Single-Peak Methods I. Diffracting Power

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Diffracting Power

We need to start with some fairly obvious definitions. Imagine we have two phases, A and B, which have diffraction patterns collected under identical conditions, as illustrated:

Phase A might be a high symmetry material (e.g. a face centred cubic metal) resulting in a relatively small number of diffraction peaks in the pattern, whereas phase B has a lower symmetry and larger number of peaks. The peaks in A will tend to be larger than those in B because A's diffracting power will all be concentrated into just a few peaks. This obvious difference must be taken into account. Let us imagine now that we have a mixture of phases A and B yielding a composite diffraction pattern as follows:

If we wished to determine the weight fractions of A and B in the mixture by using just one peak to represent each phase, we would choose peaks I and II since they are the most intense for each phase that do not overlap: peaks III and IV would not be suitable since they overlap with other peaks in the mixture and it would therefore be difficult to apportion intensity between them reliably. The required ratio of the weight fractions is simply the ratio of the intensities of peaks I and II in the mixture, I_A^mix/ I_B^mix, normalised to take into account the differing diffracting powers of A and B. Ignoring factors such as absorption, this normalisation can simply be taken as I_A^pure/ I_B^pure, the ratio between the same two peaks from pure phase patterns taken under identical conditions. So the answer simply becomes:

This kind of normalisation procedure, rather obvious in this simple case, underpins all quantitative analysis whether by single peak or whole pattern method. The normalisation ratio, 1/( I_A^pure/ I_B ^pure), will be constant for a given pair of materials. This means it only has to be measured once, after which it can be re-used over and again for any other A/B-mixture. This prompted the idea that a database of normalisation constants could be set up covering all common materials, so that the normalisation constants would not need not be determined locally, but rather the appropriate values could simply be lifted out from the database. The chosen vehicle for this information was the ICDD powder diffraction database, which was discussed earlier in Qualitative Analysis. For each listed material, the intensity of the strongest peak is compared with that of the strongest peak from a reference material, obtained under identical conditions. The reference material chosen was corundum (α-Al₂O₃, alumina) since 1) it can be obtained plentifully with high purity, small particle size and little if any preferred orientation, and 2) it is stable and relatively inert with most materials. So for the more recent entries in the ICDD powder diffraction database you will find an entry called "I/I_cor" (pronounced "I over I corundum"). This parameter is defined as the ratio of the strongest diffraction peak from the material compared to the strongest corundum peak (the 113 hexagonal reflection) from a 50:50 material:corundum mixture by weight. Indeed during the Qualitative Analysis section covered earlier in this course, you would have come across two cases when the card image displayed I/I_cor values (zinc, card 4-831, had an I/I_cor value of 3.80; while calcite, card 5-586 had an I/I_cor value of 2.00).

In the earlier days the I/I_cor values were measured using the height of the respective diffraction peaks. You will be aware by now that peak heights are unreliable measures of intensity and that it is the integrated intensity (area under the peak) that should ideally be used. It is also preferable to consider more than just one peak and to consider other reference materials besides corundum. This had led to the more general term "RIR" (Reference Intensity Ratio) of which I/I_cor is just a special case.

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