Environmental Engineering Reference
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where
f
n
k
is the Fermi-Dirac function. It is clear that a large contri-
bution to the sum is made by pairs of electronic states, separated by
q
,
one of which is occupied and the other empty, and both of which have
energies very close to the Fermi level. Consequently, parallel or
nest-
ing
regions of the Fermi surface tend to produce peaks, known as
Kohn
anomalies
, at the wave-vector
Q
which separates them, and it is believed
that the parallel sections of the webbing in the hole surface of Fig. 1.11
give rise to the maxima shown in Fig. 1.17. As we have mentioned, this
conjecture is supported by both positron-annihilation experiments and
band structure calculations but, despite extensive efforts, first-principles
estimates of
(
q
) have not proved particularly successful.
χ
(
q
)maybe
calculated quite readily from the energy bands (Liu 1978), and exhibits
the expected peaks, but the exchange matrix elements which determine
I
(
q
) are much less tractable. Lindg ard
et al.
(1975) obtained the correct
general variation with
q
for Gd, but the matrix elements were, not sur-
prisingly, far too large when the screening of the Coulomb interaction
was neglected.
The Kohn anomalies in
J
J
(
q
) Fourier transform into
Friedel oscilla-
tions
in
(
R
), and such oscillations, and the extremely long range of the
indirect exchange, are illustrated in the results of Houmann
et al.
(1979)
for Pr in Fig. 1.18. As is also shown in this figure, they found that the
anisotropic
component of the coupling is a substantial proportion of the
Heisenberg exchange. The anisotropic coupling between the moments
on two ions can be written in the general form
J
2
ij
ll
mm
K
1
mm
ll
(
J
i
)
O
m
(
ij
)
O
l
H
JJ
=
−
(
J
j
)
,
(1
.
4
.
24)
l
where the terms which appear in the sum are restricted by symmetry,
but otherwise may exhibit a large variety, depending on their origin. The
many possible causes of anisotropy have been summarized by Jensen
et
al.
(1975). They are usually associated with the orbital component of
the moment and are therefore expected to be relatively large when
L
is large. In addition to contributions due to the influence of the local-
ized 4
f
orbital moment on the conduction electrons (Kaplan and Lyons
1963), and to the magnetization and spin-orbit coupling of the latter
(Levy 1969), direct multipolar interactions and two-ion magnetoelas-
tic couplings, for which the coecients
mm
ll
depend explicitly on the
strain, may be important. A general two-ion coupling which depends
only on the dipolar moments of the 4
f
electrons is
K
2
1
H
dd
=
−
ij
J
αβ
(
ij
)
J
iα
J
jβ
.
(1
.
4
.
25)
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