Temperature (Kelvin)

1/5

Fig. 11. Temperature dependence of the magnetic (solid symbols) and spin-slip (open symbols) wave vectors for two different bulk samples studied by X-ray diffraction. The wave vectors of the spin slips broadens considerably and moves to zero when the magnetic wave vector locks to -~, as indicated by the arrow. (From

Helgesen et al. 1994.)

1 * then from the above rs 0, and thus samples in fig. 11. Note that when Tm locks to ~c , =

1 phase is explained.

the disappearance o f the slip scattering in the

Referring now to fig. 8, it may also be seen that when T m = ~7C*, then the radial width o f the spin-slip scattering at ~c* is at its narrowest. This is a recurring property o f the simple commensurable phases o f the lanthanides, namely, that the correlations among the spin-slips extend over the greatest distances in these phases. Moreover, it is only the simple commensurable phases that are observed to lock in zero field. Higher- order commensurable phases, involving non-integral ratios o f doublets and slips (and, therefore, two or more spin-slip periods), have not to our knowledge been observed in bulk samples, although they have been observed in the presence o f an applied magnetic field (Cowley et al. 1991), and in thin films (Helgesen et al. 1994). We conclude that the additional charge scattering observed in the diffraction patterns arises from magnetoelastic

28 D.E McMORROW et al.

modulations o f the lattice accompanying magnetic ordering, and that the positions, intensities, and widths of these peaks reflect the correlations among the spin slips.

So far we have developed these arguments for the case of Ho only. General expressions for the relationship between Tm and rs have been derived by Bohr et al. (1986), to which the reader is referred for more details.

3.3.1. Lattice modulations

The presence of spin slips in the magnetic structures leads through the magneto-elastic coupling to an induced modulation of the lattice. Magnetostriction is known to be significant in the lanthanides as expressed, for example, by their anomalous lattice distortions below the N6el point.

Three possible mechanisms that may produce a lattice modulation are shown in fig. 12. In fig. 12a a distance-dependent exchange interaction is sketched. At spin slip positions the turn angle differs from turn angles in the rest of the structure. A distance- dependent exchange interaction will therefore lead to lattices modulations which arise from spin slips. In fig. 12b the effect of quadrupoles in a crystal field is considered.

The different orientations of the magnetic moments relative to the crystal field give rise to local differences in the magnetostriction. The third proposed mechanism (fig. 12c) is a quadrupole-quadrupole interaction. This mechanism would also give rise to a second harmonic of the magnetic satellite. The so-modified structure of the magnetic spiral will in addition to the magnetoelastically induced lattice modulation, give rise to weak magnetic satellites at rm -4- %.

(a) (b) (c)

, • % "

T " " "

DISTANCE DEPENDENT QUADRUPOLE IN QUADRUPOLE-

EXCHANGE J ( r ) A CRYSTAL FIELD QUADRUPOLE

INTERACTION c l d ( r i j ) -- -- a V ( r i + l - r i - I ) 2

Oz S i" Sj 8z Si

Fig. 12. Schematic o f three mechanisms which can lead to lattice modulations: (a) distance dependent exchange, (b) quadrupole in a crystal field, and (c) quadmpole~lnadrupole interaction. (From Bohr et al. 1989.)

X-RAY SCATTERING STUDIES OF LANTHANIDE MAGNETISM 29 3.3.2. Neutron scattering

The fact that the existence of spin slips in Ho was first revealed by X-ray magnetic scattering can be viewed as a major coup for this technique. Inevitably, it was not very long before a new series of neutron scattering experiments were undertaken (Cowley and Bates 1988). Instead of adopting the same approach used in the X-ray study of Gibbs et al. (1985), who concentrated on the first-order rm satellite, Cowley and Bates (1988) focussed on making a detailed study of the higher-order peaks. Two data sets from their work are shown in fig. 13, where it can be seen that in addition to the first, fifth and seventh harmonics reported in earlier neutron scattering studies, there are many additional peaks.

An important aspect of their work, however, is that they developed a spin-slip model for the structure that was able not only to account for the positions of all the observed peaks, but also their intensities over nearly decades of intensity variation. In their notation, the position in Q of the spin-slip peaks is given by

Qmag = i [ q 0 + ~(q0 +Ic" )], (24)

1 ~ for Ho, I is an integer, and b is the number of lattice planes between spin where g0 = gc

slips on different sub-lattices, so that b is necessarily odd. For example, for the ~7 phase shown in fig. 13a we have that b = 9, and all of the peaks in this figure can be indexed using eq. (24). In order to account for the intensities of the magnetic peaks Cowley and Bates found it necessary to introduce a few extra ingredients into the original spin-slip model. The most significant o f these modifications was the inclusion of a Debye-Waller factor to allow both for the fact that the position of the slips was not sharply defined, but was instead represented by a distribution with a width of a few lattice planes, and that there was a distribution in the bunching angles. With these modifications it was found that a good fit to the peak intensities could be achieved for both the [00g] and [10g] data as shown in fig. 14. (It is worth noting that the higher-order magnetic satellites have so far eluded attempts to observe them by X-ray scattering techniques.)

Support for the results of the detailed modelling of the spin-slip structures by Cowley and Bates (1988) was produced by self-consistent mean-field calculations (Jensen and Mackintosh 1991). The Hamiltonian used in these calculations was essentially the same as developed to explain the magnetic excitations in Ho and its alloys (Larsen et al. 1987), with the exchange parameters adjusted to produce the correct Tm at a given temperature.

1 and ~1 phases the mean-field calculations were able to As shown in fig. 15 for the

confirm the salient features of the spin-slip structures. This modelling was later extended to calculate the magnetic excitations in the 2 , or one-spin-slip phase as it is referred to.

This showed that the effect of the spin-slips on the excitations is to open up a characteristic gap near ~ . The presence of this gap was subsequently confirmed by high-resolution neutron scattering (McMorrow et al. 1991).

The spin-slip model has also found application in the interpretation of neutron diffraction data obtained from Ho in a magnetic field. In 1986, Bohr and coworkers noted that the ~l phase of Ho possesses a ferrimagnetic component in the basal plane (due to

30 D.F. McMORROW et al.

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