Structure amplitude phase angle

Since the phase angles cannot be measured in X-ray experiments, structure solution usually involves an iterative process, in which starting from a rough estimate of the phases, the structure suggested by the electron density map obtained from Eq. (13-3) and the phase computed by Eq. (13-1) are gradually refined, until the computed structure factor amplitudes from Eq. (13-1) converge to the ones observed experimentally. 

We have collected a set of observed structure factors - hL for which we have only amplitude information. Denote the required phase angle for... 

Once the heavy-atom position has been determined, its structure factor amplitude f h and phase an can be calculated. Since the structure factor amplitudes for the native (Fp) and derivative (Fp ) are experimentally measured quantities, it is thus possible to calculate the protein phase angle ap from the following equations ... 

The Patterson function (Patterson, 1934) is a phaseless Fourier summation similar to that in Eq. 4 but employing as coefficients, thus it can be calculated directly from the experimentally measured amplitudes (Fp) without the need to determine the phase angle. In the case of macromolecules, (Fpn —Fp ) are used as coefficients in Eq. 4 to produce a Patterson map (hence the name difference Patterson). Such a map contains peaks of vectors between atoms (interatomic vectors). Thus in the case of a difference Patterson of macromolecules, it is a heavy-atom vector map. For example if a structure has an atom at position (0.25, 0.11, 0.32) and another atom at position (0.10, 0.35, 0.15), there will be a peak in the Paterson map at position (0.25-0.10, 0.11-0.35, 0.32-0.15), meaning a peak at (0.15, —0.24, 0.17). 

FW2, and so on are the structure amplitudes for these reflections, calculated from the intensities in the way described earlier in this book. If the projection has a real or apparent centre of symmetry, the phase angles with respect to this centre of symmetry are all either 0° or 180°. 

A one-dimensional Fourier synthesis can be used for the direct determination of atomic parameters along any crystal axis, provided that the phase angles ct for the various orders of reflection are known. For an example, consider the structure of sodium nitrite, already described in Chapter IX. It is body-centred, hence only even orders of 00/ appear their structure amplitudes are 8, 15, 10, 2, and 7 for 002, 004, etc., respectively. 

Figure 6.8 a shows the phase determination using a second heavy-atom derivative F h is the structure factor for the second heavy atom. The radius of the smaller circle is IF Hpl, the amplitude of F Hp for the second heavy-atom derivative. For this derivative, Fp = F Hp — F H. Construction as before shows that the phase angles of F and Fj are possible phases for this reflection. In Fig. 6.8 b, the circles from Figs. 6.7b and 6.8a are superimposed, showing that Fp is identical to F. This common solution to the two vector equations is Fp, the desired structure factor. The phase of this reflection is therefore the angle labeled a in the figure, the only phase compatible with data from both derivatives. 

The structure factor is a complex number, expressing both the amplitude Fhki and the phase angle Uhki of the scattered... 

Thus if we know the moduli (amplitudes) and the phase angles of X-ray reflections, we can calculate the distribution of electron density in the unit cell, that is, find the atomic positions. Finding Fhki from fu is straightforward, but the values of ahu are lost in all usual diffraction experiments. Determining (reconstructing) phase angles remains the most difficult part of the structure solution. 

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. 

If this atom is sufficiently heavy, it will also dominate the relative phase angles of the Bragg reflections, as shown in Figure 8.18. An electron-density map calculated with observed structure amplitudes, and phases calculated from the positions of the heavy atoms (see Figure 6.21. Chapter 6), will contain a high peaJi at the position of the heavy atom. 

FIGURE 8.23. Calculation of phase angles for a centrosymmetric crystal by the method of isomorphous replacement. Two isomorphous crystals have structure amplitudes I El I and T2. The replaceable atom M hcis calculated structure factors M = Ml -M2. From these it is possible to deduce relative phases (signs) for Fi and F2- In each ca.se the vector from Fi to F2 must be the same as the vector Fmi—Fm2-... 

The F(obs) map this Fourier summation is calculated by use of Equation 9.1.1 in Table 9.1. It can be used to determine the arrangement of the atoms in the crystal structure. The relative phase angles a hkl) used to calculate this map have been derived by one of the methods described in Chapter 8. The coefficients are generally either experimentally determined structure amplitudes, F hkl) o, structure amplitudes calculated from a model, F(hkl) c, or normalized structure factor amplitudes E(hkl). ... 

The most valuable Bragg reflections for use in difference maps are those for which the calculated structure amplitudes are much larger than those observed. The model has contributed too much to these Bragg reflections, and a difference map with - F c as coefficients, i.e., F <, - F c, with F o set to zero for these specific Bragg reflections, will indicate how to correct the model. The calculated relative phase angle is probably approximately correct. 

Preliminary three-dimensional atomic coordinates of atoms in crystal structures are usually derived from electron-density maps by fitting atoms to individual peaks in the map. The chemically reasonable arrangement of atoms so obtained is, however, not very precise. The observed structure amplitudes and their relative phase angles, needed to calculate the electron-density map, each contain errors and these may cause a misinterpretation of the computed electron-density map. Even with the best electron-density maps, the precisions of the atomic coordinates of a preliminary structure are likely to be no better than several hundredths of an A. In order to understand the chemistry one needs to know the atomic positions more precisely so that better values of bond lengths and bond angles will be available. The process of obtaining atomic parameters that are more precise than those obtained from an initial model, referred to as refinement of the crystal structure, is an essential part of any crystal structure analysis. 

As a result, the structure amplitude can be represented by its magnitude, lF(h), and phase angle, a(h), which varies between 0 and 27t. When the crystal structure is centrosymmetric (i.e. when it contains the center of inversion), each atom with coordinates (x, y, z) has a sjmimetrically equivalent atom with coordinates (-x, -y, -z). Thus, considering that cos(-y) = cos(y) and sin(-y) = -sin(y) and assuming that all atoms scatter normally, Eq. 2.103 can be simplified to... 

As a result of symmetry transformation (Eqs. 2.111 and 2.112), both the magnitude of the structure amplitude and its phase may change. Finite symmetry operations (t, tj and tj are all 0) usually affect the phase angle, while infinite operations, i.e. those which have a non-zero translational component, affect both the magnitude and the phase. 

Equation 2.114 represents the algebraic formulation of the FriedeTs law, which states that the absolute values of structure amplitudes and intensities are identical but the phase angles have opposite signs for Bragg reflections related to one another by the center of inversion. In another formulation, it states that the reciprocal space is always centrosymmetric in the absence of the anomalous scattering because lF(h) = lF(h). Friedel s law is illustrated in Figure 2.57, left. 

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