Magnetic fields in a vacuum

First, some definitions. When a current i passes through a long, empty coil of n turns and length L, as in Figure 15.1, a magnetic field is generated. The magnitude of the field, H, is given by Ampère’s¹ law as

Figure 15.1 A solenoid creates a magnetic field H; the flux lines indicate the field strength.

(15.1)

and thus has units of amps/meter (A/m). The field has both magnitude and direction—it is a vector field.

Magnetic fields exert forces on a wire carrying an electric current. A current i flowing in a single loop of area S generates a dipole moment m, where

(15.2)

with units A.m², and it too is a vector with a direction normal to the plane of S (Figure 15.2). If the loop is placed at right angles to the field H it feels a torque T (units: Newton.meter, or N.m) of

Figure 15.2 Definition of magnetic moment and moment-field interaction.

(15.3)

where μ_o is called the permeability of a vacuum, μ_o = 4π × 10⁻⁷ henry/meter (H/m). To link these we define a second measure of the magnetic field, one that relates directly to the torque it exerts on a unit magnetic moment. It is called the magnetic induction or flux density, B, and for a vacuum or non-magnetic materials it is

(15.4)

Its units are tesla², so 1 tesla is 1 HA/m². A magnetic induction B of 1 tesla exerts a torque of 1 N.m on a unit dipole at right angles to the field H.

Magnetic fields in materials

If the space inside the coil of Figure 15.2 is filled with a material, as in Figure 15.3, the induction within it changes. This is because its atoms respond to the field by forming little magnetic dipoles in ways that are explained in Section 15.4. The material acquires a macroscopic dipole moment or magnetisation, M (its units are A/m, like H). The induction becomes

Figure 15.3 A magnetic material exposed to a field H becomes magnetised, concentrating the flux lines.

(15.5)

The simplicity of this equation is misleading, since it suggests that M and H are independent; in reality M is the response of the material to H, so the two are coupled. If the material of the core is ferro-magnetic, the response is a very strong one and it is nonlinear, as we shall see in a moment. It is usual to rewrite equation (15.5) in the form

where μ_R is called the relative permeability, and like the relative permittivity (the dielectric constant) of Chapter 14, it is dimensionless. The magnetisation, M, is thus

(15.6)

where χ is the magnetic susceptibility. Neither μ_R nor χ are constants—they depend not only on the material but also on the magnitude of the field, H, for the reason just given.

Nearly all materials respond to a magnetic field by becoming magnetised, but most are paramagnetic with a response so faint that it is of no practical use. A few, however, contain atoms that have large dipole moments and have the ability to spontaneously magnetise—to align their dipoles in parallel—as electric dipoles do in ferro-electric materials. These are called ferro-magnetic and ferri-magnetic materials (the second one is called ferrites for short), and it is these that are of real practical use.

Magnetisation decreases with increasing temperature. Just as with ferro-electrics, there is a temperature, the Curie temperature T_c, above which it disappears, as in Figure 15.4. Its value for the materials we shall meet here is well above room temperature (typically 300–500 °C) but making magnets for use at really high temperatures is a problem.

Figure 15.4 Saturation magnetisation decreases with temperature, falling to zero at the Curie temperature, T_c.

Measuring magnetic properties

Magnetic properties are measured by plotting an M–H curve. It looks like Figure 15.5. If an increasing field H is applied to a previously demagnetised sample, starting at A on the figure, its magnetisation increases, slowly at first and then faster, following the broken line, until it finally tails off to a maximum, the saturation magnetisation M_s at the point B. If the field is now backed off, M does not retrace its original path, but retains some of its magnetisation so that when H has reached zero, at the point C, some magnetisation remains: it is called the remanent magnetisation or remanence M_R and is usually only a little less than M_s. To decrease M further we must increase the field in the opposite direction until M finally passes through zero at the point D when the field is −H_c, the coercive field, a measure of the resistance to demagnetisation. Some applications require H_c to be as high as possible, others as low a possible. Beyond point D the magnetisation M starts to increase in the opposite direction, eventually reaching saturation again at the point E. If the field is now decreased again, M follows the curve through F and G back to full forward magnetic saturation again at B to form a closed M–H circuit called the hysteresis loop.

Figure 15.5 A hysteresis curve, showing the important magnetic properties.

Magnetic materials are characterised by the size and shape of their hysteresis loops. The initial segment AB is called the initial magnetisation curve and its average slope (or sometimes its steepest slope) is the magnetic susceptibility, χ. The other key properties—the saturation magnetisation M_s, the remanence M_R and the coercive field H_c—have already been defined. Each full cycle of the hysteresis loop dissipates an energy per unit volume equal to the area of the loop multiplied by μ_o, the permeability of a vacuum. This energy appears as heat (it is like magnetic friction). Many texts plot not the M–H curve, but the curve of inductance B against H. Equation (15.5) says that B is proportional to (M + H), and the value of M for any magnetic materials worthy of the name is very much larger than the H applied to it, so B ≈ μ_oM and the B–H curve of a ferro-magnetic material looks very like its M–H curve (it’s just that the M-axis has been scaled by μ_o).

There are several ways to measure the hysteresis curve, one of which is sketched in Figure 15.6. Here the material forms the core of what is, in effect, a transformer. The oscillating current through the primary input coil creates a field H that induces magnetisation M in the material of the core, driving it round its hysteresis loop. The secondary coil picks up the inductance, from which the instantaneous state of the magnetisation can be calculated, mapping out the loop.

Figure 15.6 Measuring the hysteresis curve.

Magnetic materials differ greatly in the shape and area of their hysteresis loop, the greatest difference being that between soft magnets, which have thin loops, and hard magnets, which have fat ones, as sketched in Figure 15.7. In fact the differences are much greater than this figure suggests: the coercive field H_c (which determines the width of the loop) of hard magnetic materials like Alnico is greater by a factor of about 10⁵ than that of soft magnetic materials like silicon–iron. More on this in the next two sections.

Figure 15.7 Hysteresis loops. (a) A fat loop typical of hard magnets. (b) A thin, square loop typical of soft magnets.

The remanence–coercive field chart

The differences between families of soft and hard magnetic materials are brought out by Figure 15.8. The axes are remanent magnetisation M_R and coercive field H_c. The saturation magnetisation M_s, more relevant for soft magnets, is only slightly larger than M_R, so an M_s–H_c chart looks almost the same as this one. There are 12 district families, each enclosed in a colored envelope. The members of each family, shown as smaller ellipses, have unhelpful tradenames (such as ‘H FerriteYBM-1A’) so they are not individually labeled³. Soft magnets require high M_s and low H_c; they are the ones on the left, with the best near the top. Hard magnets must hold their magnetism, requiring a high H_c, and to be powerful they need a large M_R; they are the ones on the right, with the best again at the top.

Figure 15.8 Remanent magnetisation and coercive force. Soft magnetic materials lie on the left, hard magnetic materials on the right. The red and purple envelopes enclose electrically conducting materials, the yellow ones enclose insulators.

In many applications the electrical resistivity is also important because of eddy-current losses, described later. Changing magnetic fields induce eddy currents in conductors but not in insulators. The red and purple ellipses on the chart enclose metallic ferro-magnetic materials, which are good conductors. The yellow ones describe ferrites, which are insulators.

The saturation magnetisation–susceptibility chart

This is the chart for selecting soft magnetic materials (Figure 15.9). It shows the saturation magnetisation, M_s—the ultimate degree to which a material can concentrate magnetic flux—plotted against its susceptibility, χ, which measures the ease with which it can be magnetised. Many texts use, instead, the saturation inductance B_s and maximum relative permeability, μ_R. They are shown on the other axes of the chart. As in the first chart, materials that are electrical conductors are enclosed in red or purple ellipses, those that are insulators, yellow ones.

Figure 15.9 Saturation magnetisation and susceptibility for soft magnetic materials. The red and purple envelopes enclose electrically conducting materials, the yellow ones enclose insulators.

Domains

If atomic moments line up, shouldn’t every piece of iron, nickel or cobalt be a permanent magnet? Magnetic materials they are; magnets, in general, they are not. Why not?

A uniformly magnetised rod creates a magnetic field, H, like that of a solenoid. The field has a potential energy

(15.7)

where the integral is carried out over the volume V within which the field exists. Working out this integral is not simple, but we don’t need to do it. All we need to note is that the smaller is H and the smaller the volume V that it invades, the smaller is the energy. If the structure can arrange its moments to minimise its H or get rid of it entirely (remembering that the exchange energy wants neighbouring atom moments to stay parallel) it will try to do so.

Figure 15.12 illustrates how this can be done. The material splits up into domains, within which the atomic moments are parallel but with a switch of direction between mating domains to minimise the external field. The domains meet at domain walls, regions a few atoms thick in which the moments swing from the orientation of one domain to that of the other. Splitting into parallel domains of opposite magnetisation, as in (b) and (c), reduces the field a lot; adding caps magnetised perpendicular to both as in (d) kills it almost completely. The result is that most magnetic materials, unless manipulated in some way, adopt a domain structure with minimal external field—that is the same as saying that, while magnetic, they are not magnets.

Figure 15.12 Domains allow a compromise: the cancelation of the external field while retaining magnetization of the material itself. The arrows show the direction of magnetization.

How can they be magnetised? Placed in a magnetic field created, say, with a solenoid, the domains already aligned with the field have lower energy than those aligned against it. The domain wall separating them feels a force, the Lorentz⁵ force, pushing it in a direction to make the favorably oriented domains grow at the expense of the others. As they grow, the magnetisation of the material increases, moving up the M–H curve, finally saturating at M_s, when the whole sample is one domain oriented parallel to the field, as in Figure 15.13.

Figure 15.13 An applied field causes domain boundaries to move. At saturation the sample has become a single domain, as on the right. Domain walls move easily in soft magnetic materials. Hard magnetic materials are alloyed to prevent their motion.

The saturation magnetisation, then, is just the sum of all the atomic moments contained in a unit volume of material when they are all aligned in the same direction. If the magnetic dipole per atom is n_mm_B (where n_m is the number of unpaired electrons per atom and m_B is the Bohr magneton), then the saturation magnetisation is

(15.8)

where Ω is the atomic volume. Iron has the highest because its atoms have the most unpaired electrons and they are packed close together. Anything that reduces either one reduces the saturation magnetisation. Nickel and cobalt have lower saturation because their atomic moments are less; alloys of iron tend to be lower because the nonmagnetic alloying atoms dilute the magnetic iron. Ferrites have much lower saturation both because of dilution by oxygen and because of the partial cancelation of atomic moments sketched in Figure 15.11.

There are, nonetheless, good reasons for alloying and making iron into ferrites. One relates to the coercive field, H_c. A good permanent magnet retains its magnetisation even when placed in a reverse field, and this requires a high H_c. If domain walls move easily the material is a soft magnet with a low H_c; if the opposite, it is a hard one. To make a hard magnet it is necessary to pin the domain walls to stop them moving. Impurities, foreign inclusions, precipitates and porosity all interact with domain walls, tending to pin them in place (they act as obstacles to dislocation motion too, so magnetically hard materials are mechanically hard as well). And there are subtler barriers to domain-wall motion: magnetic anisotropy arising because certain directions of magnetisation in the crystal are more favourable than others, and even the shape and size of the specimen itself—small, single-domain magnetic particles are particularly resistant to having their magnetism reversed.

The performance of a hard magnet is measured by its energy product, roughly proportional to the area of the magnetisation loop. The higher the energy product, the more difficult it is to demagnetise it.

Soft magnetic devices

Electromagnets, transformers, electron lenses and the like have magnetic cores that must magnetise easily when the field switches on, be capable of creating a high flux density, yet lose their magnetism when the field is switched off. They do this by using soft, low H_c, magnetic materials, shown on the left of the chart of Figure 15.8. Soft magnetic materials are used to ‘conduct’ and focus magnetic flux. Figure 15.14 illustrates this. A current-carrying coil induces a magnetic field in the core. The field, instead of spreading in the way shown in Figure 15.3, is trapped in the extended ferro-magnetic circuit, as shown here. The higher the susceptibility χ, the greater is the magnetisation and flux density induced by a given current loop, up to a limit set by its saturation magnetisation M_s. The magnetic circuit conducts the flux from one coil to another, or from a coil to an air gap, where it is further enhanced by reducing the section of the core, where it is used to actuate or to focus an electron beam, as shown on the right of Figure 15.14. Thus, the first requirement of a good soft magnet is a high χ and a high M_s with a thin hysteresis loop like that sketched in Figure 15.15. The chart of Figure 15.9 shows these two properties. Permendur (Fe—Co—V alloys) and Metglas amorphous alloys are particularly good, but being expensive they are used only in small devices. Silicon–iron (Fe–1–4% Si) is much cheaper and easier to make in large sections; it is the staple material for large transformers.

Figure 15.14 Soft magnets ‘conduct’ magnetic flux and are concentrated if the cross-section of the magnet is reduced.

Figure 15.15 A characteristic hysteresis loop for a soft magnet.

Most soft magnets are used in AC devices, and then energy loss, proportional to the area of the hysteresis loop, becomes a consideration. Energy loss is of two types: the hysteresis loss already mentioned—it is proportional to switching frequency, f—and eddy current losses caused by the currents induced in the core itself by the AC field, proportional to f². In cores of transformers operating at low frequencies (meaning f up to 100 Hz or so) the hysteresis loss dominates, so for these we seek high M_s and a thin loop with a small area. Eddy-current losses are present but are suppressed when necessary by laminating: making the core from a stack of thin sheets interleaved with thin insulating layers that interrupt the eddy currents. At audio frequencies (f < 50 kHz) both types of loss become greater because they occur on every cycle and now there are more of them, requiring an even narrower loop and thinner laminates; here the more expensive Permalloys and Permendurs are used. At higher frequencies still (f in the MHz range) eddy-current loss, rising with f², becomes really serious and the only way to stop this is to use magnetic materials that are electrical insulators. That means ferrites (shown in yellow envelopes on both charts), even though they have a lower M_s. Above this (f > 100 MHz) only the most exotic ceramic magnets will work. Figure 15.15 summarises the loop characteristics for soft magnets; Table 15.1 lists the choices, guided by the charts.

Table 15.1 Materials for soft magnetic applications

Hard magnetic devices

Many devices use permanent magnets: fridge doors, magnetic clutches, loudspeakers and earphones, DC motors, even frictionless bearings. For these, the key property is the remanence, M_R, since this is the maximum magnetisation the material can offer without an imposed external field to keep it magnetised. High M_R, however, is not all. When a material is magnetised and the magnetising field is removed, the field of the magnet itself tries to demagnetise it. It is this demagnetising field that makes the material take up a domain structure that reduces or removes the net magnetisation of the sample. A permanent magnet must be able to resist being demagnetised by it own field. A measure of its resistance is its coercive field, H_c, and this too must be large. A high M_R and a high H_c means a fat hysteresis loop like that of Figure 15.16. Ordinary iron and steel can become weakly magnetised with the irritating result that your screwdriver sticks to screws and your scissors pick up paper-clips when you wish they wouldn’t, but neither are ‘hard’ in a magnetic sense. Here ‘hard’ means ‘hard to demagnetise’, and that is where high H_c comes in; it protects the magnet from itself.

Figure 15.16 A characteristic hysteresis loop for a hard magnet.

Hard magnetic materials lie on the right of the M_R−H_c chart of Figure 15.8. The workhorses of the permanent magnet business are the Alnicos and the hexagonal ferrites. An Alnico (there are many) is an alloy of aluminum, nickel, cobalt and iron—hence its name—with values of M_R as high as 10⁶ A/m and a huge coercive field: 5 × 10⁴ A/m. The hexagonal ferrites, too, have high M_R and H_c and, like the cubic soft-magnet ferrites, they are insulators. Being metallic, the Alnicos can be shaped by casting or by compacting and sintering powders; the ferrites can only be shaped by powder methods. Powders, of course, don’t have to be sintered—you can bond them together in other ways, and these offer opportunities. Magnetic particles mixed with a resin can be moulded to complex shapes. Bonding them with an elastomer gives magnets that are flexible or squashy—the magnetic seal of a fridge door is an example.

The chart of Figure 15.8 shows three families of hard magnets with coercive fields that surpass those of Alnicos and hexagonal ferrites. These, the products of research over the last 30 years, provide the intense permanent-magnet fields that have allowed the miniaturisation of earphones, microphones and motors. The oldest of these is the precious metal Pt–Co family but they are so expensive that they have now all but disappeared. They are outperformed by the cobalt–samarium family based on the inter-metallic compound SmCo₅, and they, in turn, are outperformed by the recent development of the neodymium–iron–boron family. None of these is cheap, but in the applications they best suit, the quantity of material is sufficiently small and the product value sufficiently high that this is not an obstacle. Table 15.2 lists the choices, guided by the charts.

Table 15.2 Materials for hard magnetic applications

Magnetic storage of information

Magnetic information storage requires hard magnets, but ones with an unusual loop shape: one that is rectangular (called ‘square’). The squareness means that a unit of the material can flip in magnetisation from +M_R to −M_R and back when exposed to fields above +H_c or −H_c. The unit thus stores one binary bit of information. The information is read by moving the magnetised unit past a read head, where it induces an electric pulse. The choice of the word ‘unit’ was deliberately vague because it can take several forms: discrete particles embedded in a polymer tape or disk, or as a thin magnetic film that has the ability to carry small stable domains called ‘bubbles’ that are magnetised in one direction, embedded in a background that is magnetised in another.

Figure 15.17 shows one sort of information storage system. A signal creates a magnetic flux in a soft magnetic core—often an amorphous metal alloy (AMA) because of its large susceptibility. The direction of the flux depends on the direction of the signal current; switching this reverses the direction of the flux. An air gap in the core allows field to escape, penetrating a tape or disk in which submicron particles of a hard magnetic material are embedded. When the signal is in one direction all particles in the band of tape or disk passing under the write-head are magnetised in one direction; when the signal is reversed, the direction of magnetisation is reversed. The same head is used for reading. When the tape or disk is swept under the air gap, the stray field of the magnetised bands induces a flux in the soft iron core and a signal in the coil, which is amplified and processed to read it.

Figure 15.17 A magnetic read/write head. The gap in the soft magnetic head allows flux to escape, penetrating the tape and realigning the particles.

Why use fine particles? It is because domain walls are not stable in sufficiently small particles: each particle is a single domain. If the particle is elongated it prefers to be magnetised parallel to its long axis. Each particle then behaves like a little bar magnet with a north (N) and a south (S) pole at either end. The field from the write-head is enough to flip the direction, turning an N–S magnetisation into one that is S–N, a binary response that is well suited to information storage. Rewritable tapes and disks use particles of Fe₂O₃ or of CrO₂, typically 0.1 µm long and with an aspect ratio of 10:1. Particles of hexagonal ferrites (MO)(Fe₂O₃)₆ have a higher coercive field, so a more powerful field is required to change their magnetisation—but having done so, it is hard to erase it. This makes them the best choice for read-only applications like the identification strip of credit and swipe cards.