Environmental Engineering Reference
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each of the two sublattices. The difference between the two phases is
approximately
π
, or approximately 0 if
Q
, within the primitive zone, is
replaced by
Q
+
b
3
. Introducing the relative magnetization
σ
by
J
η
(
Q
)
=
M
η
σ,
(7
.
4
.
7
b
)
where the matrix element is slightly dependent on the pressure (
M
η
=
1
.
026
√
10 at 1 kbar), we find that
σ
0
.
44 under the conditions of Fig.
7.15. Because
σ
is still small, it may be utilized as an expansion pa-
rameter, both in the calculation of the ordered moments and also in the
equations of motion determining the excitation spectrum. The ordering
wave-vector is close to
8
b
2
, but whether the system is commensurable or
not is not easy to decide from the experiments. In any case, this is not
important for calculating the excitation spectrum, because distinctive
effects of commensurability only appear in the order
σ
8
0
.
001. The
modulation of the length of the moments implies that the single-ion MF
susceptibilty is site-dependent, and the
ηη
-component is found to be
≈
2
n
01
(
j
)
M
α
∆cos2
θ
j
(∆
/
cos 2
θ
j
)
2
χ
ηη
(
j, ω
)=
(
hω
)
2
+
βp
01
(
j
)
M
α
sin
2
2
θ
j
δ
ω
0
,
(7
.
4
.
8
a
)
−
equivalent to eqn (7.1.9) with
J
jη
=
M
η
n
01
(
j
)sin(2
θ
j
), and
p
01
(
j
)
defined by
n
01
(
j
)
.
p
01
(
j
)=
n
0
(
j
)+
n
1
(
j
)
−
(7
.
4
.
8
b
)
∆=∆
η
(
t
11
) is here the crystal-field splitting between the ground state
|
1
a
>
at zero stress) at the particu-
lar stress considered. In the incommensurable case, the coupling matrix
determining the longitudinal component of the susceptibility tensor is
of infinite extent. The situation is very similar to that considered in
Section 6.1.2 and, as there, the coupling matrix may be solved formally
in terms of infinite continued fractions. The only difference is that, in
the present case, the single-site susceptibility is unchanged if the mo-
ments are reversed, which means that the coupling matrix only involves
terms with
n
even (where
n
is the number of the Fourier component,
as in (6.1.28)). Since the effective modulation wave-vector seen by the
longitudinal excitations is 2
Q
and not
Q
, the acoustic and the optical
modes propagating parallel to
Q
may be treated separately, as the
q
-
dependent phase factor determining the effective coupling parameters
J
11
(
q
)
0
>
and the excited state
|
1
>
(
≡|
, derived from the interactions in the two sublattices
(see Section 5.1), is not affected.
To leading order, the modulation of the moments introduces a cou-
pling between the excitations at wave-vectors
q
and
q
±
±|J
12
(
q
)
|
2
Q
, and energy
gaps appear on planes perpendicular to
Q
passing through
n
Q
.When
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