CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 369

the uniaxial pressure. The gapless excitation-spectrum in the ordered
phase implies that the linewidth effects are predicted to vary smoothly
with temperature, and to stay non-zero at T = 0, when the second-
order contributions are included. We note that the imaginary part of
the self-energy is now non-zero below the RPA-excitonic band in the
paramagnetic phase, and that it generates an appreciable low-energy
scattering at the ordering wave-vector, just above T N , changing the in-
elastic critical excitation into a diffusive mode of diverging intensity.
Hence a true 'soft-mode transition', as found in the zeroth or first order
of 1 /Z , is no longer predicted, but the low-energy effect is far too weak to
account for the observed behaviour of the neutron-diffraction satellite.
The inclusion of the second-order effects in the theory clearly improves
the agreement with the experimental results. However, even though the
1 /Z 2 -theory predicts a non-zero linewidth in the limit T 0, the effect

is so small, at energies below 1 meV, that it can be neglected in com-
parison with the contribution due to the scattering against electron-hole
pair excitations of the conduction electrons, discussed in Section 7.3.2.
The importance of this mechanism has been estimated reasonably accu-
rately, and it leads to a linewidth of the order of 0.15 meV for the optical
modes close to Q . When all contributions are included, the theory in-
dicates that the amplitude mode should have been observable at q = Q
at the lowest temperatures, in contrast to the experimental results, but
otherwise its predictions are found to agree well with the main features
of the observations.

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