Environmental Engineering Reference
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which is a simple combination of Bose Green-functions determined by
(5.2.20), with
E
q
replaced by
E
q
(
T
). Introducing these functions and
the parameters given by (5.2.34), we finally obtain
m
)
A
q
(
T
)
−
B
q
(
T
)
χ
xx
(
q
,ω
)=
J
(1
−
(
hω
)
2
,
(5
.
2
.
40
a
)
E
q
(
T
)
−
neglecting third-order terms. A rotation of the coordinate system by
π/
2 around the
z
-axis changes the sign of
B
q
(
T
), and hence we have
χ
yy
(
q
,ω
)=
J
(1
− m
)
A
q
(
T
)+
B
q
(
T
)
(
hω
)
2
.
(5
.
2
.
40
b
)
E
q
(
T
)
−
These results show that the ratio between the neutron-scattering inten-
sities due to the spin-wave at
q
, neglecting
S
zz
(
q
,ω
), in the two cases
where the scattering vector is perpendicular to the basal
y
-
z
plane and
to the
x
-
z
plane is
hω
=
±E
q
(
T
)
R
q
(
T
)=
S
xx
(
q
,ω
)
S
yy
(
q
,ω
)
=
χ
xx
(
q
,
0)
χ
yy
(
q
,
0)
=
A
q
(
T
)
− B
q
(
T
)
A
q
(
T
)+
B
q
(
T
)
.
(5
.
2
.
41)
The measured intensities from Tb, which differ substantially from those
calculated for the Heisenberg ferromagnet, agree well with this expres-
sion, especially if the correction for anisotropic two-ion coupling is taken
into account (Jensen
et al.
1975).
In the Heisenberg ferromagnet without rotational anisotropy, corre-
sponding to
B
q
(
T
) = 0, the elementary excitations at low temperatures
are
circularly
polarized spin waves, in which the local moments precess
in circles around the equilibrium direction. In the presence of anisotropy,
R
q
(
T
) differs from unity, and the excitations become
elliptically
polar-
ized spin waves. The eccentricity of the ellipse depends on the wave-
vector of the excited spin wave, and by definition
R
q
(
T
) is the square of
the ratio of the lengths of the principal axes which, at least to the order
in 1
/J
which we have considered, is equal to the ratio between the cor-
responding static susceptibility components. So the static anisotropy is
reflected, in a direct way, in the normal modes of the system. The result
(5.2.41) justifies the transformation (5.2.34) by attributing observable
effects to the parameters
A
q
(
T
)
B
q
(
T
), whereas the parameters which
are defined via the Hamiltonian alone, here
A
q
(
T
)
±
± B
q
(
T
), depend on
the particular Bose representation which is employed.
The longitudinal correlation function
S
zz
(
q
,ω
), which is neglected
above, contains a diffusive mode at zero frequency, but no well-defined
normal modes of non-zero frequency. There is inelastic scattering, but
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