Environmental Engineering Reference
In-Depth Information
particular plane is then maintained, but it suffers a lateral displacement
which follows the direction of the helical component of the moment.
The transition from the cycloidal to the cone structure in Er at 18
K reflects a shift in the balance between a number of competing effects.
At this low temperature, the entropy is not important, since most of the
moments are close to their saturation value near
T
C
,nordoesthedif-
ference between the single-ion crystal-field anisotropy energy in the two
phases play a significant role. Because of cancellation among the three
contributions, the axial anisotropy is relatively insensitive to the angle
between the
c
-axis and the moments, the average value of which does
not, in any case, change much at the transition. The small amplitude of
the basal-plane components ensures that the hexagonal-anisotropy en-
ergy also has only a minor influence. Hence the choice between the two
phases is dominated by the two-ion contributions to the energy. From
the spin-wave dispersion relation, discussed in Section 6.1, the difference
J
⊥
(
0
) is estimated to be about 0.07-0.1 meV, strongly favour-
ing a modulated structure. The tendency towards a modulation of the
c
-axis component is opposed by three effects. Firstly, the anisotropy of
the classical dipole-dipole contribution reduces
(
Q
)
−J
⊥
−J
(
0
)by0.03
meV to about 0.04-0.07 meV. Secondly, the modulated ordering of the
c
-axis component cannot take full advantage of the large value of
J
(
Q
)
(
Q
),
because of the squaring up which occurs as the temperature is decreased.
The energy due to the coupling of the longitudinal component of the mo-
ments is
J
N
U
ζζ
=
−
4
2
=
−
2
2
,
n
odd
J
(
n
Q
)
J
ζ
(
n
Q
)
NJ
(
Q
)
|J
ζ
|
(2
.
3
.
1
a
)
introducing the effective coupling parameter
J
(
Q
). At high tempera-
tures, close to
T
N
, the two coupling parameters
J
(
Q
)and
J
(
Q
)are
equal, but as the higher odd harmonics gradually develop,
J
(
Q
)de-
creases, and when the structure is close to the square wave, we find from
(2.1.36) that
π
2
J
···
.
8
(
Q
)+
9
J
J
(
Q
)
(3
Q
)+
(2
.
3
.
1
b
)
Just above the cone transition, the model calculations indicate that
J
(
Q
) is reduced by 0.02-0.03 meV, compared to
J
(
Q
), which in com-
bination with the dipolar term removes most of the energy difference
between the modulated and ferromagnetic ordering of the
c
-axis compo-
nent. The final contribution, which tips the balance into the cone phase
below
T
C
, is the magnetoelastic energy associated with the
α
-strains
−
2
−
2
U
me
=
c
66
)(
11
+
22
)
2
c
33
33
−
(
c
11
−
c
13
(
11
+
22
)
33
.
(2
.
3
.
2)
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