Environmental Engineering Reference
In-Depth Information
which shows the importance of the constant-strain contribution Λ
γ
.It
ensures that the spin-wave energy gap
E
0
(
T
), instead of going to zero
as
1
/
2
, remains non-zero, as illustrated in Fig. 5.4, when the
transition at
H
=
H
c
is approached. Such a field just cancels the macro-
scopic hexagonal anisotropy, but energy is still required in the spin wave
to precess the moments against the strain field of the lattice.
By symmetry, the
γ
-strains do not contain terms linear in (
θ
|
H
−
H
c
|
−
π
2
),
and the choice between constant-stress and constant-strain conditions
therefore has no influence on their contribution to the second derivative
of
F
with respect to
θ
,at
θ
=
π/
2. Consequently, the
γ
-strains do
not change the relation between
A
0
(
T
)+
B
0
(
T
)and
F
θθ
, given by eqn
(5.3.14). The
ε
-strains vanish at
θ
=
π/
2, but they enter linearly with
(
θ
−
2
). Therefore they have no effect on
A
0
(
T
)+
B
0
(
T
), but they
contribute to
F
θθ
. To see this, we consider the
ε
-strain part of the
Hamiltonian, eqn (2.2.29):
H
ε
=
i
2
(
J
i
)
ε
2
}
.
c
ε
(
ε
1
+
ε
2
)
Q
2
(
J
i
)
ε
1
+
Q
−
1
−
B
ε
1
{
(5
.
4
.
16)
2
The equilibrium condition is
1
c
ε
=
4
Q
2
ε
1
=
B
ε
1
H
ε
sin 2
θ
cos
φ,
in terms of the magnetostriction parameter
H
ε
. In the basal-plane fer-
romagnet,
ε
1
and
ε
2
both vanish.
The transformation (5.2.2) leads
to
Q
2
=
4
(
O
2
−
O
2
)sin2
θ
cos
φ
O
2
cos 2
θ
cos
φ
+
O
−
1
−
cos
θ
sin
φ
2
+
2
O
−
2
2
sin
θ
sin
φ,
(5
.
4
.
17)
−
2
and
Q
−
1
2
is given by the same expression, if
φ
is replaced by
φ
.This
implies that
4
c
ε
2
c
ε
B
ε
1
4
B
ε
1
J
(2)
I
5
/
2
[
σ
]
η
−
1
(
O
2
−
O
2
)
H
ε
=
=
.
(5
.
4
.
18)
+
The static
ε
-strains are zero and do not contribute to the spin-wave
parameters
A
0
(
T
)
B
0
(
T
), but they affect the second derivative of
F
,
with respect to
θ
, under zero-stress conditions and, corresponding to
(5.4.12), we have
±
=
∂
2
F
∂θ
2
1
NJσ
1
NJσ
F
θθ
,
A
0
(
T
)+
B
0
(
T
)=
=Λ
ε
+
(5
.
4
.
19)
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