Environmental Engineering Reference
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7.3 Perturbations of the crystal-field system
In this section, we shall discuss various effects of the surrounding medium
on a crystal-field system. The first subject to be considered is the mag-
netoelastic coupling to the lattice. Its contribution to the magnetic-
excitation energies may be described in terms of frequency-dependent,
anisotropic two-ion interactions, and we include a short account of the
general effect of such terms. We next consider the coupling to the con-
duction electrons, which is treated in a manner which is very parallel
to that used for spin-wave systems in Section 5.7. Finally, we discuss
the hyperfine interaction between the angular momenta and the nuclear
spins, which becomes important at the lowest temperatures, where it
may induce an ordering of the moments in an otherwise undercritical
singlet-ground-state system.
7.3.1 Magnetoelastic effects and two-ion anisotropy
The magnetoelastic interactions which, in the kind of system we are
considering, primarily originate in the variation of the crystal-field pa-
rameters with lattice strain, produce a number of observable phenomena.
The lattice parameters and the elastic constants depend on temperature
and magnetic field, the crystal-field excitation energies are modified, and
these excitations are coupled to the phonons. In addition, the magneto-
elastic coupling allows an externally applied uniaxial strain to modify
the crystal-field energies. All these magnetoelastic effects have their
parallel in the ferromagnetic system discussed in Section 5.4 and, in the
RPA, they may be derived by almost the same procedure as that pre-
sented there, provided that the spin-wave operators are replaced by the
standard-basis operators, introduced in eqn (3.5.11).
In the paramagnetic phase in zero external field, only those strains
which preserve the symmetry, i.e. the α -strains, may exhibit variations
with temperature due to the magnetic coupling. The lowering of the
symmetry by an applied external field may possibly introduce non-zero
strains, proportional to the field, which change the symmetry of the
lattice. In both circumstances, the equilibrium strains may be calculated
straightforwardly within the MF approximation. As an example, we
shall consider the lowest-order magnetoelastic γ -strain Hamiltonian
H γ =
i
2
B γ 2 O 2 ( J i ) γ 1 + O 2
( J i ) γ 2 ,
c γ ( γ 1 + γ 2 )
(7 . 3 . 1)
2
corresponding to eqn (5.4.1) with B γ 4 = 0. The equilibrium strain γ 1 ,
for instance, is determined in the presence of an external magnetic field
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