MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 109

the c -axis moments begin to order. Just below T N ,the c -component is
modulated with the wave-vector Q at which

J ( q ) has its maximum,
and only if Q = Q is the structure the tilted helix. If Q = 0 , so that the
c -axis moments are ferromagnetically ordered, the resulting structure is
the cone .
The magnetoelastic contributions require special treatment when
the structures are modulated, because of the limited ability of the lattice
to adapt to various strain configurations, when the strains are spatially
modulated. The magnetoelastic Hamiltonians considered above are only
strictly valid in the uniform case, but they may be generalized to non-
uniform structures by replacing the strains by their local values αβ ( i ), at
least in the limit where the wavelength of the modulation is much longer
than the range of the interactions. At shorter wavelengths, the form
of the magnetoelastic-interaction Hamiltonian may still be applicable,
but the effective coupling parameters may depend on the wave-vector.
This suggests that the above discussion may be largely unchanged if
the magnetic structure is modulated, provided that we take account of
the new constraints which we shall now examine. The displacement of
the i th ion, u ( R i )= R i − R i , from its equilibrium position R i may
be divided into a uniform and a non-uniform component, and the non-
uniform part may be written as a linear combination of contributions
from the normal phonon modes at various wave-vectors. It follows from
this that a displacement of the ions which varies with a certain wave-
vector should be describable in terms of the normal phonon modes at
that particular wave-vector, in order to ensure that such a displacement
is compatible with the lattice.
To be more specific, we shall consider the wave-vector to be along
the c -axis in the hcp lattice. In the double-zone representation ,which
neglects the two different displacements of the hexagonal layers, there
are only three normal modes; one longitudinal and two energetically-
degenerate transverse modes. All three modes correspond to rigid dis-
placements of the hexagonal layers. The γ -strains describe an elongation
of these layers along a certain direction in the plane. If the γ -strains are
uniform within each hexagonal layer, the magnitude or the direction of
the elongation cannot be allowed to vary from one layer to the next,
as this would destroy the crystal. Hence, even though Q 2 ( J i )

in the
equilibrium equation for γ 1 ( i ), corresponding to eqn (2.2.25), varies in
a well-defined way in a helical structure with Q along the c -axis, γ 1 ( i )
is forced to stay constant. The site-dependent version of (2.2.25) is only
valid when the right-hand sides are replaced by their averages with re-
spect to any variation along the c -axis, and these averages vanish in
the helix. This phenomenon was named the lattice clamping effect by
Cooper (1967), and further discussed by Evenson and Liu (1969). One