Molecules convolutions with lattices

Treatment of the experimental data requires that the diffraction peak be fitted to a function that characterizes the peak shape convoluted with the instrumental function a background, which may depend on Q, must also be subtracted. The choices involved in these procedures are not unequivocal. Even with intense synchrotron sources, higher-order reflections are not usually observed, so the assignment of a structure is based on the position of only one peak, i.e., on only one lattice spacing. The interpretation of the structures is made plausible by knowledge of the molecular structure of the amphiphile and of the crystal structures of similar molecules. 

Fig. 2. The crystal lattice is a series of points. When combined (convoluted) with a motif such as protein molecule, a crystal structure is obtained.

Because the diffraction pattern of a crystal is the periodic superposition (or product, or convolution) of the continuous transform of the unit cell contents with the lattice transform, other interesting consequences follow. For example, the locations of reflections in the diffraction pattern of a crystal, the net or lattice on which they fall, is entirely determined by the lattice properties of the crystal, namely the unit cell vectors. They in no way depend on the structure or properties of the molecules that fill the unit cells. On the other hand, the intensity we measure at each point in the diffraction pattern, and its associated phase, is entirely determined by the distribution of electrons, the positions of atoms xj, yj, Zj, within the unit cells. 

We have seen that the diffraction pattern of a crystal is the convolution of the contents of a unit cell with that of the crystal lattice (or product of the diffraction patterns, or Fourier transforms). As we have seen, and as illustrated in Figure 1.8 of Chapter 1, and in Figure 8.8, the lattice determines the points in reciprocal space where the transform of the molecules can be observed, and the arrangement of atoms within the unit cell specify the intensity, or value of the combined transform at each point. The asymmetric units in the unit cell are related by space group symmetry, and this symmetry is carried over, except for translations, into reciprocal space. Thus the diffraction pattern reflects the rotational symmetry elements of... 

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