Solid State Physics
297
∆E =
(
2forspin)
2
=
S
e M B g
m
(8.102)
This leads to
χ =
2
0
(
1) ,
2
3
e
S S
e B
N
k T
m
kT
m
+
µ
(8.103)
for the ions of iron group. As an example, in the case of χ for Mn
3+
(
5
D 0 ),
Eq. (8.100) predicts that χ = 0, whereas the prediction of Eq. (8.103) with S = 2
is in very good agreement with experimental observations.
It may be noted that adiabatic demagnetization of a paramagnet system can
be used for attaining low temperatures, T < 1 K. This is done as follows. A
magnetic field is applied to a paramagnetic substance in good thermal contact
with the surroundings at T 1 . The field aligns the magnetic moments along the
direction of the field. This increase in order is equivalent to a decrease in the
entropy and hence heat flows out of the system. If now the substance is insulated,
and the field removed adiabatically, the spins gradually get out of the alignment
by absorbing energy from the lattie vibration which leads to a lowering of the
temperature of the paramagnetic substance. Temperatures of the order of 10
–3
K
have been reached by this method.
Ferromagnetism
Ferromagnetism is the phenomenon in which some materials like iron, cobalt,
nickel, and some of their alloys behave like ordinary paramagnets at high
temperatures but which below a critical temperature known as the Curie
temperature T c , acquire a nonzero magnetic moment even in the absence of an
applied magnetic field. This is due to the interaction between the magnetic
ions, which is strong enough to align their magnetic moments against the disorder
introduced by thermal effects.
The interaction that aligns the magnetic moments is quantum mechanical
in origin and is due to the exchange properties of the electron wave functions.
When the wave functions of two atoms overlap, the electrons being
indistinguishable, belong to both the atoms. In such cases, the symmetry or the
antisymmetry of the wave functions will strongly influence the energy of the
system (as in the case of covalent bonding, see Chapter 5). In particular, it is the
exchange symmetry between the spins and the extent of the overlap of the wave
functions that determines the nature and the strength of the exchange interaction.
It is reasonable to represent the energy from the exchange interaction by
297
∆E =
(
2forspin)
2
=
S
e M B g
m
(8.102)
This leads to
χ =
2
0
(
1) ,
2
3
e
S S
e B
N
k T
m
kT
m
+
µ
(8.103)
for the ions of iron group. As an example, in the case of χ for Mn
3+
(
5
D 0 ),
Eq. (8.100) predicts that χ = 0, whereas the prediction of Eq. (8.103) with S = 2
is in very good agreement with experimental observations.
It may be noted that adiabatic demagnetization of a paramagnet system can
be used for attaining low temperatures, T < 1 K. This is done as follows. A
magnetic field is applied to a paramagnetic substance in good thermal contact
with the surroundings at T 1 . The field aligns the magnetic moments along the
direction of the field. This increase in order is equivalent to a decrease in the
entropy and hence heat flows out of the system. If now the substance is insulated,
and the field removed adiabatically, the spins gradually get out of the alignment
by absorbing energy from the lattie vibration which leads to a lowering of the
temperature of the paramagnetic substance. Temperatures of the order of 10
–3
K
have been reached by this method.
Ferromagnetism
Ferromagnetism is the phenomenon in which some materials like iron, cobalt,
nickel, and some of their alloys behave like ordinary paramagnets at high
temperatures but which below a critical temperature known as the Curie
temperature T c , acquire a nonzero magnetic moment even in the absence of an
applied magnetic field. This is due to the interaction between the magnetic
ions, which is strong enough to align their magnetic moments against the disorder
introduced by thermal effects.
The interaction that aligns the magnetic moments is quantum mechanical
in origin and is due to the exchange properties of the electron wave functions.
When the wave functions of two atoms overlap, the electrons being
indistinguishable, belong to both the atoms. In such cases, the symmetry or the
antisymmetry of the wave functions will strongly influence the energy of the
system (as in the case of covalent bonding, see Chapter 5). In particular, it is the
exchange symmetry between the spins and the extent of the overlap of the wave
functions that determines the nature and the strength of the exchange interaction.
It is reasonable to represent the energy from the exchange interaction by