Environmental Engineering Reference
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that the quantization axis (
z
-axis), defining the Stevens operators, co-
incides with the direction of magnetization. We shall continue to use
this convention, but must then take account of the diculty that the
crystal-field Hamiltonian in the hcp metals only has the simple form
of eqn (1
.
4
.
6
b
) if the quantization axis is chosen to be along the
c
-or
ζ
-direction. In order to distinguish between the two systems, we shall
denote the Stevens operators in the Hamiltonian defined with respect
to the crystallographic axes, i.e. in the (
ξ, η, ζ
)-coordinate system, by
Q
l
(
J
). The direction of magnetization, the
z
-axis, is specified by the
polar angles (
θ, φ
)inthe(
ξ, η, ζ
)-coordinate system, and we must intro-
duce the following transformation of the angular momentum operators
in
Q
l
(
J
):
J
ξ
=
J
z
sin
θ
cos
φ
−
J
x
cos
θ
cos
φ
+
J
y
sin
φ
J
η
=
J
z
sin
θ
sin
φ
−
J
x
cos
θ
sin
φ
−
J
y
cos
φ
(2
.
2
.
8)
J
ζ
=
J
z
cos
θ
+
J
x
sin
θ,
choosing the
y
-axis to lie in the basal-plane. By this transformation,
Q
l
is expressed as a linear combination of the Stevens operators
O
m
l
,
with the same
l
but various
m
-values. For instance, we have
Q
2
=3
J
ζ
−
J
(
J
+1)
=3
J
z
cos
2
θ
+3
J
x
sin
2
θ
+
2
(
J
z
J
x
+
J
x
J
z
)sin2
θ
−
J
(
J
+1)
=
1
2
1) +
3
2
O
2
(3 cos
2
θ
O
2
sin
2
θ
+3
O
2
sin 2
θ.
−
(2
.
2
.
9)
Carrying out the same transformation on
Q
2
we find the following rela-
tions:
Q
2
=
2
O
2
+3
O
2
)
(
−
(2
.
2
.
10)
Q
2
=
2
(
O
2
+
O
2
)
;
φ
=
pπ,
when the moment is in the basal-plane (
θ
=
π/
2). The expectation value
of
Q
2
is relevant for determining the
γ
-strain
γ
1
, as shown in (1.5.29).
According to the result (2.2.5) of Callen and Callen,
O
2
should vanish,
but in a basal-plane ferromagnet this may not occur. The eccentricity
parameter mentioned above is just defined as
b
=
O
2
O
2
/
,
(2
.
2
.
11)
which is zero, by definition, only if the anisotropy is invariant with re-
spect to a rotation about the
z
-axis.
The numerical programme sketched above has been carried through
for a model corresponding to Tb. The effective basal-plane anisotropy
is about a factor of 10 smaller than the axial anisotropy, so that
b
is
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