Results

Because of the large number of repeated measurements and the extended counting time required for each reflection, the measurements had to be concentrated upon a restricted area of reciprocal space which included one fundamental reflection, the four surrounding first-order satellites, two diffuse streaks, and the corresponding second-order reflections. The area around the (0 0 22) fundamental reflection was chosen for this study to avoid the scattering from a secondary crystallite which confuses the pattern to the right of the stronger (0 0 20) reflection (seen in Figure 3.9(a) of Chapter 3). The actual measurements at each stage consisted of detailed small area intensity maps around each of the reflections. The method allows precise location of peak positions along both reciprocal axes, and more accurate measurement of peak intensity than a simple rocking curve would allow.

The incommensurate wavevector ${\bf q}^*$ which describes the position of the satellites was measured to have a value at room temperature (which has already been presented in previous chapters) of:

${\bf q}^*$ = 0.207(1)${\bf b}^*$ + ${\bf c}^*$

From the measurements of both the first- and second-order satellites, no changes in the satellite positions either along ${\bf b}^*$ or ${\bf c}^*$ were discernible over the entire temperature range studied. This is illustrated by profiles along the ${\bf b}^*$ direction of two satellites in Figures 6.2(a) and 6.2(b): the results of measurements at 197K, 138K and 20K are overlaid, showing the constancy of the satellite positions despite the temperature related changes in intensity. The periodicity of the modulation can therefore be considered to be constant, and displays no temperature dependence, in agreement with the results of Takenaka [93]. The full width at half maximum (FWHM) of the peak profiles, such as those in figure 6.2, are limited in these measurements to that of the mosaic width of the pyrolitic graphite crystals used as monochromator and analyser. The values were nevertheless monitored for any possible increase beyond that of the graphite width which would signify a change in the coherence of the modulation. No reproducible changes in the FWHM values were observed, however.

Figure 6.1: Scans along ${\bf b}^*$ of the (0 0 22) fundamental reflection at the temperatures 200K, 120K, and 20K.

Figure 6.2: Scans along ${\bf b}^*$ for: (a) the first-order satellite at (0 0.21 23); and (b) the first-order satellite at (0 -0.21 23). For each reflection, profiles measured at the temperatures 200K, 120K, and 20K are overlaid.

The influence of temperature upon diffracted intensity can be interpreted in terms of the Debye-Waller theory, as was discussed in section 2.4.2 of Chapter 2. Any instantaneous deviation of an atom from its ideal crystallographic site, such as by thermal vibration, is observed in a diffraction experiment as a time average, and can be accounted for by the introduction of the Debye-Waller factor (from equation 2.16):

$$D= \frac{1}{2}k^2<\Delta r_m^2>$$ (8.1)

The diffraction equation, in the simplest terms using an Einstein model of the vibrations, is then:

I=I$_o$exp(-2D)

This shows the intensity of a diffraction peak to increase with decreasing temperature, and that reflections further out in reciprocal space to be affected to a greater extent. The quantity is also sensitive to any form of static distortions and the possible structural changes already discussed would be expected to reveal themselves as deviations from this idealised Debye-Waller curve.

The temperature dependence of the peak intensity for the (0 0 22) fundamental reflection and for each of the four first-order satellites is plotted relative to each other in Figure 6.3. The intensity values, although arbitrary, do indicate the relative values for each of the reflections. The slight difference in the satellites' intensities, the +ve ${\bf b}^*$ satellites being slightly higher than the -ve ${\bf b}^*$ (when ideally they should be equal), is merely due to the difference in their inclination to the incident beam. As can be seen from the constancy of these relative values, no systematic changes in the alignment are introduced by the repeated cooling and heating. The results of least-squares fitting of the data to the logarithmic curve of the Debye-Waller theory are shown as solid lines in Figure 6.3. A reasonable agreement is obtained. In particular, the slope of the satellites is almost twice that of the fundamental reflection, which is in agreement with the Debye-Waller equation considering the larger reciprocal space positions of the satellites.

Figure 6.3: The intensity as a function of temperature for the (0 0 22) and for four first-order satellite reflections. The straight lines are the result of fitting to the expected Debye-Waller behaviour.

The linear intensity changes indicated in Figure 6.3 demonstrate that there is no major change in the structure with temperature. The standard deviation of the points from the straight line fit limits any anomalous intensity change to within 5%. The scatter in the data is an indication of the uncertainty introduced by random changes in the sample position after separate heating and cooling cycles. No noticeable hysteresis was observed, the peak intensities being approximately equal after heating or cooling. Scans along ${\bf b}^*$ of the (0 0 22) fundamental reflection and the (0 0.21 23) and (0 -0.21 23) first-order satellites are shown in Figures 6.1 and 6.2. It is noted that there was often observed a very small decrease in intensity over the range 200-140K, or sometimes no change in intensity whilst at lower temperatures (100-20K) the expected increase occurred. Such a small effect was consistently observed for nearly all the reflections and reproducibly observed on both heating and cooling cycles. Such an effect is, however, so small as to be within the scatter of the data points in Figure 6.3, and hence no conclusion about this observation can be drawn with any certainty. Such a change could occur because of an anomalous disordering at $\sim$ 140K which would cause the peaks to broaden, but this would be unobserved because the instrumental resolution is greater than the intrinsic peak width. Careful examination of Figure 6.3 reveals that in most cases there is a marginal deviation below the straight line at $\sim$ 140-120K. From the results of this experiment on their own, it is uncertain whether this is a real effect or perhaps due to some small systematic misorientation of the sample. Further studies with considerably higher resolution would be needed to verify such an effect.

Figure 6.4: The temperature dependence of the integrated intensity for the diffuse streaks. The arbitrary intensity scale indicates relative values only. The straight lines are the result of fitting the points to the expected logarithmic behaviour of the Debye-Waller theory.

The diffuse streaks which exist in addition to the satellites, and were identified as important by the previous room temperature measurements were also investigated. Their much lower intensity, and considerable overlap with neighbouring satellites, make the intensity measurements much less accurate. The results, though, do indicate their temperature dependence to be similar in nature to that of the satellites, and again a loss in the intensity is observed at 140-120K.

Figure 6.5: The behaviour of the lattice parameters $b$ and $c$ plotted from values measured during the four temperature cycles of the experiment.

With each change in temperature, the calculation of the lattice parameters was undertaken, and the results of this are presented in Figure 6.5. The ${\bf b}$ axis showed a marked change between room temperature and 240K with a smaller decrease thereafter down to 20K; the ${\bf b}$ axis is 5.403(2) at room temperature, 5.387(2) at 240K and 5.383(2) at 18K. The long ${\bf c}$ axis showed a linear behaviour over the entire range, it has values of 30.905(5), 30.86(1) and 30.755(5) at 300K, 240K and 18K respectively. A common property observed in the cuprates is a small cusp in the linear expansivity close to T$_c$ [181]. In Bi-2212, a previous study has suggested a small discontinuity in the c lattice parameter, of the order of $\Delta c=0.004$, in the temperature region T=110-75K [182]. But the precision in the work here was not sufficient to determine any such small anomalous changes.