Environmental Engineering Reference
In-Depth Information
depending only on the dynamical part of the strains.
To leading order, the magnetoelastic energy is determined by the
static part (5.4.5), corresponding to eqn (2.2.27).
H
γ
influences the
equilibrium condition determining
φ
and, in the spin-wave approxima-
tion (
H
is neglected), we have
N
∂
H
γ
}
N
∂
}
1
N
∂F
∂φ
1
1
=
∂φ
{H
+
∂φ
{H
+
H
γ
(sta)
6
B
6
J
(6)
I
13
/
2
[
σ
]
η
−
15
=
−
sin 6
φ
+
gµ
B
HJσ
sin (
φ
−
φ
H
)
−
+2
c
γ
C
(
γ
1
sin 2
φ
−
γ
2
cos 2
φ
)
−
2
c
γ
A
(
γ
1
sin 4
φ
+
γ
2
cos 4
φ
)
,
(5
.
4
.
7)
or, using the equilibrium values of
γ
1
and
γ
2
,
=
gµ
B
Jσ
H
sin (
φ
−
6
H
c
sin 6
φ
,
1
N
∂F
∂φ
−
φ
H
)
(5
.
4
.
8
a
)
with the definition
gµ
B
H
c
=36
κ
6
/
(
Jσ
)=36
B
6
J
(6)
I
13
/
2
[
σ
]
η
−
15
c
γ
CA
/
(
Jσ
)
.
(5
.
4
.
8
b
)
If
H
= 0, the equilibrium condition
∂F/∂φ
= 0 determines the sta-
ble direction of magnetization to be along either a
b
-axis or an
a
-axis,
depending on whether
H
c
is positive or negative respectively.
The additional anisotropy terms introduced by
+
2
−
H
γ
and proportional
B
γ
2
Q
2
(
J
i
)
γ
1
in (5.4.5),
contribute to the spin-wave energies. Proceeding as in Section 5.3, we
find the additional contributions to
A
0
(
T
)
± B
0
(
T
) in (5.3.22), propor-
tional to the static
γ
-strains,
−
to the static strains, as for instance the term
∆
{
A
0
(
T
)+
B
0
(
T
)
}
Jσ
2
C
2
+
A
2
η
−
8
)cos6
φ
η
−
1
+
c
γ
η
−
4
CA
(2 +
η
−
8
+
η
−
4
−
=
−
−
η
−
+
Jσ
4
C
2
+4
A
2
10
CA
cos 6
φ
.
c
γ
∆
{
A
0
(
T
)
−
B
0
(
T
)
}
=
−
(5
.
4
.
9)
The contribution to
A
0
(
T
)
B
0
(
T
) is expressible directly in terms of
the strain-parameters,
C
and
A
, without the further correction factors
necessary for
A
0
(
T
)+
B
0
(
T
). By using
H
c
and the non-negative quantity
−
4
c
γ
Jσ
(
C
2
+
A
2
+2
CA
cos 6
φ
)
,
Λ
γ
=
(5
.
4
.
10)
we can write the
total
spin-wave parameter
gµ
B
H
c
cos 6
φ
+
gµ
B
H
cos (
φ
A
0
(
T
)
−
B
0
(
T
)=Λ
γ
−
−
φ
H
)
.
(5
.
4
.
11)
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