SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 287

numbers, these structures are most naturally classified as incommen-
surable, particularly as the distinction is immaterial in this case. If
the hexagonal anisotropy and possible external fields are neglected, the
translational symmetry is broken only formally, as a rigid rotation of
the moments, or of the total system, around the spiral axis costs no
energy. We then consider the longitudinally polarized phase, in which
genuine effects due to incommensurability would be expected. On the
other hand, the stronger coupling between the two periodicities increases
the tendency of the magnetic-ordering wave-vector to lock into a value
which is commensurable with the lattice. It may perhaps be questioned
whether theoretical results derived for ideal incommensurable models
are relevant to real, three-dimensional systems. However, it appears
that the essential features of systems which are classified experimentally
as incommensurable may be described theoretically as such, provided
that the analysis includes an averaging or coarse graining of the results,
of a magnitude somewhat smaller than the experimental resolution.

6.1.1 The helix and the cone
A helical ordering of the moments in an hcp lattice, with a wave-vector
Q along the c -or ζ -axis, is described by the following equations:

J iξ = J ⊥ cos ( Q · R i + ϕ )

J iη = J ⊥ sin ( Q · R i + ϕ ) (6 . 1 . 1)

J iζ =0 .

As usual, we shall be most interested in excitations propagating in the
c -direction, and hence may use the double-zone representation, corre-
sponding to the case of a Bravais lattice.

The moments of constant

length lie in the ξ - η plane perpendicular to Q , and rotate uni-

formly in a right-handed screw along the Q -vector. The elastic cross-
section corresponding to this structure is, according to (4.2.6),

J ⊥

d Ω = N hγe 2

2

dσ

| 2 e − 2 W ( κ )

2 2 (1 + κ ζ )

gF ( κ ) | J ⊥

mc 2

(2 π ) 3
υ

1
4 {

× δ ( τ + Q − κ )+ δ ( τ − Q − κ ) } . (6 . 1 . 2)

τ

In this system the molecular field in (3.5.3) changes from site to site,
as does the MF susceptibility χ o

i ( ω ) in (3.5.7). This complication may
be alleviated by transforming into a rotating ( xyz )-coordinate system
with the z -axis parallel to the moments, i.e.
J iξ = J iz cos φ i + J iy sin φ i
J iη = J iz sin φ i − J iy cos φ i (6 . 1 . 3)

J iζ = J ix ,