Environmental Engineering Reference
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where
α
=
ξ
,
η
,and
ζ
,and
Ξ(
q
,ω
)=
χ
44
(
ω
)+
α
χ
α
4
(
ω
)
χ
4
α
(
ω
)
J
αα
(
q
)
.
(7
.
3
.
9
b
)
−
χ
αα
(
ω
)
J
αα
(
q
)
1
At long wavelengths, this pole determines the velocity of the magneto-
acoustic sound waves, as measured in an ultrasonic experiment, and
expressing this velocity in terms of the corresponding elastic constant,
we find
c
66
c
66
Ξ(
q
,
0)
B
γ
2
/c
γ
,
=1
−
(7
.
3
.
10)
by combining the above relation with eqns (5
.
4
.
24
b
) and (5.4.34). This
result is valid when
q
is along the
ξ
-or
η
-axes, provided that the exter-
nal field is applied along one of the principal axes. In the general case,
it is necessary to include the coupling to the other phonon branches in
eqn (7.3.7), and also to take into account possible off-diagonal terms in
the Cartesian part of the matrices, but these complications may be in-
cluded in the above calculations in a straightforward fashion. One ques-
tion raised by (7.3.10) is whether the magneto-acoustic sound velocities,
measured at non-zero frequencies, depend on possible purely-elastic con-
tributions to the RPA susceptibilities. That these should be included
in (7.3.7), at
ω
= 0, can be seen by the argument used in deriving
(3.5.22). In the preceding section, we found that the coupling between
the angular momenta broadens the elastic RPA response into a diffusive
peak of width 2Γ, as in (7
.
2
.
11
b
), proportional to
T
1
/
2
at low temper-
atures. Unless this coupling is very weak, Γ is likely to be much larger
than the applied
hω
in an ultrasonic experiment, in which case the total
elastic contribution to Ξ(
q
,
0) in (7.3.10) should be included. A more
detailed investigation of this question is given by, for instance, Elliott
et al.
(1972), in a paper discussing systems with Jahn-Teller-induced
phase transitions.
In the paramagnetic phase without any external
magnetic
field, the
susceptibility components
χ
α
4
(
ω
) all vanish in the zero frequency limit,
due to the time-reversal symmetry of the system. Replacing
t
by
−
t
generates the transformation
χ
α
4
(
ω
)
χ
α
T
4
T
(
→
−
ω
), where the time-
reversed operators are
J
iα
=
2
(
J
i
)
T
=
O
−
2
(
J
i
). These re-
sults follows from the symmetry properties of the axial tensor operators,
discussed after eqn (5.5.14), recalling that the operators are Hermitian,
of rank
l
=1and
l
= 2 respectively. Hence, because of the time-reversal
symmetry,
χ
α
4
(
ω
)=
J
iα
,and
O
−
2
−
−
χ
α
4
(
ω
∗
)
∗
, where the last result
follows from (3.2.15), and we assume implicitly that all poles lie on the
real axis. This quantity must therefore vanish at zero frequency, and
the reactive and absorptive components are either zero or purely imag-
inary at non-zero frequencies. If there is no ordered moment and no
χ
α
4
(
−
−
ω
)=
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