Environmental Engineering Reference
In-Depth Information
H
in
additional assumption, which we have not stated explicitly, that
H
is the sum of the terms
proportional to Stevens operators
O
l
with
m
odd, and it includes for in-
stance the term 3
B
2
(
J
z
J
x
+
J
x
J
z
)cos
θ
sin
θ
associated with
B
2
Q
2
in eqn
(5.2.3).
the starting Hamiltonian (5.2.12) is negligible.
H
vanishes by symmetry if the magnetization is along a high-
symmetry direction, i.e.
θ
=0or
π/
2and
φ
is a multiple of
π/
6. In these
cases, the results obtained previously are valid. If the magnetization is
not along a high-symmetry direction,
H
must be taken into account.
The first-order contributions arise from terms proportional to (1
/J
)
1
/
2
in
H
, which can be expressed effectively as a linear combination of
J
x
and
J
y
.Inthisorder,
∂H
/∂θ
=0
by definition. For a harmonic oscillator, corresponding in this system
to the first order in 1
/J
, the condition for the elimination of terms in
the Hamiltonian linear in
a
and
a
+
coincides with the equilibrium con-
dition
∂F/∂θ
=
∂F/∂φ
= 0. Although the linear terms due to
= 0 therefore, because
J
x
=
J
y
H
can
be removed from the Hamiltonian by a suitable transformation, terms
cubic in the Bose operators remain. Second-order perturbation theory
shows that, if
H
is non-zero,
H
/∂θ
and the excitation energies in-
clude contributions of the order 1
/J
2
. Although it is straightforward to
see that
∂
H
makes contributions of the order 1
/J
2
, it is not trivial to
calculate them. The effects of
H
have not been discussed in this con-
text in the literature, but we refer to the recent papers of Rastelli
et al.
(1985, 1986), in which they analyse the equivalent problem in the case
of a helically ordered system.
In order to prevent
H
from influencing the 1
/J
2
-contributions de-
rived above, we may restrict our discussion to cases where the mag-
netization is along high-symmetry directions. This does not, however,
guarantee that
H
is unimportant in, for instance, the second deriva-
tives of
F
.Infact
∂
(1
/J
2
) may also be non-zero when
θ
=0or
π/
2, and using (5.3.4) we may write
∂
H
/∂θ
/∂θ
∝O
m
q
,b
q
F
θθ
=
∂
2
F
∂θ
2
∂
2
U
∂θ
2
(1
/J
2
)
=
+
O
(5
.
3
.
6
a
)
=
∂
2
H
2
)
+
π
2
,
(1
/J
2
)
H
0
+
H
1
+
O
θ
=0
,
∂θ
2
(
;
and similarly
F
φφ
=
∂
2
H
2
)
+
φ
=
p
π
(1
/J
2
)
∂φ
2
(
H
0
+
H
1
+
O
;
6
,
(5
.
3
.
6
b
)
where the corrections of order 1
/J
2
are exclusively due to
H
.Herewe
have utilized the condition that the first derivatives of
m
q
and
b
q
vanish
when the magnetization is along a symmetry direction.
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