MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 110

of its consequences is that the γ -strain contributions (2.2.27) to the free
energy cancel out in the helical phase. This behaviour of the γ -strains
therefore enhances the tendency of the wave-vector of the helix to jump
to one of the two commensurable values Q =0or2 π/c ,ormayin-
crease the stability of other commensurable structures which have a net
moment in the basal-plane.
The only strain modes which are allowed to vary along the c -axis
are those deriving from the transverse modes, which are ε 1 ( i )and
ε 2 ( i ), and the longitudinal component 33 ( i ). Like the γ -strains, the
α -strains 11 ( i )and 22 ( i ) must remain constant. In the longitudinally
polarized phase, the ε -strains are not affected by the ordered moment.
The uniform α -strains are determined by the average of Q l ( J i ) and, in
addition, the c -axis moments induce a non-uniform longitudinal-strain
mode 33 ( i )

2 at the wave-vector 2 Q , twice the ordering wave-
vector. The amplitude 2 Q ,in 33 ( i )= 2 Q cos (2 QR iζ ), may be de-
termined by the equilibrium conditions for the single sites, with the
magnetoelastic-coupling parameters replaced by those corresponding to
2 Q . The longitudinal strain at site i is directly related to the displace-
ment of the ion along the ζ -axis; 33 ( i )= ∂u ζ /∂R iζ and hence u ζ ( R i )=
(2 Q ) − 1 2 Q sin (2 Q · R i ). Below T N ,where

∝ J iζ

becomes non-zero, the
cycloidal ordering induces an ε 1 -strain, modulated with the wave-vector
2 Q . The presence of a (static) transverse phonon mode polarized along
the ξ -direction corresponds to ∂u ξ /∂R iζ

J iξ

= 13 ( i )+ ω 13 ( i ) =0,whereas

∂u ζ /∂R iξ ω 13 ( i )=0. Henceitis ε 1 ( i )+ ω 13 ( i ), with

ω 13 ( i )= ε 1 ( i ), which becomes non-zero, and not just 13 ( i )= ε 1 ( i ).
In these expressions, ω 13 ( i ) is the antisymmetric part of the strain ten-
sor, which in the long-wavelength limit describes a rigid rotation of the
system around the η -axis. Because such a rotation, in the absence of ex-
ternal fields, does not change the energy in this limit, the magnetoelastic
Hamiltonian may still be used for determining ε ( i ). Only when the rela-
tion between the strains and the transverse displacements is considered,
is it important to include the antisymmetric part. In helically-ordered
systems, the γ -strains are zero, due to the clamping effect, as are the
ε -strains, because the moments are perpendicular to the c -axis. Only
the α -strains may be non-zero, and because

= 13 ( i ) −

Q l ( J i )

are independent of
the direction of the basal-plane moments, the α -strains are the same as
in the ferromagnet (we neglect the possible six-fold modification due to
B 6 α in (1.4.10)). Their contributions to the axial anisotropy (2.2.33), to
be included in κ l , are also the same as in the ferromagnetic case. In the
basal-plane ferromagnet, the ε strains contribute to the axial anisotropy
1 /χ xx in eqn (2 . 2 . 19 b ):

1
N ( σJ ) 2 ∂ 2 F ε /∂θ 2 =

; θ 0 = π

− 4 c ε H ε / ( σJ ) 2

∆(1 /χ xx )=

2 , (2 . 2 . 34)