Environmental Engineering Reference
In-Depth Information
This leaves only the
axial
tensors, i.e. magnetic multipoles of odd rank
and electric multipoles of even rank. Time reversal of these tensors ef-
fects the transformation
c O
lm
→
O
l −m
, whereas Hermitian
c
∗
(
1)
l
+
m
−
conjugation gives
c O
lm
†
=
c
∗
(
O
l −m
. The requirement that
1)
m
H
JJ
should be invariant under both transformations allows only those terms
in eqn (5.5.14) for which
l
+
l
is even. The violation of time-reversal
symmetry which occurs when the system is magnetically ordered im-
plies that
−
H
JJ
should be supplemented by interactions proportional to
O
λµ
O
lm
(
J
i
)
O
l
m
(
J
j
), satisfying the condition that
λ
+
l
+
l
is even.
An obvious example is magnetoelastic contributions to the Hamiltonian
such as eqn (5.4.5). The tensor operators in (5.5.14) emanate from lo-
calized 4
f
wavefunctions with the orbital quantum number
l
f
=3,which
puts the further restriction on the phenomenological expansion of
H
JJ
that
l
and
l
cannot be larger than 2
l
f
+1 = 7, as the operator-equivalents
of higher rank than this vanish identically.
In the rare earth metals, several different mechanisms may give rise
to such anisotropic two-ion couplings, and these have been listed by,
for instance, Wolf (1971) and Jensen
et al.
(1975). We have already
considered the magnetostatic coupling of lowest rank in the magnetic
multipole expansion, namely the classical magnetic dipole-dipole inter-
action. This is of importance only because of its long range. The higher
order magnetostatic couplings are of shorter range (
(1
/r
)
l
+
l
+1
)and
have negligible effects. The electrostatic Coulomb interaction gives rise
to terms in (5.5.14) in which both
l
and
l
∝
are even.
The single-ion
contributions (
l
= 0, but
even the lowest-order electrostatic two-ion term, which contributes to
the quadrupole-quadrupole interactions, is so small that it may be ne-
glected.
Theoverlapbetweenthe4
f
wavefunctions of neighbouring ions is so
weak that it cannot generate any two-ion coupling of significance. The
dominant terms in the two-ion Hamiltonian
= 0) are of decisive importance, when
L
H
JJ
therefore arise indi-
rectly via the propagation of the conduction electrons. We have already
mentioned in Section 1.4 the most important of these, due to the ex-
change interaction between the band electrons and the 4
f
electrons, and
it will be discussed in more detail in Section 5.7. In the simplest ap-
proximation, the indirect exchange is invariant with respect to a uniform
rotation of the angular momenta, i.e. this RKKY interaction is
isotropic
.
However, the neglect of the contribution of the orbital moment in the
scattering process is not generally justified. If
L
is non-zero, the or-
bital state of the 4
f
electrons may change in an exchange-scattering
process, if the conduction electron is scattered into a state with a dif-
ferent orbital momentum relative to the ion. The leading-order correc-
tions to the isotropic RKKY interaction due to such processes have been
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