Environmental Engineering Reference
In-Depth Information
structure is reached at a field
J
[
J
(
Q
)
−J
(
0
)]
H
c
=
,
(1
.
5
.
20)
gµ
B
but there is an intermediate transition, occurring at approximately
H
c
/
2,
at which the helix transforms abruptly through a first-order transition
to a
fan structure
, in which the moments make an angle
θ
with the field
direction, given by
=
1
/
2
sin
θ
i
2
2
gµ
B
(
H
c
−
H
)
sin
Q
·
R
i
.
(1
.
5
.
21)
J
[3
J
(
Q
)
−
2
J
(
0
)
−J
(2
Q
)]
The opening angle of the fan thus goes continuously to zero at the
second-order transition to the ferromagnetic phase.
The crystal fields manifest themselves in both microscopic and ma-
croscopic magnetic properties. The macroscopic anisotropy parameters
κ
l
are defined as the coecients in an expansion of the free energy in
spherical harmonics, whose polar coordinates (
θ, φ
) specify the magne-
tization direction relative to the crystallographic axes. For hexagonal
symmetry,
F
(
θ, φ
)=
N
κ
0
(
T
)+
κ
2
(
T
)
P
2
(cos
θ
)+
κ
4
(
T
)
P
4
(cos
θ
)
+
κ
6
(
T
)
P
6
(cos
θ
)+
κ
6
(
T
)sin
6
θ
cos 6
φ
,
(1
.
5
.
22)
where
P
l
(cos
θ
)=(4
π/
2
l
+1)
1
/
2
Y
l
0
(
θ, φ
) are the Legendre polynomials.
Anisotropic two-ion coupling and magnetoelastic strains may introduce
additional higher-rank terms of the appropriate symmetry. If the Hamil-
tonian is written in a representation
H
(
θ, φ
) in which the quantization
axis is along the magnetization, the macroscopic and microscopic pa-
rameters are related by
β
ln Tr
e
−βH
(
θ,φ
)
.
1
F
(
θ, φ
)=
−
(1
.
5
.
23)
Transforming the Stevens operators to a coordinate system with the
z
-
axis along the magnetization direction, and assuming that the isotropic
exchange is the dominant interaction, we find at absolute zero
κ
2
(0) = 2
B
2
J
(2)
κ
4
(0) = 8
B
4
J
(4)
(1
.
5
.
24)
κ
6
(0) = 16
B
6
J
(6)
κ
6
(0) =
B
6
J
(6)
where
−
n
−
1
2
−
2
J
(
n
)
≡
J
(
J
)(
J
−
1)
···
(
J
)
.
(1
.
5
.
25)
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