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with
c
ε
4
Jσ
H
ε
,
Λ
ε
=
(5
.
4
.
20)
where Λ
ε
in (5.4.19) just cancels the
ε
-contribution to
F
θθ
/
(
NJσ
)de-
termined from eqn (2.2.34).
The dependence of the magnon energy gap in Tb on magnetic field
and temperature has been studied in great detail by Houmann
et al.
(1975a). They expressed the axial- and hexagonal-anisotropy energies
of eqn (5.2.44) in the form
A
0
(
T
)
±
B
0
(
T
)=
P
0
(
±
−
P
6
(
±
)cos6
φ
+
gµ
B
H
cos (
φ
−
φ
H
)(5
.
4
.
21)
)
and, by a least-squares fitting of their results, some of which are shown
in Fig. 5.4, they were able to deduce the values of the four parameters
P
0
,
6
(
), shown as a function of magnetization in Fig. 5.5. According to
eqns (5.3.22) and (5.4.9), these parameters are given at low temperatures
by:
±
P
0
(+) =
6
B
2
J
(2)
60
B
4
J
(4)
+ 210
B
6
J
(6)
+
c
γ
(2
C
2
+
A
2
)
/J
(
a
)
P
6
(+) =
6
B
6
J
(6)
+3
c
γ
CA
/J
−
(
b
)
)=4
c
γ
C
2
+
A
2
/J
P
0
(
−
(
c
)
)=
36
B
6
J
(6)
+10
c
γ
CA
/J,
(
d
)
(5
.
4
.
22)
where, for convenience, we have set the renormalization parameters
σ
and
η
±
to unity. These expressions for the parameters
P
0
,
6
(
P
6
(
−
)arede-
rived from a particular model. In general, additional contributions may
appear due to other magnetoelastic interactions, and to anisotropic two-
ion couplings. Nevertheless, within the RPA, the relations between the
spin-wave energy parameters
A
0
(
T
)
±
± B
0
(
T
) and the bulk anisotropy
parameters, (5.4.12) and (5.4.19) combined with (5.3.7), should still be
valid. The values of the anisotropy parameters, and their temperature
dependences, determine the static magnetic and magnetoelastic proper-
ties, and can thus be obtained from bulk measurements on single crys-
tals. A comparison between such static parameters and the dynamic val-
ues
P
0
,
6
(
), derived from the field dependence of the spin-wave energy
gap, can therefore elucidate the extent to which the spin-wave theory of
the anisotropic ferromagnet is complete and correct.
Such a comparison has been made by Houmann
et al.
(1975a). The
axial-anisotropy parameter
P
0
(+)+
P
6
(+), when the moments are along
the easy axis, agrees with the values deduced from torque and mag-
netization experiments, to within the rather large uncertainties of the
±
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