Relative phase angle

Figure 24.11 Variation of Trms (left axis) values measured at port 1 x/d = 4) and CO concentration (right axis) of gases sampled from port 4 x/d= 14) with relative phase angle between primary and secondary air driving in the 50-kilowatt forced combustor at China Lake...

Vector voltmeter An instrument that measures the amplitude (voltage) and relative phase angle of two signals, one of which serves as its reference. Typically, the phase difference as well as the ratio, difference, or individual values of the amplitudes of the two signals can be output in analog, digital, or visual display form. 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

FIGURE 8.1. Four waves with the same amplitude and periodicity are combined in three different ways [(a), (b), and (c)] as a result of different relative phase angles In each case the result of the Fourier synthesis (addition of waves) is different. Shown at the top of this Figure is a crystallographer with information on amplitudes and periodicities of the electron-density waves to be summed (on cardboard strips), but no information on relative phases (how to align the cardboard strips). 

The phase problem is more extensive for noncentrosymmetric structures. because then most of the structure factors have phase angles with any value between 0° and 360°. The number of relative phase combinations is, then, infinite. If, however, the relative phase angle is in the correct quadrant (0 to 90°, 90 to 180°, etc.), the map is generally interpretable. Therefore, as an approximation, the number of possibilities is reduced from infinity to 4. Of these, only one is totally correct, several are essentially correct, while the majority are incorrect. 

There are three ways commonly used to deduce the relative phase angles of a Bragg reflection, hkl, and one additional method, presently being developed. 

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. 

Note that even though the crystal structure is centrosymmetric in projection view down a), in this space group the origin is, by convention, chosen so that the relative phase angle a 0kl) equals 0, 90, 180 or 270°. 

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