Solid State Physics
299
Below T = T c , the spontaneous magnetization is to be obtained by using the
complete expression in Eq. (8.99) for M, with
a =
0
(
)
2
+ λ
e g B
M
m
(8.112)
The solutions are obtained by plotting M in Eq. (8.99) as a function of x,
and also
M =
2
,
λ
mkT x
e g
(8.113)
obtained from Eq. (8.112) with a = xkT, B 0 = 0, and looking for the intersection
of the two curves [see Fig. 8.16(a), it can be shown that the intersection at the
origin gives an unstable solution]. At T = T c the curve given by Eq. (8.113) is
tangential to the curve given by Eq. (8.99), at the origin, and there is no
spontaneous magnetization for T > T c . When T < T c , there are two equal and
opposite solutions for each T, for example corresponding to points A and A′ in
Fig. 8.16(a). One set of these spontaneous magnetizations is plotted as a function
of T (T < T c ) in Fig. 8.16(b).
For B 0 ≠ 0, the magnetization is obtained from the intersection of the curve
given by Eq. (8.99) and
M =
0
2
1
−
λ
λ
mkT x
B
e g
(8.114)
obtained from Eq. (8.112) with a = xkT. There are two solutions for
M corresponding to intersections at D and D′ in Fig. 8.16(a), for each B 0 (T <
T c ). These solutions trace the boundary of the hysteresis curve [Fig. 8.16(c)]. It
can be shown that the third solution corresponding to intersection F in Fig.
8.16(a) is unstable. This solution is the extension of the unstable solution at the
origin for B 0 = 0.
At T = 0, all the spins and the magnetic moments of the atoms are aligned
corresponding to the ground state of the system. The direction of the alignment
is introduced by the arbitrarily assumed direction of the internal field B int , and
obviously the ground state is infinitely degenerate.
For T > 0 K, some of the spins go out of alignment. As in the case of lattice
vibrations, the disturbances are correlated, and misalignments travel as waves
known as spin waves. In analogy with photons and phonons, the excitations of
spin waves are quantized into quanta known as magnons. Magnons, which obey
Bose-Einstein statistics, play a significant role in determining the behaviour of
M(T ) at low temperatures, and contribute to the specific heat and thermal
conductivity of ferromagnets.
299
Below T = T c , the spontaneous magnetization is to be obtained by using the
complete expression in Eq. (8.99) for M, with
a =
0
(
)
2
+ λ
e g B
M
m
(8.112)
The solutions are obtained by plotting M in Eq. (8.99) as a function of x,
and also
M =
2
,
λ
mkT x
e g
(8.113)
obtained from Eq. (8.112) with a = xkT, B 0 = 0, and looking for the intersection
of the two curves [see Fig. 8.16(a), it can be shown that the intersection at the
origin gives an unstable solution]. At T = T c the curve given by Eq. (8.113) is
tangential to the curve given by Eq. (8.99), at the origin, and there is no
spontaneous magnetization for T > T c . When T < T c , there are two equal and
opposite solutions for each T, for example corresponding to points A and A′ in
Fig. 8.16(a). One set of these spontaneous magnetizations is plotted as a function
of T (T < T c ) in Fig. 8.16(b).
For B 0 ≠ 0, the magnetization is obtained from the intersection of the curve
given by Eq. (8.99) and
M =
0
2
1
−
λ
λ
mkT x
B
e g
(8.114)
obtained from Eq. (8.112) with a = xkT. There are two solutions for
M corresponding to intersections at D and D′ in Fig. 8.16(a), for each B 0 (T <
T c ). These solutions trace the boundary of the hysteresis curve [Fig. 8.16(c)]. It
can be shown that the third solution corresponding to intersection F in Fig.
8.16(a) is unstable. This solution is the extension of the unstable solution at the
origin for B 0 = 0.
At T = 0, all the spins and the magnetic moments of the atoms are aligned
corresponding to the ground state of the system. The direction of the alignment
is introduced by the arbitrarily assumed direction of the internal field B int , and
obviously the ground state is infinitely degenerate.
For T > 0 K, some of the spins go out of alignment. As in the case of lattice
vibrations, the disturbances are correlated, and misalignments travel as waves
known as spin waves. In analogy with photons and phonons, the excitations of
spin waves are quantized into quanta known as magnons. Magnons, which obey
Bose-Einstein statistics, play a significant role in determining the behaviour of
M(T ) at low temperatures, and contribute to the specific heat and thermal
conductivity of ferromagnets.