SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 206

Fig. 5.3. The temperature dependence of the dispersion relations for
the unperturbed spin waves in the c -direction in Tb. Both the energy gap
and the q -dependence renormalize with temperature. The results have
been corrected for the magnon-phonon interaction, and the lines show the
calculated energies.

the magnetoelastic coupling introduces qualitatively new effects, not de-
scribable by eqn (5.3.22), to which we will return after a short digression
to summarize our understanding of the spin-wave theory.

5.3.2 The validity of the spin-wave theory
In presenting the spin-wave theory, we have neglected phenomena which
first appear in the third order of 1 /J , most importantly the finite life-
times of the excitations. In the presence of anisotropy, when B is dif-
ferent from zero, the total moment is not a conserved quantity, since

[ i J iz ,

H ] = 0, unlike in the Heisenberg model. On the microscopic