The hysteresis curve
We have concluded from our theoretical analysis that a ferromagnetic material should spontaneously become magnetized below a certain temperature so that all the magnetism would be in the same direction. But we know that this is not true for an ordinary piece of unmagnetized iron. Why isn’t all iron magnetized? We can explain it with the help of Fig. 37–4. Suppose the iron were all a big single crystal of the shape shown in Fig. 37–4(a) and spontaneously magnetized all in one direction. Then there would be a considerable external magnetic field, which would have a lot of energy. We can reduce that field energy if we arrange that one side of the block is magnetized “up” and the other side magnetized “down,” as in Fig. 37–4(b). Then, of course, the fields outside the iron would extend over less volume, so there would be less energy there. Ah, but wait! In the layer between the two regions we have up-spinning electrons adjacent to down-spinning electrons. But ferromagnetism appears only in those materials for which the energy is reduced if the electrons are parallel rather than opposite. So, we have added some extra energy along the dotted line in Fig. 37–4(b); this energy is sometimes called wall energy. A region having only one direction of magnetization is called a domain. At the interface—the “wall”—between two domains, where we have atoms on opposite sides which are spinning in different directions, there is an energy per unit area of the wall. We have described it as though two adjacent atoms were spinning exactly opposite, but it turns out that nature adjusts things so that the transition is more gradual. But we don’t need to worry about such fine details at this point. Now the question is: When is it better or worse to make a wall? The answer is that it depends on the size of the domains. Suppose that we were to scale up a block so that the whole thing was twice as big. The volume in the space outside filled with a given magnetic field strength would be eight times bigger, and the energy in the magnetic field, which is proportional to the volume, would also be eight times greater. But the surface area between two domains, which will give the wall energy, would be only four times as big. Therefore, if the piece of iron is big enough, it will pay to split it into more domains. This is why only the very tiny crystals can have but a single domain. Any large object—one more than about a hundredth of a millimeter in size—will have at least one domain wall; and any ordinary, “centimeter-size” object will be split into many domains, as shown in the figure. Splitting into domains goes on until the energy needed to put in one extra wall is as large as the energy decrease in the magnetic field outside the crystal. Actually nature has discovered still another way to lower the energy: It is not necessary to have the field go outside at all, if a little triangular region is magnetized sideways, as in Fig. 37–4(d).3 Then with the arrangement of Fig. 37–4(d) we see that there is no external field, but instead only a little more domain wall. But that introduces a new kind of problem. It turns out that when a single crystal of iron is magnetized, it changes its length in the direction of magnetization, so an “ideal” cube with its magnetization, say, “up,” is no longer a perfect cube. The “vertical” dimension will be different from the “horizontal” dimension. This effect is called magnetostriction. Because of such geometric changes, the little triangular pieces of Fig. 37–4(d) do not, so to speak, “fit” into the available space anymore—the crystal has got too long one way and too short the other way. Of course, it does fit, really, but only by being squashed in; and this involves some mechanical stresses. So, this arrangement also introduces an extra energy. It is the balance of all these various energies which determines how the domains finally arrange themselves in their complicated fashion in a piece of unmagnetized iron. Now, what happens when we put on an external magnetic field? To take a simple case, consider a crystal whose domains are as shown in Fig. 37–4(d). If we apply an external magnetic field in the upward direction, in what manner does the crystal become magnetized? First, the middle domain wall can move over sideways (to the right) and reduce the energy. It moves over so that the region which is “up” becomes bigger than the region which is “down.” There are more elementary magnets lined up with the field, and this gives a lower energy. So, for a piece of iron in weak fields—at the very beginning of magnetization—the domain walls begin to move and eat into the regions which are magnetized opposite to the field. As the field continues to increase, a whole crystal shifts gradually into a single large domain which the external field helps to keep lined up. In a strong field the crystal “likes” to be all one way just because its energy in the applied field is reduced—it is no longer merely the crystal’s own external field which matters. What if the geometry is not so simple? What if the axes of the crystal and its spontaneous magnetization are in one direction, but we apply the magnetic field in some other direction—say at $45^\circ$? We might think that domains would reform themselves with their magnetization parallel to the field, and then as before, they could all grow into one domain. But this is not easy for the iron to do, for the energy needed to magnetize a crystal depends on the direction of magnetization relative to the crystal axis. It is relatively easy to magnetize iron in a direction parallel to the crystal axes, but it takes more energy to magnetize it in some other direction—like $45^\circ$ with respect to one of the axes. Therefore, if we apply a magnetic field in such a direction, what happens first is that the domains which point along one of the preferred directions which is near to the applied field grow until the magnetization is all along one of these directions. Then with much stronger fields, the magnetization is gradually pulled around parallel to the field, as sketched in Fig. 37–5. In Fig. 37–6 are shown some observations of the magnetization curves of single crystals of iron. To understand them, we must first explain something about the notation that is used in describing directions in a crystal. There are many ways in which a crystal can be sliced so as to produce a face which is a plane of atoms. Everyone who has driven past an orchard or vineyard knows this—it is fascinating to watch. If you look one way, you see lines of trees—if you look another way, you see different lines of trees, and so on. In a similar way, a crystal has definite families of planes that hold many atoms, and the planes have this important characteristic (we consider a cubic crystal to make it easier): If we observe where the planes intersect the three coordinate axes—we find that the reciprocals of the three distances from the origin are in the ratio of simple whole numbers. These three whole numbers are taken as the definition of the planes. For example, in Fig. 37–7(a), a plane parallel to the $yz$-plane is shown. This is called a $[100]$ plane; the reciprocals of its intersection of the $y$- and $z$-axes are both zero. The direction perpendicular to such a plane (in a cubic crystal) is given the same set of numbers. It is easy to understand the idea in a cubic crystal, for then the indices $[100]$ mean a vector which has a unit component in the $x$-direction and none in the $y$- or $z$-directions. The $[110]$ direction is in a direction $45^\circ$ from the $x$- and $y$-axes, as in Fig. 37–7(b); and the $[111]$ direction is in the direction of the cube diagonal, as in Fig. 37–7(c). Returning now to Fig. 37–6, we see the magnetization curves of a single crystal of iron for various directions. First, note that for very tiny fields—so weak that it is hard to see them on the scale at all—the magnetization increases extremely rapidly to quite large values. If the field is in the $[100]$ direction—namely along one of those nice, easy directions of magnetization—the curve goes up to a high value, curves around a little, and then is saturated. What happened is that the domains which were already there are very easily removed. Only a small field is required to make the domain walls move and eat up all of the “wrong-way” domains. Single crystals of iron are enormously permeable (magnetic sense), much more so than ordinary polycrystalline iron. A perfect crystal magnetizes extremely easily. Why is it curved at all? Why doesn’t it just go right up to saturation? We are not sure. You might study that some day. We do understand why it is flat for high fields. When the whole block is a single domain, the extra magnetic field cannot make any more magnetization—it is already at $M_{\text{sat}}$, with all the electrons lined up. Now, if we try to do the same thing in the $[110]$ direction—which is at $45^\circ$ to the crystal axes—what will happen? We turn on a little bit of field and the magnetization leaps up as the domains grow. Then as we increase the field some more, we find that it takes quite a lot of field to get up to saturation, because now the magnetization is turning away from an “easy” direction. If this explanation is correct, the point at which the $[110]$ curve extrapolates back to the vertical axis should be at $1/\sqrt{2}$ of the saturation value. It turns out, in fact, to be very, very close to $1/\sqrt{2}$. Similarly, in the $[111]$ direction—which is along the cube diagonal—we find, as we would expect, that the curve extrapolates back to nearly $1/\sqrt{3}$ of saturation. Figure 37–8 shows the corresponding situation for two other materials, nickel and cobalt. Nickel is different from iron. In nickel, it turns out that the $[111]$ direction is the easy direction of magnetization. Cobalt has a hexagonal crystal form, and people have botched up the system of nomenclature for this case. They want to have three axes on the bottom of the hexagon and one perpendicular to these, so they have used four indices. The $[0001]$ direction is the direction of the axis of the hexagon, and $[1010]$ is perpendicular to that axis. We see that crystals of different metals behave in different ways. Now we must discuss a polycrystalline material, such as an ordinary piece of iron. Inside such materials there are many, many little crystals with their crystalline axes pointing every which way. These are not the same as domains. Remember that the domains were all part of a single crystal, but in a piece of iron there are many different crystals with axes at different orientations, as shown in Fig. 37–9. Within each of these crystals, there will also generally be some domains. When we apply a small magnetic field to a piece of polycrystalline material, what happens is that the domain walls begin to move, and the domains which have a favorable direction of easy magnetization grow larger. This growth is reversible so long as the field stays very small—if we turn the field off, the magnetization will return to zero. This part of the magnetization curve is marked $a$ in Fig. 37–10. For larger fields—in the region $b$ of the magnetization curve shown—things get much more complicated. In every small crystal of the material, there are strains and dislocations; there are impurities, dirt, and imperfections. And at all but the smallest fields, the domain wall, in moving, gets stuck on these. There is an interaction energy between the domain wall and a dislocation, or a grain boundary, or an impurity. So when the wall gets to one of them, it gets stuck; it sticks there at a certain field. But then if the field is raised some more, the wall suddenly snaps past. So the motion of the domain wall is not smooth the way it is in a perfect crystal—it gets hung up every once in a while and moves in jerks. If we were to look at the magnetization on a microscopic scale, we would see something like the insert of Fig. 37–10. Now the important thing is that these jerks in the magnetization can cause an energy loss. In the first place, when a boundary finally slips past an impediment, it moves very quickly to the next one, since the field is already above what would be required for the unimpeded motion. The rapid motion means that there are rapidly changing magnetic fields which produce eddy currents in the crystal. These currents lose energy in heating the metal. A second effect is that when a domain suddenly changes, part of the crystal changes its dimensions from the magnetostriction. Each sudden shift of a domain wall sets up a little sound wave that carries away energy. Because of such effects, the second part of magnetization curve is irreversible, and there is energy being lost. This is the origin of the hysteresis effect, because to move a boundary wall forward—snap—and then to move it backward—snap—produces a different result. It’s like “jerky” friction, and it takes energy. Eventually, for high enough fields, when we have moved all the domain walls and magnetized each crystal in its best direction, there are still some crystallites which happen to have their easy directions of magnetization not in the direction of our external magnetic field. Then it takes a lot of extra field to turn those magnetic moments around. So the magnetization increases slowly, but smoothly, for high fields—namely in the region marked $c$ in the figure. The magnetization does not come sharply to its saturation value, because in the last part of the curve the atomic magnets are turning in the strong field. So we see why the magnetization curve of ordinary polycrystalline materials, such as the one shown in Fig. 37–10, rises a little bit and reversibly at first, then rises irreversibly, and then curves over slowly. Of course, there is no sharp break-point between the three regions—they blend smoothly, one into the other. It is not hard to show that the magnetization process in the middle part of the magnetization curve is jerky—that the domain walls jerk and snap as they shift. All you need is a coil of wire—with many thousands of turns—connected to an amplifier and a loudspeaker, as shown in Fig. 37–11. If you put a few silicon steel sheets (of the type used in transformers) at the center of the coil and bring a bar magnet slowly near the stack, the sudden changes in magnetization will produce impulses of emf in the coil, which are heard as distinct clicks in the loudspeaker. As you move the magnet nearer to the iron you will hear a whole rush of clicks that sound something like the noise of sand grains falling over each other as a can of sand is tilted. The domain walls are jumping, snapping, and jiggling as the field is increased. This phenomenon is called the Barkhausen effect. As you move the magnet even closer to the iron sheets, the noise grows louder and louder for a while but then there is relatively little noise when the magnet gets very close. Why? Because nearly all the domain walls have moved as far as they can go. Any greater field is merely turning the magnetization in each domain, which is a smooth process. If you now withdraw the magnet, so as to come back on the downward branch of the hysteresis loop, the domains all try to get back to low energy again, and you hear another rush of backward-going jerks. You can also note that if you bring the magnet to a given place and move it back and forth a little bit, there is relatively little noise. It is again like tilting a can of sand—once the grains shift into place, small movements of the can don’t disturb them. In the iron the small variations in the magnetic field aren’t enough to move any boundaries over any of the “humps.”