MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 89

case of a helix with this particular period, the coupling induces a modu-
lation of the c -axis moments with the same wave-vector, 2 π/c − 3 Q = Q ,

causing a tilting of the plane of the helix.

2.1.6 Multiply periodic structures
We have so far only considered order parameters which are specified by
two Q -vectors ( ± Q ), or one Q plus a phase. This is a consequence of

the assumption that Q is along the c -axis. If Q is in the basal-plane,
as in the light rare earths Pr and Nd, there are six equivalent ordering
wave-vectors,

± Q 3 , where the three vectors make an
angle of 120 ◦ with each other. This leads to the possibility that the
ordered structure is a multiple - Q structure ,where

± Q 1 , ± Q 2 ,and

J i = J 1 cos ( Q 1 · R i + ϕ 1 )+ J 2 cos ( Q 2 · R i + ϕ 2 )+ J 3 cos ( Q 3 · R i + ϕ 3 )

(2 . 1 . 40)
referred to as single-, double-, or triple- Q ordering, depending on the
number of vectors J p which are non-zero. The transition at T N will
generally involve only a single real vector J p for each Q p , as implic-
itly assumed in (2.1.40). We will not therefore consider multiple- Q cy-
cloidal/helical structures, but restrict the discussion to configurations
which correspond to the type observed in Pr or Nd. We furthermore
neglect the complications due to the occurrence of different sublattices
in the dhcp crystals, by assuming the lattice to be primitive hexagonal.
This simplification does not affect the description of the main features
of the magnetic structures. On the hexagonal sites of Pr and Nd, the
ordered moments below T N lie in the basal plane. This confinement is
not primarily determined by the sign of B 2 , but is decisively influenced
by the anisotropic two-ion coupling

2

( ij ) ( J iξ J jξ − J iη J jη )cos2 φ ij +( J iξ J jη + J iη J jξ )sin2 φ ij ,

1

H an =

ij K

(2 . 1 . 41)
where φ ij is the angle between the ξ -axis and the projection of R i − R j
on the basal plane. This anisotropic coupling, which includes a minor
contribution from the classical dipole-dipole interaction, is known from
the excitation spectrum to be of the same order of magnitude as the
isotropic coupling in Pr, as we shall discuss in Chapter 7, and must
be of comparable importance in Nd. We define the coupling parameter
K ( q )= K 0 ( q )+ K 6 ( q )cos6 ψ q ,where ψ q is the angle between q (in the

basal plane) and the ξ -axis, and K 0 ( q ) ±K 6 ( q ) is the Fourier transform

of ( ij )cos2 φ ij when q is respectively parallel or perpendicular to

the ξ -axis. Introducing J p

±K

σ p , and assuming the moments to be
perpendicular to the c -axis, we find the mean-field free energy of second

= J