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in (4.1.8) with
n
even. Neglecting the (
n
= 2)-term in
K
s
, proportional
to
s
times an orbital quadrupole moment, we have
K
s
(
κ
)
j
0
(
κ
)
s
,or
)=
K
(
κ
)=
2
(
l
+2
s
)+
2
κ
j
0
(
κ
)
j
2
(
κ
)
l
.
(4
.
1
.
14)
K
(
This result for
K
(
κ
) is the basis of the dipole approximation for the
scattering cross-section. Within this approximation, it is straightfor-
wardly generalized to the case of more than one electron per atom, as
the contributions are additive, in the sense that
l
and
s
are replaced by
L
=
l
and
S
=
s
,and
R
2
(
r
) by the normalized distribution for all
unpaired electrons belonging to the atom at
R
j
.
The orbital contribution is important in the case of rare earth or
actinide ions. In transition-metal ions, the orbital momentum is fre-
quently quenched, and
K
p
may then be neglected to leading order. In
the rare earths, the spin-orbit coupling is strong and only matrix ele-
ments within the ground-state multiplet of
J
2
=(
L
+
S
)
2
contribute. In
this case, as discussed in Section 1.2,
L
+2
S
=
g
J
and
L
=(2
−
g
)
J
,
where
g
is the Lande factor, and we have
K
(
κ
)=
2
(
L
+2
S
)+
2
L
=
2
j
0
(
κ
)
j
2
(
κ
)
gF
(
κ
)
J
,
(4
.
1
.
15
a
)
where
F
(
κ
)isthe
form factor
+
1
F
(
κ
)=
j
0
(
κ
)
−
g
)
j
2
(
κ
)
,
(4
.
1
.
15
b
)
g
(2
defined so that
F
(0) = 1. When the spin-orbit interaction is introduced,
the (
n
= 2)-term in the expansion of
K
s
gives a contribution to the
dipolar part of
K
(
κ
) proportional to
j
2
(
κ
)
, but this is an order of
magnitude smaller than the orbital term in (4.1.14). A more systematic
approach, making extensive use of Racah tensor-algebra, is required to
calculate this term and to include the contributions of the higher-rank
multipoles produced by the expansion of exp(
κ
·
r
). This analysis
may be found in Marshall and Lovesey (1971), Stassis and Deckman
(1975, 1976), and references therein. Within the present approximation,
only tensors of
odd
rank give a contribution to
K
, proportional to
κ
τ−
1
,
where
τ
is the rank of the tensors (terms with
τ
= 3 appear already in
order
κ
2
). In contrast to the dipole contributions, the higher-rank tensor
couplings give rise to an angular dependence of
K
=
K
(
−
i
). The smaller
the scattering wavelength
λ
=2
π/κ
, the more the neutron senses the
details of the spin and current distributions within the atom, but as long
as
λ
is larger than approximately the mean radius
κ
of the unpaired
electrons, only the dipolar scattering is important. For rare earth ions,
r
0
.
6 A, indicating that (4.1.15) is a valid approximation as long as
κ
is smaller than about 6 A
−
1
.
r
≈
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