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Fig. 2.8. The relative orienta-
tions and magnitudes of the funda-
mental wave-vectors which describe
the quadruple- Q magnetic structure
of Nd at low temperatures. All four
periodicities are present on each type
of site, but Q 1 and Q 2 generate the
dominant structures on the hexago-
nal sites, while Q 3
and Q 4
predom-
inate on the cubic sites.
bly of magnetoelastic origin, lifts the degeneracy between the different
multiple- Q structures and stabilizes the single- Q state just below T N .
The Landau expansion can also explain the rotation of the wave-vectors
and moments away from the b -axes, with the consequent stabilization of
the double- Q configuration, and account for the observed harmonics in
this structure. A similar analysis for the quadruple- Q structure in the
low-temperature region would provide the basis for understanding the
even more complicated phenomena which are observed when a magnetic
field is applied to Nd (Zochowski et al. 1991).
2.3.2 The magnetization of Holmium
The analytical mean-field treatment by Nagamiya et al. (1962) of the
effect of a magnetic field applied in the plane of a helical structure was
mentioned in Section 1.5. As the field is increased, the helix first distorts,
giving rise to a moment along H , and then undergoes a first-order tran-
sition to a fan structure, in which the moments oscillate about the field
direction. A further increase in the field reduces the opening angle of
the fan which, in the absence of magnetic anisotropy, goes continuously
to zero, establishing a ferromagnetic phase at a second-order transition.
Hexagonal anisotropy may modify this process by inducing a first-order
transition or, if it is large enough, eliminate the fan phase entirely.
The magnetization curves measured by Strandburg et al. (1962) and
Feron (1969) behaved in accordance with this description at low tem-
peratures, but above about 40 K when the fan phase was first observed,
a further phase also appeared, manifested by a plateau corresponding to
a moment about one half of that attained in the fan phase. This extra
phase was clearly apparent in the magnetoresistance measurements of
Mackintosh and Spanel (1964), and later experiments by Akhavan and
Blackstead (1976), in which the field was changed continuously, revealed
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