Angles, real-reciprocal space

Figure 4.8 The powder difTraction experiment, (a) Reciprocal space notation. The Ewald sphere is fixed, and the lattice is rotated about all angles about the origin. Only the rotations about [100] are shown in this two-dimensional section. Intersections with the Ewald sphere define the diffracting conditions, (b) The corresponding diffracted beams in real space...

Structure solution in powder diffraction is approached by two different methodologies. One is using the conventional reciprocal space methods. The second is by real space methods where all the known details about the sample (say, molecular details such as bond distances, angles, etc., for an organic molecule, and coordination spheres such as octahedral, tetrahedral etc., in case of inorganic compounds) in question are exploited to solve the structure. 

Fourier series are used in crystal structure analysis in several ways. An electron-density map is a Fourier synthesis with measured values of F hkl) and derived values of phase angles 0 1. A Fourier analysis is the breakdown to component waves, as in the diffraction experiment. Fourier transform theory allows us to travel computationally between real space, p xyz), and reciprocal space, F hkl). 

In Table 7.1, where real and reciprocal cell dimensions, or other distances are related, an orthogonal system is assumed for the sake of simplicity. For nonorthogonal systems, the relationships are somewhat more complicated and contain trigonometric terms (as we saw in Chapter 3), since the unit cell angles must be taken into account. Rotational symmetry is preserved in going from real to reciprocal space, and translation operations create systematic absences of certain reflections in the diffraction pattern that makes them easily recognized. As already noted, because of Friedel s law a center of symmetry is always present in diffraction space even if it is absent in the crystal. This along with the absence of... 

In evaluating a structure determination for accuracy and precision, it is usually prudent to inquire as to the quality of the electron density map according to which the final model was built, and to the properties and quality of the final model. The former question can be addressed in real space, that is, how good is the electron density map, and/or in reciprocal space, that is, how well determined were the phases, and how well measured the structure amplitudes that contributed to the map. The model, of course, can be judged by how well it predicts the diffraction intensities (the R factor), by how it explains chemical and biochemical questions, and how well it agrees with canonical stereochemical properties, such as bond lengths and angles. 

At last, and whatever the wavelength, the atomic distribution in real space is not directly accessible. In addition, because now reciprocal space corresponds to a continuous content, I(s) should be recorded in the whole reciprocal space. This is impossible because the pattern is limited on one side by the width of the incident beam (the zero 9 angle is not available) on the other side, the wide angles are never accessible beyond 2/X. These two conditions amount to introducing a mask in the diffraction pattern, responsible for many artifacts. 

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