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within the (2
I
+ 1)-ground state manifold, i.e.
2
I
+1
m
I
2
m
I
M
z
A/
∆
2
δ
ω
0
=
β
3
1
δχ
zz
(
ω
)=
β
I
(
I
+1)
A
2
(2
M
z
/
∆)
2
δ
ω
0
.
(7
.
3
.
23)
This result may be straightforwardly generalized to an arbitrary level
scheme, including non-zero matrix elements of the other
J
-components,
as the different contributions are additive. The susceptibility may then
be written
χ
αβ
(
ω
)=
χ
αβ
(
ω
)+
A
2
γγ
χ
αγ
(
ω
)
χ
γγ
(
ω
)
χ
γ
β
(
ω
)
,
(7
.
3
.
24)
to leading order in
A
, which is valid as long as the general assumptions
made above are satisfied.
χ
αβ
(
ω
) is the MF susceptibility for the elec-
tronic system alone, when the extra term
δ
I
·
J
is included
in its MF Hamiltonian. In order to derive the effective MF Hamiltonian
H
I
(MF), determining the susceptibility of the nuclear spins
χ
αβ
(
ω
), we
must consider the possibility, neglected above, that
H
J
(MF) =
A
H
hf
may lift the
(2
I
+ 1)-fold degeneracy of the ground-state manifold. Calculating the
energies of the ground-state levels, in the presence of an external field,
by second-order perturbation theory, we find straightforwardly that the
equivalent Hamiltonian, describing the splitting of these levels, is
H
I
(MF) =
−g
N
µ
N
H
·
I
+
A
J
+
A
I
·χ
J
(0)
·
I
−
2
A
2
I
· χ
J
(0)
·
I
.
(7
.
3
.
25
a
)
This result can be interpreted as expressing the ability of
J
to follow
instantaneously any changes of
I
. The molecular field due to
J
is
subtracted from the response to
I
, which then instead gives rise to the
last quadrupolar term. This quadrupolar contribution is the only effect
which is missing in a simple RPA decoupling of the interactions intro-
duced through
H
hf
. f
χ
J
(0) is not a scalar, the last term gives rise
to a quadrupole-splitting of the ground-state manifold, and the zero-
frequency susceptibility is then, to leading order in this term,
3
I
(
I
+1)
β
1+
15
χ
γγ
(0)
)
3
χ
αα
(0)
χ
αα
(0) =
1
A
2
β
(
I
+
2
−
2
)(
I
−
γ
(7
.
3
.
25
b
)
if
χ
J
(0) is diagonal. The results above were first obtained and anal-
ysed by Murao (1971, 1975, 1979), except that he replaced
χ
αα
(0) in
(7.3.25) by (1
/N
)
q
χ
αα
(
q
,
0) which, according to the above inter-
pretation, is to be expected in order 1
/Z
. For the hexagonal ions in
Pr-metal,
Aχ
αα
(0) = 0
.
026 for the two basal-plane components, but is
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