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velocities are determined by
c
cc
c
66
Λ
γ
=1
−
;
q
⊥
J
.
(5
.
5
.
13)
A
0
(
T
)
−
B
0
(
T
)+4
πgµ
B
M
This modification means that only the velocity of those magneto-acoustic
modes which propagate parallel to the magnetization vanishes at the
critical field
H
I
=
H
c
where, according to (5.4.15),
A
0
(
T
)
B
0
(
T
)=Λ
γ
,
whereas the velocity of the modes propagating in the basal-plane per-
pendicular to the field remains non-zero. This difference in the velocities
of the ultrasonic modes was very clearly manifested in the ultrasonic ex-
periment on Tb discussed in Section 5.4 (Jensen and Palmer 1979). This
example shows that the dipolar coupling is able to lift the degeneracy
in the velocities of two transverse sound-waves which are related to each
other by an interchange of the directions of the wave-vector and the
polarization vector. The same effect may be produced by the torque
exerted on the moments when the local magnetic anisotropy axes are
rotated by the transverse phonons relative to the direction of an exter-
nal magnetic field. As discussed by Melcher (1972) and Dohm and Fulde
(1975) the influence of this mechanism on the sound velocities may be
derived from the principle that the total system has to be
rotationally
invariant
. Their theory has been extended by Jensen (1988b), who finds
that the dipolar-coupling contribution strongly dominates in a ferromag-
net, but that the importance of the two mechanisms may be comparable
in paramagnets.
−
5.5.2 General two-ion interactions
The two-ion couplings described by eqn (5.5.1) only involve the dipolar
moments of the 4
f
electrons. A more general two-ion Hamiltonian is
2
ij
K
mm
ll
1
(
ij
)
O
lm
(
J
i
)
O
l
m
(
J
j
)
H
JJ
=
−
mm
l
+
l
:even
}
∗
O
l −m
(
J
i
)
O
l
−m
(
J
j
)
,
(5
.
5
.
14)
expressed in terms of the
Racah operators
or
tensor operators
O
lm
(
J
i
)
introduced in Section 1.4, rather than the Stevens operators. Tables
of these operators and a discussion of their properties may be found in
Buckmaster
et al.
(1972) and in Lindg ard and Danielsen (1974). Here
we neglect the possible effects of the
polar
tensors, which vanish for the
isolated ions. In principle, these polar tensors may be non-zero in the
hcp metals, because the surroundings lack inversion symmetry, but they
occur only because of odd-parity configuration-mixing of the 4
f
wave-
functions, which should be insignificant for the ground-state multiplet.
1)
m
+
m
K
mm
ll
+(
−
{
(
ij
)
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