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as derived from (2.2.31). It is straightforward to see that we get the
equivalent contribution in the helix at q = 0 in eqn (2.2.33), except that
the coupling parameters in (2.2.34) should have the effective values at
the wave-vector Q . In the conical phase, both ε 1 ( i )and ε 2 ( i ) become
non-zero, 90 out of phase with each other, corresponding to a transverse
displacement of the planes, in a direction which follows the orientation
of the moments in the basal plane.
2.3 Magnetic structures of the elements
As we have seen, the 'exotic spin configurations' first observed by Koeh-
ler and his colleagues in the heavy rare earths may be understood as the
result of a compromise between the competing magnetic interactions to
which the moments are subjected. The complex changes which occur
as the temperature is varied stem primarily from the temperature de-
pendence of the expectation values of the terms in the MF Hamiltonian
(2.1.16). The crystal-field parameters B l are expected to change little
with temperature but, as shown in the previous section, the variation
of the expectation values
O l
of the Stevens operators may give rise
to a very pronounced temperature dependence of the anisotropy forces,
including the magnetoelastic effects. The contribution from the two-ion
coupling generally varies more slowly, since the exchange field is pro-
portional to
or σ , but changes in the magnitude and orientation of
the ordered moments alter the band structure of the conduction elec-
trons,whichinturnmodifiesthe indirect exchange
J j
J
( ij ). Hence the
Fourier transform
( q ), and in particular the value Q at which it at-
tains its maximum, may change with temperature in the ordered phase.
In addition, the possibility that anisotropic two-ion coupling may be
of importance implies that the effective parameters of the simple MF
Hamiltonian (2.1.16) may all depend on the magnitude and orientation
of the moments.
The anisotropy forces favour a set of crystallographic directions,
related by a rotational symmetry operator, along which the moments
tend to align themselves. In particular, the low-order crystal-field term
B 2
J
O 2 ( J )
gives rise to an axial anisotropy , which strives to confine
the magnetization either to the basal plane or along the c -axis, and de-
clines relatively slowly with temperature. Except for Gd, the rare earth
elements all have a
= 0 , reflecting the com-
plexities of the Fermi surface and corresponding to a periodicity which is
not generally commensurable with the lattice. Transverse and longitu-
dinal magnetic structures can accomodate both the anisotropy and the
periodicity constraints at high temperatures, with respectively uniform
helical or longitudinal-wave configurations of the moments, character-
ized by a single wave-vector. As the temperature is lowered, however,
J
( q )withamaximumat Q
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