Low Resolution

The use of large-scale QQ maps make it possible to explore broad areas of reciprocal space to determine the distribution of satellites and other features of interest. Once identified, more detailed maps, using longer counting times, can be made concentrating on the features of interest. By way of illustration, Figure 3.7 shows the result of a large-scale map of the on-axis area of the Beijing crystal encompassing some seven main reflections running down the centre of the Figure. A profusion of additional diffuse features can be seen extending out on both sides of the ${\bf c}^*$-axis, but with the highest density of scattering lying in vertical lines along the ${\bf b}^*$ positions equal to roughly $\pm$0.2 and $\pm$0.4. The weak horizontal bands across the picture are powder lines from the copper plate used to mount the sample. The contour lines are drawn here, as in all the subsequent two-dimensional maps, to follow an essentially logarithmic intensity scale to span the up to five orders of magnitude difference between the lowest background levels (typically 1 to 10 counts/sec) and the very strongest fundamental reflections. But in order to best show the positions and shapes of the peaks present in any particular map, especially those of the 1st order satellites which can be anything form two to four orders of magnitude above background, some weighting and fine tuning of the scale is always neccessary to extract the most detailed information.

Figure 3.7: A large-scale contour map of the on-axis area of reciprocal space centred around the (0 0 30) fundamental reflection; measured from the Beijing crystal.

Each of the four crystals were characterised in this way, and to illustrate the comparisons which have been made between them, an example of a detailed QQ map will now be presented for each crystal in turn. The maps are all centred around the same (0 0 20) main reflection, this being chosen for its strong intensity and relatively large Q value. The maps encapsulate examples of all of the disparate points which have been observed, and are typical of the many maps recorded around other main reflections.

The Beijing crystal, Figure 3.8(a).

The first-order satellites due to the modulation can be seen sitting on the four corners of a rectangle centred on the main reflection, visible further out on either side of the map are the much weaker second-order satellites. The picture is comparable to that illustrated in the schematic diagram in Figure 3.5, and the wavevector q has also been included here. In the structure models discussed at the start of this chapter, the satellites described by q would be the only features present. However, it is immediately obvious that the map displays a whole confusion of additional features, the majority of which are also at incommensurate positions. All of the anomalous features which were detailed at the close of the last section are distinguishable in this map. The weak features (marked) close to the forbidden fundamental positions (0 0 21) and (0 0 19) are those previously identified as violating the generally accepted space group by Eibl [91], and Zaretskii [94]. Similar diffuse features could be observed close to all other positions examined both on-axis (0 0 l) and off-axis (0 $\pm$2, l), and are also visible, for instance, in the large-scale QQ map of Figure 3.7. But the features have been resolved here into two separate peaks displaced to opposite sides of the reciprocal lattice node. This is not consistent with them originating from a symmetry property of the average structure as has been previously believed. The detailed map in Figure 3.10(a), this time close in around the (0 0 35) position demonstrates this more clearly, with the two diffuse features in incommensurate positions, lying asymmetrically on either side of the ${\bf c}^*$ axis.

The satellite reflections are also observed to be asymmetric in Figure 3.10(a) where the right satellite is stretched above, and the left below, the (0 k 35) line, giving an apparently incommensurate value for the $\gamma{\bf c}^*$ component of $\approx 0.97{\bf c}^*$. This is caused, however, by the splitting of the two peaks, just visible emerging from the resolution function in Figure 3.10(a). This is made yet more obvious in the second-order satellites of Figure 3.8(a) where the splitting has clearly separated into extra peaks along the ${\bf c}^*$ direction. As already mentioned, such asymmetry has previously been reported but without explanation.

Figure 3.8: Contour maps of the scattered intensity centred around the (0 0 20) fundamental reflection, the lines are drawn to a pseudo-logarithmic intensity scale, weighted to accentuate weaker features. In (a), the Beijing crystal, features around the forbidden fundamental positions are marked; (b) shows the Birmingham crystal.

The Birmingham crystal, Figure 3.8(b).

The map for this crystal is dominated by a broad width to all peaks, showing the crystal to have a large mosaic spread. But the pattern is somewhat simplified compared to that of the Beijing crystal. The main satellites are now more sharply defined along the ${\bf c}^*$ direction, with no evidence for any asymmetry, and as a result the splitting of the second-order satellites has completely disappeared. This clearly demonstrates the sample-dependent nature of such asymmetry and confirms the $\gamma{\bf c}^*$ component to be commensurate. The diffuse features around the odd ${\bf c}^*$ (0 0 19) and (0 0 21) positions are again present in this map and show a similar character to those of the Beijing crystal. The most prominent anomalous feature of this picture is the streaking along the ${\bf c}^*$ direction which lies between the first-order satellites. The same features are also clearly visible in Figure 3.8(a) of the Beijing crystal. These, as discussed in Section 3.2, have been commonly reported in electron diffraction studies [52,101], and in x-ray studies [92,99]. They appear to be closely related to the modulation, possessing the same 0.21${\bf b}^*$ component, and have been previously interpreted as being evidence for a second modulation wavevector, one which has no ${\bf c}^*$ element, or they have been described using the normal ${\bf q}$ modulation vector as satellites of the weak symmetry-violating reflections on the forbidden fundamental positions.

The Oxford crystal, Figure 3.9(a).

Comparing the peak widths with the previous two pictures shows this to be a crystal of a much higher quality. The pattern is also greatly simplified, looking much more like the expected schematic picture of Figure 3.5. The only point of confusion is the presence to the far right of an (0 0 20) reflection from a small secondary crystallite which is present in the sample at a slightly misaligned angle. Scattering from such secondary crystallites in the mosaic direction of the main crystal is a commonly observed feature for all the samples in these experiments. In most cases it is possible to remove such secondary features by careful vertical or horizontal translation of the sample in the beam. However, where this proved impossible to remove, as was the case here, it suggests the secondary crystallite is located at a different depth within the sample. Whatever, the resolution of the experiments is such that it is an easy task to differentiate between its contribution and the features of interest.

The crystal shows very little evidence of asymmetry in the first-order satellites but again, as in the Beijing crystal, some splitting of the second-order satellites is evident. The improvement in crystal quality has not therefore influenced this particular feature but there has been an accompanying change in the weak diffuse features on the forbidden positions. These are now completely absent in this pattern. A repeat of the detailed (0 0 35) map, shown in Figure 3.10(b), confirms their disappearance with only slight traces of diffuse scattering in the area. The result demonstrates the sample-dependent nature of these low intensity diffuse features. Their origin must therefore be interpreted as being due to some sort of defect structure which is present in crystals of a poorer quality. A possible origin could be stacking faults along the ${\bf c}^*$ axis. But they are certainly not an attribute of the fundamental structure as has been previously claimed.

Figure 3.9: Contour maps of the scattered intensity centred around the (0 0 20) fundamental reflection for the Oxford and Warwick crystals. In (a) the Oxford crystal, the peak to the far right is the (0 0 20) reflection of a small secondary crystal only slightly misaligned; in (b) the Warwick crystal, the extra diffuse peaks have been marked.

Figure 3.10: Detailed maps highlighting the scattered intensity around the forbidden fundamental position (0 0 35). (a) for the Beijing crystal showing the diffuse features lying at asymmetric positions between the two first-order satellites on either side, and (b) for the Oxford crystal which in comparison shows the complete absence of any diffuse scattering in these positions.

The Warwick crystal, Figure 3.9(b).

Comparing this with the Oxford crystal shows it to be similar in nearly all respects. In both cases, the high-quality of the crystal is such that the widths of the observed reflections are not the intrinsic widths but are rather limited by the resolution of the graphite optics. This is unlike the Birmingham crystal where, because of its inferior quality, the widths were broadened significantly beyond the instrumental resolution and were therefore a true intrinsic measurement. The forbidden odd (0 0 l) positions are again free from any scattering, confirming the Oxford result. The only point of difference between this and the Oxford crystal is the apparent absence of any splitting in the second-order satellites, and little trace of asymmetry in any of the satellite reflections. The results on this point are therefore more in accord with the Birmingham crystal, and confirms that this too is a characteristic with a strong sample-dependent element.

As in all the crystals that have been studied, the additional diffuse streaks between the satellites are once again observed here, and are now very well resolved (their position has this time been marked in the Figure). They therefore remain the sole anomalous feature of these observations which has not been found to vary between samples, and which are unaccounted for by current models of the modulated structure. Their shape, position, and intensity are consistent in pictures from all four crystals and it must therefore be concluded that they do relate to a prevalent aspect of the structure. Although much less intense, similar peaks are also seen on the equivalent second-order positions (also marked). Both first and second order examples are visibly more diffuse than their primary satellite counterparts and are broadened along the ${\bf c}^*$ direction in such a way as to almost link the two neighbouring satellites.

To investigate their characteristics further, profiles of the incommensurate satellites have been measured along both the ${\bf c^*}$ and ${\bf b}^*$ directions. The results show the diffuse streaks to be broadened in both. Examples of the ${\bf c^*}$ profile are shown in Figure 3.11 from the Oxford crystal, these clearly show the widths of the diffuse streaks to be at least twice those of the main satellites. They also possess an intrinsically different shape in the ${\bf c^*}$ direction, in some cases the diffuse streak even appears almost attached to one of the neighbouring satellites, while from the other it remains well separated. In Figure 3.12 the ${\bf b}^*$ profiles of a diffuse streak and a first-order satellite are compared. The satellite is limited to the mosaic of the graphite optics, however, the comparison shows the streak to be broadened slightly beyond this.

Figure 3.11: Two examples of profiles along the ${\bf c}^*$ direction of the first-order incommensurate satellites of the Oxford crystal. The main satellites are seen with l=25,27 and 31,33 while the associated diffuse peaks lie in-between with l=26 and 32, respectively.

Figure 3.12: A comparison of profiles in the ${\bf b}^*$ direction for the first-order satellite (0 -0.21 19) and its companion diffuse streak at (0 -0.21 20).

Figure 3.13: The contrasting pictures of comparable areas of reciprocal space from around two different fundamental reflections (a) the (0 0 24) and (b) the (0 0 30).

Figure 3.14: The increasing intensity of the diffuse streaks with value of l, plotted on a scale normalised to a Bragg intensity of 10000.

In the hope of finding a clue to the origin of the diffuse streaks, their intensity distribution relative to the satellites has been surveyed over a large area of reciprocal space. Certain points stand out from this. Around the (0 0 24), shown in Figure 3.13(b), which is a relatively weak reflection but which is surrounded by four comparatively very strong satellites, the diffuse streaks are entirely absent. The same picture is repeated around the (0 0 30) in Figure 3.13(a). In contrast here, where the satellites of the (0 0 30) are remarkably weak, the diffuse streak is actually more intense than the satellites. This evidence for an inverse dependence of satellite to streak intensity does not hold around other positions however, the (0 0 20) for instance has both strong satellites and streaks. It would seem from these observations that the diffuse streak intensities are overall closely dependent upon their fundamental reflection of origin, and that this dependence has a different nature from that of the satellite reflections but that no simple systematic relationship exists between the two. The overall intensity dependence upon increasing Q values for the streaks, after being normalised to their originating fundamental reflection, is shown in Figure 3.14. It was stated in Chapter 2 that diffuse scattering originating from displacive as opposed to compositional effects will have an increasing intensity with increasing Q value, and this is certainly demonstrated to be true for the streaks in Figure 3.14.