Environmental Engineering Reference
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dimensional Ising model,
d
∗
= 4, with pronounced modifications in-
duced by the critical fluctuations. The original treatment by Turov and
Shavrov (1965) of the
γ
-strain contributions, which prevent the uniform
magnon mode from going soft at the critical field, included only the
static-strain components. The more complete analyses, including the
phonon dynamics, were later given by Jensen (1971a,b), Liu (1972b),
and Chow and Keffer (1973).
When the wave-vector is in the
c
-direction, the
γ
-strain couplings
vanish, but instead the
ε
-strains become important. The
O
2
-term in
Q
2
,
given by eqn (5.4.17), leads to a linear coupling between the magnons
and the phonons, and proceeding as in eqns (5.4.26-27), we find the
additional contribution to
H
mp
H
mp
=
k
iW
k
(
ε
)(
α
k
+
α
−
k
)(
β
ν
k
+
β
ν−
k
)
,
∆
(5
.
4
.
41
a
)
with
c
ε
√
N
(
k
1
F
k
,
3
+
k
3
F
k
,
1
)cos
φ
+(
k
2
F
k
,
3
+
k
3
F
k
,
2
)sin
φ
−
4
W
k
(
ε
)=
A
k
(
T
)
2
H
ε
,
B
k
(
T
)
2
JσE
k
(
T
)
−
×
(5
.
4
.
41
b
)
in the long-wavelength limit. When
k
is parallel to the
c
-axis, (5.4.28)
and (5.4.41) predicts that only the transverse phonons with their polar-
ization vectors parallel to the magnetization are coupled to the magnons.
The calculation of the velocity of this coupled mode leads, by analogy
to (5.4.38), to an elastic constant
c
44
c
44
Λ
ε
A
0
(
T
)+
B
0
(
T
)
f
k
J
−
.
(5
.
4
.
42)
=1
when
The same result is obtained for the transverse-phonon mode propagating
in the direction of the ordered moments, with the polarization vector
parallel to the
c
-axis. These are the two modes which go soft in the case
of a second-order transition to a phase with a non-zero
c
-axis moment.
We have so far only considered the dynamics in the long-wavelength
limit. At shorter wavelengths, where the phonon and spin-wave energies
may be comparable, the magnon-phonon interaction leads to a strong
hybridization of the normal modes, with energy gaps at points in the
Brillouin zone where the unperturbed magnon and phonon dispersion
relations cross each other, as illustrated in Fig. 5.6. The interaction
amplitudes (5.4.28) and (5
.
4
.
41
b
) are correct only for small wave-vectors.
At shorter wavelengths, we must consider explicitly the relative positions
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