Environmental Engineering Reference
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at a non-zero frequency. Therefore the spin-wave mode at
q
=
0
per-
ceives the lattice as being completely static or 'frozen'. This is clearly
consistent with the result (5.4.12), that the spin-wave energy gap is
proportional to the second derivative of the free energy under constant-
strain, rather than constant-stress, conditions.
If the lattice is able to adapt itself to the applied constant-stress
condition, in the static limit
ω
ω
q
, then, according to (5
.
4
.
36
b
),
=
N
(
Jσ
)
2
F
φφ
Jσ
χ
yy
(
q
→
0
,
0) =
χ
yy
(
q
≡
0
,
0) =
,
A
0
(
T
)
−
B
0
(
T
)
−
Λ
γ
(5
.
4
.
39)
in agreement with (5
.
3
.
7). However, the first equality is not generally
valid. The susceptibility depends on the direction from which
q
ap-
proaches
0
. If the direction of
q
is specified by the spherical coordinates
(
θ
q
,φ
q
), then eqn (5.4.39) is valid only in the configuration considered,
i.e. for
θ
q
π
2
=0or
2
. If we assume elastically isotropic
conditions (
c
11
=
c
33
,
c
44
=
c
66
,and
c
12
=
c
13
), which is a reasonable
approximation in Tb and Dy, we find that (5.4.39) is replaced by the
more general result
=
and
φ
q
χ
yy
(
q
→
0
,
0) =
Jσ
(5
.
4
.
40)
,
Λ
γ
sin
2
θ
q
{
ξ
)sin
2
θ
q
sin
2
2
φ
q
}
A
0
(
T
)
−
B
0
(
T
)
−
1
−
(1
−
when
φ
=0or
2
0
.
3inTborDy). Therea-
son for this modification is that discussed in Section 2.2.2; the abil-
ity of the lattice to adapt to various static-strain configurations is lim-
ited if these strains are spatially modulated. If
q
is along the
c
-axis
(
θ
q
= 0), the
γ
-strains are 'clamped', remaining constant throughout
the crystal, so that the susceptibilities at both zero and finite frequen-
cies are determined by the uniform
γ
-strain contributions alone. We
note that, according to (5.4.28),
W
k
,and
ξ
=
c
66
/c
11
(
vanishes if
k
is parallel to the
c
-
= 0). The opposite extreme occurs when
θ
q
=
2
axis (
k
1
=
k
2
and
=0or
2
φ
q
. The relevant strain-mode is determined by the equilib-
rium conditions (5.4.3) at zero constant stress, but generalized to the
non-uniform case where the
y
-component of the moments has a small
modulation, with the wave-vector
q
along the
x
-direction. This strain
mode (
γ
2
(
i
)+
ω
21
(
i
)
cos (
q
·
R
i
+
ϕ
)) coincides with a phonon eigen-
state, the transverse phonon at
q
with its polarization vector in the basal
plane. This coincidence makes the equilibrium strain-mode viable, which
then explains the constant-stress result (5.4.39) obtained for
χ
yy
in this
situation.
We shall now return to the discussion of the second-order transition
occurring at
H
=
H
c
, when the field is applied along a hard direction
∝
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