MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 119

lie consistently above the theoretical. This may indicate that the per-
fect periodicity of the less stable spin-slip structures is more effectively
disturbed by imperfections.
AsmaybeseenfromTable2.1,the easy direction in Er is the c -
axis at high temperature, so the moments order in a longitudinal-wave
structure at T N . As the temperature is reduced, the structure squares
up, as discussed in Section 2.1.4. The basic wave-vector Q describ-
ing the magnetic ordering increases approximately linearly just below
T N (Atoji 1974; Habenschuss et al. 1974). This is not in accord with
the quadratic dependence predicted by (2 . 1 . 35 b ) and furthermore, since
J (3 Q ) is probably negative, the predicted change in Q also has the
opposite sign to that observed. This behaviour can only be accounted
for if

( q ) is temperature dependent, as is indicated even more clearly
at lower temperatures, where Q starts to decrease quite rapidly.

J

At

T N

52 K, a basal-plane component begins to order, through the mech-
anism described in Section 2.1.5. When the temperature is lowered fur-
ther, Q continues to decrease, exhibiting a number of plateaux, and a
rich harmonic structure is observed (Atoji 1974; Habenschuss et al. 1974;
Gibbs et al. 1986). Very detailed neutron-diffraction measurements by
Cowley (1991) have revealed a whole sequence of commensurable struc-
tures with decreasing temperature, with Q = 2/7, 3/11, 7/26, 4/15,
5/19, 6/23, and 1/4, in units of 2 π/c . At 18 K, a first-order transition to
a steep cone, with an opening angle of 30 ◦ and a wave-vector of ∼ 5 / 21,

is observed.
To explain these results, we may employ a modified version of the
model of Jensen (1976b), in which crystal fields, isotropic exchange, and
dipolar interactions are included. In addition, the anisotropic two-ion
coupling, which is required by the observed excitation spectrum and dis-
cussed in Section 6.1, is also taken into account. Mean-field calculations
then predict that the structure in the intermediate temperature range is
an elliptic cycloid, the hodograph of which at 48 K, just below the transi-
tion temperature, is shown in Fig. 2.6. As discussed in Section 2.1.5, an
additional second-order transition may occur below T N , to a phase with
a non-collinear, elliptical ordering of the basal-plane moments. In the
presence of random domains, the neutron-diffraction patterns from the
two structures are essentially indistinguishable, and if this transition oc-
curs in Er, the fluctuations expected near a second-order transition may
also be suppressed, because it is then likely that it coincides with one
of the first-order commensurable transitions. The model calculations
indicate that the non-collinear component in the basal plane is close
to becoming stable when the cycloidal phase is disrupted by the first-
order transition to the cone phase. Hence it is most probable that the
moments in Er are ordered in a planar elliptic-cycloidal structure in the