Environmental Engineering Reference
In-Depth Information
of crystal-field systems. We begin by using the RPA to analyse a num-
ber of model systems which, though oversimplified, contain much of the
essential physics of the magnetic excitations, sometimes known as
mag-
netic excitons
, observed on both the hexagonal and the cubic sites in Pr.
In the following section, it is shown how effects neglected in the RPA
modify the
energies
and
lifetimes
of these excitations. The perturba-
tions of the crystal-field system by the
lattice
,the
conduction electrons
,
and the
nuclei
are then considered. This discussion is largely parallel to
that of spin-wave systems in Chapter 5; the magnetoelastic interactions
couple the phonons to the magnetic excitations and modify the elastic
constants, and the conduction electrons limit the lifetimes of the excita-
tions, especially at small
q
, while themselves experiencing a substantial
increase in effective mass. The major effect of the
hyperfine interaction
has no counterpart in spin-wave systems, however, since it is able to
induce
collective electronic-nuclear ordering
at low temperatures, and
hence affect all magnetic properties drastically. Because the hexagonal
sites in Pr constitute an
almost-critical system
, relatively small pertur-
bations are able to drive it into a magnetically-ordered state. The effect
of the
internal
interactions with the nuclei and magnetic impurities, and
external
perturbations by uniaxial stress or a magnetic field, are consid-
ered. Finally, we discuss a number of specific aspects of the magnetic
excitations in Pr, in the paramagnetic and ordered phases.
7.1 MF-RPA theory of simple model systems
The general procedure for calculating the RPA susceptibility was out-
lined in Section 3.5. If we consider the Hamiltonian
=
2
ij
1
H
i
H
J
(
J
i
)
−
J
i
·J
(
ij
)
·
J
j
,
(7
.
1
.
1)
which includes a general two-ion coupling between the dipolar moments,
and assume the system to be in the paramagnetic state, we find the RPA
susceptibility to be
χ
(
q
,ω
)=
1
(
q
)
−
1
χ
o
(
ω
)
,
χ
o
(
ω
)
−
J
(7
.
1
.
2)
which is a simple generalization of eqn (3.5.8), as in (6.1.7). The essence
of the problem therefore lies in the calculation of the non-interacting sus-
ceptibility
χ
o
(
ω
), as determined by the single-ion Hamiltonian
H
J
(
J
i
).
In the case of a many-level system, where
J
is large, this normally re-
quires the assistance of a computer. Analytical expressions for
χ
(
q
,ω
)
may, however, be obtained for systems where the number of crystal-field
levels is small, i.e. between 2-4 states corresponding to
J
=
2
,1,or
2
.
Such small values of
J
are rare, but the analysis of these models is also
1
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