Environmental Engineering Reference
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linear combinations of the
l
-rank Stevens operators, which have their
polar axis along the
z
-axis defined by the direction of the moments. The
variational expression for the free energy then involves the calculation of
the expectation values of these Stevens operators. To leading order in the
crystal-field parameters, we may neglect the influence of the anisotropy
terms on the thermal averages in (2.2.13). This is the approximation
used by Callen and Callen, and we may utilize their result, eqn (2.2.5).
This has the consequence that, in the various linear combinations of
Stevens operators, only those terms in which
m
= 0 contribute to the
free energy, to leading order in the anisotropy parameters. From the
expansion (2.2.9) of
Q
2
, we find the following result:
δ
Q
2
δ
2
1)
O
2
(3 cos
2
θ
−
and, repeating this calculation for the other operators, we have in general
δ
Q
l
O
l
δP
l
(cos
θ
);
δ
Q
6
16
O
6
δ
sin
6
θ
cos 6
φ
.
(2
.
2
.
14)
Because
J
x
=
J
y
= 0, the Zeeman terms in (2.2.13) cancel within
this approximation, and an integration of
δF
(
θ, φ
)leadsto
f
0
+
l
P
l
(cos
θ
)+
16
B
l
O
l
B
6
O
6
sin
6
θ
cos 6
φ.
(2
.
2
.
15)
F
(
θ, φ
)
/N
Comparing this result with the free energy expression (1.5.22), and in-
troducing the anisotropy parameters
κ
l
(
T
), we obtain to a first approx-
imation
κ
l
(
T
)=
c
l
B
l
J
(
l
)
I
l
+
2
κ
6
(
T
)=
B
6
J
(6)
I
13
/
2
[
σ
]
,
[
σ
]
;
(2
.
2
.
16)
with
σ
=
σ
(
T
), which leads to eqn (1.5.24) at zero temperature (
σ
=1).
The equilibrium values of the angles in zero field are determined
by
∂F
(
θ, φ
)
/∂θ
=
∂F
(
θ, φ
)
/∂φ
= 0. In the above result for the free
energy, the
φ
-dependence is determined exclusively by
B
6
, the sign of
which then determines whether the
a
-or
b
-directions are the magneti-
cally easy or hard axes in the basal-plane (
φ
0
=
pπ/
3or
π/
2+
pπ/
3).
Because
B
6
is a sixth-rank coupling parameter, the importance of this
anisotropy decreases rapidly with the magnetization;
I
13
/
2
[
σ
]
σ
21
at
low temperatures, or
σ
6
when
σ
is small. The axial anisotropy derives
from all four parameters, and the equilibrium value
θ
0
is determined by
minimizing
∝
f
(
u
=cos
θ
)=
F
(
θ, φ
0
)
/N
−
f
0
=
2
1) +
8
κ
2
(3
u
2
κ
4
(35
u
4
30
u
2
+3)
−
−
+
1
16
κ
6
(231
u
6
315
u
4
+ 105
u
2
κ
6
|
u
2
)
3
.
(2
.
2
.
17)
−
−
5)
−|
(1
−
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