Environmental Engineering Reference
In-Depth Information
π
2
The expression for
Q
2
is then determined using
Q
2
(
θ
=
1
2
O
2
+
)=
−
2
O
2
. For the case of
Q
6
, we refer to Lindg ard and Danielsen (1974),
who have established the Bose operator expansion of the tensor operators
up to the eighth rank. Introducing these expansions into (5.2.1), and
grouping the terms together according to their order in 1
/J
,wemay
write the Hamiltonian
3
H
,
H
=
H
0
+
H
1
+
H
2
+
···
+
(5
.
2
.
12)
where
H
0
=
U
0
is the zero-order term, and
U
0
=
N
−
(
0
)
,
(5
.
2
.
13)
corresponding to (5.2.5), when we restrict ourselves to the case
θ
=
θ
H
=
π/
2.
H
1
comprises the terms of first order in 1
/J
, and is found to be
H
1
=
i
−
2
B
2
J
(2)
+
B
6
J
(6)
cos 6
φ
J
2
−
gµ
B
JH
cos (
φ
−
φ
H
)
J
Aa
i
a
i
+
B
2
+
a
i
a
i
)
−
(
a
i
a
i
(
ij
)(
a
i
a
j
−
a
i
a
i
)
,
J
J
ij
(5
.
2
.
14)
where the parameters
A
and
B
are
J
3
B
2
J
(2)
φ
H
)
1
21
B
6
J
(6)
cos 6
φ
+
gµ
B
JH
cos (
φ
A
=
−
−
(5
.
2
.
15)
J
3
B
2
J
(2)
+15
B
6
J
(6)
cos 6
φ
.
1
B
=
If we consider only the zero- and first-order part of the Hamiltonian,
i.e. assume
H
1
, it can be brought into diagonal form via
two transformations. The first step is to introduce the spatial Fourier
transforms of
HH
0
+
J
(
ij
), eqn (3.4.2), and of
a
i
:
1
√
N
1
√
N
a
i
e
−i
q
·
R
i
a
q
a
i
e
i
q
·
R
i
,
a
q
=
;
=
(5
.
2
.
16)
i
i
for which the commutators are
N
i
[
a
q
,a
q
]=
1
e
−i
(
q
−
q
)
·
R
i
=
δ
qq
.
In the case of an hcp lattice, with its two ions per unit cell, the situation
is slightly more complex, as discussed in the previous section. However,
this complication is inessential in the present context, and for simplicity
we consider a Bravais lattice in the rest of this section, so that the results
which we obtain are only strictly valid for excitations propagating in
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