Environmental Engineering Reference
In-Depth Information
γ
-strain part of the magnetoelastic Hamiltonian is given by eqn (2.2.23):
H
γ
=
i
2
c
γ
(
γ
1
+
γ
2
)
Q
2
(
J
i
)
γ
1
+
Q
−
2
−
B
γ
2
{
(
J
i
)
γ
2
}
2
(5
.
4
.
1)
(
J
i
)
γ
2
}
.
Q
4
(
J
i
)
γ
1
−
Q
−
4
4
−
B
γ
4
{
The equilibrium condition,
∂F/∂
γ
= 0, leads to eqn (2.2.25) for the
static strains
γ
. The static-strain variables are distinguished by a bar
from the dynamical contributions
γ
−
γ
. The expectation values of
the Stevens operators may be calculated by the use of the RPA theory
developed in the preceding section, and with
θ
=
π/
2 we obtain, for
instance,
2
=
J
(2)
I
5
/
2
[
σ
]
η
−
1
Q
2
(
O
2
+
O
2
)cos2
φ
+2
O
−
1
=
sin 2
φ
cos 2
φ
−
2
2
=
J
(2)
I
5
/
2
[
σ
]
η
−
1
Q
−
2
2
(
O
2
+
O
2
)sin2
φ
2
O
−
1
2
=
−
cos 2
φ
sin 2
φ.
(5
.
4
.
2)
−
H
in (5.2.12) can be ne-
glected. Introducing the magnetostriction parameters
C
and
A
via eqn
(2
.
2
.
26
a
), when
θ
=
π/
2,
O
−
1
2
We note that
vanishes only as long as
−
2
γ
1
=
C
cos 2
φ
A
cos 4
φ
(5
.
4
.
3)
γ
2
=
C
sin 2
φ
+
2
A
sin 4
φ,
Q
±
4
4
and calculating
,weobtain
1
c
γ
B
γ
2
J
(2)
I
5
/
2
[
σ
]
η
−
1
C
=
−
(5
.
4
.
4)
2
c
γ
B
γ
4
J
(4)
I
9
/
2
[
σ
]
η
−
6
A
=
−
,
−
instead of eqn (2
.
2
.
26
b
), including the effects of the elliptical preces-
sion of the moments. The equilibrium conditions allow us to split the
magnetoelastic Hamiltonian into two parts:
H
γ
(sta) =
i
2
c
γ
(
γ
1
+
γ
2
)
− B
γ
2
{Q
2
(
J
i
)
γ
1
+
Q
−
2
(
J
i
)
γ
2
}
2
(
J
i
)
γ
2
}
,
Q
4
(
J
i
)
γ
1
−
Q
−
4
4
−
B
γ
4
{
(5
.
4
.
5)
depending only on the static strains, and
H
γ
(dyn) =
i
2
c
γ
{
(
γ
1
−
γ
1
)
2
+(
γ
2
−
γ
2
)
2
}
−
B
γ
2
{
Q
4
}
(
γ
1
−
Q
2
(
J
i
)
Q
2
}
Q
4
(
J
i
)
−
+
B
γ
4
{
−
γ
1
)
γ
2
)
(5
.
4
.
6)
−
B
γ
2
{
}
(
γ
2
−
Q
−
2
2
Q
−
2
2
Q
−
4
4
Q
−
4
4
(
J
i
)
−
} −
B
γ
4
{
(
J
i
)
−
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