CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 341

by the nuclear scattering of the neutrons. In a paramagnetic system in
zero field, the p p susceptibility-matrix partitions into two indepen-

dent blocks, at zero frequency, the one depending only on the even-rank
couplings and the other only on the odd-rank couplings. If one of the
two parts of χ ( q , 0) diverges at some temperature T ∗ , it signals the oc-
currence of a second-order phase transition at this temperature. If it
is the block determined by the even-rank couplings which diverges, the
order parameter below T ∗ is associated with the quadrupole moments,
assuming the lowest-rank terms to be dominant. If there is any coupling
between this order parameter and one of the phonon modes, the transi-
tion is accompanied by a softening of these phonons, provided that the
pure quadrupolar excitations have higher energies than the phonons at
the ordering wave-vector. If this vector Q is zero, the corresponding
elastic constant vanishes at the transition. In the case where Q

×

= 0 ,the
situation corresponds to that considered in the magnetic case, and the
phonon mode shows soft-mode behaviour according as there are pure
elastic contributions to the (RPA) susceptibility or not. A quadrupolar
phase-transition involving the phonons is usually referred to as being
induced by the Jahn-Teller effect, and a more detailed discussion and
relevant examples may be found in, for instance, Elliott et al. (1972).
The presence of a non-zero quadrupole moment does not destroy the
time-reversal symmetry, and an ordering of the dipole moments may
follow only after an additional phase transition. In TmZn (Morin et al.
1980) an ordering of the quadrupole moments occurs below a first-order
transition at T Q =8 . 6 K, and this phase is disrupted by the onset of
ferromagnetic ordering at T C =8 . 1 K. In the opposite case of order-
ing of the dipole moments, the breaking of the time-reversal symmetry
allows a direct coupling between the dipole and quadrupole moments,
so that the latter are forced to order together with the dipoles, giving
rise to, for example, crystal-field-induced magnetostriction effects, and
the dipolar ordering will normally quench any tendency toward a purely
quadrupolar-ordered phase.
In this chapter, we have formulated the various RPA results in terms
of the generalized-susceptibility matrices. The results apply in param-
agnetic as well as in ordered systems, so long as the order parameter
is uniform throughout the crystal. They agree with the more explicit
results derived previously in the case of a weakly-anisotropic ferromag-
netic system. In a paramagnet or a strongly-anisotropic ferromagnet,
the results above may also be given a more transparent and explicit
form, but only if the number (2 J + 1) of different angular-momentum
states can be taken as small; else the matrix-equations themselves are
well-suited for solution by numerical methods. The reduction of the
matrix-equations in, for instance, the ( J = 1)-case is straightforward