The Structure Factor as a Product of Transforms

The last approach is for the more mathematically inclined. The expression for F was formulated above with the implicit assumption that Bragg s law was obeyed. It can be derived in another manner that explicitly contains Bragg s law. 

It was shown that the scattering function for a general scattering distribution such as the N atoms in a unit cell is 

The summation of exponential terms on the right is a Dirac delta function, a discrete function, which is everywhere zero except when the argument is zero or integral. The summation on the left is a continuous function, which determines the value of the entire transform at those nonzero points. Now d ki is normal to the set of planes of a particular family, and d ki I is the interplanar spacing. In order for dhu s = 1, s must be parallel with dhki and have magnitude 1/ Sm I that is, s = h the reciprocal lattice vector. If s h, then there is destructive interference of the waves diffracted by different unit cells, and the resultant wave from the crystal is zero. The elements of the diffraction spectra, the structure factors, for the crystal can therefore be written as 

We have now shown, by three different approaches, that if one knows the atomic coordinates xj, yj, Zj of all of the atoms j in a unit cell, and their scattering factors fj, then one can precisely predict the amplitude and phase of the resultant wave scattered by a specific family of planes hkl. We can calculate this for any and all families of planes in the crystal, hence the amplitude and phase can be calculated for every structure factor in the X-ray diffraction pattern. Given the structure of a crystal, namely the coordinates of the atoms in the unit cell, we can predict the entire diffraction pattern, the entire Fourier transform of the crystal. This is an enormously powerful statement. It means that if, by some means, we can deduce the positions of the atoms in a crystal structure, then we can immediately check the correctness of that deduction by seeing how well we can predict the relative values of the intensities in the diffraction pattern. 

2 To save paper, we are here compressing notation to the vector forms for position x = x, y, z and indexes h = h,k t, and using Euler s form of the complex numbers. We will expand them back to their full forms later. Again, it is unnecessary to actually multiply by NP, which we do not even know, since it is a constant and doesn t change the relative intensities or phases of the structure factors. 

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