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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, αα (0) = 0 . 026 for the two basal-plane components, but is
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