Environmental Engineering Reference
In-Depth Information
using the series
exp[
i
(
u
+
γ
sin 6
u
)]
=
J
0
(
γ
)
e
iu
+
J
1
(
γ
)
e
i
7
u
e
−i
5
u
+
J
2
(
γ
)
e
i
13
u
+
e
−i
11
u
+
−
···
2
e
i
7
u
e
−i
5
u
,
e
iu
+
γ
−
(2
.
1
.
29
b
)
where
J
n
(
x
) are the Bessel functions. To leading order in
γ
, the equi-
librium equation then gives
12
κ
6
(
Jσ
)
2
2
(7
Q
)
,
γ
=
(2
.
1
.
30
a
)
J
(
Q
)
−J
(5
Q
)
−J
and the free energy is reduced proportionally to
γ
2
:
(7
Q
)
γ
2
.
(2
.
1
.
30
b
)
The hexagonal anisotropy introduces harmonics, of equal magnitude,
in the basal-plane moments at the wave-vectors 6
Q
±
Q
and, in higher
order, at the wave-vectors 6
m
Q
±
Q
.If
κ
6
, and thus also
γ
, are negative,
the easy directions in the plane are the
a
-axes. In the special case
where the angle
u
i
=
π/
12, i.e. the unperturbed
i
th moment is half-way
between an easy and a hard direction, the largest change
φ
i
−
(
Jσ
)
2
2
−
2
−
8
(
Jσ
)
2
F/N
=
F
1
/N
J
(
Q
)
J
(
Q
)
−J
(5
Q
)
−J
u
i
=
γ
occurs in the orientation of the moments, and the angle to the nearest
easy direction is reduced, since
u
i
lies between 0 and
π/
6, and
κ
6
is
negative. The moments in the helix are therefore distorted so that they
bunch around the easy axes.
The above calculation is not valid if
Q
is 0 or 2
π/c
, when the hexag-
onal anisotropy may be minimized without increasing the exchange en-
ergy, as it may also if the (average) turn angle
ω
of the moments from
one hexagonal plane to the next is a multiple of 60
◦
,sothat6
Q
is an
integer times 4
π/c
. The products of the fifth and seventh harmonics
introduce additional umklapp contributions to the free energy if 12
Q
is
a multiple of the
effective
reciprocal-lattice spacing 4
π/c
, implying that
the cases where
ω
is
p
30
◦
and
p
=1
,
3
,
5 are also special. In higher
order, corrections appear whenever
m
12
Q
=
p
4
π/c
,where
m
and
p
are
integers and 0
6
m
, i.e. at any commensurable value of
Q
, but
the corrections decrease rapidly with
m
, excluding cases where
m
and
p
have common factors. In contrast to the result found above, the com-
mensurable contributions depend on the absolute phase
ϕ
in (2
.
1
.
26
b
),
and an adjustment of this phase will quite generally allow the system to
reduce the anisotropy energy through the umklapp terms. This change
in energy may compensate for the increase in the exchange energy when
the ordering wave-vector
Q
is changed from its value
Q
=
Q
0
,atwhich
≤
p
≤
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