X-Ray Diffraction from Crystalline Solids

In order to experimentally ascertain the space group and ultimate 3-D structure of a crystalline solid, one must impinge the crystal with high-energy electromagnetic radiation. For instance, when X-rays interact with a crystalline solid, the incoming 

1 = wavelength of the incoming beam 6 = angle of the incident and diffracted beams (i.e., Gin = 9out) 

Illustrated another way, if two waves differ by one whole wavelength, they will differ in phase by 360° or 27t radians. For instance, the phase difference, (p, of the (hkl) reflection resulting from a wave scattered by an arbitrary lattice atom at position(x, y, z), and another scattered from an atom at position (0, 0, 0) is  

If we now include the phase and scattering factor expressions we used in Eqs. 11 and 12, we will obtain an equation that fully describes the scattering of the wave from a lattice atom, in complex exponential form (Eq. 15)  

If we want to describe the resultant wave scattered by all atoms of the unit cell, we will need to replace the atomic scattering factor with the structure factor, F. This factor describes how the atomic arrangement affects the scattered beam, and is obtained by adding together all waves scattered by the discrete atoms. Hence, if a unit cell contains N atoms with fractional coordinates (Un, Vn, w ), and atomic scattering factors, /n, the structure factor for the hid reflection would be given by  

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