MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 81

which is then minimized with respect to q when q λ = Q ,atwhich

wave-vector ( q ) has its maximum. Hence the two-ion energy attains its

minimum when only the two Fourier components

J

J i ( ± Q ) are non-zero.

The stronger constraint that should be constant is then met only by

the helix (2.1.26). In the zero-temperature limit, this constraint derives
from the fact that the moments attain their saturation value,

| J i |

= J ,
immediately the exchange field is not identically zero, since χ αα ( σ =0)
diverges in this limit when

| J i |

H cf = 0. At elevated temperatures, it is
clear that the sum of the single-ion terms in the free energy (the A -
and B -terms in (2.1.22)) is most effectively minimized if the minimum
condition is the same for all the ions. H cf =0,thereareno

restrictions on the plane in which the moments spiral; it may be rotated
freely, without change in energy, as long as

When

is constant and all the
components vary with the wave-vector Q . This behaviour is analogous
to that of the Heisenberg ferromagnet, which may be considered as a
helically ordered system with Q = 0 .If Q is not perpendicular to the
plane in which the moments lie, the structure is called the tilted helix
(Elliott 1971; Sherrington 1972) and the extreme case, with Q in the
plane of the moments, is the cycloidal structure .When B 2 > 0, the
orientation of the plane is stabilized to be perpendicular to the c -axis,
and with Q along this axis we obtain the true helical structure.
If B 2 > 0 is the only crystal-field parameter of importance, the
regular helix is the stable structure in the whole temperature interval
between zero and T N . If the Landau expansion (2.1.22) is continued to
the sixth power in the magnetization, a term appears proportional to
B 6 , distinguishing between the a -and b -directions in the basal-plane.
Instead of using this expansion, we shall consider the alternative expres-
sion for the free energy, to leading order in B 6 ,
F

| J i |

2

+
i

1

B 6 O 6 ( J i ) F 1 −

ij J

( ij )

J i · J j

(2 . 1 . 28)

2
ij

φ j )+
i

1

( Jσ ) 2 κ 6 cos 6 φ i , = F 1 − J ( ij )cos( φ i −

where J i = Jσ (cos φ i , sin φ i , 0) and F 1 is the part independent of φ i .
The expectation values are those obtained in the limit B 6 = 0, i.e. σ
and κ 6 are assumed to be independent of the angle φ i . The presence
of the six-fold anisotropy term distorts the helix. In order to solve the
equilibrium equation
∂F/∂φ i =( Jσ ) 2
j

6 κ 6 sin 6 φ i =0 ,

J ( ij )sin( φ i − φ j ) −

we introduce the expansion
φ i = u i + γ sin 6 u i + ··· ; u i = Q · R i , (2 . 1 . 29 a )