Normalized structure factors values

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),... 

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. 

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). 

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.

Two helices are packed antiparallel in the orthorhombic unit cell. Association of the helices occurs through a series of periodic carboxylate potassium water - carboxylate interactions. An axial projection of the unit-cell contents (Fig. 23b) shows that the helices and guest molecules are closely packed. This is the first crystal structure of a polysaccharide in which all the guest molecules in the unit cell, consistent with the measured fiber density, have been experimentally located from difference electron-density maps. The final / -value is 0.26 for 54 reflections, of which 43 are observed, and it is based on normal scattering factors.15... 

Normally, if the assumed model for a crystal structure has an R value of 0.5 and resists attempts to refine to a lower residual, then the model structure is rejected as false, and a new model is tried until a fit between the observed and calculated structure factors yields an acceptable residual (R < 0.25). (Other models were tried for this complex, but they either gave Fourier maps which were uninterpretable or they converged to the present model). However, the normal crystal structure is solved with data obtained from crystals which have dimensions of the order of 0.1 mm. In the crystals available for this experiment, two of the dimensions were of the order of 0.01 mm. Thus, long exposures were required to give a small number of relatively weak diffraction spots. (Each Weissenberg photograph was exposed for five days with Cuka radiation 50 kv., 20 ma. loading, in a helium atmosphere). 

In Figure 31 we have displayed a comparison between the calculated and observed values of dfldE for benzene. As mentioned in the main text, inclusion of the structure factor leads to a large deviation between calculations and observations. Now, the structure factor influences the normalization of the wave function, which normalization involves only the asymptotic form of the wave function. Examination of the relevant formula for dfjdE shows that it is necessary that there be very accurate... 

In Fig. 22, the normalized dynamic structure factor F(q, t) /S(q) is plotted at a small wavevector value, ql = 0.1206, in the time domain where the VACF shows pronounced t 3 decay. Circles show the simulated values, and the full line is the Gaussian fit. As seen from the figure, F(q, t) is a Gaussian function of time in the q — 0 limit. This is an important observation because it provides the key to the physical origin for the slow decay of Cv t). 

The intensity variation along the rod (i.e. as a function of or /) is solely contained in the structure factor it is thus related to the z-co-ordinates of the atoms within the unit-cell of this quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements. This is the case of a reconstructed surface, for which rods are found for fractional order values of h and k, i.e. outside scattering from the bulk. 

Both atoms have been confirmed on the Fourier map and it appears that the next missing atom is located in the 4(e) site with coordinates 0,0,0.368. All interatomic distances are normal and after including this atom into the computation of structure factors and phase angles, the corresponding Rp = 32.2%. This value is quite high, but it is still worthwhile to calculate a third Fourier map, which is shown in Table 6.15. 

The summation of exponential terms on the right is a Dirac delta function, a discrete function, which is everywhere zero except when the argument is zero or integral. The summation on the left is a continuous function, which determines the value of the entire transform at those nonzero points. Now d ki is normal to the set of planes of a particular family, and d ki I is the interplanar spacing. In order for dhu s = 1, s must be parallel with dhki and have magnitude 1/ Smreciprocal lattice vector. If s h, then there is destructive interference of the waves diffracted by different unit cells, and the resultant wave from the crystal is zero. The elements of the diffraction spectra, the structure factors, for the crystal can therefore be written as... 

In the case of neutron diffraction, the radiation is scattered by the atomic nuclei, not by the electrons. It turns out that nucleons such as H and have very different scattering amplitudes. This means that isotope effects are very important in developing experimental strategies. Soper and Phillips [8] used data for the structure function obtained in mixtures of normal and heavy water to extract values of the partial structure factors for water. In this way they were able to determine all of the pair distribution functions for water from their diffraction data. These are gHnW. g oHW) and gooW- More details of their experimental results are given in section 2.10. 

For a random structure, / = 0.83 for a centric distribution and R = 0.59 for an acentric distribution (which is always the case with proteins in three dimensions) [112,113]. In a small molecule structure R values of <0.10 are routine and many have R <0.05. For proteins an R 0.30 at 2.5 A resolution usually indicates that most of the structure is correct but several errors may remain. An R < 0.2 is usually satisfactory. Luzzati [114] has shown that if the errors in position are normally distributed and that if these errors are the sole cause of differences between observed and calculated structure factors, then at 2 A resolution a mean error in atomic position of 0.2 A gives rise to an R = 0.23, and an error of 0.1 A gives rise to an R of 0.12. The Luzzati estimate of errors, which is frequently used in protein crystallography, is usually an overestimate because other sources of error also contribute to the residual R. 

Fig.44. Collective structure factor S(x,e) dotted vs x = qRg(e,N) for f = 1/2, N = 20 and various choices of the energy kBTe between monomers of different kinds, allowing for a volume fraction , = 0.2 of vacancies on the simple cubic lattice. Curves are a fit to Eq. (187), treating both % and Sg in Eqs. (187) — (189) as adjustable parameters, while the actual gyration radius is used for the normalization of the abscissa. Perpendicular straightline shows the value x = 1.945of Leibler s theory [43]. The symbols denote the choices eN = 0,1,2,3,4 and6 (from bottom to top). From Fried and Binder [325],...

The finite size also causes a discreteness of the possible values of q to (2n/L) (vx, vy, vz), where (vX) vy, vz), are integers, in the calculation of the collective structure factors, which in the simulations is normalized as... 

As this procedure is iterated x will initially decrease until it reaches an equilibrium value about which it will fluctuate. The resulting configuration should be a three-dimensional structure that is consistent with the experimental total structure factor within experimental error. Statistically independent configurations may then be collected. In MMC, configurations are normally assumed to be independent if separated by N accepted moves, but in practice we normally use at least 5TV moves. 

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