mathbf{B}$, and could be considered easily saturated in most cases. In contrast, for soft-ferromagnetic and superparamagnetic materials, $\mathbf{M}$ is strongly related to $\mathbf{B}$. Hence, and as shown in magnetization curve in Figure 1, two regions can be defined: a quasi-linear and saturation domains. Depending on the magnetic field, the magnetization $\mathbf{M}$ will be either a function of $\mathbf{H}$-field in the linear region, or constant at saturation $\Vert \mathbf{M}\Vert =M_s$.