Designing electric machines sometimes requires an understanding of the physical properties of the materials we’re working with. In this article, I’ll be exploring the magnetic properties of materials. I’m particularly interested in a diamagnetic material called pyrolytic graphite – the strongest room temperature diamagnet in existence today. My goal is to understand how magnetic and diamagnetic materials interacts with an externally applied magnetic field.
Disclaimer: This article was originally published on July 23, 2012. This is a re-posting from my old blog that was recovered and edited from the wayback machine. My original blog was hacked.
Okay, well, lets start with a simple question. What makes a material magnetic? … Well, let’s start by looking at the atomic scale. From there we can look more closely at the different kinds of magnetism and more. I’ll divide this article up as follows:
- A Model for the Atom
- The Origins of Magnetism
- Types of Magnetism
1. A Model for the Atom
Do you remember those exciting days in school when you’d be in science class … ahh yes, those were the good ol’ days … the perfect time to take a nap … errhhmmm, no no .. I mean, the perfect time to enjoy the beauty of science .. yes .. that’s right. Well … surely you remember what an atom is, right? No?! … well fear not, I’ll explain!
We typically started our studies in secondary school with the Bohr Model of the atom – which is actually a quantum physics improvement on the earlier Rutherford model. I’ll use this model as the starting point. At its most basic level, an atom consists of a dense and small central nucleus around which electrons orbit.
Orbitals, Shells and Energy
Electrons can only orbit in distinct orbitals. Each orbital has a fixed energy associated with it. We can represent this orbital energy using the variable n. Orbitals with the same energy are typically grouped into shells. In figure 1 below, we can see an (inaccurate) representation of an atom with two orbitals – each orbital in this atom has a distinct energy, n=1 and n=2. Scientist have captured images of an atom too. You can see one such image in figure 2.
So figure 1 might give you the impression that orbitals are circular. This is not at all true. In fact, there are different kinds of orbitals, typically denoted as s-orbitals, p-orbitals, d-orbitals or f-orbitals. And more over, each kind has its own shape. We can see how weird some of these shapes can be in figure 3. Each of the orbital types is preceded by the n energy level in which it exists. For instance 1s implies an s orbital with energy n=1, likewise 2p would imply a p orbital with energy n=2. If we were to fill an atom with electrons starting only from a central nucleus, figure 4 demonstrates the order in which electrons would occupy orbitals.
The s-orbital can hold a maximum of 2 electrons, the p-orbital 6 electrons, the d-orbital 10 and the f-orbital 14. This is shown in the table below:
| Electron Orbital | Num. of Electrons |
|---|---|
| s | 2 |
| p | 6 |
| d | 10 |
| f | 14 |
The orbitals are actually abstract mathematical representations of the wave-like properties of an atom. Electrons don’t necessarily orbit within these orbitals – the electrons are often described as existing as a cloud, where the cloud is a representation of how likely it is to find an electron at a particular location in the atom itself. The probability of finding an electron within an orbital tends to be high and drops off as we get further from the orbital.
Remembering that a shell is a collection of orbitals with the same energy, electrons in shells further from the nucleus tend to have higher energy than those in closer shells. The energy that an electron in a particular energy level, n, has, is measured in Electron Volts [eV]. That’s the energy that an electron gains when placed in an electric field with a one volt potential difference. We can represent an electron volt in terms of Joules [J] , such that 1eV=1.6⋅10−19J. In order to move an electron from say, energy level n=1 (the inner most shell) to energy level n=2 (the next outer level) we need to add energy. This can be done, for instance, by using light of a particular frequency. Jumping from n=1 to n=2 causes absorption of photons, and returning to a lower energy state from n=2 to n=1 causes release of photons. The lowest possible energy level that an electron can occupy is called the ground state.
Electron Spin & Orbital Motion
One additional property of note in any model for the atom is the electron spin. Electrons sitting in a particular orbital always prefer to do so in pairs. An electron can have an UP spin and a DOWN spin. Up and down spins prefer to exist together – or, at least, I would like to think so. Quantum theory suggests that this preferential togetherness minimizes the energy and leads to a more stable configuration of the atom. here is a great little online tool that provides a dynamic periodic table at: https://www.ptable.com/ . Looking at the various electron configurations with this tool seems to indicate that atoms don’t necessarily have an equal number of up and down spinning electrons.
It’s, maybe, worth mentioning that electrons can move – but not in the way we regard as motion. At the atomic scale, the wave nature of electrons plays a dominant role. This means that electron motion in a given orbital may be regarded as a vibration at a specific frequency – the frequency is such that the electron can exist as a standing wave.
We can see an abstract representation of this in figure 5. Note that in the figure to the right, the oscillation frequency does not match the energy of the orbital – the electron can’t exist in this orbital. While we can’t really think of electron motion in the traditional sense, the motion does produce a current. And as we’ll see next, this current can generate a magnetic field.
There is also another quantity called the nuclear spin . This quantity is the analogue of electron spin on the nucleus of an atom. At the present moment, I’m not really going to discuss nuclear spin.
2. The Origins of Magnetism
We can typically attribute magnetism to three atomic properties:
- Nuclear Spin: Some nuclei have a spin that creates a magnetic field.
- Electron Spin: An electron has two spin states that create magnetic fields.
- Electron Orbital Motion: An electrons motion in an atom creates a magnetic field.
I’m going to avoid talking about nuclear spin at the moment. So moving on to electron spin. There are some who suggest that magnetism is a direct consequence of electron spin. Electrons prefer pairing such that pairs consists of up and down spins. In a given orbital, under ordinary conditions, all electrons want to be paired. However, a particular atom may not have an even number of electrons, or may not have an equal number of up and down spin electrons. This variation in the number of up and down spinning electrons in a particular material’s atom is one of the causes of magnetism. Materials with at least one unpaired electron tend to be magnetic (either paramagnetic or ferromagnetic). Materials in which unpaired electron spins are oriented in the same direction tend to be strongly magnetic (i.e. ferromagnetic). Materials with paired electrons are non-magnetic. These non-magnetic materials are termed as diamagnetic.
Likewise, as demonstrated by Maxwell and other illustrious scientists, electron motion itself also generates a magnetic field. Orbital motion may be limited or may be such that the generated magnetic fields cancel each other out. And … I’m going to go out on a limb here … most of the time this is the case. Some materials may have atomic configurations where this may not be true, but I’m not sure. However, orbital motions may potentially be induced, via Lenz’ Law, by a changing external magnetic field. In such scenarios, orbital motions can create opposing magnetic fields.
At the atomic scale, the quantity known as the magnetic dipole moment is a measure of the contribution that nuclear spin, electron spin and orbital motion has on an atom’s magnetic properties. See figure 6.
There are, actually, distinct formulaic representations of this moment for each of the three causes of magnetism. We can use this idea to treat an atom, to some degree, as a tiny magnet – an atomic magnet, as it were.
Structure and State: Gases, Liquids & Solids
In addition to the intrinsic properties of atoms themselves. Atoms of different elements can be combined to form molecules. These molecules tend to have a geometry and shape, which, in-turn can exhibit distinct magnetic properties.
Gases and liquids tend to exhibit very poor magnetic characteristics compared to solids. This is because of random Brownian motion as a result of their naturally more thermally excited states. There are a few exceptions to this, of-course. For instance, liquid iron in the earth’s core spins as a result of the rotation of the earth about its own axis. This rotation causes an alignment (of the magnetic dipoles) of iron atoms such that they produce a noticeable magnetization.
As materials change from gaseous to liquid and liquid to solid states, the density of the materials also increases. This increased density means that magnetic properties are amplified proportional to the increase.
Solid materials have a structure which is the result of particular natural/geological (or artificial) processes. These processes may include, for instance, pyrolysis or crystallization. The resultant structural characteristics can effect magnetic properties. By carefully creating metal alloys one can produce materials with very specific magnetic characteristics. For instance, the steel used in transformers exhibits very low coercivities. I’ll explain what this means later on.
3. Types of Magnetism
All materials in existence today exhibit one of three kinds of magnetism. Add to that rumours of two additional types and you have five kinds of magnetism. These are listed below. Now, remember, magnetism is not something to be taken lightly … yes these terms that I’m introducing, well they’re not words to be bandied about willy nilly at cocktail parties! It’s serious stuff folks!!
- Diamagnetism
- Paramagnetism
- Ferromagnetism
- Ferrimagnetism
- Antiferromagnetism
Errhm, yes … getting back to the point. There are two things that distinguish these types of magnetism from each other. For the case of diamagnetism, paramagnetism and ferromagnetism the distinguishing property is the magnetic susceptibility.
For the case of ferrimagnetism and antiferromagnetism more than anything, it seems to be the structure of the material that’s important – Specifically the magnetic ordering of its dipoles. Its all a bit mysterious – personally, I don’t think these last two are really meaningful categories. But we’ll talk about that in a bit, and maybe … just maybe … I’ll change my mind.
Just a quick note: We’ll be talking about magnetic susceptibility a little bit. Magnetic susceptibility is a measure of the strength of a material’s response to a magnetic field. It’s often denoted as χM.
A. Diamagnetism
When looking at the atomic configuration of diamagnetic materials, we find that these materials do not have unpaired electron spins. This means they are non-magnetic.
Furthermore these materials have a value χM<0. In the presence of an externally applied magnetic field, these materials produce repulsive fields. This means when measuring the total field (including the applied field) inside the diamagnetic material it would be smaller than the externally applied field. For diamagnets the value, χM, is independent of the temperature of the material – this is not true for ferromagnets or paramagnets.
I’m particularly interested in two diamagnetic materials: Pyrolytic Graphite and Bismuth. As I mentioned at the beginning of this article, pyrolytic graphite is the strongest artificially created room temperature diamagnet to present day. And Bismuth is the strongest naturally occuring diamagnetic material. Table 1 provides a summary of some diamagnetic materials and their magnetic susceptibilities.
| Material | χM |
|---|---|
| Superconductor | −1 |
| Pyrolytic Graphite | −40.0 x 10−5 |
| Bismuth | −16.6 x 10−5 |
| Mercury | −2.9 x 10−5 |
| Silver | −2.6 x 10−5 |
| Diamond | −2.1 x 10−5 |
| Lead | −1.8 x 10−5 |
| Graphite | −1.6 x 10−5 |
| Copper | −1 x 10−5 |
| Water | −0.91 x 10−5 |
Superconductors are perfect diamagnets via the Meissner Effect. This effect is essentially Lenz’ Law in action, with the added realization that resistance inside the superconductor is zero. Hence, current continues to flow producing a continuous magnetic field in response to the external field’s brief change when it was first brought near the superconductor. In truth, superconductors operate under different principles from those of ordinary diamagnets.
Normal diamagnets will levitate in a constant magnetic field stably – this is not true for ferromagnets. With the advent of modern day magnets, this stable levitation opens up the possibility of designing passive magnetic bearings and levitation systems (i.e. those that consume no power). Like diamagnets superconductors also levitate stably in a constant magnetic field. However they do this using very different mechanisms, i.e. the Meissner Effect … superconductors also exhibit another property , specifically flux pinning that forces the levitating body to lock into a specific levitated position. The flux pinning phenomenon is a result of magnetic fields penetrating the superconducting material at locations where impurities exist.
Most everyday materials are diamagnetic. We see, from the table, that present day diamagnets generally have very small magnetic susceptibilities. This means that under most situations the effect of diamagnetism isn’t visible to us ordinary folk. New magnets, i.e. neodymium and superconducting magnets, have made it possible to visibly see diamagnetism’s repulsive force in action. Diamagnetic materials will actually levitate if the externally applied field is strong enough.
Why do Diamagnets Produce Repulsive fields?
One common explanation I’ve seen on the internet suggests that diamagnetism is a consequence of Lenz’ Law. An externally applied field causes orbital acceleration of electrons in the atom – a current. The current, in-turn, produces an opposing magnetic field. Unfortunately, this explanation doesn’t provide an answer for why a diamagnet still produces an opposing field even in the presence of a constant (static) magnetic field. Soooo …. BOOO! Boo! Lenz’s Law only applies to changing magnetic fields! …. What’s actually happening?!! … I don’t know.
In order to get a better understanding of this it may be worth while to look at the text: Magnetism in Condensed Matter by Stephen Blundell. In pyrolytic graphite, the effect of paramagnetism as a result of pi-bonds may be a dominant effect if one were to try and model the material. Okay okay … so I haven’t actually described what a pi-bond is…. At some point in a future article I’ll get to that!
Diamagnets produce repulsive fields that are linearly proportional to the applied external field. There appear to be no non-linear (e.g. saturation) effects.
B. Paramagnetism
Paramagnetic materials have a magnetic susceptibility χM>0. These materials are considered weakly magnetic. An externally applied magnetic field produces an attractive field inside the material. So, if we measured the resultant magnetic field inside a paramagnetic material we’d find that the total field would be slightly larger. This also means that paramagnetic materials are attracted to externally applied magnetic fields.
Like their diamagnetic counter parts, paramagnetic materials exhibit very very weak attractive forces. In everyday situations you wouldn’t see any visible movement of the material if a magnet was brought close to a diamagnetic material and the same is true for a paramagnet. Table 2 shows a list of some paramagnetic materials.
| Material | χM |
|---|---|
| Iron oxide (FeO) | 720 x 10−5 |
| Iron amonium alum | 66 x 10−5 |
| Uranium | 40 x 10−5 |
| Platinum | 26 x 10−5 |
| Tungsten | 6.8 x 10−5 |
| Cesium | 5.1 x 10−5 |
| Aluminum | 2.2 x 10−5 |
| Lithium | 1.4 x 10−5 |
| Magnesium | 1.2 x 10−5 |
| Sodium | 0.72 x 10−5 |
| Oxygen Gas | 0.19 x 10−5 |
| Air | 37 x 10−8 |
We can think of the atoms and molecules in a paramagnetic material as atomic magnets. When no external magnetic field is applied to these atomic scale magnets they exhibit random orientation and so the net magnetic field ends up being zero. With an external magnetic field these magnets align themselves and reinforce the external field. Higher temperature means greater thermal agitation, so it becomes harder to align these atomic magnets via an external field. Likewise a lower temperature means easier alignment. Its important to note that paramagnets don’t retain their magnetic field after the external field is removed. So, at least conceptually, we see that the χM would be dependent on temperature for these materials. Couldn’t the same then be true for diamagnets? Apparently not. Do I know why? … Don’t be silly! Of-course I do! … but … erhhmmm … yes … I’d be happy to explain … but … ummm ….. my dog ate my explanation.
Like diamagnets, another interesting thing about paramagnets, bearing in mind the temperature effect, is that the magnetic field inside the paramagnet rises linearly with the external field without any kind of saturation point. This is quite different from ferromagnets.
C. Ferromagnetism
Ferromagnetic materials have a very high magnetic susceptibility, such that χM≫0. Like paramagnetic materials, these materials are attracted to externally applied magnetic fields – the field is reinforced inside the material resulting in a much stronger magnetic field. Like paramagnets, ferromagnets exhibit alignment of their atomic magnets, however unlike paramagnets, these atomic magnets form domains. Now, just because a material is ferromagnetic does not mean that it does not exhibit paramagnetic and diamagnetic properties. It does. But the ferromagnetic domain alignments have such a significant and dominant effect that they eclipse the other effects. Table 3 demonstrates some magnetic susceptibilities associated with ferromagnets.
| Material | χM |
|---|---|
| Pure Iron | >200000 |
| Pure Nickel | >600 |
| Pure Cobalt | >250 |
| Steel (0.9% C) | ≈100 |
Domains
Even when no external field is applied, the atomic magnets in a ferromagnet tend to align themselves, forming what are called domains. A material will typically have randomly aligned domains and so their net magnetic field remains at zero without an external field. Applying an external field aligns these domains. In-fact, its possible to hear this domain alignment via the Barkhausen effect – this effect pertains to the magneto-acoustic noise generated when domains suddenly switch direction. The Barkhausen effect does not occur with paramagnetic or diamagnetic materials.
Its possible for a ferromagnetic material to have a magnetic field of its own, if a sufficient number of domains in the material are aligned in one particular direction. Application of an external magnetic field can sometimes cause this less than random persistent alignment. And so some domains may unalign, but some will continue to exist in their aligned state after the field is removed. This is very much NOT the case in paramagnets. Banging on a magnet can also disrupt domains and unalign them.
Temperature
Ferromagnetic susceptibility is effected by temperature – higher temperatures increase thermal agitation of the molecules in the material and thus its harder to align these molecules via an external field. Of-course, over time, thermal agitation, even without increasing temperatures, causes misalignment of atomic magnets and domains and so a magnet tends to lose its magnetic properties. At some point, increasing temperature causes a ferromagnetic material to loses its properties and becomes paramagnetic. The temperature at which this happens is known as the Curie temperature. At the curie temperature the individual atomic magnets begin to precess around their nominal alignment. This precession effect explains why magnet properties return (partially) when the temperature drops below the curie point. Of-course, despite this, my suspicion is that the magnet will eventual lose its magnet properties permanantly if this process is repeated indefinitely.
Magnets are commonly made through a process of heating and cooling of ferromagnetic materials in the presence of an external magnetic field. This process allows quantitatively more and stronger domain alignment producing longer lasting magnets.
Magnetostriction
An external magnetic field applied to any magnetic material causes magnetostriction. The term magnetostriction is defined as a change in shape of a material under the influence of a magnetic field. Because atoms and molecules are physically being realigned and moved, friction and heat are often associated with magentostriction. At its extreme it can cause structural stress on a material and can effect the material’s longevity – it can also fundamentally and permanently alter magnetic and physical properties.
My sense is that magnetostriction effects must also occur in diamagnetic and paramagnetic materials, however, because of the very weak magnetic susceptibilities of these materials and our limited capabilities generating strong enough magnetic fields, the resultant shape changes would be highly limited.
We can hear the effect of magnetostriction in transformer circuits as a buzzing sound. This is a result of the changing magnetic field causing vibrations of not only the transformer coils (via the lorentz force) but also of the core as it extends or contracts, with the changing magnetic field, as a result of magnetostriction.
Dipole Moment
As we discussed earlier, the magnetic susceptibility of a material, χM is a measure of its response to an external magnetic field. Typically a magnetic field (independent of medium) is denoted by the variable H. The value H provides a method of distinguishing an external material independent driving field from that produced by the material in response to this driving field.
We also looked at the idea of nuclear spin, electron spin and orbital motion and their link to magnetism via a concept called magnetic dipole moment. This magnetic moment,m⃗ , is a result of intrinsic atomic properties.
Current Loops
We can model these properties in terms of atomic currents . For orbital electron motion, we can use a simplified atomic model, i.e. Bohr’s model, to do this dipole moment calculation using the equation below. The moment is measured in units [A/m2].
Magnetic Dipole Moment (Planar Case)
\vec{m} =I \cdot \vec{A} = N \cdot I \cdot \vec{A} \tag{1} Magnetic Dipole Moment (General Case)
\vec{m} =N \cdot \frac{1}{2} \int (\vec{r} \times \vec{J}) \cdot dV \tag{1A} Figure 8 below elaborates on variables and their definitions. The value, I is a representation of the current produced by an electron (with charge q) moving at a particular velocity, v, around its orbital in units of meters/sec [m/s]. In a sense the atom can be thought of as a coil. In fact, we can calculate the magnetic dipole moment for macro-sized objects (e.g. solenoids) in the same way. Notice that the area vector A is normal to the area enclosed by the current loop and is measured in [m2]. And for the total current we must multiply the current by the number of turns in the loop.
We can also represent magnetic moment in a more arbitrary volumetric case where current distribution is more complex (i.e. current density J). That is, in cases where the electron or charge motion is not confined to a plane (equation 1a). The magnetic moment is measured in units of: [A⋅m2]. The value r is the vector position pointing to the location of the volume element, dV and J the current density at that location. J, is defined as the amount of current per unit area (measured in units [A/m2]. This area is the cross-sectional area through which the current is passing.
The velocity, v is intrinsic to the energy of the particular orbital – note that an electron does NOT move in the traditional sense of the word, so this intrinsic velocity is linked to the fundamental frequency of oscillation of this electron wave at the given orbital. Bearing this in mind, we can, represent the current as shown in the following equation. Note that we can represent the distance that the electron travels in unit time using the variable d. If we imagine the orbital as being circular (as shown in figure 8), the vector A is the normal area vector of this circular path. Because we know the shape of the orbital we can determine the value d – in the simple circular case this would simply be some fraction of the circumference of the orbit – with the radius given by r and measured in meters [m].
Atomic Current
I = \frac{q}{t} = q \cdot \frac{\vec{v}}{d} = q \cdot \frac{\vec{v}}{2\pi\vec{r}} \tag{2}But hold on a sec!!!! HOW DO YOU KNOW THE ORBITAL IS ROUND???? … excellent question! We don’t. But it is the basis of a more general representation of magnetic moment. In-fact, we’ll use the idea of angular momentum (L) of an electron to help us out … remember? No? Well here’s a quick reminder:
Angular Momentum
\vec{L} = \vec{r} \times \mathcal{M}\vec{v} \;\;\; \text{OR} \;\;\; L = r\mathcal{M}v \tag{3}The mass of the electron is given by M = 9.109 x 10−31 [Kg], and the momentum, L is in units of [Kg⋅m2⋅s−1] = [N⋅m⋅s] = [J⋅s]. We can go further and define angular momentum at the atomic scale as the quantized orbital angular momentum. The value n is the principal quantum number (electron shell in which the electron resides) and h is Planck’s constant.
Orbital Angular Momentum
L = n\;\frac{h}{2\pi} = n \hbar \tag{3A}Now, given this added information, at the atomic scale, we can represent the electron magnetic dipole moment (the moment resulting from electron spin) by using the Bohr Magnetron. Here’s a formula to show you how from equations 1, 2 and 3.
Magnetic Dipole Moment
\vec{m} = I \cdot \vec{A} \tag{1} \begin{aligned}
\vec{m} &= q \cdot \frac{\vec{v}}{2\pi\vec{r}} \cdot \vec{A}\\ \; \\
&=q \cdot \frac{v}{2\pi r} \cdot A \cdot \hat{A}\\ \; \\
&= q \cdot \frac{r \mathcal{M} v}{2\pi \mathcal{M} r^2} \cdot \pi r^2 \cdot \hat{A}\\ \; \\
&=q \cdot \frac{L}{2 \mathcal{M}} \cdot \hat{A}
\end{aligned}Bohr Magnetron
\vec{m} =\frac{q \hbar}{2 \mathcal{M}} \cdot \hat{A}Notice that q = 1.602 x 10−19 [C] is the elementary charge of an electron in coulombs, ℏ = 6.626 x 10−34/2π [J/s] is a modified Planck’s Constant in joules per second and M = 9.109 x 10−31 [Kg] is the mass of an electron. The resultant electron magnetic moment is thus m = 9.27400968 x 10−24 [J/T].
Magnetic Charges
Another method of modeling properties that collectively result in the magnetic dipole moment of an atom is to model an atom not as a current loop, but as two magnetic charges or monopoles. This is sometimes referred to as the Gilbert Model. Clearly, magnetic monopoles don’t exist. But sometimes its mathematically more convenient to pretend that they do. This is especially true when trying to determine the magnetic field at some distance from the given dipole. Using this representation we can arrive at the same end results for the magnetic dipole moment, as we would if we were to model as current loops.
Magnetic Field
\vec{H} = \frac{1}{r^3}\;\left(3\;(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m} \right) \tag{3B}For now, I won’t get into the derivation of this magnetic charge model (though one is provided in: https://www.physicsinsights.org/dipole_field_1.html). However, given that the magnetic dipole moment is known and the dipoles are small enough that their shape has no effect on the neighbouring dipoles, then we can construct a representation of the magnetic field, H at some distance, r, as shown in the equation above. The units for this field work out to [A/m]. Note that the unit vector r hat is unit-less. We’ll discuss magnetic fields in more detail in the sections below.
Magnetization
We can also quantize the magnetic moment per unit volume of material by representing the total magnetic moment using the term: Magnetization. Magnetization is often denoted with the vector M and represents the total magnetic moment per unit volume of the material – in some sense it represents the density of the magnetic dipole moments in the material. Note that magnetization is measured in amperes per metre [A/m]. Equation 3 below provides a definition for magnetization. In the equation, N represents the average number of magnetic moments in the material (this would be proportional to the number of atoms) and V represents the material’s volume. Notice that sometimes, the value N, is absorbed by m ⋅ M – this may originate from the fact that it is sometimes represented as N current loops: m⋅M=N ⋅I ⋅A.
Magnetization
\vec{M\;\;} = \frac{N}{V} \cdot \vec{m} = \sum_{i}^{N}{\frac{1}{V} \cdot \vec{m}_{\; i}} = \frac{1}{V} \cdot \vec{m} \tag{4}Normally, in diamagnets and paramagnets, the random orientation of individual magnetic moments means that the overall magnetization of these materials is zero. This changes with the application of an external material independent magnetic field, H. The magnetization then becomes linearly proportional to the external field, with the proportionality constant being the magnetic susceptibility, χM. The next equation demonstrates this. In the equation, χM is sometimes represented in terms of the relative magnetic permeability, μr such that χM=μr−1. And μr, itself, can be combined with another value μ0, the permeability of free space (vacuum), and represented as μ.
Magnetization & Magnetic Field
\vec{M} = \chi_{\small{M}} \cdot {\vec{H}} \tag{4A}Now the interesting thing is that ferromagnetic materials already contain oriented magnetic dipoles in the form of domains. These domains have all their dipoles already aligned in a particular direction. This means that there is already an existing net magnetization for ferromagnets. And in-fact, ferromagnets exhibit hysteresis. Let’s go on and discuss hysteresis. But before that, a few additional quick notes about magnetic fields, magnetization, and magnetic susceptibility
Magnetic Fields
Before getting to hysteresis, I have few more quick notes about magnetic fields. The actual magnetic response field inside a material, B is a combination of H and M as shown below in equation 5.
Magnetic Field Inside a Material
\vec{B} = \mu_{0} \cdot (\vec{H} + \vec{M}) \tag{5}Now, confusingly, our H is also known as the magnetic field strength/intensity and is measured in amperes per meter [A/m]. And likewise, B is sometimes called the magnetic flux density and is measured in Teslas [T]. Typically when generating our applied material independent field, we utilize coils. These coils have a certain current density – which is why our field strength units are [A/m]. As the external material independent driving field (H) enters a material, the material properties have an amplifying or attenuating effect on the field as a result of their permeability or magnetization. To distinguish this new field we denote it the flux density.
There is another important quantity, called the magnetic flux, ϕB. This is distinct from our B in that it is a scalar quantity that represents the quantity of B going through some surface in our system. In fact we can represent this mathematically as shown in the following equation.
Magnetic Flux
\phi_{B} = \int_{S}{\vec{B} \cdot d\vec{A}} \tag{6}The value, S, is the surface through which the B field is penetrating. The vector dA is a small surface element pointing in the normal direction. The dot product of the two means that only those components of the B field that are parallel to the normal of the surface are contributors to the flux. The flux, like the magnetic flux density is measured in [T].
Incidentally, if a magnetic dipole is put inside an external magnetic field, then we can associate a magnetic torque with it (figure 9). As the external field passes through the dipole it takes the form of a material dependent external field, B acting on the intrinsic moment, m. This results in equation 7 below. I’ve used B to represent the external field specifically because the torque is dependent on the actual response field at the dipole location. Let’s suppose that the dipole is completely immersed in iron. The response field, B, will be much much greater in magnitude than the material independent H field and so will produce a much greater torque on the dipole.
Magnetic Torque
\vec{\tau}_{\small{M}} = \vec{m} \times \vec{B} = V \cdot \vec{M} \times \vec{B} \tag{7}For macroscopic materials, the dipole moment would be replaced with the magnetization of the material – that is, the cumulative effect of individual moments per unit volume. This idea can be taken even further and develop an expression for the magnetic potential energy of the dipole under the same external material dependant field B as shown below:
Magnetic Potential Energy
U_{\small{M}} = - \vec{m} \cdot \vec{B} = - V \cdot \vec{M} \cdot \vec{B} \tag{8}Now from this we see that when them⃗ is aligned with the field external to the dipole, B, we get the minimum potential energy from the system. And the maximum potential energy when the two vectors are facing opposite directions. This chimes well with what we see happen with real magnets, i.e. north is attracted to south and vice versa. As we see from the equation, clearly if we are talking about macroscopic materials the potential energy would be computed based on the magnetization of the material.
We can take all this one small step further and talk about the energy density. This energy density is a concept strongly related to Poynting’s Theorem. Poynting’s theorem is an analogue to the work-energy theorem in classical mechanics which states that the average sum of the total potential and kinetic energy in a system is conserved. Let’s start by looking at our energy UM once more.
Magnetic Potential Energy
U_{\small{M}} = \int_{V}{u_{\small{M}} \cdot dV} \tag{8A}The value uM is our volumetric energy density. I’m going to leave this topic for another time because Poynting’s theorem requires a bit of effort to write about and is more relevant to an article I will write at a later time.
Magnetic Force
Ok. Let’s stop for a moment. If we look at the real world, the only measurable quantity is force. It is also the most important quantity when designing real machines. We can represent magnetic force as F ⋅ M. This magnetic force can be defined in one of two ways, via the lorentz force (equation 9) or via the gradient of the potential energy (equation 9A):
Magnetic Force
\vec{F}_{\small{M}} = Q \cdot \vec{v} \times \vec{B} = \vec{J} \times \vec{B} \tag{9}\vec{F}_{\small{M}} = \nabla (\vec{m} \cdot \vec{B}) \tag{9A}Lets consider for an instant that a material is immersed in an external magnetic field, H. This external field will cause a material response B. In these equations, F⋅M is the force on the material, applied over the entire volume of the material. The ∇ in this particular case, represents the gradient. In the first representation, we need to link the concept of current (I or J), charge (Q) and velocity (v) from electrostatics to determine this force. In the second representation we use the concept of magnetic moments. In practice, a solenoid with a known current and thus a known magnetic field, H can be used as a basis for determining a magnetic material’s magnetization (and moments).
Hysteresis
Figure 10 below demonstrates a property called hysteresis, a property which only exists in ferromagnetic materials. The hysteresis curve below models the response of a ferromagnetic material to a material independent externally applied field, H. Initially the material has never been exposed to a magnetic field. This leads us to position (a) in the curve. As the external magnetic field increases, magnetic domains in side the material align with and reinforce the external field. If we were to measure the field, B inside the material we would see it rise along with H along what’s known as the virginal curve. Yes, yes, I know … physicists have dirty minds …
At some point, all the domains and dipoles in the ferromagnetic material have been aligned with the external H field. At this point, (b), the material is said to be saturated and the B increases linearly with the external applied field, H … so the material starts to behave more like a diamagnetic or paramagnetic material.
At this point if we remove the external field, the internal magnetic field drops off to point (c) in the curve. At this point the material has become magnetized and is said to have a magnetic remanence. We can link this to the idea of magnetization. The net magnetic dipole moments of the material without an external field are non-zero at this remanence point.
In order to remove this remnant magnetism we must now apply a field in the opposite direction … so if we continue, the B, will continue to drop and at some point will reach a value of zero. This point, (d), is known as the coercivity of the material. At this point continuing to increase H in the negative direction eventually leads to saturation again (point (e)). And again the material starts responding linearly to increases in H. We can then continue this process going through points (f) and (g) by changing the direction of the external field again.
De-magnetization
But, how do we now completely de-magnetize the material again and return to our virginal curve? Well, one way would be to take a hammer and hit the material, randomizing the magnetic dipole moments and their associated domains. … Me big and strong! Me smash!! … very caveman-esque .. no? There is an alternative, a hammer can after all damage the material. You could try heating it up-to its curie temperature and beyond … this would effectively scramble its domains. This effect can be temporary unless the material is hot enough … I’m not sure why it is temporary, I think it has something to do with structural bonding mechanisms between atoms and molecules. Heating again can cause damage to the material.
Applying an oscillating H field that oscillates from negative to positive and starts from very high amplitude and drops to very low amplitude can effectively demagnetize an object. Don’t understand? … well, here’s a picture (figure 11). This is the technique libraries and stores sometimes use to demagnetize their theft protection mechanism when you buy something.
D. Ferrimagnetism
Ferrimagnetic materials are materials that exhibit a particular structure and alignment of their atomic magnets. Figure 12 shows a representation of the atomic scale magnets of a ferrimagnetic material. Like, ferromagnets and paramagnets the magnetic susceptibility of these materials is such that: χm>0. These materials, like ferromagnets, have a curie temperature – and their χm value is also temperature dependent. Ferrimagnets still exhibit ferromagnetic properties. I have very little idea of why you’d want a ferrimagnetic material or of how they are used.
E. Antiferromagnetism
The properties of an antiferromagnetic material are similar to those of a ferrimagnetic material. Figure 13 shows the ordering of atomic magnets in this material. The key difference lies in the strength of the opposing dipoles. God knows what anti-ferromagnets are useful for or why anyone really cares.
Well, that’s it for now folks!
