Environmental Engineering Reference
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presence of moderate anisotropy. However, it is necessary to be aware
that the renormalization itself may cause special effects not expected
in the harmonic approximation, as the amount of renormalization may
change when the system is perturbed by an external magnetic field or
pressure, or when the temperature is altered.
There have been attempts (Lindg ard 1978, and references therein)
to construct an analytical spin-wave theory starting with a diagonaliza-
tion of the MF Hamiltonian. In principle, this should be an appropriate
starting-point, since the ground state is closer to the MF ground-state
than to the fully polarized state, as soon as the planar anisotropy be-
comes significant. As in the model calculations discussed above, the MF
Hamiltonian can be diagonalized numerically without diculty, but in
this form the method is non-analytical and the results are not easily
interpretable. In order to diagonalize the MF Hamiltonian analytically,
one is forced to make a perturbative expansion, unless
J
is small. If
the MF Hamiltonian is expressed in the
|J
z
>
-basis, the natural ex-
pansion parameter is
at
T
= 0. The use of this
expansion parameter and the 1
/J
-expansion considered above lead to
identical results in the limit 2
J
∼|
B
q
o
/A
q
o
|
2
J
|
b
o
|
1 (Rastelli and Lindg ard 1979).
However, the expansion parameter is not small when the anisotropy is
moderately large (2
J
|
b
o
|
0
.
35 in Tb at
T
= 0), which severely limits
the usefulness of this procedure as applied by Lindg ard (1978, 1988)
to the analysis of the spin waves in the anisotropic heavy rare earths.
It gives rise to a strong renormalization of all the leading-order spin-
wave-energy parameters, which are thus quite sensitive, for example,
to an external magnetic field, and it is extremely dicult to obtain a
reasonable estimate of the degree of renormalization. In contrast, the
1
/J
-expansion leads, at low temperatures, to results in which only the
high-rank terms (which are quite generally of smaller magnitude than
the low-rank terms) are renormalized appreciably, and the amount of
renormalization can be determined with fair accuracy. In the numerical
example corresponding to Tb, the
B
6
-term is renormalized by
|
b
o
|
38% at
T
= 0, according to the spin-wave theory, which agrees with the value
obtained by diagonalizing the MF Hamiltonian exactly, as indicated in
Fig. 2.3.
To recapitulate, we have developed a self-consistent RPA theory for
the elementary excitations in a ferromagnet, i.e. the spin waves, valid
when the magnetization is close to its saturation value. The major com-
plication is the occurrence of anisotropic single-ion interactions, which
were treated by performing a systematic expansion in 1
/J
.Tofi t
order in 1
/J
, the theory is transparent and simple, and it is straightfor-
wardly generalized to different physical situations. Much of the simplic-
ity is retained in second order, as long as the magnetization is along a
−
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