Environmental Engineering Reference
In-Depth Information
Section 5.4.2, and the appearance of an energy gap at long wavelengths.
This gap has its origin in the magnetic anisotropy. Even though
the exchange energy required to excite a magnon vanishes in the long-
wavelength limit, work is still required to turn the moments away from
the easy direction against the anisotropy forces. If we neglect the small
terms due to the sums over
k
in (5.2.38), the dispersion relation along
the
c
-axis in zero field becomes, from eqns (5.2.36-38),
E
q
(
T
)=
[
A
0
(
T
)+
B
0
(
T
)+
J
z
{J
(
0
)
−J
(
q
)
}
]
(5
.
2
.
43)
]
2
.
×
[
A
0
(
T
)
−
B
0
(
T
)+
J
z
{J
(
0
)
−J
(
q
)
}
For an arbitrary direction in the zone, this relation is generalized anal-
ogously to eqn (5.1.9), giving rise again to acoustic and optical modes.
From the dispersion relations, the
magnon density of states
and
(
q
)
may readily be determined and hence, by a Fourier transform, the nom-
inal Heisenberg exchange interaction
J
(
ij
) between moments on differ-
ent atomic sites (Houmann 1968). The energy gap at zero wave-vector
is given by
E
0
(
T
)=
[
A
0
(
T
)+
B
0
(
T
)][
A
0
(
T
)
J
B
0
(
T
)]
2
,
−
(5
.
2
.
44)
and as we shall see in the next section, it is proportional to the geo-
metrical mean of the axial- and hexagonal-anisotropy energies. We shall
return to the dependence of this energy gap on the temperature and the
magnetoelastic effects in the following two sections.
5.3 The uniform mode and spin-wave theory
The spin-wave mode at zero wave-vector is of particular interest. In
comparison with the Heisenberg ferromagnet, the non-zero energy of
this mode is the most distinct feature in the excitation spectrum of the
anisotropic ferromagnet. In addition, the magnitude of the energy gap
at
q
=
0
is closely related to the bulk magnetic properties, which may
be measured by conventional techniques. We shall first explore the con-
nection between the static magnetic susceptibility and the energy of the
uniform mode, leading to an expression for the temperature dependence
of the energy gap. In the light of this discussion, we will then consider
the general question of the validity of the spin-wave theory which we
have presented in this chapter.
5.3.1 The magnetic susceptibility and the energy gap
The static-susceptibility components of the bulk crystal may be deter-
mined as the second derivatives of the free energy
1
β
ln
Z.
F
=
U
−
TS
=
−
(5
.
3
.
1)
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