The Results

A large intensity contour map calculated using a simplified computational approach, suitable for the unmodulated structure only, is shown in Figure 7.1. The structure model was of 200x200x20 unit cells generated from the coordinates of the fundamental structure in Table 7.1, without any modulation. The result can be seen to include only scattering located at the fundamental positions. Artefacts of the calculation which are related to the size of the crystal model used are always present, however, this calculation demonstrates that the size of the model in this case is more than sufficient to reduce such features to below significant levels; the difference between maximum and background values is eight orders of magnitude. The difference between the intensity of a reflection calculated from the fundamental structure and that calculated from the structure when the modulation has been applied is shown in Figure 7.2. The difference in the two intensities is effectively the term $<({\bf k}.\Delta{\bf r}_m)^2>$ in equation 2.14, and it is this reduction in intensity which distinguishes the scattering from the average structure as opposed to the fundamental structure. Both these profiles were calculated using the parallel code described in the previous section. The size of the model was ($N_x$x$N_y$x$N_z$)=(1x60x2). This value for the ${\bf b}$ dimension was found to be a suitable compromise size, with the artefact features still being satisfactorily suppressed.

Figure 7.1: A contour map around the three fundamental reflections calculated from the unmodulated atomic coordinates.

The advantage of one-dimensional scans such as those in Figure 7.2, as opposed to the two-dimensional maps, is that the size of the model may be reduced appropriately in the two other directions, and this of course dramatically reduces the computing time. Only approximate timings of the run times for code were made, but it was immediately apparent from the runs that it would require unfeasibly long periods of processor time to generate the two-dimensional maps of Figure 7.1 using this method. The one-dimensional scans, however, could be quickly generated, with the processor time on the full Connection Machine, for one of the profiles in Figure 7.2 being less than 20 minutes, which involved the calculation of the intensity at 21 k-points.

Figure 7.2: Profiles along the (0 $l$ 0) direction of the (0 0 20) reflection. They are both calculated from models of the structure with ($N_x$,$N_y$,$N_z$)=(1x60x2), but in one the modulation is absent, and in the other it is present.

The profiles in Figures 7.3 and 7.4 demonstrate the success of the simulation at reproducing the incommensurate satellite features. The calculations were again performed on a model of size (1x60x2), modulated with a wavevector of ${\bf q}=0.211{\bf b}^* + {\bf c}^*$, identical to that measured experimentally. Figure 7.3 profiles two first order satellites, one at (0 -0.21 19) and the other at (0 0.21 21). In Figure 7.4, the effect upon a first order satellite of varying the ${\bf b}^*$ component of the modulation wavevector is tested. The value of wavevector applied to the structural model is 0.21${\bf b}^*$ in the first, and 0.26${\bf b}^*$ in the second. The Figure demonstrates that the simulation correctly reproduces this in the change of the coordinates of the satellite reflection.

Figure 7.3: The profiles of two first order satellites located at (0 -0.21 19) and (0 0.21 21).
Figure 7.4: Two (0 $l$ 0) scans through the same reciprocal space coordinates, calculated in one case when ${\bf q}=0.211{\bf b}^*$ and in the second, when ${\bf q}=0.261{\bf b}^*$.

A profile of the satellites in the ${\bf c}^*$ direction, running across the (0 0.21 $l$) position, is shown in Figure 7.5. The calculation was made from the same model as that used in the previous figures. The widths of the reflections are, of course, now significantly broadened due to the restricted c axis dimension of the model crystal. Weak second harmonics of the satellites are also produced due to this, lying at values of $l$=0.25; these would disappear if the c dimension of the model was to be extended further than $N_z$=2. What is made apparent in this Figure is the presence of a minimum in the calculated intensity at the $l$=20 position (the even value of $l$) as is to be expected due to the symmetry of the modulation. This is just the position at which a maximum in the diffuse scattering is observed experimentally due to the diffuse streaks. This is even more clearly illustrated in Figure 7.6 which contrasts a ${\bf b}^*$ scan through a second order satellite position at (0 0.42 20), with that through the position at (0 0.42 21). Correctly, according to the symmetry of the modulation, the second order satellite is reproduced for $l$=20, and is entirely absent for $l$=21. Where only background scattering is calculated for the $l$=21 position (slowly sloping away from the fundamental position), experimentally a second order diffuse streak would be observed, with an intensity comparable to that of the second order satellite. These two results prove that the streaks are not in any way accounted for by the refined models of the modulated structure, such as Yamamoto's used here [85], and that the program is producing sensible and consistent results.

Figure 7.5: A ${\bf c}^*$ scan through the position of two satellites at (0 0.21 19) and (0 0.21 21).
Figure 7.6: Two scans along the ${\bf b}^*$ direction through second order positions. In one, a second order satellite is realised correctly at (0 0.42 21), while in the other only background scattering is found through the (0 0.42 20) position which is where experimentally second order diffuse streaks are observed, at least as intense as the second order satellite.