Environmental Engineering Reference
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general two-ion coupling. Referring to (5.5.14), in which is introduced a
general two-ion Hamiltonian in terms of the tensor operators
O
lm
(
J
i
),
we may write
2
ij
1
J
i
·
J
j
,
p
(
ij
)
H
JJ
=
−
·J
(7
.
3
.
11)
, O
lm
,
(
J
x
,J
y
,J
z
,O
−
2
where
J
p
≡
···
) is a generalized
p
-dimensional mo-
2
ment operator, and the
-set of operators comprises the tensor cou-
plings from the original Hamiltonian, except those between the first
four components. It is then immediately clear that the final RPA sus-
ceptibility is given by an expression equivalent to (7.3.7), in terms of
the
p
{
lm
}
p
αβ
(
q
), except that
×
p
susceptibility-matrix with
J
αβ
(
q
,ω
)=
J
p
44
(
q
). If
the frequency is not near a pole in
D
(
q
,ω
), the effect of the coupling
to the phonons on the magnetic excitations is therefore similar to that
stemming from the corresponding quadrupole-quadrupole interaction.
If
J
44
(
q
,ω
)=
N
(
iqF
q
B
γ
2
/
2)
2
D
(
q
,ω
)+
(at long wavelengths)
J
p
44
(
0
) is non-zero, the ultrasonic velocities are influenced by this cou-
pling, as we now have
J
c
66
c
66
1
−
Ξ(
q
,
0)
J
44
(
0
,
0)
Ξ(
q
,
0)
44
(
0
)
B
γ
2
/c
γ
,
=
=1
−
(7
.
3
.
12)
p
44
(
0
)
p
1
−
Ξ(
q
,
0)
J
1
−
Ξ(
q
,
0)
J
where the sum over
α
in (7
.
3
.
9
b
) comprises all the (
p
−
1) components
= 4, under the same condition that
χ
o
(
ω
)and
for which
α
J
(
q
,ω
)
= 4. In general,
χ
4
α
(0) may be non-zero, in
the paramagnetic phase in zero magnetic field, if the
α
-component is
an even-rank tensor, and these interactions may contribute to Ξ(
q
,
0),
whereas the odd-rank couplings are prevented from affecting the phonons
in the zero-frequency limit by time-reversal symmetry.
In our discussion of crystal-field excitations, we have only been con-
cerned with the excitation spectrum derived from the time variation of
the
dipole
moments. There are two reasons for this. Most importantly,
the coupling between the dipolar moments expressed in eqn (7.1.1) is
normally dominant in rare earth systems, so that the collective phenom-
ena are dominated by the dipolar excitations. The other reason is that
the
magnetic
response, including the magnetic susceptibility and the
(magnetic) neutron scattering cross-section, is determined exclusively by
the upper-left 3
are both diagonal for
α
3partof
χ
(
q
,ω
), in the generalized
p
-dimensional vec-
tor space introduced through eqn (7.3.11). However, strong quadrupolar
interactions may lead to collective effects and to an ordered phase of the
quadrupole moments. The quadrupolar excitations are not directly vis-
ible in neutron-scattering experiments, but may be detected indirectly
via their hybridization with the dipole excitations, in the same way as
the phonons, or via their hybridization with the phonons, as measured
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