Symmetry and the strategy of collecting data

Strategy of data collection is guided not only by the unit cell s dimensions but also by its internal symmetry. If the cell and its contents are highly symmetric, then certain sets of crystal orientations produce exactly the same reflections, reducing the number of crystal orientations needed in order to obtain all of the distinct or unique reflections. 

As mentioned earlier, the unit-cell space group can be determined from systematic absences in the the diffraction pattern. With the space group in hand, the crystallographer can determine the space group of the reciprocal lattice, and thus know which orientations of the crystal will give identical data. All reciprocal lattices possess a symmetry element called a center cf symmetry or point of inversion at the origin. That is, the intensity of each reflection hkl is identical to the intensity of reflection -h k -1. To see why, recall from our discussion of lattice indices (Section II.B) that the the index of the (230) planes can also be expressed as (-2 -3 0). In fact, the 230 and the —2 -3 0 reflections come from opposite sides of the same set of planes, and the reflection intensities are identical. (The equivalence of Ihkl and l h k l is called Friedel s law,but there are exceptions. See Chapter 6, Section IV.) This means that half of the reflections in the reciprocal lattice are redundant, and data collection that covers 180° about any reciprocal-lattice axis will capture all unique reflections. 

In producing an image of molecules from crystallographic data, the computer simulates the action of a lens, computing the electron density within the unit cell from the list of indexed intensities obtained by the methods described in Chapter 4. In this chapter, I will discuss the mathematical relationships between the crystallographic data and the electron density. 

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

Chapter 5. From Diffraction Data to Electron Density 

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