Crystallite Size Estimation

Basics and Background

Match! can estimate the crystallite size in your sample using Scherrer's formula[1,2]:

Crystallite size (average in Å) = K λ / (FWHM * cos θ)

where K is the so-called "Scherrer constant" (typically =0.94 for FWHM of spherical crystals with cubic symmetry), λ is the wavelength of the radiation and θ is the diffraction angle of the peak. As you can see, the main "input" to the formula are the FWHM values (FWHM = Full Width at Half Maximum) of the peaks.

Before the FWHM values from the diffraction pattern can be correlated to the crystallite size, it is necessary to separate the contribution of the instrument to the peak broadening (FWHM) from the actual contribution of the sample. The FWHM values of the latter are then used in the Scherrer formula to calculate an estimate for the crystallite size of the corresponding phase/compound.

In order to separate the instrument contribution to the FWHM of the experimental peaks from the sample contribution, you first have to create a so-called instrumental standard that contains your instrument's contribution to peak broadening. For this, you have to record a powder diffraction pattern from a sample/compound for which it is known that it does not contribute to the diffraction peak profile (so that the peak broadening is almost entirely caused by the instrument). In this compound/sample, the crystallite size should be larger than 500 nm while the particle size should be less than 10 μm. In addition, the standard material should be defect and strain free. Finally, the standard should have a similar mass absorption coefficient than the nanocrystalline specimen as well as a similar atomic weight and packing density [3].

A standard as it is used in Match! consists of a list of peaks from the diffraction pattern of a suitable standard compound (e.g. LaB6, CeO2, Si, etc.) that has been recorded using the same instrument and setting than the compound for which you would like to determine the crystallite size. The most important data are the peak positions and their FWHM values, both of which should have been obtained from a profile fit calculation of the peak data to the standard's diffraction pattern. At least one peak position and FWHM value must be entered in order to create a valid standard.

Instructions

In order to run the crystallite size estimation calculation, please make sure first that a suitable instrumental standard is already present. You can do so by running the "Crystallite size estimation..." command from the "Tools" menu. If no instrumental standard is present up to now, Match! will ask you to add a new one. Once you have checked the presense of a suitable standard in the settings, please close the "Estimate crystallite size" dialog again for the current.

You now have to prepare your sample's diffraction pattern for the analysis:

Import the diffraction pattern of your sample for which you would like to perform the analysis, run the peak searching (or add the peaks manually), then mark (click on) the individual peaks and perform profile fitting calculations for them until the agreement between the experimental diffraction profile and the one calculated from the experimental peak is fairly good. You can display the resulting peak parameters in the peak list if you like. As has already been mentioned above, the main input to the crystallite size estimation calculation (Scherrer formula) are the FWHM values of the peaks along with the peak positions.

Now you are ready to perform the crystallite size calculation. In order to do so, simply run the "Crystallite size estimation..." command from the "Tools" menu. Match! will open the corresponding dialog, automatically subtract the instrumental peak broadening contribution (using the default instrumental standard mentioned above), and then present a table of the individual peak data and the resulting crystallite size estimation values.

Let's take a closer look at the elements and buttons that can be used: At the top left you can select the instrumental standard that shall be used in the calculation. To the right of the corresponding drop down list there is the Add button using which you can open the Add instrumental standard dialog in order to add a new standard. The Edit button can be used to modify the standard that is currently selected at the top left (also using the dialog just mentioned).

On the upper right-hand side you can modify the Scherrer constant K to be used in the calculation. By pressing the Default button you can select the current settings of instrumental standard and Scherrer constant to be used by default for future crystallite size calculations.

Using the three buttons below, you can mark or unmark peaks for being included in the calculation of the average crystallite size:

The main part of the dialog is covered by the Peak data table in which you can see the peak data like peak positions, intensity, FWHM values (both instrumental and sample contribution), the name(s) of correlated phase(s) from the match list (if any), the crystallite size calculated from each peak's data, and finally a checkbox in the right-most column Use.

At the bottom right of the peak data table, the average crystallite size value of all peaks for which the checkbox Use is marked in the table above is displayed. When you open the "Estimate crystallite size" dialog to perform the analysis for the first time, Match! will automatically mark the checkboxes for all peaks in the range 30-50° (which is typically most suitable for crystallite size analysis) for which all required FWHM data are available.

Finally, you can Copy the peak data table to the clipboard as well as Print or Export the table, using the buttons below it. Pressing the Help button will open this page.

References

[1] P. Scherrer, “Bestimmung der Größe und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen", Nachr. Ges. Wiss. Göttingen 26 (1918), 98-100.
[2] J.I. Langford and A.J.C. Wilson, “Scherrer after Sixty Years: A Survey and Some New Results in the Determination of Crystallite Size", J. Appl. Cryst. 11 (1978), 102-113.
[3] Scott A Speakman, "Estimating Crystallite Size Using XRD", http://prism.mit.edu/xray/documents/5a%20Estimating%20Crystallite%20Size%20Using%20XRD.pdf