X-Ray diffraction is used to evaluate a number of material properties regarding the crystal structure of a material. For this project you will develop a MATLAB code to analyze an X-Ray data set provided to you electronically. The data was measured on a silicon sample crushed down to a fine powder by a planetary ball mill, see Figure 1. Following the steps provided below you will determine the sample’s crystallite size, micro-strain, and lattice parameter. The following steps will require you to use several of the MATLAB codes developed over the course of this semester in a single practical application. The data provided to you was obtained on a Bruker D-8 Advance configured in the Bragg-Brentano configuration, see Figure 2. The x-axis values represent the angle of the diffractometer called the 2θ angle, the y-axis values represent the intensity of the X-Ray detector. The sample was measured over a 2θ range of 10-80◦ with a scan rate of 3.5◦ min−1 . The Bruker D-8 Advance uses Cu kα radiation with a wavelength of λ = 0.1541 nm. The peaks in the scan represent reflections in different crystallographic directions in silicon. A crystallographic direction is represented by three indexes named h, k, and l commonly called the (hkl) direction. For example the large peak at 28.426◦ corresponds to a reflection from the h = 1, k = 1, l = 1 or simply the (111) direction in silicon. The different directions can be determined from the silicon database entry provided to you. The section of the document named ’Peak List’ contains a listing of all the (hkl) directions and their corresponding 2θ locations. For this project you only need to be concerned with the following five peaks 28.426, 47.275, 56.089, 69.086, and 76.329◦ . The remaining peaks are either too small, or outside of the range of our data.
The data set contains a background curve which must be subtracted from the data. For our analysis we would like the region between peaks to be at a height of 0 intensity. To subtract the background, curve fit a Lagrange interpolating polynomial to points between the peaks and subtract the resulting polynomial from the original data. You may use any order polynomial you feel fits the background best, but as a guide some suitable 2θ values to use would be around 10, 20, 35, 60, and 80◦ . Select a high enough order polynomial to remove the background and result in a data set with 0 intensity at all places other than near the 5 clear peaks. Plot the data with the background removed, we will use this background removed data set for the remainder of the project.
The lattice parameter is the size of the crystal’s most basic repeating unit, or unit cell. Silicon forms a cubic unit cell and therefore has only one lattice parameter. Bragg’s Law relates a cubic lattice parameter to the location of a particular (hkl) reflection’s θ value as follows: