MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 85

mQ c = p (4 π/c ) and additional umklapp terms contribute to the free
energy. Again these contributions depend on the absolute phase ϕ ,and
there will always be values of Q c close to Q leading to a lower free energy
than that obtained in the incommensurable case. In the low-temperature
limit, the modulation of the c -axis moment is therefore locked to the
lattice periodicity. This tendency is already apparent close to T N .In
the expansion of the free energy considered above for m =4,umklapp
terms modify the fourth-power coecient, and analogous effects occur in
higher powers of the magnetization. This indicates that the system may
stay commensurable even near T N although, in the close neighbourhood
of T N , the critical fluctuations neglected here may oppose this tendency.
The optimal value of Q c may change as a function of temperature, in
which case the system will exhibit a number of first-order, or possibly
continuous, transitions from one commensurable structure to another.
Of these structures, those for which Q c =3 Q c =5 Q c =

, i.e. Q c =0
or 2 π/c , are particularly stable, as they only involve one wave-vector,
so that f (0) =

···

1

2 J 2

H cf − J ( Q c ) (in this connection, we note that

1+ 9 +

1
25 +

= π 2 / 8). The anisotropic Ising-model with competing
interactions, the so-called ANNNI model, is a simplified version of the
above, and it shows a rich variety of different incommensurable, com-
mensurable, and chaotic ordered structures as a function of temperature
and the coupling parameters (Bak 1982).

···

2.1.5 Competing interactions and structures
The complex behaviour of the longitudinally ordered phase is a conse-
quence of the competition between the single-ion part of the free energy,
which favours a structure in which the magnitude of the moments varies
as little as possible, particularly at low temperature, and the two-ion
contributions, which prefer a single- Q ordering. When B 2 is positive,
helical ordering satisfies both tendencies without conflict. This points
to another alternative which the longitudinal system may choose. Al-
though χ ζζ ( Q ) decreases below T N , the two perpendicular components
continue to increase, and they may therefore diverge at a lower temper-
ature T N . Assuming the expansion (2.1.24) of the free energy still to be
valid at T N , and neglecting the third and higher harmonics of J iζ ,we

may write it:

J 2
α = ξ,η

2 A ξ −J

} σ α

f = f ( σ Q )+ 4

( Q )+ B ξζ ( Jσ Q ) 2

{ 2+cos2( ϕ α − ϕ )

J 4 B ξξ 3 σ ξ +3 σ η +2 σ ξ σ η . + 8

{ 2+cos2( ϕ ξ − ϕ η ) } (2 . 1 . 37)

± 2 The effective coecient of σ α ( α = ξ or η ) is smallest when ϕ α = ϕ

,
meaning that the basal-plane moments appearing just below T N ,where