Environmental Engineering Reference
In-Depth Information
This parameter does not obey the relation (5.3.14) with the second
derivative
F
φφ
ofthefreeenergy. Adifferentiation
∂F/∂φ
,asgiven
by (5.4.8), with respect to
φ
shows that (5.3.14) accounts for the last
two terms in (5.4.11), but not for Λ
γ
. A calculation from (5.4.7) of the
second derivative of
F
, when the strains are kept constant, instead of
under the constant (zero) stress-condition assumed above, yields
=
∂
2
F
∂φ
2
1
NJσ
1
NJσ
F
φφ
,
A
0
(
T
)
−
B
0
(
T
)=
=Λ
γ
+
(5
.
4
.
12)
which replaces (5.3.14).
−
B
0
(
T
), was based on a calculation of the frequency dependence of the
bulk susceptibility and, as we shall see later, it is the influence of the
lattice which invalidates this argument. The Λ
γ
term was originally sug-
gested by Turov and Shavrov (1965), who called it the 'frozen lattice'
contribution because the dynamic strain-contributions were not consid-
ered. However, as we shall show in the next section, the magnon-phonon
coupling does not change this result.
The modifications caused by the magnetoelastic
γ
-strain couplings
are strongly accentuated at a second-order phase transition, at which
F
φφ
vanishes.
The relation (5.3.14), determining
A
0
(
T
)
Let us consider the case where
H
c
is positive,
H
c
=
| H
c
|≡
H
c
, i.e. the
b
-axis is the easy axis. If a field is applied along an
a
-axis,
φ
H
= 0, then the magnetization is pulled towards this direction,
as described by eqn (5.4.8):
6sin
φ
=
H
c
1
sin
4
φ
cos
φ,
sin 6
φ
−
1
3
sin
2
φ
+
1
3
H
=
H
c
(5
.
4
.
13)
as long as the field is smaller than
H
c
. At the critical field
H
=
H
c
,
the moments are pulled into the hard direction, so that
φ
=0andthe
second derivative of the free energy,
F
φφ
=
Ngµ
B
{
H
cos
φ
−
H
c
cos 6
φ
}
Jσ,
(5
.
4
.
14)
vanishes. So a second-order phase transition occurs at
H
=
H
c
,andthe
order parameter can be considered to be the component of the moments
perpendicular to the
a
-axis, which is zero for
H
H
c
. An equally good
choice for the order parameter is the strain
γ
2
, and these two possibili-
ties reflect the nature of the linearly coupled magnetic-structural phase
transition. The free energy does not contain terms which are cubic in
the order parameters, but the transition might be changed into one of
first-order by terms proportional to cos 12
φ
,e.g.if
σ
or
η
±
, and thereby
H
c
, depend suciently strongly on
φ
(Jensen 1975). At the transition,
eqn (5.4.11) leads to
≥
A
0
(
T
)
−
B
0
(
T
)=Λ
γ
at
H
=
H
c
,
(5
.
4
.
15)
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