Normalization, structure factors

Figure 13. Principle of direct methods using triplet relations. As shown in the lower right-hand image the trial structure eonsists of atoms which are located at the eomers of the unit eell. Aeeording to the Z2 formula (Sayre equation) a strict phase relation exists within a eertain set of three reflections (a triplet) with large normalized structure factor amplitudes Eu. Sueh a triplet or origin invariant sum is defined as hiEli + + h k h = 0 or hiEli +...

Many people have recognized that the rotation function suffers from some drawbacks and have tried to improve the score by using origin-removed Patterson functions, normalized structure factors E-values, etc. (Briinger, 1997). 

As will be described in Section 9.3, direct methods are techniques that use probabilistic relationships among the phases to derive values of the individual phases from the experimentally measured amplitudes. In order to take advantage of these relationships, a necessary first step is the replacement of the usual structure factors, F, by the normalized structure factors (Hauptman and Karle, 1953),... 

Table 9.2 Normalized structure-factor magnitude statistics for the peak-wavelength data for methylmalonyl-coA epimerase (1JC4)...

The phase problem of X-ray crystallography may be defined as the problem of determining the phases ( ) of the normalized structure factors E when only the magnitudes E are given. Since there are many more reflections in a diffraction pattern than there are independent atoms in the corresponding crystal, the phase problem is overdetermined, and the existence of relationships among the measured magnitudes is implied. Direct methods (Hauptman and Karle, 1953) are ab initio probabilistic methods that seek to exploit these relationships, and the techniques of probability theory have identified the linear combinations of three phases whose Miller indices sum to... 

In DLS computations, constant atomic distances are used and have been very useful. Therefore, it appears worthwhile to constrain the positional parameters of the framework with respect to the known distances for Si-O, Al-O, and 0-0 for normal structure factor least-squares computations. Constrained refinement essentially reduces the number and/or variability of the parameters and can be helpful for work with limited data sets (e.g., for powder diffraction). Constrained refinement has been discussed by Pawley (26, 27). 

Direct methods is the most widely used techinque for getting a trial structure in small molecule crystallography, and has increasing applications in macromolecular crystallography as well (36). The problem is one of finding a set of approximate phases 0 to assign the observed normalized structure factor magnitudes... 

DM uses the normalized structure factors Em obtained from the observed structure factors Fhtd and the individual theoretical atomic scattering factors /, ... 

In direct methods it is usual to replace this expression by one for that of point atoms, so that X-ray scattering is now essentially independent of sind/X. This is done by dividing F(hkl) by a function of fj that eliminates any fall-off of intensity as a function of sinO/X (see Figure 8.4). The resulting normalized structure factor, E[hkl) is ... 

Normalized structure factors E are calculated from the observed magnitudes F of the structure factors. Only high values e.g., I l > 1.5) are used, because they will be the main contributors to the E map. 

Epsilon factor, e This is a weighting factor used in the calculation of normalized structure factors. It takes into account the fact that, depending on which of the 32 crystal classes the crystal belongs to, there will be certain groups of reflections that... 

Normalized structure factor The ratio of the value of the structure amplitude I F I to its root-mean-square expectation value. This ratio is denoted E(hkl). 

This is now generally replaced by a normalized structure factor (q.v.). Vector superposition map See Superposition method. 

Karle, J. Direct methods for structure determination origin specification, normalized structure factors, formulas, and the symbolic-addition procedure for phase determination. In International Tables for X-ray Crystallography. Volume IV. Revised and Supplementary Tables. Section 6. (Eds., Ibers, J. A., and Hamilton, W. C.) pp. 339-358. Kynoch Press Birmingham (1974). 

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). ... 

In certain applications, e.g. when the normalized structure factors should be calculated (see section 2.14.2), the knowledge of the approximate scale factor is required before the model of the crystal structure is known. This can be done using various statistical approaches [e.g. see A.J.C Wilson, Determination of absolute from relative x-ray intensity data,... 

The normalized structure factor is commonly denoted as Ehu and it is calculated from the conventional structure factor as follows ... 

Once scale and average thermal factors have been determined, normalized structure factors IFhl can be calculated as follows ... 

Furthermore, the distributions of the normalized structure factors are strongly affected by pseudotranslational symmetry, and for powder data also by preferred orientation effeets. The above information can also be used as prior information to perform a better powder pattern decomposition, so improving the efficieney of DM. 

Figure 8.3 Probability distribution function of normalized structure factor ampli tudes for centrosymmetric (centric) and non centro symmetric (acentric) structures.

Fig. 5.1 Values of the normalized structure factor kDH(kD) for 2 m solutions of the alkali metal chlorides in water as a function of reciprocal distance /cd [3]. The data for KCl, NaCl, and LiCl have been shifted vertically by 20, 40, and 60nm respectively, for the sake of clarity.

The commonly used methods for solving the phase problem required for structure solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets 9 = The probability of 0 0 increases with the magnitude of the product of the normalized structure factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

It was in 1953 that Hauptman and Karle established the basic concepts and the probabilistic foundations of direct methods. Of major importance was the introduction of the normalized structure factor, given by the equation... 

The tangent formula provides a powerful equation for establishing relations between structure factors and phases and for refining a probabilistic set of phases till self-consistence this process is usually known as convergence. With the knowledge of a sufficiently large subset of phases, an electron density map can be calculated using eqn [11] to reveal the structure. These maps are called E-maps because they use the normalized structure factor, E hkl), as coefficients of the Fourier series rather than the structure factor, F hkl). 

The Hughes form of the Sayre equation (34) is for the centrosymmetric case in the noncenlro-symmetric case, at least as far as physical information is concerned, it corresponds to the tangent formula. In contrast to Equation (33), here the. sum is weighted with the normalized structure factors, and all pairs of structure factors possible in a special case are taken care of ... 

The intensity data collected from an experiment can be transformed into normalized structure factor magnitudes by... 

The structure factors in inequality (18) can be replaced by quasi-normalized structure factors , which represent point atoms (to an approximation when atoms of unequal atomic numbers are present) rather than atoms with electron distributions. 

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