MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 88

two. If the η -component is locked at the same wave-vector as the two
other components, and if the ellipse is tilted just such an amount that
σ η = σ ξ , the structure is a helix superimposed on a modulated c -axis
moment. If a transition to the tilted cycloidal structure has occurred,
and the hexagonal anisotropy is small, it might be favourable for the
system at a lower temperature to pass directly, via a first-order transi-
tion, to this helical structure in which the c -axis component is no longer
phase-locked to the basal-plane moments.
Instead of basing our analysis on the Hamiltonian (2.1.1), we may
use symmetry arguments for deriving the most general behaviour of the
magnetic ordering in hcp crystals. We have already indicated that J ( q )

may differ from ( q ) and mentioned some of the consequences. The

assumption that the c -axis is effectively a six-fold axis of the lattice leads
to the strong restriction that the expansion of the free energy, (2.1.22) or
(2.1.24), only involves even powers of each of the Cartesian components,
when q is along this axis. This has the consequence, for example, that
all the main transitions, at T N or T N , are predicted to be of second
order, excluding those involving changes of the same component, i.e.
transitions between different commensurable structures. However, there
are two-ion terms which reflect the fact that the c -axisisonlyathree-fold
axis. The term of lowest rank has the form

J ⊥

1) s K 3 ( J iζ − 1 ) O − 3

3

( J s− 1 ) O − 3

3

H 3 ( i ∈ s 'th plane) = ( −

2

J iζ ( J s +1 ) −

J s− 1 ,ζ ,

) J s +1 ,ζ −

− 2

+( O − 3
3

O − 3
3

(2 . 1 . 39)

( J i ) ( J i )

in the MF approximation, where only interactions between neighbouring
planes are included. O − 3
3

=( J + −

J 3
−

) / 2 i ,and J s± 1 denotes a moment

in the ( s

±

1)th plane. The contribution of this coupling to the expansion

(2.1.22) of the free energy to the fourth power is found by adding i H 3
to F , using the approximation O − 3

3

( J i ) ∝ J iη (3 J iξ

2

− J iη

2 )=

3 sin 3 φ i . One remarkable effect is that this coupling introduces a
term linear in

J ⊥

in the helix. If the basal-plane moments are ordered
with the wave-vector Q , they induce a c -axis moment modulated with
a wave-vector along the c -axis of length 2 π/c

J iζ

3 Q , provided that 6 Q
is not a reciprocal lattice vector. In the elliptic cycloidal structure, this
coupling induces an ordering of the η -component at the two wave-vectors
of length 2 π/c

−

3 Q , when the ellipse is assumed to lie in
the ξ - ζ plane and only the fundamental at Q is considered. Although
this additional coupling may not change the nature of the transitions at
T N or T N , it has qualitative consequences for the magnetic structures,
and it may introduce new effects associated with commensurability. For
instance, the three-fold symmetrical interaction will favour the commen-
surable structure with Q = π/ 2 c (an average turn angle of 45 ◦ ). In the

− Q and 2 π/c −