MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 87

the most important axial-anisotropy parameter in the low-temperature
limit, the helix is still a stable structure at T = 0 whereas, in the lon-
gitudinally polarized case, the tendency to minimize the variation of
the lengths of the moments may result in two different paths. Either
the system stays in the longitudinally polarized phase, ending up as a
(commensurable) square-wave structure at T = 0, or it goes through
a transition to an elliptic cycloidal structure. The path which is cho-
sen depends on the magnitude of B 2 ; if the effective axial anisotropy
− B 2 O 2 is suciently large, the ordering of the basal-plane moments

is quenched. It has already been mentioned in Section 1.5 that this
anisotropy depends on the magnetization, being proportional approxi-
mately to σ 3 . We shall discuss this renormalization in more detail in
the next section, but it is worth mentioning here that this behaviour
of the effective anisotropy-parameter means that there is an intermedi-
ate range of B 2 for which the system makes a transition to the elliptic
cycloidal structure, but leaves it again at a lower temperature, by re-
turning to the longitudinally polarized phase when B 2 O 2 becomes

large enough. When B 4 and B 6 are included, a more realistic situation
may occur, in which the low-temperature anisotropy favours an orienta-
tion of the moments making an angle θ with the c -axis, which is neither
0or π/ 2 but some, temperature-dependent, intermediate value. In the
case of the helix, this means that there will be a critical temperature
T N (below T N ) where the effective axial anisotropy parameter vanishes,
andbelowwhichthe c -axis moments are ordered. If the ordering wave-
vector for the c -axis component is the same as the helical wave-vector,
the structure adopted is the tilted helix. However the two-ion coupling
between the c -axis moments,

−

c -axis, is not restricted
by any symmetry argument to be equal to the coupling between the
basal-plane moments,

J ( q )with q

J ⊥ ( q )= J ( q )withitsmaximumat q = Q 0 .

If the maximum of = Q 0 ,the c -component will or-

der at this wave-vector and not at Q 0 , as the extra energy gained by
the c -component by locking to the basal-plane moments is very small,
being proportional to {B 6 O 6 / ( Jσ ) 2

J ( q ) lies at a q

2 .When B 2 is negative, a
non-zero value of θ favours the elliptic cycloidal structure, compared to
the longitudinally polarized phase. If the system is already in the cy-
cloidal phase, it may undergo a new second-order transition, in which
the plane of the ellipse starts to tilt away from the ξ - ζ plane, in close
correspondence with the behaviour of the helix. Referring back to eqn
(2.1.37), we observe that this transition occurs when the coecient of
σ η ,with ϕ η = ϕ (+ π )= ϕ ξ ±

J ( Q ) }

π/ 2, becomes zero. The phase-locking en-
ergy, comprising the terms in (2.1.37) involving ϕ η , is more important in
this case than in the helix, but it is nevertheless possible that the third
component may order at a wave-vector different from that of the other