Environmental Engineering Reference
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Fig. 1.18.
The indirect-exchange interaction between ions on the
hexagonal sites in Pr, deduced from measurements of the magnetic exci-
tations at 6 K. The circles represent the isotropic interaction
J
(
R
)be-
tween an ion at the origin and those at different sites. The filled sym-
bols are for pairs of ions in the same hexagonal plane, and the open
symbols for pairs in different planes. The former are reasonably well
described by the simple free-electron model of Section 5.7.1, with an ef-
fective value of 1.1 A
−
1
for 2
k
F
, as shown by the full curve. In addition,
the exchange incorporates an anisotropic component
K
(
R
), discussed in
Section 2.1.6, which is smaller, but of comparable magnitude. Its values
between pairs of ions in the plane are indicated by the squares. The calcu-
lated uncertainties in the exchange interactions are, at the most, the size
of the points.
The dispersion relations for the magnetic excitations provide extensive
evidence for anisotropy of this form. A special case is the classical
dipole-
dipole interaction
for which
2
J
αβ
(
ij
)=(
gµ
B
)
2
3(
R
iα
−
R
jα
)(
R
iβ
−
R
jβ
)
−
δ
αβ
|
R
i
−
R
j
|
.
|
R
i
−
R
j
|
5
(1
.
4
.
26)
Although it is very weak, being typically one or two orders of magnitude
less than the exchange between nearest neighbours, the dipole-dipole
coupling is both highly anisotropic and extremely long-ranged, and may
therefore have important effects on both magnetic structures and exci-
tations. Apart from this example, the anisotropic two-ion couplings are
even more dicult to calculate than are the isotropic components, so
the strategy which has generally been adopted to investigate them is to
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