Environmental Engineering Reference
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plane, this means that the number of spin-wave excitations, i.e. magnons,
is not necessarily conserved in a scattering process. In contrast to the be-
haviour of the isotropic ferromagnet, the linewidths do not therefore van-
ish at zero temperature, although energy conservation, combined with
the presence of an energy gap in the magnon spectrum, strongly limit
the importance of the allowed decay processes at low temperatures.
The two-ion interactions are assumed to involve only tensor op-
erators of the lowest rank, so that these terms in the 1
/J
-expansion
only have small numerical factors multiplying the Bose operator prod-
ucts. Therefore, if
J
is large, as in heavy rare earth-ions, the third-order
terms due to the exchange coupling, which are neglected in the spin-
wave theory, are expected to be small, as long as the number of excited
magnons is not very large. The weak influence, at low temperatures,
of the higher-order contributions of the exchange coupling is also indi-
cated by a comparison with the low-temperature expansion of Dyson
(1956) of the free energy in a Heisenberg ferromagnet with only nearest-
neighbour interactions, also discussed by Rastelli and Lindgard (1979).
If
A
=
B
= 0, the results derived earlier, to second order in 1
/J
,are
consistent with those of Dyson, except that we have only included the
leading-order contribution, in the Born approximation or in powers of
1
/J
,tothe
T
4
-term in the magnetization and in the specific heat. The
higher-order corrections to the
T
4
-term are significant if
J
=
1
2
, but if
J
= 6 as in Tb, for example, they only amount to a few per cent of this
term and can be neglected.
If only the two-ion terms are considered, the RPA decoupling of
the Bose operator products (5.2.29) is a good approximation at large
J
and at low temperatures. However, this decoupling also involves an
approximation to the single-ion terms, and these introduce qualitatively
new features into the spin-wave theory in the third order of 1
/J
.For
example, the
C
3
-term in (5.2.26) directly couples the
|
J
z
=
J>
state
and
4
>
, leading to an extra modification of the ground state not
describable in terms of
B
or
η
±
. Furthermore, the Bogoliubov trans-
formation causes the (
J
x
,J
y
)-matrix elements between the ground state
and the third excited state to become non-zero. This coupling then
leads to the appearance of a new pole in the transverse susceptibilities,
in addition to the spin-wave pole, at an energy which, to leading order,
is roughly independent of
q
and close to that of the third excited MF
level, i.e. 3
E
q
o
(
T
), with
q
o
defined as a wave-vector at which
|
J
−
(
q
o
)=0.
A qualitative analysis indicates that the third-order contribution to e.g.
χ
xx
(
0
,
0), due to this pole, must cancel the second-order contribution of
H
to
F
θθ
in the relation (5
.
3
.
12
b
) between the two quantities. Hence
the approximation
F
θθ
J
∂
2
/∂θ
2
H
, used in (5.3.22), corresponds to
the neglect of this additional pole.
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