Technical Article

# Diamagnetic, Paramagnetic, and Ferromagnetic Materials Explained

May 22, 2024 by Dr. Steve Arar

## This article examines three different types of magnetic materials and how they react to an external magnetic field.

Magnetic materials can be roughly classified into three main groups: diamagnetic, paramagnetic, and ferromagnetic. A thorough understanding of how these materials interact with an external field requires a knowledge of quantum theory. However, we can still use simplified explanations to gain a basic understanding of these materials’ properties.

### Key Concepts

In the previous article in this series, we discussed some basic concepts of magnetism in matter. We learned that for a material placed in a uniform magnetic field (B0), the total magnetic field inside the material (B) is given by:

$$B~=~(1~+~\chi)B_{0}~=~\mu_r B_{0}$$

##### Equation 1.

where the Greek letter χ is a proportionality factor called magnetic susceptibility. In this equation, χ appears as part of the factor (1 + χ). This factor, known as relative permeability, is commonly denoted by μr.

Depending on the type of material, the magnetic susceptibility can be either positive or negative. When it’s positive, the external field is reinforced inside the material. On the other hand, a negative susceptibility means that the magnetic response of the material opposes the applied field, which results in the total field being slightly smaller than the external field.

We also learned that the magnetic response of a material is produced by the interaction between its atomic magnetic moments and the external magnetic field. But how do these interactions actually take place? Let’s start by examining diamagnetism, since it’s the easiest to explain without quantum mechanics.

### Diamagnetism

Diamagnetic materials produce magnetic moments that oppose the external field. Figure 1, which shows the classical model of the atom, can help us understand this behavior. In this model, an electron orbits an atomic nucleus. The electron moves with constant speed v0 in a circular orbit of radius r; the electron’s angular velocity is represented by ⍵0.

##### Figure 1. Classical model of an electron orbiting an atomic nucleus. Image used courtesy of Steve Arar

We know from high-school physics that acceleration equals force divided by mass ($$a~=~\frac{F}{m}$$). We also know that the acceleration vector for a uniform circular motion always points towards the center of the circle and is given by $$a_c~=~\frac{v^{2}}{r}$$. This is called the centripetal (meaning center-seeking) acceleration.

Based on the above, the centripetal acceleration of the electron in Figure 1 is given by:

$$a_c~=~\frac{v_{0}^{\; 2}}{r}~=~\frac{F_{c}}{m}$$

##### Equation 2.

where:

v0 is the speed of the electron

r is the radius of the electron’s circular orbit

m is the electron’s mass

Fc is the centripetal force acting on the electron.

The acceleration associated with the electron’s orbital motion is produced by the Coulomb force between the nucleus and electron. Equating the centripetal force (Fc) with the Coulomb force gives us the electron’s angular velocity in the absence of an external field.

Now suppose that we apply an external magnetic field (B0) in the direction of the z-axis. Figure 2 shows our new model.

##### Figure 2. The external field, represented here by orange arrows, perturbs the electron’s orbital motion. Image used courtesy of Steve Arar

The external field applies an additional force on the electron. This magnetic force is proportional to the cross-product of the velocity factor and the magnetic field vector. It can be found using the magnetic force equation:

$$\overrightarrow{F} ~=~ q \overrightarrow{v} ~\times~ \overrightarrow{B}$$

##### Equation 3.

where:

q is the charge

$$\overrightarrow{v}$$ is the velocity vector

$$\overrightarrow{B}$$ is the magnetic field vector.

To find the direction of $$\overrightarrow{F}$$, we can use the right-hand rule. Just point the fingers of your right hand in the direction of $$\overrightarrow{v}$$, curl them toward $$\overrightarrow{B}$$, and your thumb will point in the direction of $$\overrightarrow{F}$$. If the charge is negative, the direction of $$\overrightarrow{F}$$ is opposite to your thumb instead.

The right-hand rule tells us that the magnetic force in Figure 2 is oriented toward the center of the circle. This means that the external field increases the centripetal force acting on the electron. The additional centripetal force changes the electron’s velocity and/or its radius of rotation. We can see this relationship in Equation 2 by using the following logic:

1. Centripetal acceleration increases in direct proportion to centripetal force ($$a_{c}~=~\frac{F_{c}}{m}$$).
2. Since $$a_{c}~=~\frac{v^{2}}{r}$$, an increased value for ac means either an increase in speed or a decrease in radius (or both).
3. Therefore, the addition of more centripetal force must produce an increase in speed or a decrease in radius.

For simplicity’s sake, let’s assume that the radius remains constant. It follows that the electron’s speed increases in response to the external magnetic field.

If we regard the electron’s orbit as a current loop, a higher speed means that the electron crosses any point on the orbit more frequently in unit time. An increase in the electron’s speed corresponds to an increase in the current associated with the electron’s orbital motion. In turn, the increase in current corresponds to an increase in the magnetic field produced by the current loop.
To summarize:

1. Adding an external magnetic field increases centripetal force.
2. Increased centripetal force means that the electron’s speed increases.
3. Increased electron speed means that the current associated with the electron’s orbit increases.
4. Increased current means that the magnetic field produced by the electron’s orbit increases.

Since the charge of the electron is negative, the current flow is in the opposite direction of the electron’s orbital motion. The induced magnetic field is therefore directed along the minus z-axis—the direction antiparallel to the external field. In this way, diamagnetic materials produce magnetic moments that oppose the external field.

Note that diamagnetism is a temporary phenomenon—it only occurs when an external magnetic field is applied. If we remove the external field, the electrons’ orbital motions become randomly oriented again, and the average orbital magnetic moment of the whole material becomes zero. Therefore, a diamagnetic material doesn’t have any permanent magnetism of its own. This is in contrast to paramagnetic and ferromagnetic materials, as we’ll discuss later in the article.

#### Diamagnetic Materials and Levitating Frogs

Diamagnetism is present in all matter. However, it’s relatively weak. Where paramagnetism or ferromagnetism is also present, the material will exhibit one of the stronger types of magnetism, rather than diamagnetism. Some materials that exhibit diamagnetism are:

• Water.
• Living tissues.
• Metals that have many core electrons, such as copper, bismuth, mercury, and gold.

These materials are weakly repelled by the external field because they produce a field that opposes it.

If you want to see this effect in action, there are many videos on the internet that demonstrate diamagnetic materials being pushed away by a magnet. One of the most fascinating examples of this genre is the magnetic levitation of a frog, which is possible because the frog's body is diamagnetic. However, this experiment may not be very pleasant for the frog, as it experiences a strong magnetic force on every atom of its body.

Table 1 lists the magnetic susceptibility of some diamagnetic and paramagnetic materials. Comparing the paramagnetic and diamagnetic values should give you a feel for how weak these magnetization effects are.

##### Table 1. Magnetic susceptibility of some materials at 300 K. Data used courtesy of D. Halliday and R. Resnick
 Paramagnetic Substance χ Diamagnetic Substance χ Aluminum 2.3 × 10–5 Bismuth –1.66 × 10–5 Calcium 1.9 × 10–5 Copper –9.8 × 10—6 Chromium 2.7 × 10–4 Diamond –2.2 × 10–5 Lithium 2.1 × 10–5 Gold –3.6 × 10–5 Magnesium 1.2 × 10–5 Lead –1.7 × 10–5 Niobium 2.6 × 10–4 Mercury –2.9 × 10–5 Oxygen 2.1 × 10–6 Nitrogen –5.0 × 10–9 Platinum 2.9 × 10–4 Silver –2.6 × 10–5 Tungsten 6.8 × 10–5 Silicon –4.2 × 10–6

As you can see, the susceptibility of most paramagnetic materials is on the order of 10-5 to 10-4. Diamagnetic materials have a negative susceptibility. For the diamagnetic materials shown in the table, the susceptibility is on the order of –10-5 or less.

### Paramagnetism

Diamagnetism arises mainly from the orbital motion of the electrons. Paramagnetism, on the other hand, is primarily caused by the spin magnetic moment of electrons. This makes paramagnetism more difficult than diamagnetism to describe in classical terms.

Unlike diamagnetic materials, paramagnetic materials have atoms or ions with net magnetic moments that experience a torque in the presence of an external field and align themselves with it. However, thermally induced random motions oppose this tendency of the atomic magnetic moments to line up. As a result, paramagnetism changes significantly with temperature.

The following equation, known as Curie’s law, shows that magnetization of a paramagnetic material (M) is inversely proportional to temperature (T):

$$M~=~C \frac{B_0}{T}$$

##### Equation 4.

where:

B0 is the applied field

C is the Curie constant. The value of the Curie constant depends on the specific material.

The equation also shows that paramagnetism is temporary—removing the external field (B0 = 0) causes the magnetization of the material to disappear (M = 0). While the atoms or ions of the material have permanent magnetic moments, these magnetic moments have random orientations in the absence of an external field.

As mentioned above, paramagnetic materials are attracted to the external field. A common demonstration of this effect is how liquid oxygen is pulled by a powerful magnet as a result of its paramagnetism. When liquid oxygen is poured between the magnetic poles, it becomes attracted to the poles and remains suspended between the poles until it evaporates.

### Ferromagnetism

Ferromagnetism is observed in substances such as iron, cobalt, nickel, gadolinium, and dysprosium. Of all the magnetic materials discussed in this article, those with ferromagnetic properties are the most important to electrical engineering applications.

Like the atoms of a paramagnetic material, each atom in a ferromagnetic material has a net nonzero magnetic moment due to spinning electrons. However, the atoms of a ferromagnetic material interact strongly with their neighbors, producing tiny regions where the magnetic moments of the atoms all point in the same direction. These regions of alignment, known as magnetic domains, are separated from one another by a transition region called the domain wall. The domain wall is about 100 atoms thick.

According to quantum theory, the alignment of a domain’s dipole moments result from strong coupling forces between the domain’s atoms. These atomic interactions decrease with distance, keeping the domains small in size. The magnetic domains of iron, for example, are less than 1 mm in width or length.

Ferromagnetic domains exist even when no external magnetic field is present. Figure 3(a) and Figure 3(b) show the domains in the absence and presence of an external field, respectively. The direction of the magnetic moments within each domain is indicated by an arrow.

##### Figure 3. Ferromagnetic domains in the absence of an external field (a) and when a field of moderate strength is applied in the direction of the blue arrow (b). Image used courtesy of Yeon Ho Lee

As we see in Figure 3(a), the domains are randomly oriented with respect to one another when there’s no external magnetic field. The magnetic effects of the domains therefore cancel each other out.

When we apply an external magnetic field, however, the domains tend to align themselves with it. If we look at Figure 3(b), we see that the borders of the domains actually change—the domains that are in parallel with the external field grow larger, while the others shrink. Therefore, the material is strongly magnetized when placed in an external field.

#### Saturation and Demagnetization

The field shown in Figure 3(b) is only moderately strong. A sufficiently large external field will eventually make all the magnetic domains align with the applied field. This condition, where nearly all domains of the material are aligned, is called saturation.

Beyond the point of saturation, increasing the external magnetic field can’t produce more magnetization in the material. Core saturation can have a detrimental impact on the performance of inductors and transformers. In most applications, designers must be careful to avoid it.

Temperature is another thing to pay attention to. Increasing the temperature increases the random thermal motion of the atoms, which tends to randomize the magnetic domains. Magnetized ferromagnetic materials can therefore lose their magnetism at high temperatures.

The temperature at which ferromagnetic materials lose their magnetism completely is known as the Curie temperature or the Curie point. Like the Curie constant (Equation 4), the Curie temperature is different for different materials. Iron, for instance, has a Curie temperature of 1,043 K.

Finally, it’s worthwhile to note that striking a magnetized material with a hammer can randomize its magnetic domains, making it lose some or all of its magnetism.

#### Permeability of Ferromagnetic Materials

Ferromagnetic materials are nonlinear and exhibit hysteresis: their magnetization depends on the history of the substance as well as the applied external field. This means that the relationship between the total field inside the material and the applied field is also nonlinear. Depending on the magnetization cycle the material has undergone, the same external field can produce different internal field values.

Because of this nonlinearity, Equation 1 doesn’t really apply to ferromagnetic materials. In its place, there are several different definitions of permeability that we can use to characterize various aspects of a ferromagnetic material’s behavior. In the next article in this series, we’ll discuss the concept of complex permeability, which describes a ferromagnetic material’s magnetization as well as its losses. We’ll then use this parameter to explore how the properties of ferromagnetic materials influence the high-frequency performance of various magnetic components.

Featured image used courtesy of Tony R. Kuphaldt