Direct methods of relative phase determination

How is it possible to derive phase information when only structure amplitudes have been measured An answer can be found in what are called direct methods of structure determination. By these methods the crys-tallographer estimates the relative phase angles directly from the values of F hkl) (the experimental data). An electron-density map is calculated with the phases so derived, and the atomic arrangement is searched for in the map that results. This is why the method is titled direct. Other methods of relative phase determination rely on the computation of phase angles after the atoms in a trial structure have been found, and therefore they may be considered indirect methods. Thus, the argument that phase information is lost in the diffraction process is not totally correct. The phase problem therefore lies in finding methods for extracting the correct phase information from the experimental data. 

FIGURE 8.3. Two conditions restricting relative phases. Left hand side nonnegativity of the electron-density map. Right hand side isolated atoms. 

It is somewhat surprising that, in spite of what has been said in earlier chapters about the loss of phase information in the X-ray diffraction experiment, the intensity distribution contains information on whether the structure is centrosymmetric or noncentrosymmetric. This information is important when there is a spaoe group ambiguity in which one possibility is centrosymmetric, while the other is not (see Chapter 4. Arthur J. C. Wilson - noted that, while the intensities of Bragg reflections on the average depend only on the atomic contents of the unit cell, the distribution of intensities is different depending on whether the structure is centrosymmetric or noncentrosymmetric. The intensities 

TABLE 8.1. Values of the e factor for three crystal systems. 

Note that is a factor needed to account for the enhancement of intensities of certain classes of Bragg reflections with at least one index zero, i.e., certain classes of Bragg reflections that have an average intensity higher than the average. 

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