Environmental Engineering Reference
In-Depth Information
imental observations. In the case of the nearly saturated ferromagnet, it
is straightforward to take into account the effects of
H
JJ
on the ground-
state properties and the spin-waves. The Racah operators, defined with
reference to the (
ξ, η, ζ
)-coordinate system, may be expanded in terms
of the spin deviation operators. When the moments in the basal-plane
(
θ
=
π/
2) are close to their saturation value (
J
z
J
),
4
π
2
l
+1
1
/
2
J
(
l
)
Y
lm
(
θ
=
π
O
lm
2
,φ
)=
J
(
l
)
Γ
lm
e
imφ
,
(5
.
5
.
15
a
)
where
⎧
⎨
m
)!]
1
/
2
1)
(
l
+
m
)
/
2
[(
l
+
m
)!(
l
−
(
−
,l
+
m
even
(
l
+
m
)!!(
l
−
m
)!!
Γ
lm
=
(5
.
5
.
15
b
)
⎩
0
,
l
+
m
odd
.
Utilizing the equivalence between the Racah operators and the spherical
harmonics, and the connection between the spin-wave energies and the
angular derivatives of the expectation values (which leads to the relation
(5.3.14)), we have to first order in 1
/J
(Jensen
et al.
1975):
O
lm
(
J
i
)=
1
+
a
i
a
i
)
2
m
2
m
√
2
J
(
a
i
−
l
(
l
+1)
2
J
l
(
l
+1)
−
a
i
a
i
−
(
a
i
a
i
−
a
i
)
−
4
J
J
(
l
)
Γ
lm
e
imφ
,
×
(5
.
5
.
16
a
)
if
l
+
m
is even, and if
l
+
m
is odd
O
lm
(
J
i
)=
(
l
+1)
2
m
2
1
/
2
1
a
i
a
i
)
J
(
l
)
Γ
l
+1
m
e
imφ
.
m
2
J
(
a
i
a
i
−
√
2
J
(
a
i
−
+
a
i
)
−
(5
.
5
.
16
b
)
Introducing these expressions into the two-ion Hamiltonian, we may
derive the spin-wave energies, to leading order in 1
/J
.Thenumberof
terms in eqn (5.5.14) which contribute to the excitation energies, in this
order, may be reduced by the symmetry elements of the lattice which
leave the
q
-vector unchanged. In the simplest case, where
q
is along
the
c
-axis, the three-fold symmetry about this axis plus the mirror-
plane perpendicular to the
ξ
-axis (i.e. the
a
-axis) ensure that only terms
with
m
+
m
=3
p
,where
p
is an integer, contribute, and that their
contribution is proportional to cos (3
pφ
). The terms in which
p
is an
odd integer couple the acoustic and optical magnons, but they do not
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