Environmental Engineering Reference
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calculated and its
dynamic
manifestation in the
magnon-phonon
inter-
action is discussed.
Anisotropic two-ion coupling
between the moments
alters the form of the dispersion relations, both quantitatively and, on
occasions, qualitatively. The classical
dipole-dipole interaction
, though
weak, is highly anisotropic and long-ranged, and may therefore have
an important influence at long wavelengths. Since its form is known
exactly, we can calculate its effects in detail, but we can say much less
about the two-ion anisotropy in general. Its possible origins and symme-
try are however discussed, and examples of observable effects to which it
gives rise are presented. The mutual solubility of the rare earths allows
the formation of an enormous variety of
binary alloys
, with magnetic
properties which may be adjusted with the concentration. We show how
the excitation spectrum of such systems can be calculated by the
virtual
crystal approximation
and the
coherent potential approximation
, and il-
lustrate the phenomena which may be observed by experiments on Tb
alloys. Finally, we consider the interaction between the conduction elec-
trons and the localized 4
f
moments, and its influence on both the spin
waves and the conduction electrons themselves. The
indirect-exchange
interaction
is derived more rigorously than in Section 1.4, and the life-
time of the magnons due to electron scattering is deduced. The
mass
enhancement
of the conduction electrons is determined, and the effects
of magnetic ordering on the band structure, and of magnetic scattering
on the conductivity, are discussed.
5.1 The ferromagnetic hcp-crystal
In Chapter 3, we considered the linear response of a system of magnetic
moments placed on a Bravais lattice and coupled by the Heisenberg
interaction. We shall now generalize this treatment to the hexagonal
close-packed crystal structure of the heavy rare earth metals, in which
there is a basis of two ions per unit cell, constituting two identical sub-
lattices which, for convenience, we number 1 and 2. The surroundings of
the atoms belonging to each of the two sublattices are identical, except
for an inversion. Introducing the following Fourier transforms:
J
ss
(
q
)=
(
ij
)
e
−i
q
·
(
R
i
−
R
j
)
j∈s
−subl.
J
;
i
∈
s
-sublattice,
(5
.
1
.
1
a
)
we have, for an hcp crystal,
J
1
(
q
)
≡J
11
(
q
)=
J
22
(
q
)
(5
.
1
.
1
b
)
J
21
(
q
)
,
J
2
(
q
)
≡J
12
(
q
)=
J
21
(
−
q
)=
where
J
1
(
q
) is real. Defining the four Fourier transforms
χ
ss
(
q
,ω
)of
the susceptibility tensor equivalently to (5
.
1
.
1
a
), we obtain from the
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