## Atoms

### X-ray Diffraction

The Bragg (father and son) pioneered the discovery to discern the structure of crystals in early 20th century. They took the diffraction pattern resulting from the interaction between the atoms (in the crystal) and X-ray and developed a formula to find out the inter-atomic distance. The method is

#### Figure 13-04b XRD Pattern

now widely used in molecular biology and biochemistry as well. The following provides a brief explanation for the process.
1. Incident X-ray is collimated into plane wave and hits the crystal either in one piece or in powdery form. X-ray is used because its wavelength is in the same order of magnitude of the inter-atomic distance.

2. The atoms become polarized into dipoles. Each dipole oscillates under the influence of the electromagnetic waves. The oscillations in term re-emit radiation in all directions in the form of spherical waves. The un-scattered incident waves appear as a bright spot (black in a photographic plate) at the center (Figure 13-04a).

3. At certain incident angles to the crystal planes (Bragg planes) the scattered beams add together constructively to form bright
4. spots around the central point. The formula for such constructive interference is :
2d sin() = n
where d is the spacing between Bragg planes, is the incident angle, is the wavelength, and the integer n is the order of the scattered beam, e.g., higher number of n corresponds to bright spot further away from the incident direction. The angular range of the diffractometer usually restricts n to be 1.

#### Figure 13-04c Miller Indices [view large image]

5. The spacing between the Bragg planes "d" is related to the inter-atomic distance "a" by the formula:
a = d (h2+k2+l2)1/2
where the Miller indices (h k l) are defined as the reciprocals of the fractional intercepts (see Figure 13-04c for a graphical explanation).

6. The diffraction pattern for a piece of amorphous material (such as glass) usually appears as concentric rings around the un-scattered beam image. For crystal with regular spacing between atoms, the ring breaks up into spots as shown in Figure 13-04a. Another method is to grind the specimen into powder (to the size of about 10-4 cm) and put them in a holder. In this way
7. it is not necessary to orient the crystal in various positions to obtain diffraction patterns for different Bragg planes. Figure 13-04b plots the intensity of the scattered waves versus twice the incident angles obtained by the diffractometer similar to the one shown in Figure 13-04a. The graph shows the intensity variation produced by the various Bragg planes. The International Centre for Diffraction Data maintains JCPDS (Joint Committee on Powder Diffraction Standards) cards for about 500,000 powder diffraction patterns (as of 2006), which can be used to identify substances in a given diffraction pattern such as shown in Figure 13-04d (CPS stands for counts of X-ray photons per second).

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