Elements of Modern Physics
302
=
0
2
,
µ
>
+
A
c
c
C T T
T T
(8.118)
with T c = λ A C A , λ A = λ B , C A = C B , where T c is the Néel temperature T N . At T = T c ,
an additional solution to Eq. (8.116) exists, M A = – M B even for B 0 = 0. Thus, the
sets of atoms A and B are spontaneously magnetized for T < T c though the net
magnetization is zero. Such materials are known as antiferromagnets. In
antiferromagnets, the crystal field aligns the magnetic moments along a preferred
direction below T c . It can be shown that a field applied perpendicular to this
direction is associated with a susceptibility χ ⊥ which is essentially independent
of T(T < T c ) whereas a field applied parallel to the preferred direction, is associated
with a susceptibility χ 11 which is equal to χ ⊥ at T = T c but decreases to zero as
T → 0 K.
If the atoms A and B are magnetically inequivalent, Eq. (8.116) can be
solved for M A and M B and they lead to a susceptibility
χ =
0 [ (
)
(
)]
(
)(
)
µ
+
−
λ +λ
−
+
A
B
A B
A
B
c
c
T C
C
C C
T T T T
(8.119)
where T c = (C A C B λ A λ B )
1/2
. Thus, χ tends to infinity as T → T c , which implies that
there is a net spontaneous magnetization for T < T c (M A and M B are opposite in
sign but M A + M B ≠ 0). Such meterials are called ferrimagnets, Fe 3 O 4 being a
well known example. An important class of ferrimagnets is the ferrites which
have the formula M
2+
Fe 2
3 +
O 4
2–
where M is a member of the first transition
group. They are of great technical importance since they may have large
magnetization at room temperature, and high resistivity. They are therefore more
suitable than ferromagnets for use at high frequencies when eddy current losses
are a serious problem. They are also used for memory storage in computers.
8.7 DIELECTRIC PROPERTIES
In this section, the properties of solids in the presence of an external electric
field are discussed. These are important in the propagation of electromagnetic
waves in material media, and in the development of many devices such as
capacitors, microphones, etc.
An electric field E causes a relative displacement of the positive and negative
charges of a material. This induces an electric dipole moment which is expressed
in terms of the polarization P defined as the dipole moment per unit volume.
For ordinary electric fields, the polarizability is linear in E (it is nonlinear for
strong laser fields),
302
=
0
2
,
µ
>
+
A
c
c
C T T
T T
(8.118)
with T c = λ A C A , λ A = λ B , C A = C B , where T c is the Néel temperature T N . At T = T c ,
an additional solution to Eq. (8.116) exists, M A = – M B even for B 0 = 0. Thus, the
sets of atoms A and B are spontaneously magnetized for T < T c though the net
magnetization is zero. Such materials are known as antiferromagnets. In
antiferromagnets, the crystal field aligns the magnetic moments along a preferred
direction below T c . It can be shown that a field applied perpendicular to this
direction is associated with a susceptibility χ ⊥ which is essentially independent
of T(T < T c ) whereas a field applied parallel to the preferred direction, is associated
with a susceptibility χ 11 which is equal to χ ⊥ at T = T c but decreases to zero as
T → 0 K.
If the atoms A and B are magnetically inequivalent, Eq. (8.116) can be
solved for M A and M B and they lead to a susceptibility
χ =
0 [ (
)
(
)]
(
)(
)
µ
+
−
λ +λ
−
+
A
B
A B
A
B
c
c
T C
C
C C
T T T T
(8.119)
where T c = (C A C B λ A λ B )
1/2
. Thus, χ tends to infinity as T → T c , which implies that
there is a net spontaneous magnetization for T < T c (M A and M B are opposite in
sign but M A + M B ≠ 0). Such meterials are called ferrimagnets, Fe 3 O 4 being a
well known example. An important class of ferrimagnets is the ferrites which
have the formula M
2+
Fe 2
3 +
O 4
2–
where M is a member of the first transition
group. They are of great technical importance since they may have large
magnetization at room temperature, and high resistivity. They are therefore more
suitable than ferromagnets for use at high frequencies when eddy current losses
are a serious problem. They are also used for memory storage in computers.
8.7 DIELECTRIC PROPERTIES
In this section, the properties of solids in the presence of an external electric
field are discussed. These are important in the propagation of electromagnetic
waves in material media, and in the development of many devices such as
capacitors, microphones, etc.
An electric field E causes a relative displacement of the positive and negative
charges of a material. This induces an electric dipole moment which is expressed
in terms of the polarization P defined as the dipole moment per unit volume.
For ordinary electric fields, the polarizability is linear in E (it is nonlinear for
strong laser fields),