Structure-factor calculations

The first was not the structure of brookite. The second, however, had the same space-group symmetry as brookite (Ft,6), and the predicted dimensions of the unit of structure agreed within 0.5% with those observed. Structure factors calculated for over fifty forms with the use of the predicted values of the nine parameters determining the atomic arrangement accounted satisfactorily for the observed intensities of reflections on rotation photographs. This extensive agreement is so striking as to permit the structure proposed for brookite (shown in Fig. 3) to be accepted with confidence. 

Incident beam normal to (110). Structure factors calculated for u = 0.292,... 

In contrast to single-crystal work, a fiber-diffraction pattern contains much fewer reflections going up to about 3 A resolution. This is a major drawback and it arises either as a result of accidental overlap of reflections that have the same / value and the same Bragg angle 0, or because of systematic superposition of hkl and its counterparts (-h-kl, h-kl, and -hkl, as in an orthorhombic system, for example). Sometimes, two or more adjacent reflections might be too close to separate analytically. Under such circumstances, these reflections have to be considered individually in structure-factor calculation and compounded properly for comparison with the observed composite reflection. Unobserved reflections that are too weak to see are assigned threshold values, based on the lowest measured intensities. Nevertheless, the number of available X-ray data is far fewer than the number of atomic coordinates in a repeat of the helix. Thus, X-ray data alone is inadequate to solve a fiber structure. 

Structure factors corresponding to 3,195 reflections between lOA and 1.7A were calculated for each of 50 coordinate sets at each temperature. Only the 246 heavy atoms of the hexamer were included in the structure factor calculations hydrogen atoms were not included in the refinement. 

Fig. 51. Neutron spin echo spectra obtained from the star 18 IIAA. The lines are a result of a fit with the dynamic structure factor calculated by Dubois Violette and de Gennes [34]. (Reprinted with permission from [150]. Copyright 1987 The American Physical Society, Maryland)...

In the same paper (Yamamoto 1996) an authoritative description is given of several interrelated topics such as super-space group determination, structure determination, indexing of diffraction patterns of quasicrystals, polygonal tiling, icosahedral tiling, structure factor calculation, description of quasicrystal structures, cluster models of quasicrystals. 

An example of this procedure is shown in Fig. 1. This example shows the build-up of the 2D potential of Ti2S projected along the short c axis, but the principle is the same for creating a 3D potential. The potential is a continuous function in real space and can be described in a map (Fig. 1). On the other hand, the structure factors are discrete points in reciprocal space and can be represented by a list of amplitudes and phases (Table 1). In this Fourier synthesis we have used the structure factors calculated from the refined coordinates of Ti2S °. 

Once the phase problem is solved, then the positions of the atoms may he relined by successive structure-factor calculations (Eq. 21 and Fourier summations (Eq. 3) or by a nonlinear least-squares procedure in which one minimizes, for example, )T u ( F , - F,il(, )- with weights w lakcn in a manner appropriate to the experiment. Such a least-squares refinement procedure presupposes that a suitable calculalional model is known. 

The expressions 2.7-2.12 which define the Leibler structure factor have been widely used to interpret scattering data from block copolymers (Bates and Fredrickson 1990 Mori et al. 1996 Rosedale et al. 1995 Schwahn et al. 1996 Stiihn et al. 1992 Wolff et al. 1993). The structure factor calculated for a diblock with / = 0.25 is shown in Fig. 2.39 for different degrees of segregation JV. Due to the Gaussian conformation assumed for the chains (Leibler 1980), the domain spacing in the weak segregation limit is expected to scale as d Nm. 

The output is the integrated intensity value for that particular reflection. FHKL - This is the structure factor calculating program. The input is a list of hkl s and intensity values. The output consists of E values and phase angles to be used as input to the electron density program. ELECDEN - Calculates the electron density and contours the E-map on a Tektronix 4662 digital plotter. PATTERSON - Used to calculate three-dimensional Patterson maps. 

Figure 6 shows the respective data plotted according to (21) for a number of blends with different degrees of polymerization. The left plot shows the Soret coefficients as measured and the right one after normalization to the mean field static structure factor calculated from the Flory-Huggins model, cf. (7). Although the structure factors and the Soret coefficients of the different samples vary by more... 

The single-chain structure factors calculated in the previous sections correspond to the infinite dilution limit. This limit also corresponds to zero scattering intensity and is not useful so that concentration effects have to be included in the modeling of polymer solutions. First, Zimm s single-contact approximation [5] is reviewed for dilute polymer solutions then, a slight extension of that formula which applies to semidilute solutions, is discussed. 

According to Drits (private communication, 1979) structure-factor calculations for the hM reflections of tochilinite I show that the reflections with k = 5n and k = 6n ought to be among the strongest reflections because of the subperiodidties of metal atoms in the two component structures. On the X-ray powder patterns and rotation photographs... 

Peaks occur in a difference map in positions in the unit cell where the model did not include enough electron density valleys appear in places where the model contained too much electron density. This information may be used to obtain more precise atomic positions, atomic displacement parameters, or atomic numbers. For example, in the last category, the identities of atoms (carbon or nitrogen) in a tricyclic molecule were established by setting all atoms to one type (carbon in this case) in the structure factor calculation. A difference map was calculated with the calculated phases and examined for excess electron density at atomic positions (Table 9.2). It was found to be possible to distinguish between nitrogen (seven electrons) and carbon (six electrons), even though these atoms are adjacent in the Periodic Table. 

Refinement also may be accomplished in real space by calculating difference Fourier syntheses using as coefficients A F = F i,s — Fcaic and phases caic derived from the trial structure. In general, if an atom has been incorrectly placed near its true location, a negative peak will appear at its assigned position and a positive peak will appear at its proper location. That is, in the difference Fourier we have subtracted electron density from where the atom isn t, and not subtracted density from where it is. The atom is then shifted by altering x, y, z, improved atomic parameters are included in a new round of structure factor calculations, and another difference Fourier computed. The process is repeated until the map is devoid of significant features. 

This structure factor calculation confirms the geometric argument of (001) extinction in a body-centered crystal. Practically, we do not have to calculate the structure extinction using Equation 2.11 for simple crystal structures such as BCC and face-centered cubic (FCC). Their structure extinction rules are given in Table 2.2, which tells us the detectable crystallographic planes for BCC and FCC crystals. From the lowest Miller indices, the planes are given as following. 

The reader may have noticed in the previous examples that some of the information given was not used in the calculations. In (a), for example, the cell was said to contain only one atom, but the shape of the cell was not specified in (b) and (c), the cells were described as orthorhombic and in (d) as cubic, but this information did not enter into the structure-factor calculations. This illustrates the important point that the structure factor is independent of the shape and size of the unit cell. For example, any body-centered cell will have missing reflections for those planes which have ft + k + 1) equal to an odd number, whether the cell is cubic, tetragonal, or orthorhombic. The rules we have derived in the above examples are therefore of wider applicability than would at first appear and demonstrate the close connection between the Bravais lattice of a substance and its diffraction pattern. They are summarized in Table 4-1. These rules are subject to... 

Simplify further, as necessary. In all structure-factor calculations the aim is to obtain a set of general equations that will give the value of F for any value of hkl. 

The unit cell thus contains a total of 16 molecules of water, corresponding to the composition M3[M (CN)6]a 12 HgO. The analytical data and the densities indicate a water content of 11 to 14 molecules per formula unit. The degree of hydration is very sensitive to changes in humidity and temperature. This rather broad range of hydration follows quite naturally from the zeolitic nature of part of the water. The structure factor calculations have been carried out on the basis of 12 water molecules per formula unit. Attempts to locate additional water have not been fruitful. 

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