Environmental Engineering Reference
In-Depth Information
In the general case, the MF susceptibility
χ
o
(
ω
) is determined
by three distinct diagonal components, plus the two off-diagonal terms
χ
xy
(
ω
)=
χ
yx
(
ω
), with the same analytical properties, (3
.
5
.
24
b
)and
(5.2.42), as in the Heisenberg ferromagnet. It may be seen that
χ
xy
(
ω
),
for instance, is imaginary by recalling that the MF Hamiltonian is in-
dependent of
J
y
, in which case the eigenvectors in the
J
z
-representation
can all be chosen to be real, so that the products of the matrix ele-
ments of
J
x
and of
J
y
are imaginary. The vanishing of the other four
off-diagonal terms follows from the two-fold symmetry about the
z
-axis
of the MF Hamiltonian. In spite of this reduction, the analytical ex-
pression for
χ
(
q
,ω
) is still quite formidable. However, in most cases of
interest, the single-ion anisotropy is relatively weak, and the inelastic
modifications due to
χ
zz
(
ω
) can be neglected, so that, for
ω
−
=0,
χ
xx
(
q
,ω
)=
χ
xx
(
ω
)
|J
yy
(
q
)
/D
(
q
,ω
)
χ
o
(
ω
)
−|
(6
.
1
.
8
a
)
χ
xy
(
q
,ω
)=
χ
xy
(
ω
)+
|J
xy
(
q
)
/D
(
q
,ω
)
,
χ
o
(
ω
)
|
and the same relations hold with
x
and
y
interchanged. Here
χ
o
(
ω
)
χ
αβ
(
ω
)
D
(
q
,ω
)=1
−
J
βα
(
q
)+
|
||J
(
q
)
|
,
(6
.
1
.
8
b
)
αβ
χ
o
(
ω
)
where
α
or
β
are
x
or
y
,and
|
|
or
| J
(
q
)
|
are the determinants
of the 2
2 matrices. In the weak-anisotropy limit, we may to a good
approximation use the result (5.2.42) derived in Section 5.2, and to first
order in 1
/J
,wehave
×
A
B
+
h
ex
E
ex
−
−
χ
xx
(
ω
)=
J
z
(
hω
)
2
A
+
B
+
h
ex
E
ex
−
χ
yy
(
ω
)=
J
z
(6
.
1
.
9
a
)
(
hω
)
2
i hω
E
ex
−
χ
xy
(
ω
)=
χ
yx
(
ω
)=
−
J
z
(
hω
)
2
and
χ
zz
(
ω
)
β
(
δJ
z
)
2
δ
ω
0
. The only modification is that the exchange
field, given in eqn (6.1.5), is now
a d
E
ex
=(
A
+
h
ex
)
2
B
2
.
h
ex
=
J
z
J
(
Q
)
−
(6
.
1
.
9
b
)
There are inelastic contributions to
χ
zz
(
ω
), but they are of the order
A/
2
JE
ex
, relative to the other inelastic terms, and can be neglected.
The parameters
A
and
B
are the same as those derived in Section
5.2, when
H
J
in (6.1.4) is replaced by the usual crystal-field Hamilto-
nian, except that we here neglect explicitly the hexagonal anisotropy
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