Environmental Engineering Reference
In-Depth Information
the inelastic response, as well as the elastic mode, are purely of second
order in 1
/J
and we shall not consider the longitudinal fluctuations
further here.
The method developed in this section may be utilized, essentially
unchanged, to calculate the MF susceptibility
χ
o
(
ω
)ofthe
single
sites.
The result to first order in 1
/J
is:
A
B
+
h
ex
E
ex
−
−
χ
xx
(
ω
)=
J
z
(
hω
)
2
A
+
B
+
h
ex
E
ex
−
χ
yy
(
ω
)=
J
z
(5
.
2
.
42
a
)
(
hω
)
2
ihω
E
ex
−
χ
xy
(
ω
)=
−χ
yx
(
ω
)=
J
z
(
hω
)
2
,
where
is the MF expectation value of
J
z
,
h
ex
is the exchange field,
and
E
ex
is the energy of the first excited MF state:
J
z
E
ex
=(
A
+
h
ex
)
2
B
2
.
h
ex
=
J
z
J
(
0
)
;
−
(5
.
2
.
42
b
)
Introducing this expression for
χ
o
(
ω
) into the RPA equation (3.5.8), we
may derive
χ
(
q
,ω
) by the same method as was used for the Heisenberg
ferromagnet in Section 3.5.2, in which case
A
=
B
=0. Theresultsfor
the
xx
-and
yy
-components are then found to agree with eqn (5.2.40)
to leading order in 1
/J
. To the next order in 1
/J
, the parameters are
replaced by renormalized values, but this procedure is not here easily
generalized so as to become fully self-consistent. However, most of the
corrections may be included by substituting
A
0
(
T
)
±
B
0
(
T
)for
A
±
B
in the expression for
χ
o
(
ω
), and the self-consistent value of
for its
MF value. The only terms which are not included in
χ
(
q
,ω
)bythis
procedure, as we may see by a comparison with eqn (5.2.40), are the
q
-dependent contributions to
A
q
(
T
)
J
z
B
q
(
T
) determined by the
k
-sums
in (5.2.38). At low temperatures, these contributions are small and
may safely be neglected in systems with long-range interactions. This
formulation therefore represents a valid alternative, which is useful for
generalizing the linear spin-wave theory to the hcp structure, discussed
in Section 5.1, or to the helically or conically ordered systems which we
will consider in Chapter 6.
As an example of the
magnon dispersion relations
for the anisotropic
basal-plane ferromagnet, we show in Fig. 5.2 experimental measurements
on Tb at 4 K (Mackintosh and Bjerrum Møller 1972). The principal
differences between these results and the corresponding excitations for
Gd in Fig. 5.1 are the pronounced interactions which are observed be-
tween the magnons and phonons, which we shall discuss in some detail in
±
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