CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 339

Fig. 7.5. The field dependence of the elastic constant c 66 in Pr at
4 K, relative to the value at zero field. The elastic constant was deter-
mined from the velocity of the transverse sound waves propagating in an
a -direction, and the open and closed symbols indicate the experimental
results when the field was applied respectively in the a -ortheperpen-
dicular b -direction. The solid lines show the calculated field dependence.
the phonons are further discussed by Thalmeier and Fulde (1975), Fulde
(1979), and Aksenov et al. (1981).
The coupling (5.4.50), quadratic in the magnon operators, also has
its counterpart in crystal-field systems. Such interactions arise when, in-
stead of applying the RPA decoupling in the first step, as in eqn (7.3.4),
we proceed to the next step in the hierarchy of Green functions. The
most important effect of these terms is to replace the crystal-field param-
eters by effective values, which might be somewhat temperature depen-
dent, corresponding to an averaging of the effective crystalline field expe-
rienced by the 4 f electrons over the finite volume spanned by the thermal
vibration of the ions. As in the spin-wave case, these extra higher-order
contributions do not lead to the kind of hybridization effects produced
by the linear couplings. However, if the density of states of the phonons,
weighted with the amplitude of the coupling to the crystal-field exci-
tations, is particularly large at certain energies, resonance-like bound-
states due to the higher-order terms may be observed in the magnetic
spectrum. The dynamic Jahn-Teller effect observed in CeAl 2 (Loewen-
haupt et al. 1979) seems to be due to these higher-order effects, according
to the calculation of Thalmeier and Fulde (1982).
The expression (7.3.7) for the interaction of the crystal-field system
with the phonons has essentially the same form as that derived from any