MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 93

The explicitly sixth-order contribution to the free energy, propor-

tional to (1 /N ) i (

) 3 , is somewhat smaller than the estimated
value of ∆ f , and it leads to energy differences between the different
multiple- Q structures which are a further order of magnitude smaller.
The hexagonal-anisotropy term, which also appears in this order, is
minute compared to the anisotropy introduced by

J i · J i

( Q )inPrandNd,
and its influence on the turn angles ψ and θ should be negligible. The
only other new effect in this order is the appearance of higher harmon-
ics. The mechanism is identical to that discussed in Section 2.1.4 for
the longitudinally-polarized phase, but in addition to the occurrence of
third harmonics at the wave-vectors 3 Q p ,equivalentlyto(2 . 1 . 35 a ), they
also appear at all possible combinations of 2 Q p ± Q p

K

= p )inthe
multiple- Q structures. In the triple- Q structure, one might expect third
harmonics also at Q 1 ± Q 2 ± Q 3 , but the new wave-vectors derived from
this condition are either 0 , which changes the symmetry class of the
system, or twice one of the fundamental wave-vectors, which are ener-
getically unfavourable because they do not contribute to the 'squaring
up'. These extra possibilities in the triple- Q case are not therefore real-
ized. The appearance of the higher 'odd' harmonics is not important for
the energy differences between the different multiple- Q structures, but
they may provide an experimental method for differentiating between
the various possibilities (Forgan et al. 1989). In a neutron-diffraction
experiment, the scattering intensity at the fundamental wave-vectors in
a multi-domain single- Q structure, with an equal distribution of the
domains, is the same as that produced by a triple- Q structure. These
structures may then be distinguished either by removing some of the
domains by applying an external field, or by using scattering peaks at,
for instance, 2 Q 1 ± Q 2 to exclude the possibility of a single- Q structure.
The discussion of this section has been based exclusively on the MF
approximation, which neglects the important dynamical feature that a
system close to a second-order phase-transition will show strong corre-
lated fluctuations in the components which order at the transition. A
discussion of the effects of the critical fluctuations is beyond the scope
of this topic, and we refer instead to the recent introduction to the field
by Collins (1989), in which references may be found to the copious lit-
erature on the subject. Although the MF approximation does not take
into account the contributions to the free energy from the critical fluctua-
tions, it gives a reasonable estimate of the transition temperatures in the
rare earth metals, which can all be characterized as three-dimensional
systems with long-range interactions. The fluctuations contribute to
the free energy on both sides of the transition, and they only suppress
the transition temperature by a few per cent in such systems. The
Landau expansion considered above does not predict the right critical

( p