The Concept of Peak Shape

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The Concept of Peak Shape

Powder diffraction peaks should ideally be sharp. But however sharp a peak may be it will still have a distinctive shape. If the peak is very sharp then it simply means that we must scan with small steps (typically 6 to 10 steps as discussed earlier) through the region of 2θ within which the peak resides in order to have sufficient sampling to define its shape. The next figure shows the 211 and 886 peaks of yttrium oxide using a laboratory powder diffractometer to scan slowly through the two peaks, counting for 10 seconds every 0.01° 2θ for the 211 and 20 seconds every 0.01° 2θ for the 886. The experimental procedures for examining peak shapes are obviously a little different: here we have spent 45 minutes collecting just two peaks, whereas this would normally be sufficient to collect nearly the whole pattern. In point of fact you will notice the high angle peak is not as clear and smooth as the low angle peak. This is because it is much less intense (a peak height 37 times smaller) yet has only been scanned for twice as long. In order for the profile data to be statistically equivalent we would have to have collected for far longer, resulting in a collection time of many hours for this peak alone.

Low Angle Peak (hkl = 2 1 1)	High Angle Peak (hkl = 8 8 6)

In this section, the following discussion relates to angle-dispersive powder diffraction. One immediately recognises that, regardless of height, the peak shape changes somewhat with 2θ angle but nevertheless some of its intrinsic features remain. Being able to describe and use features of the peak shape (or "line profile" as it is also called) is the subject of this section. First we must define some basic parameters used to describe the shape of a diffraction peak (below):

The relevant parameters are:

• Peak maximum (or peak height), denoted as I_max, is fairly self explanatory, however it should not be used as a measure of diffracted intensity.
• Peak area is indicated (in yellow) as the area under the curve (and above the background should it not have been subtracted). The peak area is generally taken to be synonymous with "peak intensity" since the area represents the true sum of all the diffracted X-ray photons (or neutrons) that have been detected regardless of the peak shape.
• Peak width is a measure of the broadness of the peak. For example at low angles a small value (e.g. 0.05°) would indicate a sharp peak whereas a large value (e.g. 0.2°) would indicate a broad peak. Several parameters can be used but the most common is the FWHM (often denoted as H) which stands for full width at half the maximum; to obtain this parameter, one draws as indicated a horizontal line at half the maximum peak height (0.5 I_max) and measures its full width residing within the peak bounds. An alternative parameter is the integral breadth, β, which is defined as the ratio of the first two parameters above (peak area/peak maximum). In some of the following discussion it is not important which definition is used and in these cases the term B will be used to denote either.
• The tails represent the extremities at each side of the peak as it asymptotically approaches zero intensity (or the background if not subtracted); although they might not contribute very much to the total intensity (i.e. the peak area) they can provide a sensitive test of how well a peak is fitted by a mathematical function or model.
• Peak asymmetry is related to the mirror plane (or lack of) that passes vertically through the peak maximum; if the LHS (left hand side) to this "mirror line" is an identical reflection of the RHS, the peak is said to be "symmetrical"; if the two sides are not reflections of each other, the peak is said to be "asymmetrical". Peak asymmetry, for example, is particularly noticeable with powder patterns collected from instruments on neutron spallation sources.

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