Solid State Physics
301
and finally all the domains will have magnetization along a preferred or an
‘easy’ direction (determined by the crystal structure) nearly parallel to the
direction of the external field. A further increase in the field brings the alignment
closer to the direction of the external field. The progress of magnetization as
the external field increases is described by the broken line in Fig. 8.16(c). If the
external field is now removed, some of the magnetization is retained [point A in
Fig. 8.16(c)]. The variation of magnetization with the external magnetic field
now produces the well-known hysteresis curve, similar to that in Fig. 8.16(c)
except that usually B is plotted against H. The magnetization can be destroyed
either by heating or by mechanical shocks. The reality of the domains, which
are so successful in describing ferromagnets, can be demonstrated by scattering
finely divided iron on the surface of a ferromagnet, which collects along the
domain boundaries where the field is the strongest.
Antiferromagnetism and Ferrimagnetism
In most substances other than ferromagnets, the exchange energy in Eq. (8.104)
is positive when the neighbouring spins are parallel. This leads to a ground
state in which the neighbouring spins are antiparallel and hence to a cooperative
alternating alignment below temperature T N known as the Néel temperature.
The problem can be analysed in terms of alternating lattice points occupied by
atoms A and B with antiparallel spins.
Let the average magnetic moments of A and B sets of atoms be M A and M B
respectively. The total magnetic fields at sites A and B (for antiparallel alignment)
are:
B A = B 0 – λ A M B ,
B B = B 0 – λ B M A
(8.115)
Substituting these expressions in Eq. (8.99), for temperature above the Néel
temperature, the magnetic moments are:
M A =
0
(
) ,
− λ
A
A B
C B
M
T
(8.116)
M B =
0
(
)
− λ
B
B
A
C B
M
T
where
C A,B =
2
,
,
,
,
(
1 )
2
3
+
A B
A B
A B
A B
e g
J
J
N
m
k
(8.117)
If the atoms A and B are magnetically equivalent, then
χ = (M A + M B )/H
301
and finally all the domains will have magnetization along a preferred or an
‘easy’ direction (determined by the crystal structure) nearly parallel to the
direction of the external field. A further increase in the field brings the alignment
closer to the direction of the external field. The progress of magnetization as
the external field increases is described by the broken line in Fig. 8.16(c). If the
external field is now removed, some of the magnetization is retained [point A in
Fig. 8.16(c)]. The variation of magnetization with the external magnetic field
now produces the well-known hysteresis curve, similar to that in Fig. 8.16(c)
except that usually B is plotted against H. The magnetization can be destroyed
either by heating or by mechanical shocks. The reality of the domains, which
are so successful in describing ferromagnets, can be demonstrated by scattering
finely divided iron on the surface of a ferromagnet, which collects along the
domain boundaries where the field is the strongest.
Antiferromagnetism and Ferrimagnetism
In most substances other than ferromagnets, the exchange energy in Eq. (8.104)
is positive when the neighbouring spins are parallel. This leads to a ground
state in which the neighbouring spins are antiparallel and hence to a cooperative
alternating alignment below temperature T N known as the Néel temperature.
The problem can be analysed in terms of alternating lattice points occupied by
atoms A and B with antiparallel spins.
Let the average magnetic moments of A and B sets of atoms be M A and M B
respectively. The total magnetic fields at sites A and B (for antiparallel alignment)
are:
B A = B 0 – λ A M B ,
B B = B 0 – λ B M A
(8.115)
Substituting these expressions in Eq. (8.99), for temperature above the Néel
temperature, the magnetic moments are:
M A =
0
(
) ,
− λ
A
A B
C B
M
T
(8.116)
M B =
0
(
)
− λ
B
B
A
C B
M
T
where
C A,B =
2
,
,
,
,
(
1 )
2
3
+
A B
A B
A B
A B
e g
J
J
N
m
k
(8.117)
If the atoms A and B are magnetically equivalent, then
χ = (M A + M B )/H