Pitfalls

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Pitfalls

Data Collection

So what are the pitfalls in indexing powder diffraction data? And are there any strategies than can improve one chances of success?

Attempting to index poor data is a non-starter. To quote the original comment by J.W. Visser at the start of his zone-indexing program:

This comment makes indirect reference to the fact that Debye-Scherrer cameras have non-focussing X-ray optics, and consequently poor spatial (i.e. 2θ) resolution of the powder diffraction lines as discussed earlier. (Although much of the very early data in the ICDD PDF database was obtained using such cameras, the materials examined were typically metals and binary compounds with small unit cells. Much of the early data has since been remeasured using modern powder diffractometers, in order to improve its precision and quality.)

So one strategy is to collect high-resolution high-quality data: the ultimate of course here is to collect the powder diffraction data at a synchrotron. However, even with synchrotron data, indexing may be a problem. Impurities in the sample can give extra peaks, especially for larger d spacings since these are usually the most intense peaks in a powder pattern. In the previous example, the weak peaks were deliberately omitted. This is a good strategy if one suspects that an impurity may be present, but if the peaks are genuine, then the indexing may fail when weak, but important reflections are left out.

Pseudo Symmetry

Pseudo-symmetry is also potentially a big problem. Pseudo-symmetry occurs when certain lattice parameters have values that result in the symmetry of the lattice appearing to be higher than reality, for example, when the monoclinic angle β approaches 90°, the metric tensor for the lattice will be similar to that of an orthorhombic lattice. Another common instance occurs when the unit cell has monoclinic symmetry, but with a ≈ c and γ close to 120°; the symmetry of the lattice is then pseudo-hexagonal. Pseudo-symmetry can sometimes be recognized by collecting data at very high resolution, or with angle-dispersive neutron diffractometers, using long wavelength neutrons. Close examination of peaks widths may show that some peaks are not single lines.

Instrumental Errors

Click on the icon for a demonstration of a more difficult indexing problem based on the diffraction data of solid carbon tetrafluoride at 65 K collected at the ILL, Grenoble. No satisfactory solution will be obtained when you first click on the submit button. Now rerun with the program, but with the 2θ zero error set to -0.08°¶. The correct solution is now the first one listed with M₂₀ = 71 and is C-centred monoclinic with a = 8.50 Å, b = 4.38 Å, c = 8.35 Å, and β = 119.0°.

This above example demonstrates how sensitive indexing can be to diffractometer alignment, even in this case when the first reflection is at relatively high angle due to the use of a very long wavelength (3 Å). With the shorter X-ray wavelengths used on laboratory diffractometers it is critical that the instrument is well-aligned so that errors of this sort are avoided. Can these problems be detected?

The answer is yes, at least for some data sets. Suppose that we have a low-angle d spacing with unknown indices hkl, which is observed at angle 2θ_hkl. Using the Bragg equation, one can write:

The reflection with indices 2h,2k,2l will have half the d spacing value and will be observed at, say, 2θ_{2h 2k 2l}:

Combining these two equations yields an equation with one unknown, namely the 2θ zero offset of the diffractometer:

This equation cannot be solved analytically, but it is trivial for a computer to solve it numerically. Given a low-angle reflection, then the angle of a reflection with half the d spacing value can be calculated assuming no instrumental error. Taking the nearest observed value as the one with half the d spacing value, the value of 2θ_zero in the above equation is varied from -ve to +ve, the correct value being obtained when the left-hand side of the above equation reaches zero.

A similar method has been used for angle-dispersive power neutron diffraction data. The instrument D1A at the ILL when used with λ equal to approximately 3 Å has a small impurity wavelength λ / 3. The positions of two peaks with the same d spacing but due to the two different wavelengths can be compared and a value for the instrumental zero error estimated using a similar equation to that given above:

Dominant Zones

There is another potential problem in the indexing of powder diffraction patterns, which is due to so-called dominant zones. This is readily demonstrated using the hkl-generating program. Click on the icon, and examine the reflections obtained for the unit cell that was specially chosen here.

Impurities and Other Phases

Other potential problems can be due to samples that change phase during the data acquisition. The simple and effective strategy here is to collect the data set twice in quick succession so that any changes with time can be noted. The option of running a sample twice can also be an effective strategy for picking up on sample impurities; samples from two or more difference batches may well have different levels of impurity phases present thus enabling impurity peaks to be detected.

Much more could be written on the pitfalls of indexing. However, to finish this section, a demonstration of the effect of "model data" is provided: the same peaks as given previously are used, but this time with accurate and precise 2θ values. This demonstration just shows the figures of merit that are possible under ideal conditions!

¶The data set used here is slightly bizarre in that if you re-run the program with the zero error set to +0.1° then you will get a similar solution, though with a lower figure of merit! (This strange behaviour is probably due to the pseudo-trigonal symmetry of the system.)

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