SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 224

in the basal plane. From (5 . 4 . 36 a ), we see that χ xx ( q → 0 , 0) does
not show an anomaly at the transition. The critical behaviour is con-
fined to the yy -component of the static susceptibility. At the transition,
A 0 ( T )

B 0 ( T )=Λ γ , according to eqn (5.4.15), and (5.4.40) then pre-
dicts a very rapid variation of χ yy ( q → 0 , 0) with the direction of q ,with
a divergent susceptibility in the long wavelength limit in the two cases
where q is along the z -orthe y -axis, both lying in the basal plane, paral-
lel or perpendicular to the magnetic moments. These divergences reflect
a softening of two modes in the system, the transverse phonons propa-
gating parallel to either of the two axes ( θ q =

−

π
2

and φ q = p 2 ), with

their polarization vectors in the basal plane. Equation (5.4.38) predicts
that the velocity of these modes is zero, or c 66 =0,at H = H c ,atwhich
field the dispersion is quadratic in q instead of being linear. The soften-
ing of these modes was clearly observed in the ultrasonic measurements
of Jensen and Palmer (1979). Although the ultrasonic velocity could not
be measured as a function of magnetic field all the way to H c , because
of the concomitant increase in the attenuation of the sound waves, the
mode with q parallel to the magnetization could be observed softening
according to (5 . 4 . 38 b ), until the elastic constant was roughly halved. On
the other hand, as discussed in the next section, the dipolar interaction
prevents the velocity of the mode in which the ionic motion is along the
magnetization from falling to zero, and (5 . 4 . 38 b ) is replaced by (5.5.13).
When they took this effect into account, Jensen and Palmer (1979) could
fit their results over a wide range of fields and temperatures with the
RPA theory, without adjustable parameters or corrections for critical
phenomena, using the bulk values of the three basal-plane anisotropy
parameters C , A ,and H c ,
The absence of such corrections may be explained by the behaviour
of the critical fluctuations , which is the same as that found in a pure
structural phase-transition in an orthorhombic crystal, where c 66 is again
the soft elastic constant (Cowley 1976; Folk et al. 1979). The strong
bounds set by the geometry on the soft modes in reciprocal space con-
strain the transition to exhibit mean-field behaviour. The marginal dim-
ensionality d ∗ , as estimated for example by Als-Nielsen and Birgeneau
(1977), using a real space version of the Ginzburg criterion, is d ∗ =2
in this kind of system. Whenever the dimensionality d of the system is
larger than d ∗ , as in this case, Wilson's renormalization group theory
predicts no corrections to Landau's mean-field theory. The transition
at H = H c is thus profoundly influenced by the magnetoelastic effects.
Without them, i.e. with C = A = 0, the spin-wave energy gap would van-
ish at the transition, and the critical fluctuations, the long-wavelength
magnons, would not be limited to certain directions in q -space. Under
such circumstances, the system would behave analogously to a three-