A photograph of a commercial highspeed maglev train that connects the
Shanghai, China airport to the city
center. Maglev trains do not run on
wheels; instead, they are lifted from
the track and propelled by magnetic
fields. The highest speed for a train
is 581 km/h set by a maglev train
on an experimental track in
Yamanashi, Japan.
© Photo Japan / Alamy
The goal of this chapter is to understand
●●
The method for creating magnetic fields
●●
The response of materials to magnetic fields
●●
The intensity of magnetization of materials
●●
Diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, and ferrimagnetic materials
●●
Hard and soft magnetic materials
●●
Magnetic materials for information storage
●●
Magnetoresistance and applications to magnetic data storage systems
Chapter
17
Magnetic Materials
17.1 Introduction
The first recorded description of magnetic materials was in approximately 800 bce, when
the Greeks wrote that lodestone (Fe3O4) could attract iron. The first known use of magnetic
materials was in China as early as 4000 bce, when the Chinese utilized elongated lodestones
to orient important buildings with a south facing entrance. European explorers used an
iron needle as a compass as early as 1190 ce for navigation, because it always pointed to
magnetic north. Modern applications of magnetic materials include permanent magnets for
industrial and electronic applications, magnetic materials in motors, electrical generators,
electronic devices, magnetic recording tape, and hard disk memory and readout in
computers. Superconducting materials also allow for interesting magnetic applications, such
as magnetic levitation of trains.
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CHAPTER 17
17.2 Magnetic fields
and material response
Our first experience with magnetism comes from small magnets or by using a compass. The magnetic
needle of a compass is always attracted to the magnetic north pole of the earth. We find by experiment
that a magnet has a north pole and a south pole. The combination of a north pole and a south pole of
a magnet is called a magnetic dipole. The north pole of one magnet attracts the south pole of another
magnet, and the similar poles of two magnets repel each other. The magnetic poles act in a way similar
to positive and negative electrical charges. The magnetic poles create a magnetic field, just as positive and
negative electrical charges create an electrical field. The effect of the magnetic field of a magnet on the
environment is observed by sprinkling iron powder onto a magnetic material, as shown in Figure 17.1a.
The iron powder forms a pattern in response to the magnetic field that is shown for a bar magnet in
Figure 17.1b. The magnetic field created by a current in a single loop is shown in Figure 17.1c. A coil that
has N loops wound over a length L, and carries a current I, produces a magnetic field whose strength has
a magnitude H, as shown in Equation 17.1.
H5
NI
L
17.1
The magnetic field strength produced by a coil is a vector (H) whose direction is given by the righthand rule, with the fingers curling in the direction of the positive current and the thumb pointing in the
direction of the magnetic field. In this chapter, the magnitude of a vector is printed in italics and a vector
N
S
(a)
(b)
(c)
Figure 17.1 (a) A magnet with iron filings respond to a magnetic field. (b) A schematic of the magnetic field
lines produced by a bar magnet. (c) A schematic of the magnetic field lines produced by a conducting wire loop
with a current I. ((a) Magnet0873.png. (b) Based on http://en.wikipedia.org/wiki/File:VFPt_cylindrical_magnet_thumb.svg
(c) Based on http://www.wikipremed.com/image.php?img=010403_68zzzz141400_37603_68.jpg&image_id=141400)
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Magnetic Materials
is printed in bold-Roman type. The SI unit for magnetic field strength is amperes/meter (A/m). The term
N/L is the number of turns per meter, and N/L has the units of m21.
Example Problem 17.1
Calculate the magnetic field strength inside a coil made of copper wire that has 100 turns over a
length of 10 cm and carries a current of 10 A.
Solution
All of the information necessary to calculate the magnetic field strength with Equation 17.1 is given.
H5
IN 10A s100d
5
5 10,000 A/m
L
0.1 m
In Figure 17.1a, it is not the magnetic field (H) that is observed; what is observed is the response
of the environment to the magnetic field. A mechanical analogy is that when a force is applied to a
material, the deformation of the specimen is observed. The applied force is resisted by the atoms, but the
force between the atoms cannot be seen by looking at the material. The machine that applies the force
to a material can be seen, and the coil that produces the magnetic field also can be seen. The response of
a material to an applied magnetic field (H) is the flux density (B). The flux density (B) is also called the
magnetic induction, because B is the response induced into the material. If the material is a vacuum, the
flux density (B0), shown in Figure 17.2a, is given by Equation 17.2:
B0 5 0H
17.2
where 0 is the permeability of a vacuum (0 5 4 3 1027 Wb/A ? m). The SI unit for the flux density is
webers/m2 (Wb/m2), or a tesla (T). The vector H is the same in Figures 17.2a and 17.2b, but the vectors
B0 and B are different in Figures 17.2a and 17.2b. The coil of wire in Figures 17.2a and 17.2b are also
called solenoids.
B0 = 0 H
B = H
I
I
L
N
L H
H
I
N TURNS
(a)
I
(b)
Figure 17.2 (a) A coil in a vacuum, producing a magnetic field of H, with the response of the vacuum resulting
in a magnetic flux density of B0. (b) A coil with a material inside produces a magnetic flux density of B. (Based on
Askeland, D.R., Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford,
CT. (2011), p. 770.)
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CHAPTER 17
Example Problem 17.2
What is the magnitude of the flux density or inductance created by the coil in Example Problem 17.1
in a vacuum?
Solution
Because the coil is in a vacuum, the flux density is given by Equation 17.2.
B0 5 0H 5 4 3 1027
1
2
A
Wb
Wb
10,000
5 12.6 3 1023 2 5 12.6 3 1023 T
A?m
m
m
If a material is placed in a magnetic field of strength H, as shown in Figure 17.2b, the flux density (B)
inside the material is given by Equation 17.3:
B 5 H 5 0H 1 0M
17.3
where is the magnetic permeability of the material. The intensity of magnetization of the material
(M) is the magnetic field strength created by the material as a result of the applied magnetic field H. The
magnetic field strength (H) and intensity of magnetization (M) have the same units of A/m. The flux
density ( B) with a material is the sum of the flux density resulting from the applied magnetic field in a
vacuum without the material ( B0) and the flux density resulting from the intensity of magnetization of
the material (0M ). The relative magnetic permeability of a material is given by Equation 17.4.
r 5
0
17.4
Values of the relative magnetic permeability for some materials are presented in Table 17.1. The magnitude
of the relative magnetic permeability depends upon the temperature, purity, and treatment of the material.
Observe that the relative magnetic permeability goes from values slightly less than 1 for diamagnetic
materials to a million for a ferromagnetic material. In this chapter, we explain the reason for the difference
in relative magnetic permeability of different materials. The value of the relative magnetic permeability
is high if the value of M is greater than H. In the magnetics literature the flux density (B) is emphasized,
because flux density (B) is what is measured by instruments. If we study the flux density (B), the material
and vacuum responses are combined, and the material response is not immediately obvious. In this
textbook, our interest is in understanding the response of materials to the applied magnetic field H, and
the material response is the intensity of magnetization of the material (M); as a result our focus is on M.
Example Problem 17.3
The ferromagnetic material supermalloy listed in Table 17.1 is inserted into the coil in Example
Problem 17.1. What is the flux density created by the coil with this material inserted?
Solution
The flux density is given by Equation 17.3, and μ is calculated from Equation 17.4.
1
5 r 0 5 106 4 3 1027
B 5 H 5 4 3 1021
1
2
Wb
Wb
5 4 3 1021
A?m
A?m
2
Wb
A
Wb
10,000
5 12.6 3 103 2 5 12.6 3 103 T
A?m
m
m
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Magnetic Materials
Table 17.1Materials, the Type of Magnetic Response, and the Relative Permeability at
Room Temperature
Substance
Group type
Relative permeability r
Bismuth
Diamagnetic0.99983
Silver
Diamagnetic0.99998
Lead
Diamagnetic0.999983
Copper
Diamagnetic0.999991
Water
Diamagnetic0.999991
Vacuum
Nonmagnetic
Air
Paramagnetic1.0000004
Aluminum Paramagnetic
Palladium
Paramagnetic1.0008
2–81 Permalloy powder (2 Mo, 81 N i ) † Ferromagnetic
Cobalt
Ferromagnetic250
Nickel
Ferromagnetic600
Ferroxcube 3 (Mn-Zn-ferrite)
Ferromagnetic
1500
Mild Steel (0.2 C)
Ferromagnetic
2000
Iron (0.2 impurity)
Ferromagnetic
5000
Silicon iron‡ (4 Si)
Ferromagnetic
7000
78 Permalloy (78.6 Ni)
Ferromagnetic
100,000
Purified iron (0.05 impurity)
Ferromagnetic
200,000
Supermalloy (5 Mo, 79 Ni)
Ferromagnetic
1,000,000
1 (by definition)
1.00002
130
† Percentage composition. Remainder is iron and impurities.
‡ Used in power transformers.
Based on data from Kraus, J. D., Electromagnetics, McGraw-Hill, N. Y. (1953), p. 208.
Because the intensity of magnetization of the material (M) is created by the magnetic applied field
strength (H), for many materials there is a proportionality between M and H called the magnetic
susceptibility (), as shown in Equation 17.5.
M 5 H
17.5
The magnetic susceptibility ( ) in Equation 17.5 is dimensionless, because the intensity of magnetization
(M) and the magnetic field strength (H) have the same units of A/m. The dimensionless susceptibility
is also called the relative susceptibility, and is called the volume susceptibility, because as shown in
Equation 17.11 the intensity of magnetization (M) is also the dipole moment per unit volume. There are
a variety of other magnetic susceptibilities utilized in the literature of magnetism.
Magnetic materials are classified by their magnetic susceptibility as follows:
5 21: A perfect diamagnet or a superconductor
, 0 and small in magnitude: Diamagnetic
5 0: A perfect vacuum
. 0 and small: Paramagnetic and antiferromagnetic
. 0 and large: Ferromagnetic and ferrimagnetic
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CHAPTER 17
The magnetic susceptibility is determined from the relative magnetic permeability with Equation 17.6.
5 r 2 1
17.6
As with relative magnetic permeability, the magnetic susceptibility depends upon the temperature, purity,
and treatment of the material.
Example Problem 17.4
What is the magnitude of the intensity of magnetization of the supermalloy in Example Problem 17.3?
Solution
The intensity of magnetization is given by Equation 17.5, and the magnetic susceptibility is given by Equation 17.6.
5 r – 1 5 106 – 1 ø 106
M 5 H 5 106104 A/m 5 1010 A/m
17.3 Diamagnetic Materials
The magnetic susceptibility of diamagnetic materials is small and negative, as demonstrated by
subtracting 1 from the relative magnetic permeability in Table 17.1 for the diamagnetic materials. The
negative magnetic susceptibility of diamagnetic materials indicates that the intensity of magnetization in
the material (M) is in the opposite direction to the applied magnetic field (H), as shown by Equation 17.5.
The negative magnetic susceptibility is due to Lenz’s law that in a material subject to a magnetic field,
a current is induced so that it always opposes the change in the applied magnetic flux. How are these
currents induced in materials, and how do the currents result in an intensity of magnetization that is
opposed to the applied magnetic field?
The intensity of magnetization of a material (M) results from magnetic dipoles that are created in a
volume of material as a result of applying the magnetic field H. The magnetic dipole moment (pm) is the
pole strength (m and 2m) times the separation of the two poles (d), as shown by Equation 17.7.
pm 5 md
17.7
The pole strength (m) in a magnetic material is analogous to an electrical charge in electrostatics. The
direction of the vector pm is in the direction from 2m to m (south pole to north pole), and the units of
a magnetic dipole moment are A ? m2. If the dipole moment is the dipole moment of an atom ( pa ), and
the material is made up of Nv identical atoms per unit volume, then the intensity of magnetization of the
material is given by Equation 17.8.
M 5 Nv pa
17.8
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Magnetic Materials
pe = Ie Ae
Electron
pm = IA
Loop
Ie
Area A
Core
Ae
I
(b)
(a)
Figure 17.3 (a) An electron orbiting an atom, creating a current loop Ie in the reverse direction to the electron
orbit enclosing an area Ae, producing a magnetic dipole moment of magnitude pe. (b) A current (I ) enclosing an
area (A), producing a magnetic dipole moment of magnitude pm.
All materials have a diamagnetic contribution in their response to an applied magnetic field (H), because
all atoms have orbiting electrons. As shown in Figure 17.3a, the magnitude of the dipole moment created
by a single electron orbiting an atom ( pe) is given by Equation 17.9:
17.9
pe 5 IeAe
where Ie is the current associated with the orbiting electron and Ae is the area enclosed by the electron orbit.
The direction of the vector pe is given by the right-hand rule, pointing the fingers in the direction of the
positive current Ie so that the thumb points in the direction of the dipole moment. The current direction
is opposite to that of the orbiting electron. When there is no magnetic field applied to a diamagnetic
material, the sum of the dipole moments on an atom created by the orbiting electrons is equal to 0, as
indicated by the absence of any net magnetic moment (no arrow) in the individual atoms in Figure 17.4a.
However, when a magnetic field is applied to any atom, the electron orbits on the atom are altered to
oppose the applied magnetic field in compliance with Lenz’s law, and an effective dipole moment ( pa )
is created on each atom, as shown in Figure 17.4b. The value of the dipole moment per atom ( pa ) is the
sum of the values of pe from all of the electrons on the atom, and pa is in the direction opposite to the
applied magnetic field (H). The intensity of magnetization ( M ) due to this diamagnetic effect is given
by Equation 17.8. In a diamagnetic material, when the applied magnetic field ( H ) is reduced to 0, the
intensity of magnetization (M) also returns to 0, as is shown in Figure 17.4c, because pa is equal to 0.
H=0
(a)
H
(b)
H=0
(c)
Figure 17.4 (a) In a diamagnetic material with no applied magnetic field (H 5 0), there is no dipole moment
on individual atoms, and the intensity of magnetization (M) is 0. (b) When a magnetic field H is applied to a
diamagnetic material, the response of the electrons on the atoms is to produce dipole moments that oppose the
applied field H, resulting in a negative intensity of magnetization. (c) When H is removed from a diamagnetic
material, the dipole moments on the individual atoms return to 0, and the intensity of magnetization (M) returns
to 0. (Based on Askeland, D.R., Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning,
Stamford, CT. (2011), p. 774)
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CHAPTER 17
Some of the diamagnetic materials listed in Table 17.1 are metals, and metals have an additional
diamagnetic contribution to the magnetic susceptibility. The free electrons in a metal form current loops
(eddy currents) in compliance with Lenz’s law that oppose an applied magnetic field. Eddy currents
are also discussed in Section 11.2.10 as a technique of nondestructive testing. Equation 17.10 gives the
magnitude of the dipole moment (pm) created by a current (I) in amperes passing through a loop that
encloses an area (A), as shown in Figure 17.3b.
17.10
pm 5 IA
The intensity of magnetization is the vector sum of all the dipole moments (pm) in the material divided
by the volume (V ), as shown in Equation 17.11.
op
m
M5
17.11
V
From Equation 17.11 the intensity of magnetization (M) is also the total dipole moment per unit volume
of material.
The eddy currents created in a metal by a rapidly alternating magnetic field result in joule heating.
Induction furnaces or heaters use high-frequency alternating magnetic fields to induce eddy currents
that heat and can even melt a metal. The orbital electron contribution to diamagnetism described by
Equation 17.9 does not contribute to joule heating, because there is no resistance associated with the
electron orbital.
17.3.1 Superconductors and Diamagnetism
In a superconductor, there is no resistance to electrical current flow, and eddy current loops that are
established as a result of applying a magnetic field ( H ) to a superconductor produce a magnetic field
equal and opposite to the applied magnetic field ( H ). A superconductor is a diamagnet that has a
magnetic susceptibility of 21, resulting in a value of M equal to 2H.
If a magnet is placed over a superconducting material, the magnet is suspended in air or levitates,
as shown in Figure 17.5. Magnetic levitation occurs because the magnet induces in the superconductor
magnetic field equal but opposite to itself. If a permanent magnet is oriented such that its south pole
is pointing down toward the superconductor, the superconductor produces a magnetic field that has
its south pole pointing up, as shown in Figure 17.6. The south pole of the magnet and the south pole
induced in the superconductor repel each other.
Magnetic levitation (maglev) is used to lift and propel trains, allowing them to ride without any contact
with the rails. The introductory photograph in this chapter is of commercial high-speed maglev train that
connects the Shanghai, China airport to the city center, and Japan has the high-speed Linimo line. Lowspeed maglev trains are also under construction in Beijing, China and in Seoul, South Korea. Maglev
trains are known for their high reliability and low maintenance.
There is a limit to the magnitude of the magnetic field that can be applied to a superconductor and still have
zero electrical resistivity. In a superconducting metal, the free electron spins are coupled into Cooper pairs
of electron spin up and down, as discussed in Section 16.4. However, when a magnetic field (H) is applied to
an electron, there is a torque that rotates the electron spin axis into the direction of H. This torque is similar
to the torque produced on a small magnet by a large magnet that aligns the small magnet’s field to that of
the large magnet. If the applied magnetic field (H) rotates the electron spins into the direction of H, then the
coupling of Cooper electron pairs of the superconductor is destroyed, and the superconductivity is destroyed.
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Magnetic Materials
Figure 17.5 A photograph of a magnet levitating above a superconductor immersed in liquid nitrogen.
(© Stonemeadow Photography / Alamy)
N
Magnet
S
S
Superconductor
N
Figure 17.6 A schematic showing levitation of a magnet due to the magnetic field created in a superconductor
that is equal and opposite to that of the original magnet.
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CHAPTER 17
17.4 Paramagnetism
Paramagnetic materials have a small and positive magnetic susceptibility, typically from 1023 to 1025, as
demonstrated by subtracting 1 from the relative magnetic permeability of the paramagnetic materials
in Table 17.1. Paramagnetism is not of significant engineering use; however, the concepts developed
for paramagnetic materials help us understand ferromagnetic materials that are of significant
engineering importance. Why do diamagnetic materials have a negative magnetic susceptibility,
whereas paramagnetic materials have a positive susceptibility? One reason for a positive magnetic
susceptibility is that some atoms have a permanent net magnetic dipole moment (pm). One way for
an atom to have a net permanent dipole moment is for the atom to have a net electron spin. Hund’s
rule states that electrons on an atom with the same quantum numbers n and l tend to have their
electron spins parallel. As electrons are added to the 3d shell with increasing atomic number, the
first five electrons in the 3d shell all have parallel spins, as shown in Table 17.2. Table 17.2 is for free
atoms in a vapor. In solid or liquid metals, the number of parallel unpaired electron spins on atoms
can be different than the number in Table 17.2, because the bonding has an effect on the electrons.
One electron spinning about its axis creates a magnetic dipole moment, as shown in Figure 17.7,
of 1 Bohr magneton (μB), where μB 5 9.27 3 10224 A ? m2 and the direction of μB is in the direction
of the spin axis. Titanium has two electrons with parallel spins in the 3d shell, so each titanium atom
produces a dipole moment (pa) of two Bohr magnetons. When titanium atoms are subject to an
applied magnetic field (H), the torque discussed in Section 17.3.1 rotates the unpaired electron spins or
dipole moments on the titanium atoms into the direction of H. The dipole moments on the individual
titanium atoms due to the electron spins then produce an intensity of magnetization (M) that is in
the same direction as H. Since M and H are in the same direction, this results in a positive magnetic
susceptibility. Equation 17.12 is used to calculate the magnitude of intensity of magnetization (M)
resulting from parallel spins on an atom:
M 5 nBBNv
17.12
Table 17.2Atoms in the First Transition Series and the Spin Orientation of the 3d and
4s Electrons
Metal3d Electron Spin Orientations
4s
Titanium
↑
↑ ↑↓
Vanadium
↑
↑
↑ ↑↓
Chromium
↓↑
↑
↑
↑
↑
↑
Manganese
↓
↑
↑
↑
↑
↑↓
Iron
↓↑
↑
↑
↑
↑
↑↓
Cobalt
↓↑
↑↓
↑
↑
↑
↑↓
Nickel
↓↑
↑↓
↑↓
↑
↑
↑↓
Copper
↓↑
↑↓
↑↓
↑↓
↑↓
↑
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Magnetic Materials
N
or
e2
e2
N
Figure 17.7 A schematic of electron spin indicated by the curved arrow creating a dipole moment. (Based on
Askeland, D.R., Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford, CT.
(2011), p. 768.)
where nB is the average number of Bohr magnetons of magnitude B per atom pointing in the direction
of H, and Nv is the number of atoms per unit volume.
For a paramagnetic material, the flux density (B) is linearly proportional to the applied magnetic
field (H), as shown in Equation 17.3, and the intensity of magnetization (M) is linearly proportional
to the applied magnetic field (H), as shown in Equation 17.5. Figure 17.8a shows that M 5 0 when
H 5 0 for a paramagnetic material, because there is a random orientation of electron spins on different
atoms. As H is increased, the electron spins on the different atoms become oriented in the direction
of H, as shown in Figure 17.8b, and the value of M increases linearly with H. When the H field
is reduced, the intensity of magnetization follows the same slope as during the increase in H. When
H is reduced to 0, M is equal to 0, as indicated by the return of the random orientations of the
electron spins as shown in Figure 17.8c. The spins are randomized by lattice vibrations when H is
reduced to 0.
The magnetic susceptibility for a paramagnetic material decreases with increasing temperature, as
expressed in the Curie law shown in Equation 17.13:
5
C
T
17.13
where C is a constant and the temperature (T ) is in kelvin. The susceptibility decreases with increases in
temperature, because thermal vibrations disorder the electron spins.
H=0
H
H=0
(a)
(b)
(c)
Figure 17.8 (a) In a paramagnetic material with no applied magnetic field, the dipole moments on individual
atoms are in random directions, and there is no intensity of magnetization (M). (b) When a magnetic field (H) is
applied, the atoms respond by producing dipole moments that are in the same direction as the applied field H,
resulting in a positive intensity of magnetization (M). (c) When H is removed, the dipole moments on individual
atoms return to random orientations, and there is no net intensity of magnetization (M) in the material. (Based on
Askeland, D.R., Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford, CT.
(2011), p. 774)
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CHAPTER 17
Example Problem 17.5
Titanium has a room-temperature magnetic susceptibility of 1.81 3 1024. Titanium is placed in a
magnetic field of 1 3 106 A/m at room temperature.
(a) Determine the magnitude of the intensity of magnetization (M).
(b) What is the average dipole moment per atom (pa ) in titanium that is parallel to the H of
1 3 106 A/m?
(c) The dipole moment per atom calculated in (b) corresponds to how many Bohr magnetons
per atom?
Solution
a)
M is calculated from the room-temperature susceptibility ( ) and H using Equation 17.5.
M 5 H
M 5 1.81 3 1024(1 3 106) A/m 5 1.81 3 102 A/m
b)
pa is calculated from Equation 17.8. The number of atoms per unit volume (Nv ) is calculated from the density
(4.51 3 103 kg/m3 ) and the molar mass of titanium (47.88 3 1023 kg/mole).
Nv 5
1
2
4.51 3 103 kg/m3
atoms
atoms
6.02 3 1023
5 0.567 3 1029
47.88 3 1023 kg/mole
mole
m3
pa 5
M
1.81 3 102A/m
A ? m2
227
5
5
3.19
3
10
Nv 0.567 3 1029 atoms/m3
atom
c)Since one Bohr magneton (B) is equal to 9.27 3 10224 A ? m2, the number of Bohr magnetons (nB) per atom
is calculated as follows:
nB 5
pa
B
5
Bohr magnetons
3.19 3 10227A ? m2/atom
5 0.344 3 1023
atom
9.27 3 10224A ? m2/Bohr magneton
Only 3.4 electron spins in 10,000 are parallel to the applied magnetic field H. Since titanium has two Bohr
magnetons per atom, only a small fraction of the electron spins on titanium atoms are oriented in the
direction of the applied magnetic field H.
17.5 Ferromagnetism
The word ferro in Latin means “iron.” Iron was the first known ferromagnetic metal. There are other
ferromagnetic metals, such as nickel, cobalt, and the rare earth metals gadolinium and neodymium. In
ferromagnetic metals, the magnetic susceptibility is large and positive, as demonstrated by subtracting
1 from the relative permeability of ferromagnetic materials in Table 17.1. It follows from Equation 17.5
that the intensity of magnetization is much greater than the applied magnetic field. The material develops
more magnetic field than what is applied. How is this possible?
Iron is a transition metal with six 3d electrons. Since there are a total of ten possible 3d electrons,
according to Hund’s rule the first five electrons have parallel spins, and the sixth 3d electron is antiparallel
with the five parallel spins, as shown in Table 17.2. As a result, an iron atom has four parallel electron
spins that are unpaired. The difference in a paramagnetic and ferromagnetic material is demonstrated
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Magnetic Materials
M
Mr
Ms
2Hs
2Hc
O
Hs
Hc
H
Ms
2Mr
Figure 17.9 A schematic of a magnetic hysteresis loop for a ferromagnetic material, showing the magnitude of
the intensity of magnetization (M ) as a function of the magnitude of the applied magnetic field (H ).
by subjecting the ferromagnetic material to a magnetic field. Figure 17.9 shows that for a ferromagnetic
material the relationship between the applied magnetic field and the intensity of magnetization is
not linear. In a ferromagnetic material, the intensity of magnetization increases rapidly as the applied
magnetic field increases. The intensity of magnetization is greater than the applied magnetic field because
the unpaired electron spins are rotated into the direction of H, and the unpaired electron spins produce
an M that adds to H to produce a higher magnetic field. At some applied magnetic field, the intensity
of magnetization is saturated at Ms. At Ms, the maximum number of individual atom unpaired electron
spins is parallel to H, as shown in Figure 17.10a. Since the ferromagnetism of iron comes from oriented
H = Hs
H=0
H = 2Hc
(a)
(b)
(c)
H = 2Hs
H=0
H = Hc
(d)
(e)
(f )
Figure 17.10 The orientation of unpaired electron spins under various conditions in a ferromagnetic material.
(a) In magnetic domains at saturation. (b) In magnetic domains with H 5 0. (c) In magnetic domains with H 5 2Hc.
(d) At reverse saturation. (e) In magnetic domains with H 5 0. (f) In magnetic domains with H 5 Hc. (Based on Askeland,
D.R., Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford, CT. (2011), p. 774.)
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CHAPTER 17
Figure 17.11 A micrograph showing magnetic domains in a single grain of unmagnetized steel. The direction of
magnetization is indicated by the arrows. (S. Zurek, C. Vardon, CC-BY-3.0, Encyclopedia Magnetica)
unpaired electron spins on atoms, the average number of unpaired electron spins oriented in the direction
of H is calculated from the saturation magnetization using Equation 17.12.
If the applied magnetic field is reduced to 0 from saturation in a ferromagnetic material, the intensity of
magnetization only reduces to the remanent magnetization (Mr ). A remanent magnetization indicates that
the unpaired electron spins in the ferromagnetic material remain ordered in the former direction of H;
however, there is some relaxation of unpaired electron spin orientations, as shown in Figure 17.10b. The
remanent magnetization (Mr ) is a result of the magnetic field created by the oriented unpaired electron
spins, and this magnetic field is sufficient to maintain the unpaired electron spins on the atoms oriented
in the initial direction of the applied magnetic field. Figure 17.9 shows that it is necessary to reverse the
magnetic field to –Hc, the coercive field, to reduce the intensity of magnetization to 0.
In a demagnetized material, when M is forced to 0 from the saturated state, the unpaired electron spins are
not disordered. The unpaired electron spins are ordered into magnetic domains that have parallel unpaired
electron spin orientations, and the sum of the unpaired electron spins in different domains cancel each other,
as shown schematically in Figures 17.10c and in Figure 17.11, resulting in a net magnetization of 0.
Figure 17.12 shows that in a polycrystalline ferromagnetic material, the domain can be a single small grain, or
a grain can contain several domains. Figure 17.11 is an image of an unmagetized steel produced with a Kerr
effect optical microscope. A Kerr effect microscope is capable of analyzing the polarization and intensity of
light reflected from a surface that is dependent upon the direction of magnetization in the material.
Small grain with
a single domain
A grain with
domains
Figure 17.12 A schematic of magnetic domains in a ferromagnetic polycrystal. (Based on Askeland, D.R., Fulay, P.P.,
and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford, CT. (2011), p. 777.)
Magnetic Materials
Domain
Bloch
wall
Domain
Direction of
magnetic
moments
Figure 17.13 The unpaired electron spin orientations change direction at the Bloch wall. (Based on Askeland, D.R.,
Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage Learning, Stamford, CT. (2011), p. 777.)
The border between domains where unpaired electron spins rotate from one orientation to another is a
Bloch wall, as shown in Figure 17.13. Bloch walls can move under the effect of a magnetic field to allow a
change in the net orientation of unpaired electron spins. When a change in H forces a change in M, Bloch
wall motion is one way that the unpaired electron spin orientations can change.
When the applied magnetic field is increased in the reverse direction until the intensity of magnetization
reaches saturation (2Ms), the maximum number possible of magnetic domains are oriented with their
magnetization in the reverse direction, as shown in Figure 17.10d. As the applied H is reduced again
to 0, the intensity of magnetization changes to 2Mr, and the magnetic domains continue to have their
magnetization parallel to the reverse direction of H, as shown in Figure 17.10e. Increasing H again in
the forward direction results in magnetic domains with magnetization oriented in the direction of H. At
Hc in the forward direction, M is equal to 0, and the magnetic domains are equally balanced to cancel
each other, as shown in Figure 17.10f. The cycle from 1Ms to 2Ms and again to 1Ms is a hysteresis loop.
Example Problem 17.6
The saturation magnetization (Ms) of iron measured at room temperature is 1.744 3 106 A/m.
Calculate the average number of Bohr magnetons per iron atom, knowing that iron has a
BCC structure at room temperature with a lattice parameter of 0.286 nm.
Solution
The saturation magnetization is related to the number of Bohr magnetons in Equation 17.12:
Ms 5 nB B Nv
At room temperature the lattice parameter of iron is 0.286 nm, and the lattice is BCC with an atom at the
position 0, 0, 0. There are two atoms per BCC unit cell. The number of atoms per unit volume (Nv) is then
calculated as follows:
Nv 5
2 atoms
2 atoms
atoms
5
5 0.0855 3 1030
230
3
210
3
23.3 3 10 m
m3
s2.86 3 10 md
Since B 5 9.27 3 10224 A ? m2, everything in Equation 17.12 is known except the number of Bohr magnetons (nB).
nB 5
Ms
B Nv
5
9.27 3 10
Bohr magnetons
1.744 3 106A/m
5 2.20
atom
A ? m /Bohr magneton s0.855 3 1029atoms/m3d
224
2
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CHAPTER 17
On the average atom in metallic iron at room temperature, there are 2.20 parallel unpaired electron
spins, as shown in Example Problem 17.6. Table 17.2 shows that an iron atom has four parallel unpaired
electron spins in the 3d shell for a free iron atom in the vapor at a temperature of absolute zero. The
metallic bond in iron changes the unpaired electron spin orientations of the 3d electrons relative to those
of the free atom, and thermal vibrations of the metal atoms at room temperature further reduce the
number of atoms with unpaired electron spins parallel to the applied magnetic field relative to those at
absolute zero.
Example Problem 17.7
The rare earth element terbium (Tb) has one of the highest number of Bohr magnetons per atom of
9.34. Calculate the saturation magnetization for terbium.
Solution
The saturation magnetization and the number of Bohr magnetons are related through Equation 17.12.
Ms 5 nB B Nv
Ms and Nv are unknown, and we know both nB and B. Appendix B does not have any information about the
crystal structure of terbium, but we can calculate the number of atoms per unit volume from data in the periodic
table by using the molar mass of 0.15892 kg/mole, Avogadro’s number, and the density of 8.27 3 103 kg/m3.
Nv 5
1
2
8.27 3 103 kg/m3
atoms
atoms
6.02 3 1023
5 315 3 1026
0.1589 kg/mole
mole
m3
Now substitute Nv into Equation 17.12 and solve for Ms.
Ms 5 nB B Nv 5 9.34
1
Bohr magnetons
A ? m2
9.27 3 1024
atom
Bohr magneton
Ms 5 26.84 3 105
2 13.15 3 10
28
atoms
m3
2
A
m
Even though the number of Bohr magnetons in terbium is 4.2 times the number in iron, the saturation
magnetization of terbium is only 1.5 times that of iron. The reason is the low number of atoms per unit volume
in terbium.
The saturation magnetization in a ferromagnetic material is a function of temperature, as shown in
Figure 17.14. At very low temperatures the unpaired electron spins are parallel, and at high temperatures
thermal vibrations disorient the unpaired electron spins. At sufficiently high temperatures the unpaired
electron spins cannot be oriented, because of thermal vibrations. At the Curie temperature (Tc )
ferromagnetic materials change to paramagnetic with increasing temperature. Table 17.3 presents the
Curie temperature of some ferromagnetic and ferrimagnetic materials.
Is it possible to eliminate magnetic domains from a ferromagnet that has been in a strong magnetic
field, and to demagnetize the material? Figure 17.14 gives one answer. Heating the ferromagnet to above
the Curie temperature in the absence of a magnetic field disorders the unpaired electron spins. On an
individual atom, the unpaired electron spins will remain parallel; however, the magnetic domains are
Magnetic Materials
Table 17.3
The Curie Temperature of Some Magnetic Materials
Material
Curie Temperature (°C)
Gadolinium 16
Nd2Fe12B 312
Nickel 358
Co5Sm 747
Iron 771
Alnico 1 780
Cunico 855
Alnico 5 900
Cobalt1117
Based on data from Askeland, D.R., Fulay, P.P., and Wright, W.J., The Science and Engineering of Materials, 6th ed., Cengage
Learning, Stamford, CT. (2011), p. 779.
eliminated by the heating. Subsequently cooling the ferromagnet in the absence of a magnetic field results
in a demagnetized material without magnetic domains.
Iron, cobalt, and nickel are ferromagnetic metals. Because they are metals, they have free electrons.
If a ferromagnetic metal is subject to a rapidly alternating magnetic field (H), the free electrons in the
metal form current loops (eddy currents) that create a diamagnetic response in the metal that opposes the
application of H. The eddy currents produce joule heat in the metal, increasing the temperature. In an
induction furnace, an alternating magnetic field is used to induce eddy currents that melt metals such as
aluminum, copper, gold, iron, and silver.
If the metal temperature in a magnet used in an alternating magnetic field exceeds the Curie
temperature, the ferromagnetism is destroyed. Therefore, metallic ferromagnets are not suitable for
applications in high frequency magnetic fields.
Ms
T
Tc
Figure 17.14 A schematic for a ferromagnetic material of the magnitude of the saturation magnetization (Ms)
for a magnetic field pointing to the right and the relative order in the unpaired electron spin orientations as a
function of temperature up to the Curie temperature (Tc).
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CHAPTER 17
17.6 Ferrimagnetism
Ferrimagnetic materials, or ferrites, such as Fe3O4, have hysteresis loops and a magnetic susceptibility that
are both similar to iron. However, ferrimagnetic materials are insulators. Whereas in metallic iron the two
4s electrons are free electrons, the two 4s electrons in Fe3O4 are transferred to the oxygen, forming O22
and Fe21. There are no free electrons in the insulator Fe3O4. The Fe21 ion has four parallel unpaired
electron spins after removal of the two 4s electrons from the Fe atom, as shown in Table 17.2. For Fe3O4
in a high-frequency alternating magnetic field, there is no joule heating due to induced eddy currents,
because there are no free electrons. The ferrites are based upon Fe3O4. Substitution of NiO or CoO for
FeO in Fe3O4 forms NiO:Fe2O3 or CoO:Fe2O3. Ferrimagnetic materials are utilized in high-frequency
applications, such as televisions, communications equipment, and microwave devices. Table 17.4 lists the
magnetic properties of some ferrites.
Table 17.4Some Soft Magnetic Materials and the Chemical Composition, Initial Relative
Permeability, Maximum Relative Permeability, Coercivity, Retentivity, Maximum
Inductance, and Electrical Resistivity of Each
Permeability r
Coercivity Retentivity Bmax Resistivity
Name
Composition
InitialMaximum Hc(A ? m21) Br (T) (T) (V ? m)
Ingot Iron
99.8% Fe
150
5000
2.14
0.10
Low-carbon steel
99.5% Fe
200
4000
100
80
2.14
1.12
Silicon iron, unoriented
Fe23% Si
270
8000
60
2.01
0.47
Silicon iron,
Fe23% Si
1400
50,000
2.01
0.50
7
0.77
1.20
grain-oriented
4750 alloy
Fe248% Ni
11,000
80,000
2
1.55
0.48
4-79 permalloy
Fe24% Mo279% Ni
40,000
200,000
1
0.80
o.58
SuperalloyFe25% Mo280% Ni
80,000
450,000
0.4
0.78
0.65
0.4
0.78
0.65
2.30
0.40
2V-PermendurFe22% V249% Co
800
450,000
SupermendurFe22% V249% Co
100,000
Metglas 2605SC
300,000
3
Metglasa 26052S2Be78B13Si9
600,000
2
1.35 1.561.37
a
Fe81B13.5Si3.5C2
16
2.00
1.46 1.611.35
MnZn Ferrite
H5C2b
10,000
7
0.09 0.401.5 3 105
MnZn Ferrite
H5Eb
18,000
3
0.12
0.44 5 3 104
NiZn Ferrite
K5
290
80
0.25
0.33 2 3 1012
b
a
Allied Corporation trademark.
b
TDK ferrite code.
(Based on data from "Magnetic Materials: An Overview, Basic Concepts, Magnetic Measurements, Magnetostrictive Materials," by
G.Y. Chin et al. In R. Bloor, M. Flemings, and S. Mahajan (eds.), Encyclopedia of Advanced Materials, Vol. 1, Peragamon Press. (1994),
p. 1424, Table 1.)
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Magnetic Materials
17.7 Soft and hard magnetic
materials
Both ferromagnetic and ferrimagnetic materials can be either soft or hard. Soft magnetic materials have
a low Hc and hard magnetic materials have a high Hc, as shown in the hysteresis loops in Figure 17.15.
In a soft magnet, Bloch wall motion is easy, and the intensity of magnetization changes by the motion
of the Bloch walls to increase the size of the domains oriented in the direction of the applied magnetic
field. In hard magnetic materials, the Bloch wall motion is very difficult. If the Bloch walls do not move
in response to the applied magnetic field, the unpaired electron spins must flip within the domains into
the direction of the applied magnetic field to change the magnetization. Tables 17.4 and 17.5 list some
soft and hard magnetic materials, respectively.
In Table 17.4 the retentivity (Br ) is also called the remanence, and Br is equal to μ0Mr. Bmax in
Table 17.4 is equal to the maximum relative permeability (μr) times μ0Ms. The electrical resistivity
listed in Table 17.4 is important for soft magnetic materials that are to be utilized in alternating
magnetic fields. A high electrical resistivity suppresses eddy currents that heat the magnetic material
in an alternating magnetic field. The ferrites have the highest electrical resistivity of any materials
in Table 17.4.
In soft magnetic materials, Bloch wall motion is made easy by having a uniform material
without defects that would impede this motion. An example of a soft magnetic material is iron with
3 weight percent silicon. This material has a very low carbon content to prevent the formation of iron
carbide. The silicon is added to increase the electrical resistivity. A high electrical resistivity reduces
the magnitude of the eddy currents in a cyclic magnetic field. Iron-silicon alloys are used extensively in
electrical motors and transformer cores that have the magnetic field reversed at a frequency of 60 Hz.
Hard magnetic materials are used in applications where the intensity of magnetization is permanent,
such as in magnetic separators, control devices for electron beams, and magnetic holding devices. In
hard magnetic materials, domain walls are pinned by placing obstacles in the path of wall motion. The
same defects that make metals mechanically hard also make metals magnetically hard. In hard magnetic
M
Hard
Soft
H
Figure 17.15 A schematic of the magnitude of the intensity of magnetization (M ) as a function of the
magnitude of the applied magnetic field (H ) for hard and soft ferromagnetic or ferrimagnetic materials.
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CHAPTER 17
Table 17.5 Some Hard Magnetic Materials and the Retentivity (Br ), coercivity (Hc),
(BH)max, and Curie Temperature (Tc) for each
Material
Br (Tesla)
Hc (105A/m)
Co-steel1.07
0.16
BHmax (kJ ? m23)
6
Tc(˚C)
887
Alnico-51.05
0.477
44
880
BaFe12O190.42
2.47
34
469
SmCo50.87
6.37
144
723
Nd2Fe14B1.23
9.63
290–445 312
Based on data from from Permanent Magnetism, by R. Skomski and J. M. D. Coey. Edited by J. M. D. Coey and D. R. Tilley,
Institute of Physics Publishing. (1999), p. 23, Table 1.2.
Magnetic
flux density
materials, second-phase particles, such as borides in Nd2Fe14B or Ni3Al in alnico, are barriers to domain
wall motion. Alnico is an acronym for a family of iron-based ferromagnetic alloys, that contain Al,
Ni, and Co. Some of the hard magnetic materials, such as alnico, are of similar composition to hightemperature and high-mechanical-strength metal alloys. The magnetic properties of some hard magnetic
materials are presented in Table 17.5.
Comparing the soft magnetic materials in Table 17.4 to the hard magnets in Table 17.5, the main
difference is the value of the coercivity. The hard magnetic materials have a coercivity from three to five
orders of magnitude larger than that of soft magnetic materials. The high coercivity is important for
permanent magnets. In Table 17.5, the term (BH )max is a measure of the energy density that is induced
into a magnetic material by an applied magnetic field, as shown in Figure 17.16. An area in a plot of B
versus H using SI units is energy density in joules/m3. This is analogous to the area under a stress-strain
plot that is also energy density. The term (BH )max is the area of the maximum-size rectangle that can
be drawn in the second or fourth quadrant of the B versus H plot. Strong permanent magnets have a
high (BH )max.
( BH ) max
Magnetic
field
Figure 17.16 A schematic plot of the magnetic flux density (B) versus the applied magnetic field (H) and (BH)max.
(Based on Permanent Magnetism, by R. Skomski and J. M. D. Coey. Edited by J. M. D. Coey and D. R. Tilley. Institute of Physics
Publishing. (1999), p. 25, Fig. 1.15.)
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Magnetic Materials
17.8 Magnetic recording
Magnetic materials are utilized for recording information on magnetic recording tapes and hard disks for
computers. Magnetic recording tapes are made in two ways. Small magnetic particles are embedded in
a polymer tape, and the second method is by depositing thin films of magnetic material onto a polymer
backing. The magnetic materials for thin film coatings are similar to the materials for hard disks that are
discussed later in this section. The metal coating is deposited onto a polymer tape made of polyethylene
terephthalate (PET) known by the trade name of MylarTM. An advantage of magnetic particle tapes is
that they are less expensive to produce. The cost of particle magnetic tape data storage is approximately
$0.01 per gigabyte (GB). Some of the magnetic recording particle materials are shown in Table 17.6. A
recent magnetic particle is barium hexaferrite (BaFeO) that has a formula such as BaFe12O19; however,
several other compositions are also observed. The BaFeO particles in Table 17.6 are thin hexagonal
platelets with a diameter of approximately 0.08 μm. With BaFeO tapes it is possible to store 30 3 109
bits of data per square inch of tape. Magnetic tape materials have values of retentivity, as shown in
Table 17.6, similar to both hard and soft magnetic materials and the coercivity of magnetic tape materials
is higher than for soft magnetic materials but less than for hard magnetic materials. Magnetic tape is used
for archival data storage, and the magnetic stripe on a credit card is a magnetic tape.
The data are recorded onto the tape by applying a magnetic field that orients the unpaired electron spins
along the long axis of the particle in the forward or reverse direction. This corresponds to a 1 or a 0 for
digital data storage. The data storage is permanent, because after writing the data by producing saturation
magnetization (Ms ) in the material, the magnetization remains at the remanent magnetization (Mr ) after
removal of the magnetic field. The coercive field of the magnetic particles (Hc ) must be sufficiently large
that the unpaired electron spin orientations are not altered by stray magnetic fields; however, Hc also must
be sufficiently small that it can be changed by the magnetic field in the write heads of the data storage
system. It is possible to erase the data by reversing the applied magnetic field and to rewrite new data
by applying a new magnetic field. One disadvantage of tape is that tape must be wound to find the data;
therefore, it is best utilized for archiving of data. Hard disks allow data to be located more quickly.
The permanent data storage in most computers is on hard disks. The hard disk is a magnetic recording
medium made of a fine-grained (approximately 30 nm) thin film of metal alloy such as Co-Cr-Pt or
Co-Cr-Ta, and each grain of the metal film is a potential recording location. On a hard disk, magnetic
thin film is deposited on a nonmagnetic substrate material, such as an aluminum alloy, and for magnetic
Table 17.6 Properties of Some Magnetic Recording Particles, Including the Particle Length (L),
Aspect Ratio (L/d ) Retentivity (Br), Coercivity (Hc), and Curie Temperature (Tc)
Material
L (μm)L/dBr (tesla)
Hc (kA ? t/m)Tc (˚C)
g-Fe2O3 0.205/1 0.44
22–34
600
Co-g-Fe2O3
0.206/1 0.48
30–75
700
CrO2
0.2010/1 0.50
30–75
125
Fe
0.1510/1 1.40
56–176
770
BaFeO — —0.40
56–240350
Based on data from The Complete Handbook of Magnetic Recording, Fourth Edition, by F. Jorgensen, The McGraw-Hill
Companies. (1996), p. 324, Table 11.1.
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CHAPTER 17
tape similar alloys are deposited on polymer tape. The direction of magnetization determines if the data
is a 1 or a 0. The first hard disks introduced in 1956 could store 3 MB of information in a device the size
of two refrigerators. Modern hard disks can store 3 TB in a device with a volume of approximately one
cubic inch, and in the year 2012 each bit of data required approximately 800,000 atoms.
17.9 Antiferromagnetism
Materials such as Cr, Mn, MnO, MnS, and NiO are antiferromagnetic. From Table 17.2 it might be
expected that Cr and Mn would be ferromagnetic; however, they are antiferromagnetic at temperatures
below the Neel temperature. At the Neel temperature, antiferromagnetic materials change to paramagnetic
upon heating. In antiferromagnetic materials the unpaired electron spins on different atoms align
antiparallel, and the magnetic moment from the paired spins cancel each other, as shown in the
2-D schematic in Figure 17.17.
In the simplest antiferromagnetic materials, such as BCC Cr and Mn, there are two sublattices A
and B. The A sublattice atoms are at the BCC lattice cube corners, and the B sub lattice atoms are
at the body centered atom positions. The unpaired electron spins in the A lattice behave as though
they are ferromagnetic, and the unpaired electron spins on the B lattice also behave as though they are
ferromagnetic. However, the electron spins on the A and B lattices oppose each other and cancel out the
intensity of magnetization. The magnetic susceptibility ( ) for an antiferromagnetic material is positive
and small. is positive because the magnetic field rotates electron spins into the direction of the applied
magnetic field, and is small because of the opposed unpaired electron spins.
in the antiferromagnetic state decreases as temperature decreases, because the unpaired electron
spins on the two lattices more perfectly cancel each other out at lower temperatures. In the paramagnetic
state, the magnetic susceptibility decreases as temperature increases, because thermal vibrations disorder
the unpaired electron spins. At the Neel temperature, the magnetic susceptibility is at its maximum.
Scientists at IBM have recently demonstrated the possibility of storing data with as few as 12 atoms of
iron that are antiferromagnetic. Figure 17.18 shows how 12 atoms of iron are aligned in 2 rows of 6 atoms
each, in a stable cluster of iron atoms on a copper nitrate substrate. In the configuration shown, the iron
atoms are antiferromagnetic and stable. If the magnetic state is changed to ferromagnetic, the clusters
are not stable. Although iron is ferromagnetic at room temperature, it is possible for the magnetic state of
materials to change with temperature and with lattice strain. For example, Fe2O3 is ferrimagnetic at room
temperature, and it is antiferrimagnetic at temperatures below 260 K.
For computer information storage, the stable antiferromagnetic state of 12 antiferromagnetic Fe atoms
could represent a 1, and the unstable ferromagnetic state could represent a 0. Each cluster of 12 Fe atoms
would be one bit of information. Although this form of memory may never be practical, memory based
upon parallel and antiparallel unpaired electron spins is already in use in magnetoresistance systems,
discussed next in Section 17.10.
Figure 17.17 The orientation of unpaired electron spins in 2-D schematic of an antiferromagnetic material. The
oriented unpaired electron spins sum to 0.
Magnetic Materials
Figure 17.18 An STM image of eight clusters of iron atoms in an antiferromagnetic state, with each cluster
containing 12 iron atoms in two rows of six atoms, on a copper nitrate surface. Each cluster of 12 atoms could be
one bit of memory. The randomly placed atoms are xenon. (IBM)
17.10 Magnetoresistance
Magnetoresistance is when a change in the magnetization of a magnetic material changes the electrical
resistivity. Disordered unpaired electron spin orientations on atoms in a metal scatter conduction
electrons more frequently than do ordered unpaired electron spins. A ferromagnetic metal with unpaired
electron spins all oriented in the same direction has a lower electrical resistivity than the same metal with
unpaired electron spins oriented in different directions.
Giant magnetorestistance (GMR) have been found in some layered magnetic materials, where it is
possible to produce changes of 10% in the electrical resistivity by changing the magnetic state of the
material. In one GMR material, layers of a ferromagnetic material (FM), such as Fe, and nonmagnetic
(NM) spacers, such as of Cr, alternate, as shown schematically in Figure 17.19. The layered material is
produced by repeatedly depositing thin layers of material, and each layer is approximately 10-atoms thick.
If a saturation magnetic field (Hs ) is applied to the GMR material, all of the unpaired electron spins
in the FM layers are oriented parallel, and the direction of magnetization (M) in each layer is parallel
as shown by the large arrows in the FM material in Figure 17.19a. The conduction electrons in the FM
metal Fe are primarily 4s electrons, and their spins are paired up and down. The up and down spin
directions of the 4s electrons are indicated by the small up and down arrows on the left of each figure.
Conduction electrons with spin opposite to M of the FM material are scattered more frequently by the
vibrating iron atoms, as indicated by the irregular line crossing Figure 17.19, and conduction electrons
with spin parallel to M are conducted, as indicated by the straight line. If the applied magnetic field is
reversed to the coercive field (2Hc ), then the direction of M alternates in the layers, as shown in
Figure 17.19b. With layers of FM material having directions of M that alternate, both spin-up and spindown conduction electrons are scattered, and this material has a higher electrical resistivity than with
parallel M. The GMR effect is an application of spintronics. Spintronics is the use of electron spin to
produce electronic devices.
Several applications of spintronics have been developed. GMR materials are used as read heads
for magnetic recording systems. At the time of this writing, GMR read heads can read approximately
1.6 3 1015 bits per square meter. The magnetic field of a recorded bit that produces parallel unpaired
electron spin layers in the GMR read head results in a low electrical resistivity, and a recorded bit that
produces antiparallel unpaired electron spins in the read head results in a high electrical resistivity. The
low and high electrical resistivity correspond to reading a 1 or a 0, respectively, from the hard disk.
W-91
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CHAPTER 17
Spin
FM
NM
(a)
FM
Spin
FM
NM
FM
(b)
Figure 17.19 A schematic of the structure of a GMR material, showing layers of ferromagnetic (FM) material
and nonmagnetic (NM) material. In (a) the FM layers have parallel intensity of magnetization (M), as indicated by
the large bold arrows. The path of conduction electrons is indicated by arrows crossing the layers, and the spin
of the conduction electrons is indicated by the small arrows to the left of the figure. Conduction electrons with spin
that is opposite to M are more likely scattered by vibrating FM atoms than conduction electrons with spin parallel
to M. In (b) with alternating M, electrons with spin up and spin down are scattered in the FM material resulting in
a higher resistivity than the material with parallel M. (Based on http://en.wikipedia.org/wiki/File:Spinvalvescattering.png)
The GMR effect is also used for memory in magnetoresistive random access memory (MRAM). In
MRAM the magnetic state of parallel unpaired electron spins, shown in Figure 17.19, is a 1; and the state
with antiparallel unpaired electron spins is a 0. The magnetic state is switched with an applied magnetic
field (H), and the magnetic state of the MRAM is determined by electrical resistivity.
Summary
●●
●●
●●
A magnetic field is created by passing an electrical current through loops of conducting wire.
The response of the environment to a magnetic field is called the magnetic flux density. The
intensity of magnetization is the magnetic field created in a material by an external applied
magnetic field. The intensity of magnetization of a material is a result of free electrons forming
current loops, of the alteration of the electron orbitals on atoms, and of the orientation of
unpaired electron spin. The ratio of the intensity of magnetization to the applied magnetic field
is the magnetic susceptibility.
Diamagnetic materials, other than superconductors, have a small negative magnetic susceptibility.
All atoms have a diamagnetic response when a magnetic field is applied. The electron orbits on
the atom are altered to oppose the applied magnetic field in compliance with Lenz’s law. In a
metal, free electrons form current loops or eddy currents in compliance with Lenz’s law, which
oppose an applied magnetic field.
Superconductors have a magnetic susceptibility of 21. An applied magnetic field induces in a
superconductor a magnetic field equal and opposite to the applied magnetic field. The induced
Magnetic Materials
●●
●●
●●
●●
●●
●●
●●
●●
magnetic field in a superconductor results in interesting applications such as the magnetic
levitation of trains.
Paramagnetic materials have a small positive magnetic susceptibility that results from orienting
electron spin in the direction of the applied magnetic field.
Ferromagnetic materials are metals that have a large magnetic susceptibility dependent upon the
magnitude of the applied magnetic field. Ferromagnetism results when there is a large number
of parallel unpaired electron spins on the atoms. Applied magnetic fields (H) rotate the unpaired
electron spins to produce an intensity of magnetization (M) that adds to H. At saturation, all
unpaired electron spins are oriented parallel to H. When H is subsequently reduced to 0, there
is a remanent M. To reduce M to 0 requires the application of the coercive magnetic field in the
reverse direction. When M is reduced to 0, the unpaired electron spins are parallel within magnetic
domains, and the sum of the unpaired electron spins over all of the magnetic domains is 0. When
a ferromagnetic metal is heated above the Curie temperature, the metal becomes paramagnetic.
An alternating magnetic field induces current loops in a ferromagnet that heat the metal.
Ferrimagnetic materials, such as Fe3O4, have hysteresis loops similar to those of a ferromagnetic
material, but they are insulators. Ferrimagnetic materials are utilized in high-frequency
applications, such as televisions, communications equipment, and microwave devices.
Soft magnetic materials have a small coercivity, and hard magnetic materials have a large
coercivity. Soft magnetic materials are used in applications with alternating magnetic fields, and
hard magnetic materials are used in applications requiring permanent magnets.
Magnetic recording tapes are made with magnetic particles that are embedded in a polymer
tape and by depositing a thin film of magnetic material on the tape. The magnetic particles are
usually elongated with an aspect ratio of 5/1 to 10/1, and the magnetic particles are so small that
each particle is a single magnetic domain.
The hard disk for computer permanent memory is a magnetic recording medium made of a
fine-grained (approximately 30 nm) thin film metal alloy such as cobalt-chromium-platinum or
cobalt-chromium-tantalum, and each grain of the metal film is a potential recording location.
In antiferromagnetic materials, such as Cr, Mn, and MnO, the unpaired electron spins on
different atoms align antiparallel below the Neel temperature, and above the Neel temperature
the materials are paramagnetic. The magnetic susceptibility for an antiferromagnetic material is
positive and small.
Magnetoresistance is when a change in the magnetic state of a material changes the electrical resistivity
of the material. Giant magnetorestistance (GMR) has been found in some multilayered magneticspacer materials, where it is possible to produce very large changes, such as 10%, in the electrical
resistivity by changing the magnetic state of the material. GMR materials are now used for the read
heads in magnetic recording systems and in magnetoresistance random access memory (MRAM).
Supplemental Reading: Subjects and Authors
Full references are listed at the end of the book.
General:
Askeland, Fulay, and Wright
Electromagnetics:Kraus
Magnetic materials:
Chikazumi; Crangle
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Homework
Concept Questions
1. The flux density divided by the applied magnetic field strength is equal to the magnetic ________________.
2. The intensity of magnetization divided by the applied magnetic field strength is equal to the magnetic
________________.
3. The dipole _____________ of a magnet is the pole strength times the separation distance of the two poles.
4. The relative magnetic permeability of a vacuum is equal to _____________.
5. If the magnetic susceptibility of a material is 25 3 1026, this material is _______________.
6. A superconductor at temperatures below the critical temperature has a magnetic susceptibility equal to
________.
7. If the magnetic susceptibility of a material is 5 3 1026, and the susceptibility decreases with an increase
in temperature, this material is _______________.
8. A positive magnetic susceptibility is due to an applied magnetic field orienting electron __________ in the
direction of the applied magnetic field.
9. In a paramagnetic material, if a large magnetic field is first applied and then reduced to 0, the intensity of
magnetization is then equal to ___________.
10. The applied magnetic field necessary to reduce the intensity of magnetization of a ferromagnetic material
to 0 after saturation in the forward direction is the _____________ field.
11. In a ferromagnetic material, regions of parallel unpaired electron spin are magnetic ______________.
12. To demagnetize a ferromagnetic metal, heat it to above the ___________ temperature in the absence of a
magnetic field.
13. Materials that have a large magnetic susceptibility and are insulators are _________________.
14. Ferromagnetic materials with a large ______________ are hard magnetic materials.
15. A(n) ______________________ material has a small positive magnetic susceptibility that approaches 0 at
0 K, increases with increasing temperature, reaches a maximum, and then is paramagnetic with further
increases in temperature.
16. At a __________ wall, the unpaired electron spins orientation changes from one orientation to another.
17. _________________ occurs when there is a change in electrical resistivity in a ferromagnetic material
resulting from an application of a magnetic field.
Engineer in Training–Style Questions
1.
The magnetic field produced in a material resulting from an applied magnetic field is called the
(a) Intensity of magnetization
(b) Magnetic flux density
(c) Magnetic permeability
(d) Magnetic susceptibility
Magnetic Materials
2.
If the relative magnetic permeability of a material is equal to 0.99999, this material is
(a) Diamagnetic
(b) Paramagnetic
(c) Ferromagnetic
(d) Antiferromagnetic
3.
Superconductivity is destroyed by a high magnetic field because the magnetic field
(a) Heats the superconductor
(b) Induces current in the superconductor
(c) Rotates Cooper pair electron spins into the same direction
(d) Alters the orbits of electrons on the atoms
4.
In a paramagnetic material the magnetic susceptibility is
(a) Linearly dependent upon temperature
(b) Independent of temperature
(c) A nonlinear function of temperature
(d) Inversely dependent upon temperature
5.
A given metal has a magnetic susceptibility of 100,000. This metal is
(a) Diamagnetic
(b) Antiferromagnetic
(c) Ferromagnetic
(d) Paramagnetic
6.
When the applied magnetic field on a ferromagnetic material is reduced to 0 after saturation, the intensity
of magnetization is equal to
(a) 0
(b) The saturation magnetization
(c) The remanent magnetization
(d) The coercive field
7.
The heating of a ferromagnet in an alternating magnetic field is due to
(a) Induced free-electron current loops
(b) Alterations of atomic electron orbitals
(c) Switching of electron spin orientations
(d) Friction created by Bloch wall motion
8.
Which of the following materials is most appropriate for a design requiring a material with a high
magnetic susceptibility that will be subjected to an alternating magnetic field of 106 Hz?
(a) Fe23 weight percent Si
(b) MnO
(c) Alnico25
(d) Fe3O4
9.
Which of the following materials is most appropriate for a design requiring a material with a high
magnetic susceptibility that will be subjected to an alternating magnetic field of 60 Hz?
(a) Fe23 weight percent Si
(b) MnO
(c) Alnico25
(d) Fe3O4
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10. Which of the following materials is most appropriate for a design requiring a material with a permanent
high intensity of magnetization?
(a) Fe23 weight percent Si
(b) MnO
(c) Alnico25
(d) Fe3O4
11. The main difference between hard and soft magnetic materials is their
(a) Remanence
(b) Saturation magnetization
(c) Coercivity
(d) Electrical resistivity
12. Which of the following materials is most appropriate for a design of the magnetoresistance random
access memory for a computer?
(a) Layers of Fe and Ni
(b) Layers of Fe and Cr
(c) Layers of Fe3O4 and MnO
(d) Fe23 weight percent Si
Problems
Problem 17.1
(a) Calculate the magnetic field created by a coil made of copper wire that has 200 turns in 5 cm
and carries a current of 5 A. (b) What is the magnitude of the flux density or inductance created
by this coil in a vacuum?
Problem 17.2 You are asked to design a coil that will produce a magnetic field of 1 3 104 A/m. The maximum
current for the coil wire is 1 A, and the space available for the coil limits the coil length to 1 cm.
How many loops are required for the coil?
Problem 17.3 You are asked to design a coil to operate in air that will produce a flux density of 0.01 T. The
maximum current for the coil wire is 1 A, and the space available for the coil limits the length
to 1 cm. How many loops are required for this coil? Round your answer to the nearest whole
number.
Problem 17.4 A coil to be operated in air has 200 loops in a length of 3 cm. What current is required to
produce a flux density of 0.02 T?
Problem 17.5 A solenoid made of copper wire has 100 turns in 3 cm and carries a current of 3 A in air. The
ferromagnetic material 78 Permalloy (See Table 17.1) is inserted into the coil. What is the flux
density created by the coil with this material inserted?
Problem 17.6 A solenoid is available that has 100 loops in 2 cm of length with a supermalloy core (See
Table 17.1), and the maximum current for the solenoid wire is 1A. What flux density is possible
for this solenoid in air?
Magnetic Materials
Problem 17.7 A solenoid must be produced that has a magnetic flux density of 25 Wb/m2. The coil for the
solenoid has 200 loops in a distance of 2 cm, and the maximum current for the wire is 1A.
Select a suitable material for this solenoid from the materials listed in Table 17.1 that you expect
to be cost effective.
Problem 17.8 A solenoid must be produced that has a magnetic flux density of 25,000 T. The coil for the
solenoid has 200 loops in a distance of 1 cm, and the maximum current for the wire is 1 A.
Select a suitable material for this solenoid from the materials listed in Table 17.1. For this
design, performance is more important than cost.
Problem 17.9 A magnetic field of 104 A/m is produced by a coil. What is the intensity of magnetization of the
following materials placed into the coil?
(a) Silver
(b) Vacuum
(c) Air
(d) Palladium
(e) Nickel
Problem 17.10 Prove that x 5 r 2 1.
Problem 17.11 Copper is usually the material of choice for the wire in a coil to produce a magnetic field and
in wiring for electric motors. Copper wire is placed in a magnetic field (H) of 1 3 106 A/m. The
magnetic susceptibility of copper is 29.7 3 1026.
(a) Determine the intensity of magnetization (M) in the copper. Is this M in the same direction
or opposed to H?
(b) What is the average dipole moment per atom (pa) in copper that is parallel or antiparallel
to H when subjected to the applied magnetic field?
Problem 17.12 Sodium metal is paramagnetic with a dimensionless magnetic susceptibility ( ) of 8.48 3 1026.
The paramagnetism of sodium metal is due to the rotation of some of the spins of the metal’s
free electrons into the direction of the applied magnetic field. The applied magnetic field is
equal to 1 3 105 A/m. Sodium metal is BCC with a lattice parameter of 0.428 3 1029 m, and
sodium is from Group 1 A of the periodic table.
(a) What is the intensity of magnetization or the dipole moment per unit volume?
(b) For sodium, what is the number of atoms per unit volume, and what is the number of free
electrons per unit volume?
(c) What is the dipole moment per sodium atom?
(d) Assuming that all of the magnetic susceptibility is due to the orientation of free-electron
spins, what is the net number of free electron spins per atom oriented in the direction of
the applied magnetic field?
Problem 17.13 The number of Bohr magnetons per atom for cobalt is 1.72. Calculate the saturation
magnetization for cobalt.
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Problem 17.14 The saturation magnetization of nickel measured at room temperature is 5.1 3 105 A/m.
(a) Calculate the average number of Bohr magnetons on a nickel atom, knowing that nickel
has an FCC structure with an atom at 0, 0, 0. At room temperature, the lattice parameter is
0.352 nm. (b) Compare your answer to part (a) with the theoretical number of Bohr magnetons
based upon the atomic electron structure of nickel, and explain any possible difference.
Problem 17.15 The rare earth element gadolinium is ferromagnetic, and it has a saturation magnetization
of 20 3 105 A/m at low temperatures. (a) Calculate the number of Bohr magnetons
per atom of gadolinium, knowing that the atomic mass is 157.2 g/mole and the density is
7.89 g/cm3. (b) What is the average number of parallel unpaired electron spins in the 4f shell
per gadolinium atom?