# Crystal Structure And X-Ray Diffraction Essay

Introduction

The purpose of this experiment is to study the relationship between the geometry of a sample and its diffraction patterns through an exploration of reciprocal space. In this experiment, both a single crystal piece of Si with a (004) reflection plane and a Si powder sample are scanned, as well as the peaks the locations of the diffraction peaks on the single crystal. The collected data is then analyzed to determine the criteria for planes to produce a spike in energy by plotting in reciprocal space. This experiment serves to illustrate the importance of reciprocal space. It also introduces the ideas of glancing incidence and exit.

Theory

The basis of x-ray diffraction

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While a single crystal only contains one plane that translates into one intensity peak in a diffraction scan, a powder sample is polycrystalline and thus contains many different planes results in many peaks in diffraction. This difference in number of peaks is also responsible for the difference is the regime of intensity, with single crystals have one, high intensity peaks, and polycrystalline samples have multiple smaller peaks.

Based on this background knowledge, there does not appear to be a way to access the other planes that produce intensity peaks in a single crystal sample. However, the planes can be accessed through tilting the sample stage. This is known as glancing incidence or glancing exit. Figure 1: Diagrams of glancing incidence (left) and glancing exit

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4)

Where (h1 k1 l1) are the miller indices of the fixed plane and (h2 k2 l2) are the miller indices of the desired plane

Since all the planes can be accessed, the other major concept of this experiment is reciprocal space. This concept utilizes reciprocal distance to form an easier representation of diffraction geometries. It can be calculated using Eq. 5 and analogous equation for the other axes. b_1=(a_2×a_3)/(a_1∙(a_2×a_3)) (Eq. 5)

Where a1, a2, a3 are the axes of the crystal in real space and b1 is the axis that corresponds to a1 in reciprocal space

In reciprocal space, the interplanar spacing has been translated into its reciprocal value, while the angles and crystal structure has been preserved. Since it is a two-dimensional representation of a three-dimensional phenomenon, not all the plane are represented at once.

Using reciprocal space, diffraction can then be visualized with the help of an Ewald sphere. An Ewald sphere (as see in Figure 2a) is constructed using the geometry of diffraction tied to the origin of the crystal in reciprocal space.

Figure 2a: Ewald sphere Figure 2b: Limiting