Phase informative value

Once a set of initial phases has been chosen, it must be refined against the set of structure invariants whose values are presumed to be known. So far, two optimization methods (tangent refinement and parameter-shift reduction of the minimal frmction) have proven useful for extracting phase information from the structure invariants. The tangent formula (Karle and Hauptman, 1956),... 

In order to deduce the product P(0)R(0) in these experiments from a simple inversion of V z), it is necessary to measure it as a complex-valued function, i.e. with both amplitude and phase information. This can be done, though the requirements for thermal, mechanical, and electronic stability should not be underestimated. If the phase information is not available, then it must be reconstructed. 

Thermochemical data for the solvation of ions as used in the preceding calculations are difficult to measure and even to estimate. Therefore this kind of calculation of AH° for ionic reactions involving organic molecules in solution usually cannot be made. As a result, we have considerably fewer possibilities to assess the thermodynamic feasibility of the individual steps of polar reactions in solution than we do of vapor-phase radical processes. Bond energies are not of much use in predicting or explaining reactivity in ionic reactions unless we have information that can be used to translate gas-phase AH°. values to solution AH° values. Exercise 8-3 will give you a chance to see how this is done. 

FIGURE 6.16. Summing Fourier transforms of the Fourier components of an electron-density map to get a representation of a diffraction pattern, (a) The Fourier transforms of three electron-density waves (200, 300, and 500). (b) The sum of the electron-density waves in (a) (with relative phases 200, a ici — 180°, 300, a/,ki = 0°, and 500. Ohjci = 180°) give a diffraction pattern with intensity at 200, 300, and 500, but no phase information. The sign of the results of the Fourier transformation are lost when the values of F hkl) are calculated. 

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. 

The equivalent Cartesian form is given by Eq. (15) of ). The spherical tensor form [Eq. (9)] corrects the corresponding Eq. (24) of ). Drs. R. D. Peacock, B. Stewart and B. R. Judd are thanked for the correction. The values of 12 reported in ) were generally obtained from the square modulus of the first order electric dipole moment [Eq. (5)]. The individual components of the latter were calculated in order to retain the phase information, required for the investigation of the optical activity and the terms of the Faraday effect. The phase information is lost in the direct calculation of 12 through Eq. (9). 

Phase equilibrium information characterizes partitioning between phases for a system and is important for describing separation processes. For equilibrium-limited processes, these values dictate the limits for separation in a single stage. For mass transfer-limited processes, the partitioning between phases is an important parameter in the analysis. The data can be presented in tabular form. But this approach is restricted in application, since an analysis typically requires phase equilibrium values that are not explicitly listed in the table. So, graphical representation and computational methods are usually more useful. 

It should be emphasized that the system under consideration completely describes the process of transmission of the phase information between the field and the atom. Initially the atom is in the ground state with the angular momentum 0 and has no SU(2) atomic phase at all. Absorption of photons induces an atomic phase that coincides with the phase difference between the two coherent components of the field. This can be concluded from a direct calculation of the expectation value of the atomic cosine operator (43) over the state (96) ... 

The location of the heavy-atom sites and the refinement of positions, occupancies, and temperature factors are critical steps in the crystallographic analysis. Although numerous R factors are given, they are of little value to the general reader and the only trustworthy indicator that all is well is a reasonably flat residual Fourier map for each heavy-atom derivative.4 A useful empirical indicator of the quality of phase determination provided by a particular heavy-atom derivative is the ratio of the rms lack of closure error over the mean heavy-atom scattering. The phase information is of marginal value if the ratio exceeds one.5 In favorable cases the ratio is around 1/3 to 1/2. These statistics are usually provided in the protein structure paper. 

When the scattered waves from all of the atoms in a unit cell are added, the form of the resultant wave will depend, therefore, upon both the scattering power of the atoms involved and the individual phases of all of the separate waves. A clear picture of the relative importance of the scattering from each of the atoms in a unit cell, including the phase information, and how they add together to give a final value of F(hkl) can be... 

Use of multinuclear NMR H, 7Li and 19F) has corroborated evidence from DSC towards phase information. Thus faster relaxation times (T2) have been associated with crystalline phases and while longer T2, values have been attributed to amorphous solid solutions [59]. 

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