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conditions develop which favour commensurable structures, including
the ferromagnet. The hexagonal anisotropy distorts the helical struc-
ture, while the development of higher harmonics, assisted by the axial-
anisotropy forces, favours commensurability in the longitudinal struc-
ture. The higher-order axial-anisotropy terms may also tend to pull the
moments away from their planar or axial orientations. The application
of a magnetic field requires further compromises, until it is so great that
it coerces all the moments into alignment.
The variation of temperature and field thus reveals a rich variety
of intermediate phase transitions to different structures. Most of these
transitions are discontinuous, but occasionally a second-order transition
is observed. In the following, we will discuss the relation between the
interactions , and their variation, and the magnetic structures in the rare
earths. We shall give a summary of the rather complete understand-
ing which has been attained of the heavy elements, followed by a brief
discussion of the complex structures of Nd, which is the only light rare
earth which has been studied in comparable detail. The effect of a mag-
netic field will be exemplified by a description of the magnetization of
Ho. Finally we will consider the new features which emerge when one
dimension of the magnetic lattice is bounded, illustrated by some of the
results from the rapidly developing study of thin films and superlattices.
2.3.1 Bulk magnetic structures
The manner in which the competing interactions express themselves is
very well illustrated by the heavy hcp rare earths. In their magnetically
ordered phases, all the moments in a particular plane normal to the c -
axis are aligned, but their relative orientations may change from plane to
plane. Fig. 1.19 illustrates some of the simpler of these structures, while
the transition temperatures T N and T C to ordered states, respectively
without and with a net moment, are given in Table 1.6.
Gd is magnetically by far the simplest of the rare earths. The ex-
change favours ferromagnetism and the 4 f charge-cloud is spherically
symmetric, so that the crystal-field interactions (1.4.4) are zero. How-
ever there is a residual magnetic anisotropy, which causes the moments
to point preferentially along the c -axis just below T C . At lower temper-
atures, the easy axis begins to deviate towards the basal plane, reaching
a maximum tilt angle of 60 at 180 K before decreasing to just below 30
at 4.2 K (Corner and Tanner 1976). The anisotropy parameters are typ-
ically two or three orders of magnitude smaller than those of the other
heavy rare earths (Mishima et al. 1976). Since the c/a ratio of Table 1.2
is less than the ideal value, the dipolar coupling induces an anisotropy,
discussed in Section 5.5.1, which tends to hold the moments along the
c -direction and has roughly the observed magnitude (Brooks and Good-
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