MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 124

Such is the complexity of the observed neutron diffraction patterns, how-
ever, that it is only recently that a reasonably complete delineation of
the ordered moments has been attained (Zochowski et al. 1991). At the
Neel temperature of 19.9 K, a weakly first-order transition leads to a
longitudinal-wave structure propagating in a b -direction on the hexag-
onal sites of the dhcp structure, with an incommensurable periodicity
given by Q h =0 . 13 b 1 . The moments on neighbouring hexagonal lay-
ers are ordered antiferromagnetically. Simultaneously, a c -axis moment
(plus a small component in the basal plane) with the same Q is induced
on the cubic sites by the anisotropic two-ion coupling. The moments on
neighbouring cubic layers are also ordered antiferromagnetically. As the
temperature is further lowered, another first-order transition at 19.2 K
establishes a double- Q structure, with wave-vectors Q 1 and Q 2 aligned
approximately along a pair of b -axes but canted slightly, so that the
angle between them is somewhat less than 120 ◦ . The polarization vec-
tors of the moments in the two waves are also canted away from the
corresponding b -axes and towards each other, but by a different amount
from the wave-vectors, so that the waves are no longer purely longitudi-
nal. Compared with the single- Q structure, this arrangement increases
the average ordered moment, which is further augmented, as the tem-
perature is lowered, by a squaring-up of the structure, which generates
harmonics in the neutron-diffraction pattern. Simultaneously, the period
gradually increases. At 8.2 K, the planar components of the moments on
the cubic sites begin to order, and after undergoing a number of phase
transitions, the structure at low temperatures is characterized by the
four Q -vectors illustrated in Fig. 2.8. Although all four periodicities are
present on each type of site, Q 1 and Q 2 , which are now aligned pre-
cisely along b -axes, but have different magnitudes 0.106 b 1 and 0.116 b 1 ,
generate the dominant structures on the hexagonal sites, while Q 3 and
Q 4 , which have lengths 0.181 b 1 and 0.184 b 1 and are canted towards each
other, predominate on the cubic sites. The different types of Q -vector
are interrelated; within the experimental uncertainty Q 3 + Q 4 =2 Q 1 ,
and the canting of Q 3 and Q 4 is related to the difference in length

between Q 1 and Q 2 .
The explanation of these structures from first principles in terms
of the elementary magnetic interactions is clearly a formidable task
but, as we have seen in Section 2.1.6, the ordering on the hexagonal
sites at high temperatures can be satisfactorily accounted for by a phe-
nomenological Landau expansion of the free energy in terms of the or-
der parameters, and the role of the different interactions thereby clar-
ified. The anisotropic two-ion coupling between the dipoles confines
the moments to the basal plane and tends to favour the longitudinal-
wave structure.

Two-ion coupling between the quadrupoles, proba-