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Chapter 8
MAGNETIC MATERIALS
8.1
Introduction
As the origin of dielectric properties is in the electric dipoles associated with atoms and molecules,
magnetic properties of materials are due to intrinsic magnetic dipoles of atoms. Some materials
exhibit a particularly strong response to an externally applied magnetic …eld. The most familiar
magnetic material is iron (Fe) which is found in transformers, motors, generators, magnetic tapes,
magnetic disks, and so on. The microscopic origin of ferromagnetism is in the alignment of atomic
magnetic dipoles which is a quantum mechanical e¤ect. The magnetic dipole moment of an atom
is caused by electron orbiting and electron spin. The alignment can be destroyed by thermal
agitation, and the ferromagnetism indeed disappears when the temperature exceeds a critical value
known as the Curie point. In contrast to dielectric properties which are in most cases linear in
the sense that the permittivity is independent of the electric …eld, most magnetic materials exhibit
strong nonlinearity. The e¤ective permeability 7m 1has strong dependence on the magnetic …eld.
Furthermore, in a ferromagnetic material, the magnetic …eld often becomes multivalued, that is,
the curve of magnetic …eld as a function of external magnetizing current does not trace itself when
the current is decreased. Even in the absence of any external current, a magnetic …eld can remain
in iron (permanent magnet). This phenomena, known as hysteresis, is the key element for magnetic
materials to be useful as switching and memory devices.
1
I
δ
δ
δ
Figure 8.1: Collection of magnetic dipole moments to model a permanent magnet. There is no
conduction current. However, there is an e¤ective surface current.
8.2
Magnetic Field due to a Collection of Magnetic Dipoles
Let us model an atom having a dipole moment m (A m2 ) by a cubic volume of side
carrying a
surface current I (A). When many such dipoles are packed, together with all of the dipole moments
aligned in one direction, the currents in the volume all cancel out, and only the surface current
appears at the surface of the volume. The magnetic …eld in the volume is found in analogy to the
case of a long solenoid,
B=
I
0
=
m
(8.1)
0 3
where I= (A/m) is the equivalent surface current density. However, since n = 1=
3
is the number
of dipoles in unit volume, we may rewrite the magnetic …eld as,
B=
0 nm
=
0M
(8.2)
where M (A/m2 = A m2 /m3 ) is the magnetic dipole moment density which corresponds to the
electric dipole moment density P in a dielectric. The vector M is called the magnetization vector.
It is important to note that the dipole moment m and magnetization M are not caused by
conduction currents. Rather they have their origin in either orbiting or spinning motion of electrons.
(All elementary particles have a spin and associated magnetic dipole moment. This is true even for
a neutron because of the di¤erence in positive and negative charge distributions. It appears that
in a neutron the negative charge extends radially more than the positive charge. Consequently, a
2
neutron has a negative magnetic dipole moment, "negative" in the sense that the magnetic dipole
moment is opposite to the mechanical angular momentum.) However, when a large number of
dipoles are collected, dipoles in the volume do not really contribute to the magnetic …eld, and the
magnetic …eld is determined by an equivalent surface current.
A more formal derivation of the magnetic …eld due to distributed magnetic dipoles is as follows.
In Chapter 7, we have learned that the vector potential due to a single magnetic dipole is given by,
A=
0
m
4
(r r0 )
jr r0 j3
(8.3)
where r0 is the location of the ”point” dipole m. If there is a continuous distribution of dipoles,
this may be generalized as,
A=
0
4
Z
M (r r0 )
dV 0
3
0
jr r j
(8.4)
Then, the magnetic …eld can be calculated as,
B=r
A =
=
=
0
rr
Z
0
rr
rr
0
rr rr
4
4
4
M(r0 )
jr
Z
(r r0 )
dV 0
r0 j3
M(r0 )
1
dV 0 Note rr
=
0
jr r j
jr r0 j
Z
Z
M(r0 )
M(r0 )
0 2
0
dV
r
dV 0
r
jr r0 j
4
jr r0 j
r
jr
r0
r0 j3
(8.5)
However,
er
r
rr
0
(8.6)
Therefore, the …rst integral in RHS vanishes. The second integral is equal to
0
4
=
0
r2r
Z
Z
M(r0 )
dV 0
jr r0 j
M(r0 ) (r
r0 )dV 0 =
0 M(r)
(8.7)
where use has been made of the identity,
r2
1
jr
r0 j
=
4
(r
r0 )
B=
0
Thus,
B=
0M
or
r
3
r
M
(8.8)
Comparing this with the Maxwell’s equation (Ampere’s law),
r
B=
(8.9)
0J
we observe that
r
(A=m2 )
M = Jm
is equivalent to a current density (magnetization current). If a conduction current Jc is also present,
the total current density is therefore given by
J = Jc + Jm
= Jc + r
(8.10)
M
and the Maxwell’s equation becomes,
r
B=
0 (Jc
+r
M)
(8.11)
It is customary to introduce a new vector de…ned by
H=
B
M
(8.12)
(conduction current only)
(8.13)
0
so that
r
B
0
M
=r
H = Jc
The relationship between B and H is reminiscent to that between E and D. In some cases, separating the conduction current will be useful because the conduction current is the quantity externally
controllable (and readily measurable, too).
In vacuum or nonmagnetic materials, M = 0, and
B=
(M = 0)
0H
(8.14)
In magnetic materials with a magnetization M, this is modi…ed as,
B =
0 (H
+ M)
(8.15)
=
H
4
where
is the permeability of the material. For
to be independent of H or B, M must be linearly
proportional to H and B,
0
M=
H=
mH
(8.16)
0
where
is a proportional constant (dimensionless) and called the magnetic susceptibility. In prac-
tice, however, all ferromagnetic materials exhibit strong nonlinearity, and B is not linearly proportional to H. As magnetization increases, the magnetic …eld B tends to saturate. This is due
to an upper limit in magnetization allowed in magnetic materials. Magnetic materials consist of
so-called magnetic domains which may be regarded as tiny permanent magnets. Typical size of
the magnetic domains is of order 10 m and can be directly observed with the aid of an electron
microscope. In the absence of an external magnetic …eld, these magnetic domains are randomly
oriented, and there is no net (or macroscopic) magnetic …elds. When an external …eld is applied,
the domains orient themselves in the direction of the …eld, and the net magnetic …eld much larger
than the external …eld is produced. However, as the …eld further increases, all domains are aligned,
and this corresponds to the maximum available magnetization. The magnetic …eld B becomes
saturated and is insensitive to a further increase in the external magnetization H.
There is another nonlinear feature of ferromagnetic materials. The magnetic …eld B is usually
a double-valued function of H, as shown in Fig. 8.3. This phenomenon is called hysteresis. The
magnetic …eld B starts at zero only when a material is completely demagnetized. When magnetization (H) increases, B also increases, though not linearly. As magnetization decreases, the B
H
curve does not trace the original one, and even when external magnetization (H) is zero, a residual
magnetic …eld B remains. This is how a permanent magnetic is realized. In some applications
(switching, magnetic memory), strong hysteresis is required, and several alloys and compounds
(such as Alnico, Ferrite) have been developed particularly for strong hysteresis.
However, in devices for which linearity is desired (transformer is a typical example), hysteresis is
an undesirable feature and should be minimized. This is because hysteresis causes energy dissipation
within magnetic materials called hysteresis loss. To evaluate the loss, let us recall that the magnetic
energy density in a nonmagnetic material is given by,
um =
1
2
1
B2 = H B
2
0
5
(J/m3 )
(8.17)
Figure 8.2: B-H curve of a few iron materials. (From Introduction to Electric Circuits, 5th ed.,
H.W. Jackson, Prentice Hall)
6
This is valid only when the permeability is independent of the …elds. For magnetic materials
which exhibit strong nonlinearity between H and B, the above expression should be modi…ed as,
Z B
H dB
(8.18)
um =
0
since the change in the magnetic energy density is given by,
@um
@B
=H
@t
@t
or
dum = H dB
(8.19)
where the time varying magnetic …eld @B=@t induces an electric …eld as we will see in Chapter 9.
Note that for a nonlinear material, H dB 6= dH B. Therefore, if the magnetization is started
at B = H = 0, increased to the maximum, and returns to H = 0 (back to zero external current),
the shaded area in Fig. 8.3 appears as the net hysteresis loss per unit volume. For a sinusoidal
excitation current (thus H), the whole area inside the hysteresis curve indicates the hysteresis loss
per unit volume per unit cycle. The mechanism of energy dissipation is a kind of ohmic dissipation
of microscopic nature. Any change in magnetization is caused by the change in the alignment of
magnetic domains. When a domain changes its direction, a microscopic electric …eld is induced,
and ohmic dissipation J E occurs. Of course, in magnetic materials, which are often conductors,
ohmic loss due to conduction current can also take place. Transformers usually have laminated
magnetic materials to minimize the conduction ohmic loss by insulating one layer from the neighbouring layer.
8.3
Boundary Conditions for B and H
The magnetic …eld B satis…es
r B =0
I
or
B dS = 0
(8.20)
under any circumstances. Therefore, on a surface interfacing two media with di¤erent permeabilities, the normal components of B must be continuous,
Bn1 = Bn2
(8.21)
This is analogous to the continuity in the normal components of the displacement vector D in the
absence of free surface charges.
7
Bn1
B1
dS1
µ1
µ2
B2
dS2
Bn2
Ht1dl
µ1
µ2
Figure 8.3: r B = 0 ! (Bn1
-Ht2dl
Bn2 )dS = 0: Normal components of B are continuous. For H
…eld, the tangential components are continuous if there is no conduction current since r
! (Ht1
Ht2 ) dl = 0: If there is a surface conduction current Js (A/m), n
n is the unit normal vector at the surface.
8
(H1
H=0
H2 ) = Js where
The vector H in static cases satis…es,
r
H = Jc
In the integral form, we have,
I
(conduction current only)
H dl =
Z
(8.22)
Jc dS
(8.23)
Applying this to a rectangular closed loop in Figure 8.4(b), we …nd,
(H1t
where
H2t )dl = Jc dl
is the width of the rectangle. In the limit
! 0; Jc can remain …nite only if the current
density is singular, and in the form of a surface curent,
lim Jc = Js ;
!0
(A/m)
Therefore, the boundary condition for H is,
(Ht1
n
Ht2 ) = Js
(8.24)
where n is the normal unit vector directed from medium 2 to medium 1. In the
Fig. 8.4 Fields at a boundary separating two media with
1
and
2.
(a) Normal components
of B, (b) Tangential components of H,
absence of conduction surface current, Js = 0, we …nd that the tangential components of H are
continuous,
Ht1 = Ht2
If the medium 2 has a permeability
2
and the medium 1 has a permeability
(8.25)
1,
the normal
components of the magnetic …eld are given by,
Bn1 =
1 Hn1
(8.26)
Bn2 =
2 Hn2
The continuity in Bn thus yields,
1 Hn1
=
9
2 Hn2
(8.27)
µ0
10µ0
air
iron
Figure 8.4: B …eld pro…le at a surface of magnetic material with
= 10
0:
Combining with the continuity in the tangential components of H, we …nd,
tan
tan
where
1
and
2
1
1
=
2
(8.28)
2
are the angles between the magnetic …eld B and the normal vector n in each
medium.
If
angle
2
1.
1 ; tan 1
becomes large, and the angle
In practice, however, the angle
1
2
approaches =2, regardless of the ”incident”
is not a controllable quantity. When
2
1,
the
tangential component of H becomes negligibly small. Otherwise, the magnetic …eld B in the
medium 2 diverges since
B2t =
What happens in practice is that when
2
1,
2 H2t :
the magnetic …eld lines at the boundary all fall
almost normal to the highly permeable medium, as if they were strongly absorbed. The situation
is similar to static electric …eld lines falling (or leaving) normally on a conductor surface. Consider
a straight current placed parallel to the ‡at surface of iron as shown in Figure 8.6(a). Except
for one singular point (called stagnation or separatrix point), all magnetic …eld lines fall or leave
normal to the iron surface. The magnetic …eld pro…le in the air is equivalent to the …eld due to
two parallel currents. That is, the image of a current parallel to a surface of a highly permeable
body is in the same direction as the current itself. In contrast, the image of a current vertical to a
highly permeable surface is opposite to the current. This is because the tangential component of
the magnetic …eld (B in this case) should vanish on the surface.
Let us now consider the magnetic …eld near a superconducting surface. A superconductor is
10
I
µ0
µ>>µ0
I
image
Figure 8.5: Image of a current parallel to a highly permeable slab. Iimage = I: The magnetic …eld
lines fall normal to the surface except at the midpoint where the magentic …eld vanishes.
I
I
B
superconductor
superconductor
-I
I
Figure 8.6: Left: image of a current parallel to a highly permeable slab. Right: magnetic …eld lines
fall normal to the surface except at the midpoint where B = 0:
characterized by the complete absence of a magnetic …eld in its body. This e¤ect is known as
the Meissner e¤ect. Then, the current in a superconductor must ‡ow entirely on its surface. The
width of the surface current depends on materials, and is typically of order 1 m. Since B = 0 in
a superconductor, the normal component of the magnetic …eld must vanish at the surface. The
tangential component of the magnetic …eld is related to the surface conduction current through
n
H = Js
(8.29)
since in Eq. (8.24, the …eld in the superconductor is zero. A superconducting body simply does
not allow a magnetic …eld to penetrate, and on its surface, only the tangential component of the
magnetic …eld can exist.
11
Figure 8.7 shows the image current of a current placed near a superconducting plane in three
di¤erent cases, parallel, perpendicular and arbitrary orientation. The magnetic …eld pro…le of the
parallel current is identical to that due to two opposite currents, and the image in this case is
opposite to the current in contrast to Fig. 8.6 which is for highly permeable material (
0 ).
The image of a perpendicular current is in the same direction. This is also in contrast to the case
of highly permeable material.
The images shown in Figure 8.7 are applicable even when the conducting plane is not ideally
superconducting. For dc currents, the plane must be a superconductor. However, for high frequency
ac currents, similar images appear as long as the magnetic …eld does not penetrate well into the
conductor. This occurs when the skin depth de…ned by
=p
1
f
0
is small. Here, f (Hz) is the oscillation frequency of the current, and
(S/m) is the conductivity
of the plane. Skin e¤ects will be studied in Chapter 9.
Magnetic Shielding
Inside a volume covered with a ferromagnetic shell, the magnetic …eld is greatly reduced because
the magnetic …eld lines are strongly shielded by the material. In order to illustrate this magnetic
shielding, which is similar to electrostatic shielding (known as the Faraday shield), let us consider
a cylindrical highly permeable tube placed normal to an external magnetic …eld, as shown in Fig.
8.8. Since there are no conduction currents in the region of interest, the vector H satis…es,
r
H=0
(8.30)
everywhere. This allows us to introduce a magnetic scalar potential
H=
r
such that
(8.31)
The potential formally has the dimensions of current, but the magnetic potential introduced here
is only for mathematical convenience and has no physical signi…cance.
From r B = 0, we …nd
r ( r ) = 0:
Therefore, in a medium with uniform permeability r = 0,
12
(8.32)
should satisfy the Laplace equation,
r2
which we know how to handle. To be precise,
r2
However, when
= 0:
(8.33)
satisi…es
=r M
(8.34)
is uniform, r M is non-vanishing only at boundaries of di¤erent magnetic
materials. r M is analogous to an induced surface charge on the surface of a dielectric. It causes
a discontinuity in the normal component of H, but the potential
itself is continuous, which is
equivalent to the continuity in the tangential component of H.
In the cylindrical coordinates, with axial (z) dependence suppressed, the Laplace equation
becomes,
@2
1 @
1 @2
+
+
2
@ 2
@
@ 2
( ; )=0
(8.35)
This has general solutions of the form,
( ; )=
X
An
n
+
Bn
n
(Cn cos n + Dn sin n )
(8.36)
n
If the angle
is measured from the direction of the external magnetic …eld, then the system is
symmetric with respect to , and the sin n term drops out.
The uniform magnetic …eld externally applied, Bx = B0 ex may be described by a potential
0
B0
=
x=
0
Bo
cos
(8.37)
0
This …eld prevails everywhere. Therefore, in order to satisfy the boundary conditions for B and
H, the potential perturbation caused by the magnetic cylinder must also have the same angular
dependence, cos n . Imposing the boundedness of the potential everywhere, we may assume the
following solutions for each region,
Outside
( ; )=
B0
1
cos + c1 cos
;
>b
(8.38)
0
In the magnetic material
( ; )=
B0
cos + c2 +
0
13
c3
cos
; a <
<b
(8.39)
Inside
B0
( ; )=
cos + c4 cos
; 0<
<a
(8.40)
0
Then, the vector H in each region becomes,
H=
=
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
r
B0
=
+
0
B0
e
c1
2
c2 +
B0
cos e
c1
c3
2
sin e ;
2
0
0
B0
e @
@
+
@
@
B0
cos e
c3
c2
2
0
B0
c4 cos e
0
>b
sin e ;
c4 sin e ;
a<
<b
(8.41)
<a
0
The magnetic …eld in the air is given by,
0H
(8.42)
B= H
(8.43)
B=
and in the magnetic material (a <
< b),
The continuity in the tangential ( ) components of H at the outer and inner surfaces yields,
c1 = c3 + b2 c2
(8.44)
c3
a2
c4 = c2 +
(8.45)
The continuity of the normal ( ) components of B yields,
B0
0
+
0
B0
0
c1
b2
=
c2 +
c3
a2
0
B0
c2 +
0
=
B0
0
c3
b2
(8.46)
c4
(8.47)
0
Solving these simultaneous equations for c4 , which is required for the internal magnetic …eld we
are interested in, we …nd
c4 = H0
b2
4H0
0 2
b
2
1+
0
14
2
a2
1
0
(8.48)
Figure 8.7: Magnetic shielding by a highly permeable cylindrical tube.
Consequently, the magnetic …eld inside the cylinder ( < a) is given by,
b2
Bi = 4B0
0 2
b
a2 1
1+
0
0
which is uniform and directed along the external …eld, B0 ex . If
it should be. If
0,
(8.49)
2 ez
2
=
0,
we recover Bi = B0 ex as
the interior …eld approaches,
Bi ' 4B0
a2
0
b2
a2
which is much smaller than the external …eld B0 . For
(8.50)
ez
= 104
0,
the …eld inside the cylinder
remains smaller than the external …eld as long as,
b
a (thickness of the cylinder)
10
4
a:
High permeability sheets are commercially available speci…cally for the purpose of shielding the
earth magnetic …eld which is often required for sensitive electronic equipment to work properly.
Uniformly Magnetized Sphere
This problem is analogous to the uniformly polarized sphere worked out in Chapter 4. Consider
a ferromagnetic sphere of radius a uniformly magnetized by a magnetization M = M ez . There are
no conduction currents. Therefore, we may introduce the magnetic scalar potential
(r; ), which
satis…es Laplace equation,
@2
2 @
1 1 @
+
+ 2
@ 2
@
sin @
15
sin
@
@
=0
(8.51)
The potential outside the sphere is of dipole type, and we assume,
e
=A
a
r
2
cos
(r > a)
(8.52)
Inside, the potential should be of the form
r
= A cos
a
i
(r < a)
(8.53)
The continuity in the potential at the sphere surface r = a is satis…ed by the assumed potential.
The …eld H is therefore given by,
H=
r
8
>
>
a2
a2
>
>
< 2A 3 cos er + A 3 sin e
r
r
=
>
>
>
A
A
>
:
cos er + sin e
a
a
(r > a)
(r < a)
Note that the …eld H in the sphere is uniform and is directed in -z direction. Since the magnetization
in M is also uniform, the magnetic …eld B is uniform too,
Bz =
0 (Hz
+ M)
(r < a)
(8.54)
The radial component of B at the sphere surface should be continuous. The interior …eld is,
Bir =
0
A
a
M
cos
(8.55)
and the exterior …eld at r = a is,
Ber = 2
0A
1
cos
a
(8.56)
Equating Eq. (8.55) to Eq. (8.56), we …nd,
A=
a
M
3
(8.57)
Therefore, the interior magnetic …eld is,
B=
2
3
0M
(8.58)
1
M
3
(8.59)
but the vector H is opposite to M and B,
H=
16
In general, the vector H is opposite to the magnetic …eld B in a permanent magnet. This is
not surprising since in the absence of conduction currents,
r
imposes a condition
H = Jc = 0;
I
H dl = 0:
C
This can be satis…ed only if either H is identically zero everywhere (which is not the case here) or
it changes signs along the integration path C (which is the case here).
If we further pursue the analogy with the uniformly polarized dielectric sphere, the surface
charge on the dielectric sphere corresponds to the "magnetic charge" on the magnetized sphere.
However, this is too much of contrivance, because the absence of magnetic charges is very well
imposed by
r B = 0;
and also because of no experimental evidence for supporting the existence of magnetic charges
(magnetic monopoles).
Figure 8.8: Spherical permanent magnet. The B pro…le is shown qualitatively in the left …gure
and H pro…le in the right …gure. Note the discontinuity in the H …eld lines at the surface.
17
8.4
Diamagnetism (
m
< 0)
In contrast to the electric susceptibility of matter which is always positive for dc …elds, the
magnetic susceptibility can be negative in some materials. Materials which exhibit a negative
magnetic susceptibility is called diamagnetic materials. The origin of diamagnetism may be seen
as follows. Consider a free electron undergoing cyclotron motion in an external magnetic …eld,
B = Bez , as shown in Figure 8.10. The magnetic moment due to the electron is
r2
m =
=
!ce
ez
2
2
mv?
ez
2B
(8.60)
which is opposite to the external …eld. Here, rc is the electron cyclotron radius, and ! c = eB=m
is the electron cyclotron frequency. They are related by rc = v? =! c with v? the electron velocity
perpendicular to the magnetic …eld. (See Chapter 7.) Note that the e¤ective current associated
with the cyclotron motion is
e! c =2 . Therefore, if there are n electrons per unit volume, the
magnetization M becomes
M=
2
nmv?
ez
2B
(A/m)
(8.61)
Such diamagnetism due to free electrons (and ions) occurs in a plasma con…ned in a magnetic …eld.
Similar diamagnetism occurs in gases and solids consisting of atoms and molecules having even
number of electrons, such as He and H2 gases, although the e¤ect is much weaker than in a plasma.
For simplicity, let us consider one of two electrons in a helium atom. In a classical picture, it
revolves about the nucleus having a charge Ze (Z = 2 for two protons), with a frequency
! 20 =
Ze2
4 0 r3
(8.62)
where r is the orbit radius. If the atom is placed in a magnetic …eld so that the angular momentum
of the electron is in the same direction as the magnetic …eld, the magnetic Lorentz force
ev
B
is radially inward, that is, in the same direction as the Coulomb force. (Figure 8.11) Since the orbit
radius cannot change because of quantum mechanical constraints, the orbiting frequency must
18
increase to compensate the increase in the centripetal force. From the radial force balance,
mr! 2 =
=
Ze2
+ eBv
4 0 r2
Ze2
+ eBr!
4 0 r2
(8.63)
! 20 = 0
(8.64)
we obtain,
!2
!! c
where ! c = eB=m is the electron cyclotron frequency. Solving for !, we …nd
! =
1h
!c
2
p
4! 20 + ! 2c
1
' !0 + !c
2
(! c
i
!0)
(8.65)
where the negative sign has been discarded because the frequency ! must be larger than the original
orbiting frequency ! 0 . If the angular momentum is opposite to the magnetic …eld, the frequency
decreases
1
!c
2
! = !0
(8.66)
In both cases, the change in the magnetic moment is opposite to the external …eld. In gases,
molecules are randomly oriented, and the net magnetization is given by,
M =
n0
Ze! c 2
r ez
24
Zn0 e2 0 2
r H
24 m
=
(8.67)
where n0 is the molecule density, and m here is the electron mass. The diamagnetic susceptibility
can be de…ned as,
m
=
Zn0 e2 0 r2
24 m
=
Z! 2p 2
r
24 c2
(8.68)
! 2p =
n0 e2
0m
(8.69)
where
19
is the square of the plasma frequency. For a helium gas, Z = 2; n ' 2:7
r ' 10
10 m
= 1A. Then,
m
is of order 10
10 ,
1025 =m3 at STP, and
a rather small (but measureable) e¤ect.
The change in the harmonic oscillation frequency in the presence of magnetic …eld is known as
the Zeeman e¤ect. It can be observed spectroscopically. Atoms emit electromagnetic waves (photons) at discrete frequencies determined by discrete energy levels which follow quantum mechanical
rules. For example, the energy levels allowed for the electron in a hydrogen atom are,
En =
where m = 9:1
10
31
1
(4
2
2
2 me4
h2
0)
1
(J) =
n2
kg is the electron mass, h = 6:63
13:6
10
1
(eV)
n2
34
(8.70)
J sec is Planck’s constant, and n
is an integer. (Note that the energy is negative. This is because the total energy of the electron
is the sum of a kinetic energy and potential energy. As shown in Chapter 4, the potential energy
of a hydrogen atom is negative. Its magnitude is twice of the electron kinetic energy, and the
total energy , kinetic plus potential, is negative. This means that a positive energy is required to
dismantle (or ionize) the hydrogen atom.) A photon can be emitted when an electron at a higher
energy level falls to a lower energy level. The photon energy is given by,
E = hv = 13:6
1
n22
1
n21
(eV)
(8.71)
if the electron falls from the state n1 to the state n2 (< n1 ).
When a magnetic …eld is present, a single spectroscopic line ! 0 in the absence of the …eld will
split into three lines,
!0 ; !0
1
!c;
2
when observed in the direction perpendicular to the magnetic …eld. (In the direction of the magnetic
…eld, the center line ! 0 disappears. Why?) This is because of the change in the harmonic oscillation
frequency caused by the magnetic …eld. The original line ! 0 also appears since those electrons
orbiting in a plane parallel to the magnetic …eld experiences no net magnetic e¤ects.
8.5
Paramagnetism (
m
> 0)
Paramagnetism is exhibited in general by atoms and molecules having an odd number of electrons.
Examples are alkali metals, aluminum, tungsten. The exception to this rule is oxygen which shows
20
particularly strong paramagnetism for a gas. An electron has its own intrinsic magnetic moment
(Bohr magneton) given by
m=
eh
4 m
A m2 =
J
Tesla
(8.72)
This is the smallest unit of magnetic moment and due to electron spin. Therefore, it is understandable that atoms having an odd number of electrons exhibit paramagnetism, which is a weak form
of ferromagnetism.
The reader must be wondering why atoms having an even number of electrons exhibit diamagnetism while atoms having an odd number of electrons exhibit paramagnetism. (There are, of
course, exceptions to this rule.) To answer this question, we again have to resort to quantum mechanical e¤ects, the Pauli exclusion principle. According to this principle, two electrons at the same
energy level in an atom cannot be spinning in the same direction. If one electron has a spin +~=2,
the other must have a spin
~=2, that is, the two spins must be antiparallel. Therefore, an atom
having an even number of electrons should not exhibit strong paramagnetism because magnetic
moments of the electrons cancel each other. The lowest order magnetic e¤ect of a helium atom is
therefore diamagnetism caused by the change in the harmonic oscillator frequency, as discussed in
the preceding Section.
It is somewhat surprising that nonmagnetic metals exhibit paramagnetism, for electrons in
metals are "free" to move around, as those in a plasma which is diamagnetic as we have seen.
Electrons in metals are not really free, however, because they su¤er frequent collisions with atoms.
The collision frequency is much higher than the cyclotron frequency even in the strongest magnetic
…eld available in the laboratory. Therefore, diamagnetism as in ordinary gases is not expected
in metals. The paramagnetism of conductors is due to incomplete cancellation between the two
spin states ~=2 and
~=2. Before the advent of quantum mechanics, the theory developed by
Langevin for paramagnetism in metals was considered applicable. This theory is essentially an
analogy of the dielectric susceptibility discussed in Chapter 4, and based on the assumption that
each electron has a permanent magnetic dipole moment. The magnetic susceptibility predicted
by Langevin is inversely proportional to the temperature as the dielectric susceptibility. This has
been disproved by experiments. Furthermore, the predicted susceptibility is too large (by a factor
more than 100) to explain those experimentally observed. Quantum mechanical calculations based
21
on the Pauli exclusion principle later gave a satisfactory explanation for experimentally-observed
magnetic susceptibilities of metals. For more details, the reader is referred to a textbook on solid
state physics (e.g., Kittel, Introduction to Solid State Physics).
22