MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 91

where γ = 0, 1/4, or 1/3 respectively, in a single-, double-, or triple-
Q structure. If only an isotropic two-ion coupling and the crystal-field
terms are included, 2 u = v or w = 0, and the different multiple- Q
structures are degenerate to the fourth power of the order parameter.
This situation is not changed by the anisotropic dipole coupling

K ( q )

introduced above (as long as σ p is parallel to Q p ). However, two-ion

quadrupole couplings may remove the degeneracy. For example, the

coupling K 2 ( ij ) J i + J j− makes a contribution proportional to

w ∼ 3 K 2 ( 0 )+ K 2 (2 Q ) − 2 K 2 ( Q ) − 2 K 2 ( Q 1 − Q 2 ) . (2 . 1 . 46)

Depending on the detailed q -dependence of this coupling, it may lead
to a positive or a negative contribution to w .If w is positive, the single-
Q structure is stable, and conversely a negative w leadstoatriple- Q
structure just below T N . The Landau expansion for this case has been
discussed by Forgan (1982), Walker and McEwen (1983) and McEwen
and Walker (1986), who all take the possible contributions to w as being
of magnetoelastic origin. In Pr, the dominating magnetoelastic interac-
tion is known to be due to the γ -strain coupling, and a rough estimate
(including both the uniform and modulated γ -strain) indicates that v
is unaffected, whereas the reduction of u proportional to B γ 2 /c γ ,with
the parameters of (1.5.27), is about 10%, corresponding to a positive
contribution to w of about 0 . 2 u , or to an energy difference between the
single- and double- Q structures of 0 . 05 uσ 4 . If the other quadrupolar

contributions are unimportant, as is indicated by the behaviour of the
excitations in Pr (Houmann et al. 1979), we should expect the single- Q
structure to be favoured in Pr and Nd, at least close to T N .
If w is relatively small, the single- or triple- Q structures may only be
stable in a narrow temperature range below T N , because the sixth-order
contributions may assume a decisive influence. A number of new effects
appear in this order, but the most important stems from the possibility
that the moments and the wave-vectors may rotate away from the b -
directions, as first considered by Forgan (1982). The (

∼

σ p · σ p ) 2 -term in
(2.1.43) may drive such a rotation, because it favours an orthogonal con-
figuration of the different σ p vectors, since B is positive. This term does

not appear in the single- Q structure, whereas in the triple- Q case, f 4 ( σ p )
is reduced quadratically with θ p ,where θ p is the angle between J p and
the nearest b -direction. However, the much larger quadratic increase of
f 2 ( σ p ), due to

( Q ), will eliminate any tendency for θ p to become non-
zero. In contrast, f 4 ( σ p ) depends linearly on θ p in the double- Q struc-
ture, and the free energy can always be reduced by allowing the two com-
ponents

K

σ 1 and σ 2 (with σ 3 = 0 ) to rotate towards each other. Defining

J 6 ( Q )equivalentlyto J 6 ( Q )cos6 ψ Q ,and

using the constraint that the change of ψ Q for the p th component must

K 6 ( Q ), i.e. J ( Q )= J 0 ( Q )+