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Unformatted text preview: 37 Magnetic Materials 37—1 Understanding ferromagnetism In this chapter we will discuss the behavior and peculiarities of ferromagnetic
materials and of other strange magnetic materials. Before proceeding to study
magnetic materials, however, we will review very quickly some of the things about
the general theory of magnets that we learned in the last chapter. First, we imagine the atomic currents inside the material that are responsible
for the magnetism, and then describe them in terms of a volume current denSIty
jmag = V X M. We emphasize that this is not supposed to represent the actual
currents. When the magnetization is uniform the currents do not really cancel
out precisely; that is, the whirling currents of one electron in one atom and the
whirling currents of an electron in another atom do not overlap in such a way
that the sum is exactly zero. Even within a Single atom the distribution of
magnetism is not smooth. For instance, in an iron atom the magnetization
is distributed in a more or less spherical shell, not too close to the nucleus and
not too far away. Thus, magnetism in matter is quite a complicated thing in its
details; it is very irregular. However, we are obliged now to ignore this detailed
complexity and discuss phenomena from a gross, average point of view. Then
it is true that the average current in the interior region, over any ﬁnite area that
is big compared With an atom, 18 zero when M = 0. So, what we mean by
magnetization per unit volume and jmug and so on, at the level we are now
considering, IS an average over regions that are large compared with the space
occupied by a single atom. In the last chapter, we also discovered that a ferromagnetic material has the
following interesting property: above a certain temperature it is not strongly
magnetic, whereas below this temperature it becomes magnetic. This fact is
easily demonstrated. A piece of nickel wire at room temperature is attracted by a
magnet. However, if we heat it above its Curie temperature With a gas ﬂame, it
becomes nonmagnetic and is not attracted toward the magnet—even when brought
quite close to the magnet. If we let it lie near the magnet while it cools off, at the
instant its temperature falls below the critical temperature it is suddenly attracted
again by the magnet! The general theory of ferromagnetism that we will use supposes that the spin
of the electron is responsible for the magnetization. The electron has spin onehalf
and carries one Bohr magneton of magnetic moment ,u = ,uB = qeh/Zm. The
electron spin can be pointed either “up” or “down.” Because the electron has a
negative charge, when its spin is “up” it has a negative moment, and when its spin
is “down” it has a posmvc moment. With our usual conventions, the moment a
of the electron lS opposne its 5pm. We have found that the energy of orientation
of a magnetic dipole in a given applied ﬁeld B is —p  B, but the energy of the
spinning electrons depends on the neighboring spin alignments as well. In iron,
if the moment of a nearby atom is “up,” there is a very strong tendency that the
moment of the one next to it Will also be “up.” That is what makes iron, cobalt,
and nickel so strongly magnetic——the moments all want to be parallel. The ﬁrst
question we have to discuss is why. Soon after the development of quantum mechanics, it was noticed that there
is a very strong apparent force—not a magnetic force or any other kind of actual
force, but only an apparent force—trying to line the spins of nearby electrons
opposite to one another. These forces are closely related to chemical valence forces.
There is a principle in quantum mechanics—called the exclusion principle—that 37—1 371 Understanding ferromagnetism
37—2 Thermodynamic properties
37—3 The hysteresis curve 37—4 Ferromagnetic materials 37—5 Extraordinary magnetic
materials References: Bozorth, R. M, “Magne tism,” Encyclopaedia Bri
tannlca, Vol. 14, 1957,
pp. 636—667. Kittel, C., Introduction to
Solid State Physzcs, John
Wiley and Sons, Inc., New
York, 2nd ed., 1956. two electrons cannot occupy exactly the same state, that they cannot be in exactly
the same condition as to location and spin orientation.* For example. if they are
at the same point, the only alternative is to have their spins opposite. So, if there
is a region of space between atoms where electrons like to congregate (as in a chem
ical bond) and we want to put another electron on top of one already there, the
only way to do it is to have the spin of the second one pointed opposite to the spin
of the ﬁrst one. To have the spins parallel is against the law, unless the electrons
stay away from each other. This has the effect that a pair of parallelspin electrons
near to each other have much more energy than a pair of oppOSitespin electrons;
the net effect is as though there were a force trying to turn the spin over. Some
times this spinturning force is called the exchange force, but that only makes it
more mysterious—it is not a very good term. It is Just because of the exclusion
principle that electrons have a tendency to make their spins opposite. In fact,
that is the explanation of the lack of magnetism in almost all substances! The
spins of the free electrons on the outside of the atoms have tremendous tendency
to balance in opposite directions. The problem is to explain why for materials
like iron it is just the reverse of what we should expect. We have summarized the supposed alignment effect by adding a suitable term
in the energy equation, by saying that if the electron magnets in the neighborhood
have a mean magnetization M, then the moment of an electron has a strong
tendency to be in the same direction as the average magnetization of the atoms in
the neighborhood. Thus, we may write for the two possible spin onentationsj Spin “up” energy = +# (H + ’
60C AM
”“ (H + ' When it was clear that quantum mechanics could supply a tremendous spin
orientating force—~even if, apparently, of the wrong sign—it was suggested that
ferromagnetism might have its origin in this same force, that due to the complexi
ties of iron and the large number of electrons involved, the Sign of the interaction
energy would come out the other way around. Since the time this was thought of—
in about 1927 when quantum mechanics was ﬁrst being understood—many people
have been making various estimates and semicalculations, trying to get a theoretical
prediction for A. The most recent calculations of the energy between the two elec
tron spins in iron—assuming that the interaction is a direct one between the two
electrons in neighboring atoms~still give the wrong sign. The present understand
ing of this is again to assume that the complexity of the situation is somehow
responSible and to hope that the next man who makes the calculation with a more
complicated situation will get the right answer! It is believed that the upspin of one of the electrons in the inside shell, which
is making the magnetism, tends to make the conduction electrons which ﬂy around
the outside have the opposite spin. One might expect this to happen because the
conduction electrons come into the same region as the “magnetic” electrons. Since
they move around, they can carry their prejudice for being ups1de down over to
the next atom; that is, one “magnetic” electron tries to force the conduction elec
trons to be opposite, and the conduction electron then makes the next “magnetic”
electron opposite to it. The double interaction is equivalent to an interaction which
tries to line up the two “magnetic” electrons. In other words, the tendency to make
parallel spins is the result of an intermediary that tends to some extent to be op
pos1te to both. This mechanism does not requ1re that the conduction electrons be
completely “upSIde down.” They could Just have a slight prejudice to be down,
just enough to load the “magnetic” odds the other way. This is the mechanism that (37.1) Spin “down” energy H * See Chapter 43. T We write these equations With H = B  M/eoc‘“) instead of B to agree With the work
ofthe last chapter. You might prefer to write U : inBa : in(B + h’M/eorZ), where
N = )x — 1. It’s the same thing. 37—2 the people who have calculated such things now believe is responsible for ferro
magnetism. But we must emphasize that to this day nobody can calculate the
magnitude of x simply by knowing that the material is number 26 in the periodic
table. In short, we don’t thoroughly understand it. Now let us continue with the theory, and then come back later to discuss a
certain error involved in the way we have set it up. If the magnetic moment of a
certain electron is “up,” energy comes both from the external ﬁeld and also from
the tendency of the spins to be parallel. Since the energy is lower when the spins
are parallel, the eﬁect is sometimes thought of as due to an “effective internal
ﬁeld.” But remember, it is not due to a true magnetic force; it is an interaction
that is more complicated. In any case, we take Eqs. (37.1) as the formulas for the
energies of the two spin states of a “magnetic” electron. At a temperature T, the
relative probability of these two states is proportional to e“"“c’gy’”, which we
can write as e”, with x = “(H + xM/eoc2)/kT. Then, if we calculate the
mean value of the magnetic moment, we ﬁnd (as in the last chapter) that it is M = Np. tanh x. (37.2) Now we would like to calculate the internal energy of the material. We note
that the energy of an electron is exactly proportional to the magnetic moment,
so that the calculation of the mean moment and the calculation of the mean energy
are the same—except that in place of ,u in Eq. (37.2) we would write —uB, which
is —pt(H + xM/eoc2). The mean energy 18 then (U)av = —N,t (H + M1) tanh x. 6062
Now this is not quite correct. The term xM/eoc2 represents interactions of
all pOSSible pairs of atoms, and we must remember to count each pair only once.
(When we consider the energy of one electron in the ﬁeld of the rest and then the
energy of a second electron in the ﬁeld of the rest, we have counted part of the
ﬁrst energy once more.) Thus, we must divide the mutual interaction term by two,
and our formula for the energy then turns out to be )xM (U>.w = —Nu (H 36—0?) tanh x. (37.3) In the last chapter we discovered an interesting thing—that below a certain
temperature the material ﬁnds a solution to the equations in which the magnetic
moment is not zero. even With no external magnetizing ﬁeld. When we set H = 0 in Eq. (37.2), we found that
M To M
Msat — tanh M3313) q where Mm = Nit, and Tc = posat/keocz. When we solve this equation
(graphically or otherwise), we ﬁnd that the ratio M/Msat as a function of T/TC is
a curve like that labeled “quantum theory” in Fig. 37—1. The dashed curve marked
“cobalt, nickel” shows the experimental results for crystals of these elements.
The theory and experiment are in reasonably good agreement. The ﬁgure also
shows the result of the classical theory in which the calculation is carried out
assuming that the atomic magnets can have all possrble orientations in space.
You can see that this assumption gives a prediction that is not even close to the
experimental facts. Even the quantum theory deViates from the observed behavior at both high
and low temperatures. The reason for the deviations is that we have made a rather
sloppy approximation in the theory: We have assumed that the energy of an
atom depends upon the mean magnetization of its neighboring atoms. In other
words, for each one that is “up” in the neighborhood of a given atom, there Will
be a contribution of energy due to that quantum mechanical alignment effect.
But how many are there pointed “up”? On the average, that is measured by the 37—3 Fig. 37—1. from Encyclopaedia Britannica] (b) Fig. 37—2. The energy per unit vol
ume and speciﬁc heat of a ferromagnetic
crystal. The spontaneous magne
tization (H = 0) of ferromagnetic crystals
as a function of temperature. [Permission gggﬁil l 0 2 0 3 0 A 0 5 0 6 0 7 0.0 0 9 I T/Tc .0 magnetization M—but only on the average. A particular atom somewhere might
ﬁnd all its neighbors “up.” Then Its energy will be larger than the average. Another
one might ﬁnd some up and some down, perhaps averaging to zero, and it would
have no energy from that term, and so on. What we ought to do is to use some more
complicated kind of average, because the atoms in different places have different
environments, and the numbers up and down are different for different ones.
Instead of just taking one atom subjected to the average inﬂuence, we should
take each one in its actual situation, compute its energy, and ﬁnd the average
energy. But how do we ﬁnd out how many are “up” and how many are “down”
in the neighborhood? That is, of course, just what we are trying to calculate——
the number “up” and “down”—so we have a very complicated interconnected
problem of correlations, a problem which has never been solved. It is an intriguing
and exc1ting one which has exxsted for years and on which some of the greatest
names in physics have written papers, but even they have not completely solved it. It turns out that at low temperatures, when almost all the atomic magnets are
“up” and only a few are “down,” it is easy to solve; and at high temperatures, far
ab0ve the Curie temperature Tc when they are almost all random, it is again easy.
It is often easy to calculate small departures from some Simple, idealized situation,
so it is fairly well understood why there are deviations from the simple theory at
low temperature. It IS also understood physrcally that for statistical reasons the
magnetization should deviate at high temperatures. But the exact behavior near
the Curie point has never been thoroughly ﬁgured out. That’s an interesting
problem to work out some day if you want a problem that has never been solved. 9" 37—2 Thermodynamic properties In the last chapter we laid the groundwork necessary for calculating the
thermodynamic properties of ferromagnetic materials. These are, naturally, related
to the internal energy of the crystal, which includes interactions of the various
spins, given by Eq. (37.3). For the energy of the spontaneous magnetization below
the Curie point, we can set H = 0 in Eq. (37.3), and—noticing that tanhx =
M/Msat—we ﬁnd a mean energy proportional to M 2: NMM2 _ 260C2Msat I <U>av =
If we now plot the energy due to the magnetism as a function of temperature, we
get a curve which is the negative of the square of the curve of Fig. 37—l, as drawn
in Fig. 37—2(a). If we were to measure then the speciﬁc heat of such a material
we would obtain a curve which is the derivative of 37—2(a). It is shown in Fig. 37—4 37—2(b). It rises slowly with increasing temperature, but falls suddenly to zero at
T = Tc. The sharp drop is due to the change in slope of the magnetic energy and
is reached right at the Curie point. So without any magnetic measurements at
all we could have discovered that something was going on inside of iron or nickel
by measuring this thermodynamic property. However, both experiment and
improved theory (with ﬂuctuations included) suggest that this simple curve is
wrong and that the true situation is really more complicated. The curve goes
higher at the peak and falls to zero somewhat slowly. Even if the temperature is
high enough to randomize the spins on the average, there are still local regions
where there is a certain amount of polarization, and in these regions the spins still
have a little extra energy of interaction—which only dies out slowly as things get
more and more random with further increases in temperature So the actual curve /// // J/ / ./ /
looks like Fig. 37—2(c). One of the challenges of theoretical physics today is to
ﬁnd an exact theoretical description of the character of the speciﬁc heat near the Curie transition—an intriguing problem which has not yet been solved. Naturally,
this problem is very closely related to the shape of the magnetization curve in the o 0
same region. Now we want to describe some experiments, other than thermodynamic ones, ELEEITNRSON which show that there is something right about our interpretation of magnetism
When the material is magnetized to saturation at low enough temperatures, M is
very nearly equal to ngt—nearly all the spins are parallel, as well as their mag
netic moments. We can check this by an experiment. Suppose we suspend a bar
magnet by a thin ﬁber and then surround it by a coil so that we can reverse the . . . n V
3‘ magnetic ﬁeld Without touching the magnet or putting any torque on it. This is a
‘v‘svery difficult experiment because the magnetic forces are so enormous that any Fig, 37_3. When the magnehzaﬁon
irregularities, any lopSidedness, or any lack of perfection in the iron will produce of a bar of iron is reversed, the bar is
aCCidental torques. However, the experiment has been done under careful con given some angular velocity. ditions in which such accidental torques are minimized. By means of the magnetic
ﬁeld from a c011 that surrounds the bar, we turn all the atomic magnets over at
once. When we do this we also change the angular momenta of all the spins from
“up” to “down” (see Fig. 37—3). If angular momentum is to be conserved when the
spins all turn over, the rest of the bar must have an oppOSite change in angular
momentum. The whole magnet Will start to spin. And sure enough, when we do
the experiment, we ﬁnd a slight turning of the magnet. We can measure the
total angular momentum given to the whole magnet, and this is simply N times h,
the change in the angular momentum of each spin. The ratio of angular momentum
to magnetic moment measured this way comes out to within about 10 percent of
what we calculate. Actually, our calculations assume that the atomic magnets are
due purely to the electron spin, but there is, in addition, some orbital motion also in
most materials. The orbital motion is not completely free of the lattice and does
not contribute much more than a few percent to the magnetism. As a matter of
fact, the saturation magnetic ﬁeld that one gets taking Mm = Np. and using the
dens1ty of iron of 7.9 and the moment ,u of the spinning electron is about 20,000
gauss. But according to experiment, it is actually in the neighborhood of 21,500
gauss. This is a typical magnitude of error—5 or 10 percent——due to neglecting
the contributions of the orbital moments that have not been included in making
the analySis. Thus, a slight discrepancy with the gyromagnetic measurements is
quite understandable. 37—3 The hysteresis curve We have concluded from our theoretical analysis that a ferromagnetic material Fig. 37—4. The formation of domains
should spontaneously become magnetized below a certain temperature so that in 0 Single CFYSl0 01‘ iron [me Charles
all the magnetism would be in the same direction. But we know that this is not true Ki’fe': ’""°d”C“°" ’° 5°”d S’O’e Phys’csl
for an ordinary piece of unmagnetized iron. Why isn’t all iron magnetized? We “M Wiley and sons' Inc" New York' 2nd
can explain it with the help of Fig. 37—4. Suppose the iron were all a big Single ed" 1956']
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 37—5 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 upspinning
electrons adjacent to downspinning 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 de
scribed it as though two adjacent atoms were spinning exactly oppos1te, 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 szze 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, “centimetersize” 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 outs1de the crystal. Actually nature has discovered still another way to lower the energy: It is not
necessary to have the ﬁeld go outs1de at all, if a little triangular region is magnetized
sideways, as in Fig. 37—4(d).* Then With the arrangement of Fig‘.”§,37—4(d) we see
that there is no external ﬁeld, but instead only a little more domairi'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
eﬁect 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 elemen
tary 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 oppos1te to the ﬁeld.
As the ﬁeld continues to increase, a whole crystal shifts gradually into a single * You may be wondering how spins that have to be either “up” or “down” can also
be “sideways”! That’s a good question. but we won’t worry about it right now. We’ll
simply adopt the classical point of View, thinking of the atomic magnets as clas51cal
dipoles which can be polarized Sideways. Quantum mechanics reqUires considerable
expertness to understand how things can be quantized both “upanddown,” and “right
andleft,” all at the same time. 37—6 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°? 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° with respect to one of the axes. Therefore, if we apply a mag
netic 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 an
other 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 yzplane 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 xdirection and
none in the y— or zdirections. The [110] direction is in a direction 45° from the
x— and yaxes, as in Fig. 37—7(b); and the [l l 1] direction is in the direction of the
cube diagonal, as in Fig. 37—7(c). 300 400 500H 600 700 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 377 Fig. 37—6.
allel to H, for different directions of H
(with respect to the crystal axes). [From F. Bitter, Introduction to Ferromagnetism,
McGrawHill Book Co., lnc., 1937.] Fig. 37—5. A magnetizing field H at
an angle with respect to the crystal axis
will gradually change the direction of the
magnetization without changing its magni
tude. The component of M par Fig. 37—7. The way the crystal planes are labeled. Fig. 37—8. Magnetization curves for
single crystals of iron, nickel, and cobalt.
[From Charles Kittel, Introduction to Solid
State PhySics, John Wiley and Sons, Inc.,
New York, 2nd ed., 1956.] 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 “wrongway”
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 itjust go right up to satura
tion? 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 MS“, with all the elec
trons lines up. Now, if we try to do the same thing in the [110] direction—which is at 45°
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 [l 10] curve extrapolates back to the vertical axis
should be at 1/\/§ of the saturation value. It turns out, in fact, to be very, very
close to l/x/i. Similarly, in the [111] direction—which is along the cube diagonal
—we ﬁnd, as we would expect, that the Curve extrapolates back to nearly l/\/§
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 [000]] 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 diﬁerent 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 crystal
line 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 °§§§§ M/‘nreocz (gauss) L»
T'
on 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 inter
action 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 loose energy in heating the metal. A second effect is that when a domain
suddenly changes, part of the crystal changes its dimensions from the magneto—
striction. 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 back
ward~snap—produces a different result. It’s like “jerky” friction, and it takes
energy. Eventually, for hiin 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 an ordinary polycrystalline materials, such as the one shown in Fig. 37—10,
rises a little bit and reversrbly at ﬁrst, then rises irreverSibly, and then curves over
slowly. Of course, there is no sharp breakpoint 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 c0il 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 grams 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. 37—9 Fig. 37—9. The microscopic structure
of an unmagnetized ferromagnetic ma
terial. Each crystal grain has an easy
direction of magnetization and is broken
up into domains which are spontaneously
magnetized (usually) parallel to this
direction. Fig. 37—l0. The magnetization curve
for polycrystalline iron. saucou
CO'L STEEL srme SPEAKER Fig. 37—1]. The sudden changes in
the magnetization of the steel strip are
heard as clicks in the loudspeaker. “800 '400 O 400 800 H
Jr (gauss)
1
ﬁg 37—12. The hyﬂereﬁs curve of
Ahﬁco V. 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 backwardgoing 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.” 37—4 Ferromagnetic materials Now we would like to talk about the various kinds of magnetic materials that
there are in the technical world and to conSider some of the problems involved in
designing magnetic materials for dilTerent purposes. First, the term “the magnetic
properties of iron,” which one often hears, is a misnomer—there is no such thing.
“Iron” is not a welldeﬁned material—the properties of iron depend critically on
the amount of impurities and also on how the iron is formed. You can appreCIate
that the magnetic properties will depend on how easily the domain walls move and
that this is a gross property, not a property of the indiVidual atoms. So practical
ferromagnetism is not really a property of an iron atom—it is a property of solid
iron in a certain form. For example, iron can take on two different crystalline
forms. The common form has a bodycentered cubic lattice, but it can also have
a facecentered cubic lattice, which is, however, stable only at temperatures above
1100°C. Of course, at that temperature the bodycentered cubic Structure is
already past the Curie pomt. However, by alloying chromium and nickel With
the iron (one possible mixture is 18 percent chromium and 8 percent nickel) we
can get what is called stainless steel, which, although it is mainly iron, retains the
facecentered lattice even at low temperatures. Because its crystal structure 18
different, it has completely different magnetic properties. Most kindsif‘tainless
steel are not magnetic to any appreciable degree, although there are some kinds
which are somewhat magnetic—it depends on the composnion of the alloy. Even
when such an alloy is magnetic, it is not ferromagnetic like ordinary iron—even
though it is mostly Just iron. We would like now to describe a few of the spe01al materials which have been
developed for their particular magnetic properties. First, if we want to make a
permanent magnet. we would like material with an enormously Wide hysteresis
loop so that, when we turn the current off and come down to zero magnetizmg
ﬁeld, the magnetization will remain large. For such materials the domain bounda
ries should be “frozen” in place as much as possible One such material is the re
markable alloy “Alnico V” (51% Fe, 8% Al, 14%> Ni, 24% Co, 3% Cu). (The
rather complex compOSition of this alloy 18 indicative of the kind of detailed effort
that has gone into making good magnets. What patience it takes to mix ﬁve things
together and test them until you ﬁnd the most ideal substance!) When Alnico
solidiﬁes, there is a “second phase” which preCipitates out, making many tiny grains
and very high internal strains. In this material, the domain boundaries have a
hard time movmg at all. In addition to havmg a preCise composition, Alnico is
mechanically “worked” in a way that makes the crystals appear in the form of
long grains along the direction in which the magnetization lS gomg to be. Then
the magnetization will have a natural tendency to be lined tip in these directions
and Will be held there from the anisotropic effects. Furthermore, the material is
even cooled in an external magnetic ﬁeld when it is manufactured, so that the grains
Will grow With the right crystal orientation. The hystereSis loop of Almco V is
shown in Fig 37—12. You see that it is about 500 times Wider than the hysteresis
curve for soft iron that we showed in the last chapter in Fig. 36—8. Let’s turn now to a different kind of material. For building transformers and
motors, we want a material which is magnetically “soft”—one in which the mag—
netism is eaSily changed so that an enormous amount of magnetization results
from a very small applied ﬁeld. To arrange this, we need pure, wellannealed
material which Will have very few dislocations and impurities so that the domain 37—10 walls can move easily. It would also be nice if we could make the anisotropy
small. Then, even if a grain of the material sits at the wrong angle With respect to
the ﬁeld, it Will still magnetize easily. Now we have said that iron prefers to mag
netize along the [100] direction, whereas nickel prefers the [l l 1] direction; so if
we mix iron and nickel in various proportions, we might hope to ﬁnd that With
just the right proportions the alloy wouldn‘t prefer any direction—the [100] and
[l l 1] directions would be equivalent. It turns out that this happens with a mixture
of 70 percent nickel and 30 percent iron. In addition—possibly by luck or maybe
because of some phySical relationship between the anisotropy and the magneto
striction eﬁects—it turns out that the magnetostrzction of iron and nickel has the
opposite Sign. And in an alloy of the two metals, this property goes through zero
at about 80 percent nickel. So somewhere between 70 and 80 percent nickel we get
very “soft” magnetic materials—alloys that are very easy to magnetize. They are
called the permalloys. Permalloys are useful for highquality transformers (at low
signal levels), but they would be no good at all for permanent magnets. Perm
alloys must be very carefully made and handled. The magnetic properties of a
piece of permalloy are drastically changed if it is stressed beyond its elastic limit—~it
mustn’t be bent. Then. its permeability is reduced because of the dislocations, slip
bands, and so on, which are produced by the mechanical deformations. The
domain boundaries are no longer eaSy to move. The high permeability can, how
ever, be restored by annealing at high temperatures. It is often convenient to have some numbers to characterize the various
magnetic materials. Two useful numbers are the intercepts of the hysteres1s loop
with the B and Haxes, as indicated in Fig. 37—12. These intercepts are called the
remanent magnetic field B, and the cacrcrvc force H,. In Table 37~l we list these
numbers for a few magnetic materials. Table 37—1 Properties of some ferromagnetic materials B, H,
Re51dual CoerCive
magnetic force
ﬁeld (gauss)
Material (gauss)
Supermalloy (~ 5000) 0 004
Silicon steel
(transformer) 12,000 0 05
Armco iron 4000 0 6
Alnico V 13000 550. (0) (b)
Fig. 37—13. Relative orientation of
+ f electron spins in various materials: (a)
l I  ferromagnetic, (b) antiferromagnetic, (c)
I l ferrite, (d) yttriumiron alloy. (Broken
arrows show direction of total angular
(c) (d) momentum, including orbital motion.) 37—5 Extraordinary magnetic materials We would now like to discuss some of the more exotic magnetic materials.
There are many elements in the periodic table which have incomplete inner electron
shells and hence have atomic magnetic moments For instance, right next to the
ferromagnetic elements iron, nickel, and cobalt you Will ﬁnd chromium and manga
nese. Why aren’t they ferromagnetic? The answer is that the )\ term in Eq. (37.l)
has the opposztc sign for these elements. In the chromium lattice, for example, the
spins of the chromium atoms alternate atom by atom, as shown in Fig. 37—13(b).
So chromium IS' “magnetic” from its own pomt of View, but it is not technically
interesting because there are no external magnetic eﬁects. Chromium, then, is an
example of a material in which quantum mechanical effects make the spins alter
nate. Such a material is called amiferromagneric. The alignment in antiferromag
rietic materials is also temperature dependent. Below a critical temperature, all
the spins are lined up in the alternating array, but when the material is heated above
a certain temperature—which is again called the Curie temperature~the spins
suddenly become random. There is, internally, a sudden tranSition. This tranSition
can be seen in the speCiﬁc heat curve. Also it shows up in some special “magnetic”
effects. For instance, the existence of the alternating spins can be veriﬁed by scatter—
ing neutrons from a crystal of chromium. Because a neutron itself has a spin 37—11 Fig. 37—l4. Crystal structure of the
mineral spinel (MgAlgotl; the Mg+2 ions
occupy tetrahedral sites, each surrounded
by four oxygen ions; the Al+3 ions occupy
octahedral sites, each surrounded by six
oxygen Ions. [From Charles Kittel, lnfro
ducfion to Solid State Physics, John Wiley
and Sons, Inc , New YOrk, 2nd ed., 1956] (and a magnetic moment), it has a different amplitude to be scattered, depending on
whether its Spin 18 parallel or opposite to the spin of the scatterer. Thus, we get a
different interference pattern when the Spins in a crystal are alternating than we
do when they have a random distribution There is another kind of substance in which quantum mechanical effects make
the electron spins alternate, but which is nevertheless ferromagnetic—that is, the
crystal has a permanent net magnetization. The idea behind such materials is
shown in Fig. 37—14. The ﬁgure shows the crystal structure of spine], a magneSium
aluminum oxide, Which—as it is Shown—is not magnetic. The ox1de has two kinds
of metal atoms: magnesium and aluminum. Now if we replace the magnesium
and the aluminum by two magnetic elements like iron and Zinc, or by zinc and
manganese—in other words, if we put in magnetic atoms instead ofthe nonmagnetic
ones—an interesting thing happens. Let’s call one kind of metal atom a and the
other kind of metal atom b; then the followmg combination of forces mNe
conSidered. There is an ab interaction which tries to make the a atoms and the
h atoms have oppOSite Spins~because quantum mechanics always gives the oppo
Site Sign (except for the mysterious crystals of iron, nickel, and cobalt). Then,
there is a direct aa interaction which tries to make the a’s opposue, and also a
hh interaction which tries to make the b’s opposne. Now, of course we cannot
have everything opposne everything else—a opposite b, a opposite a, and l) 0p
poSite b Presumably because of the distances between the a’S and the presence of
the oxygen (although we really don’t know Why), it turns out that the ab inter
action iS stronger than the aa or the bb. So the solution that nature uses in this
case is to make all the a’S parallel to each other. and all the b’S parallel to em h other,
but the two systems opposite That gives the lowest energy because of the stronger
ab interaction. The result: all the as are spinning up and all the b’S are Spinning
down—or Vice versa, of course. But if the magnetic moments of the atype atom
and the btype atom are not equal, we can get the Situation shown in Fig. 37—13(c),
and there can be a net magnetization in the material. The material Will then be
ferromagnetic—although somewhat weak Such materials are called ferrttes.
They do not have as high a saturation magnetization as iron—for obvious reasons
—So they are only useful for smaller ﬁelds. But they have a very important differ
ence—they are insulators; the ferrites are ferromagnetic insulators. In high
frequency ﬁelds, they Will have very small eddy currents and so can be used, for
example, in microwave systems. The microwave ﬁelds Will be able to get inside
such an insulating material, Whereas they would be kept out by the eddy currents
in a conductor like iron. There is another class of magnetic materials which has only recently been
discovered—members of the family of the orthosilicates called garnets. They are
again crystals in which the lattice contains two kinds of metallic atoms, and we
have again a Situation in which two kinds of atoms can be substituted almost at
Will. Among the many compounds of interest there 15 one Which is completely
ferromagnetic. It has yttrium and iron in the garnet structure, and the reason it is
ferromagnetic is very curious. Here again quantum mechanics is making the
neighboring spins oppOSite, so that there is a lockedin system of Spins With the
electron spins of the iron one way and the electron Spins of the yttrium the oppOSite
way But the yttrium atom is complicated. It is a rareearth element and gets a
large contribution to its magnetic moment from orbital motion of the electrons.
For yttrium, the orbital motion contribution iS opposite that of the spin and also
is bigger. Thus, although quantum mechanics, working through the exclusion
principle, makes the spins of the yttrium opposne those of the iron, it makes the
total magnetic momcnt of the yttiium atom parallel to the iron because of the
orbital effect—as sketched in Fig. 37—13(d) The compound is therefore a regular
ferromagnet. Another interesting example of ferromagnetism occurs in some of the rare
earth elements. It has to do with a still more peculiar arrangement of the spins.
The material is not ferromagnetic in the sense that the spins are all parallel. nor is
it antiferromagnetic in the sense that every atom is opposite. In these crystals all
of the Spins in one layer are parallel and lie in the plane of the layer. In the next 37—12 layer all spins are again parallel to each other, but point in a somewhat diﬁerent
direction. In the following layer they are in still another direction, and so on.
The result is that the local magnetization vector varies in the form of a spiral—the
magnetic moments of the successive layers rotate as we proceed along a line
perpendicular to the layers. It is interesting to try to analyze what happens when a
ﬁeld is applied to such a spiral—all the twistings and turnings that must go on in
all those atomic magnets. (Some people like to amuse themselves with the theory
of these things!) Not only are there cases of “ﬂat” spirals, but there are also cases
in which the directions the magnetic moments of successive layers map out a cone,
so that it has a spiral component and also a uniform ferromagnetic component
in one direction! The magnetic properties of materials, worked out on a more advanced level
than we have been able to do here, have fascinated physicists of all kinds. In the
ﬁrst place, there are those practical people who love to work out ways of making
things in a better way—they love to design better and more interesting magnetic
materials. The discovery of things like ferrites, or their application, immediately
delights people who like to see clever new ways of doing things. Bes1des this,
there are those who ﬁnd a fascination in the terrible complexity that nature can
produce using a few basic laws. Starting with one and the same general idea,
nature goes from the ferromagnetism of iron and its domains, to the antiferro
magnetism of chromium, to the magnetism of ferrites and garnets, to the spiral
structure of the rare earth elements, and on, and on. It IS fascinating to discover
experimentally all the strange things that go on in these speCial substances. Then,
to the theoretical physicists, ferromagnetism presents a number of very interesting,
unsolved, and beautiful challenges. One challenge is to understand why it exists
at all. Another is to predict the statistics of the interacting spins in an ideal lattice.
Even neglecting any poss1ble extraneous complications, this problem has, so far,
deﬁed full understanding. The reason that it is so interesting is that it is such an
easily stated problem: Given a lot of electron spins in a regular lattice, interacting
with suchand—such a law, what do they do? It is Simply stated, but it has deﬁed
complete analysis for years. Although it has been analyzed rather carefully for
temperatures not too close to the Curie point, the theory of the sudden transition the Curie point still needs to be completed. Finally, the whole subject of the system of spinning atomic magnets—in
ferromagnetic, or in paramagnetic materials and in nuclear magnetism, has also
been a fascinating thing to advanced students in physics. The system of spins can
be pushed on and pulled on with external magnetic ﬁelds, so one can do many
tricks With resonances, with relaxation effects, with spinechoes, and With other
effects. It serves as a prototype of many complicated thermodynamic systems.
But in paramagnetic materials the situation is often fairly simple, and people
have been delighted both to do experiments and to explain the phenomena theo
retically. We now close our study of electricity and magnetism. In the ﬁrst chapter,
we spoke of the great strides that have been made since the early Greek observation
of the strange behaviors of amber and of lodestone. Yet in all our long and in
volved discussion we have never explained why il Is that when we rub a piece of
amber we get a charge on it, nor have We explained why a lodestone is magnetzzed’
You may say, “Oh, we Just didn’t get the right sign.” No, it is worse than that
Even if we dld get the right sign, we would still have the question: Why is the piece
of lodestone in the ground magnetized? There is the earth’s magnetic ﬁeld, of
course, but where does the earth’s ﬁeld come from? Nobody really knows—there
have only been some good guesses. So you see, this phySics of ours is a lot of
fakery~we start out with the phenomena of lodestone and amber, and we end up
not understanding either of them very well. But we have learned a tremendous
amount of very eXCiting and very practical information in the process! 37—13 ...
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 Spring '09
 LeeKinohara
 Physics

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