Environmental Engineering Reference
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lift the degeneracy of the modes at A on the Brillouin-zone boundary
of Fig 1.4. When
q
is parallel to the
c
-axis, a direct calculation of the
spin-wave energies (Jensen
et al.
1975) shows that the two-ion terms in
H
JJ
lead to the following modifications of the earlier results (5.2.38) and
(5.3.22):
(i) The two-ion anisotropy may contribute to the parameters
A
q
(
T
)
B
q
(
T
) at zero wave-vector.
(ii)
B
q
(
T
) becomes dependent on
q
to leading order in 1
/J
.
(iii) The
q
-dependent parts of
A
q
(
T
)
±
B
q
(
T
) may change when
the direction of magnetization is changed.
There are no direct ways of separating the single- and two-ion con-
tributions to the energy gap at zero wave-vector. However, a strong
q
-dependence of
B
q
(
T
) is only possible if the two-ion Hamiltonian is
anisotropic. One way to determine
B
q
(
T
) is to utilize the dependence of
the neutron cross-section on this parameter, given by eqn (5.2.41). This
method requires accurate intensity measurements and is not straightfor-
ward. The other possibility is to measure the field dependence of the
spin-wave energies since, from (5.2.38) or (5.3.22),
±
α
q
(
T
)
≡ ∂E
q
(
T
)
/∂
(
gµ
B
H
)
2
A
q
(
T
)
,
(5
.
5
.
17)
when the field is parallel to the magnetization. This relation is only true
to first order in 1
/J
, and corrections have to be made for the influence
of any field-dependent changes of the correlation functions
σ
and
η
±
.
Both
A
q
(
T
)and
B
q
(
T
) may be determined from the energies and initial
slopes, since
2
±
2
4
E
q
(
T
)]
2
.
[
α
q
(
T
)
A
q
(
T
)
±
B
q
(
T
)
α
q
(
T
)
−
(5
.
5
.
18)
This method was used by Jensen
et al.
(1975) for a comprehensive study
ot the two-ion anisotropy in Tb. The values of
A
q
(
T
)and
B
q
(
T
), de-
duced from eqn (5.5.18), were parametrized in various ways, and clearly
the best least-squares fit was obtained with expressions of the form
(
A
q
+
B
q
)
−
(
A
0
+
B
0
)=
I
(
q
)+
K
(
q
)
−C
(
q
)cos6
φ
(5
.
5
.
19)
(
A
q
−
B
q
)
−
(
A
0
−
B
0
)=
I
(
q
)
−K
(
q
)
−D
(
q
)cos6
φ,
where
A
0
±
B
0
were taken from the simultaneous measurements of
the magnetic anisotropy at
q
=
0
, discussed in the previous section.
The low-temperature isotropic coupling
I
(
q
),whichintheabsenceof
anisotropy would just be
J
[
J
(
0
)
−J
(
q
)], and the
φ
-independent two-
ion anisotropy
K
(
q
)areshowninFig.5.10. The
φ
-dependent axial
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