Environmental Engineering Reference
In-Depth Information
B
6
. The result above may be generalized to include most of the renor-
malization effects appearing in the second order of 1
/J
, by replacing
A
B
0
(
T
), in accordance with the discussion at the end
of Section 5.2. After the transformation to the rotating coordinates, the
system becomes equivalent to the basal-plane ferromagnet, except that
the hexagonal anisotropy is neglected and the
γ
-strains vanish, due to
the lattice-clamping effect discussed in Section 2.2.2. Hence we may take
A
±
B
by
A
0
(
T
)
±
±
B
to be
A
0
(
T
)
±
B
0
(
T
), given by eqn (5.3.22), with
B
6
=0and
H
=0.
In the present situation, where
H
J
in (6.1.4) only depends on
J
x
,
A
=
B
and (6.1.9) implies, for instance, that
χ
yy
(
ω
=0)=1
/J
(
Q
).
This result is quite general and may be derived directly from (6.1.5); the
addition of a small rotating field
h
y
in the
y
-direction, perpendicular
to the exchange field, only has the consequence that the direction of
the angular momentum is rotated through the angle
φ
,wheretan
φ
=
h
y
/h
ex
, and hence
δ
J
y
=
J
z
tan
φ
=
{
1
/
J
(
Q
)
}
h
y
.
Substituting
(6.1.9) with
A
=
B
into (6.1.8), we obtain
A
q
−
B
q
A
q
+
B
q
E
q
−
χ
xx
(
q
,ω
)=
J
z
;
χ
yy
(
q
,ω
)=
J
z
(
hω
)
2
,
(6
.
1
.
10
a
)
E
q
−
(
hω
)
2
with
E
q
=
A
q
−
B
q
1
/
2
(6
.
1
.
10
b
)
and
A
q
+
B
q
=2
A
+
J
z
{J
(
Q
)
−J
(
q
)
}
(
q
−
Q
)
,
(6
.
1
.
10
c
)
−
2
J
−
2
J
A
q
−
B
q
=
J
z
J
(
Q
)
(
q
+
Q
)
neglecting
χ
zz
(0). The absorptive components of
χ
t
(
q
,ω
)are
A
q
−
2
χ
xx
(
q
,ω
)=
π
B
q
A
q
+
B
q
2
J
z
{
δ
(
E
q
−
hω
)
−
δ
(
E
q
+
hω
)
}
A
q
+
B
q
A
q
− B
q
2
χ
yy
(
q
,ω
)=
π
2
J
z
{
δ
(
E
q
−
hω
)
−
δ
(
E
q
+
hω
)
}
,
(6
.
1
.
11)
demonstrating that the scattered intensities due to the two components
are different, if
B
q
is non-zero. The neutron cross-section
d
2
σ/dEd
Ω,
(4.2.2), is proportional to
κ
α
κ
β
)
χ
αβ
(
κ
ζ
)
χ
ζζ
(
,ω
)+(1+
κ
ζ
)
χ
ηη
(
(
δ
αβ
−
κ
,ω
)=(1
−
κ
κ
,ω
)
,
αβ
(6
.
1
.
12
a
)
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