Structure factors and intensities

The graphical method gives a lucid picture of scattering from a unit cell, but is impractical as a method for calculation of the intensities of diffracted beams. The pictorial summation must be expressed algebraically for this purpose. The simplest way of carrying this out is to express the scattered wave as a complex amplitude, (see Appendices 5 and 6)  

Wave from atom A at x yz Wave from atom A at origin (b) (c) 

Equation (6.3) can be written in an equivalent form as a complex number  

The intensity scattered into the hkl beam by all of the atoms in the unit cell, IQ(hkl), is given by V(hkl) 2, the modulus of V kki squared, (see Appendix 6)  

In this form, the total scattering from a unit cell containing N different atoms, is simply  

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The structure factor and intensity of scattering (5.25) have a spherically symmetric Lorentzian form centered at the zero wavevector q = 0. The fuU width oti the half a maximum (FWHM) is equal to 2k=2/. ... 

When we describe structure factors and electron density as Fourier series, we find that they are intimately related. The electron density is the Fourier transform of the structure factors, which means that we can convert the crystallographic data into an image of the unit cell and its contents. One necessary piece of information is, however, missing for each structure factor. We can measure only the intensity Ihkl of each reflection, not the complete structure factor Fhkl. What is the relationship between them It can be shown that the amplitude of structure factor Fhkl is proportional to the square root of... 

The atomic form factor accounts for the internal structure of the different atoms or molecules. It will also be different for X-rays and neutrons, since the former probe the electron distribution of the target, while the latter interact with the nuclei of the atoms. Therefore, the analysis of the positions of the reflexes indicates mainly the lattice constants and angles. The intensity of the reflexes contains mainly information about the atomic configuration within an unit cell (structure factor) and the scattering behavior of the single atoms (form factor). 

For molecules within periodic crystal lattices, the measured intensity data of the X-rays reflected from the (hkl) plane of the crystal can be converted into the F(,ki structure factors, and the observed electronic charge density p(p) can be obtained by a technique called Fourier synthesis ... 

As briefly mentioned in the previous chapter, the determination of a crystal structure may be considered complete only when multiple pattern variables and crystallographic parameters of a model have been fully refined against the observed powder diffraction data. Obviously, the refined model should remain reasonable from both physical and chemical standpoints. The refinement technique, most commonly employed today, is based on the idea suggested in the middle 1960 s by Rietveld. The essence of Rietveld s approach is that experimental powder diffraction data are utilized without extraction of the individual integrated intensities or the individual structure factors, and all structural and instrumental parameters are refined by fitting a calculated profile to the observed data. 

In calculating the value of R for a particular diffraction line, various factors should be kept in mind. The unit cell volume v is calculated from the measured lattice parameters, which are a function of carbon and alloy content. When the martensite doublets are unresolved, the structure factor and multiplicity of the martensite are calculated on the basis of a body-centered cubic cell this procedure, in effect, adds together the integrated intensities of the two lines of the doublet, which is exactly what is done experimentally when the integrated intensity of an unresolved doublet is measured. For greatest accuracy in the calculation of F, the atomic scattering factor f should be corrected for anomalous scattering by an amount A/ (see Sec. 13-4), particularly when Co Ka radiation is used. The value of the temperature factor can be taken from the curve of Fig. 4-20. 

Knowing the crystalline structure of a phase makes it possible to calculate the structure factors and the eell volume of that phase. Based on this information, the phases we are trying to qnantitatively analyze can be compared in pairs by calculating the ratios between the characteristic peak intensities of the various... 

If we assume as we did before that the cell contains n vacancies, the probability of an atom s presence on a site is (N - n)/N and the diffracted intensity of each peak will be decreased by a factor corresponding to this term. Furthermore, secondary peaks associated with the atomic displacements appear. The structure factor and hence the diffracted intensity are modulated (see Figure 5.3a). 

Reduction of the temperature of the crystal decreases the thermal motion of the atoms, which can be expressed in terms of the temperature factor (B). A reduction in B enhances the atomic scattering factors (Fig. 3.1.2) and thus the corresponding structure factors and measured intensities (Table 3.1.1), particularly at higher Bragg angles. 

As mentioned previously, a crystal will diffract x-rays with an intensity proportional to the square of the structure factor and is described by Eq. (33). The abrupt termination of the lattice at a sharp boundary (i.e., a surface) causes two-dimensional diffraction features termed crystal truncation rods (CTRs). Measurements of CTRs can provide a wealth of information on surface roughness and may be useful in the determination of crystallographic phase information. ... 

We have shown that the electron density can be expressed as a Fourier series with the structure factors as coefficients. In an analogous way, the structure factors can be expressed in terms of the electron density. There is a mathematical way of expressing these analogies, and it involves Fourier transforms (Fig. 15). The electron density is the Fourier transform of the structure factor, and the structure factor is the Fourier transform of the electron density. If the electron density can be expressed as the sum of cosine waves, that is, a Fourier series, its Fourier transform gives a function with high values at positions corresponding to intensity in the diffraction pattern. The Fourier transform provides the possibility of using a... 

D20.4 The phase problem arises with the analysis of data in X-ray diffraction when seeking to perform a Fourier synthesis of the electron density. In order to carry out the sum it is necessary to know the signs of the structure factors however, because diffraction intensities are proportional to the square of the structure factors, the intensities do not provide information on the sign. For non-centrosymmetric crystals, the structure factors may be complex, and the phase a in the expression F/m = F w e is indeterminate. The phase problem may be evaded by the use of a Patterson synthesis or tackled directly by using the so-called direct methods of phase allocation. 

The structural parameters are the atomic positions, Debye-Waller factors, electron structure factors, and electron diffraction parameters, which include the absorption potential, sample thickness, and crystal orientation. Not all parameters can be refined together. Diffraction patterns that are sensitive to certain parameters are collected and they are often refined independently. Figure 9 shows an example of a structure factor measurement by fitting CBED intensities recorded in the systematic orientation where one row of reflections are set at or near the Bragg conditions. Details about this method and its applications for structure factor measurement and atomic structure refinement are given in Ref. 18 and 43. 

The X-ray diffraction patterns showed significant changes in the relative intensities of the diffraction lines of the exchanged zeolites. Furthermore, a strong decrease of the diffraction peak intensities was observed on Cs-exchanged zeolites. These results can be attributed to changes in the structure factors and in X-ray absorption coefBcients of the parent zeolites caused by the incorporation of a cation bigger than Na [10]. Nevertheless, the XRD structure of the parent zeolites was preserved. 

Most experimental structures are solved by X-ray crystallographic methods. The standard procedure for the determination of the 3-D arrangement of atoms involves the raw diffraction data (the intensities of the reflected X-ray beams or structure factors) and the refinement by a theoretical model based on an atomic model of the molecule in the case of proteins, this is the primary structure. The introduction of constraints is necessary for most diffraction experiments, in particular in the area of protein crystallography. Only with these can a chemically reasonable structure with the positions of all atoms in the molecule be obtained [288]. 

Structures of powdered P-rhombohedral boron and amorphous boron were investigated with pulsed neutron diffraction techniques (Delaplane et al. 1988). To avoid intensive neutron absorption by °B nuclei, samples were "B isotopically enriched up to 97.1% and 99.1%, respectively. Earlier neutron diffraction studies based on nuclear reactor data did not permit the derivation of a meaningful radial distribution of atoms in amorphous material due to limited range of the neutron wave vector (<10.8 A" ). The obtained static structural factor and derived radial distribution function supported a structural model of amorphous boron based on building blocks of B,2 icosahedra resembling those found in p-rhombohedral boron, but with disorder occupying in the linking between ico-sahedral subunits. The intensity data indicated that amorphous samples contained 5% of a mixture of crystalline a- and p-rhombohedral boron. 

After constructive interference, this radiation produces experimental intensities, and from these experimental magnitudes the structure factor modulus, F hkl), is obtained. A Fourier analysis based on structure factors and a calculated set of phases determines with precision the atomic arrangement in space and provides an image of the internal structure of the crystal. 

Figure 5.5 Calculated equatorial scatterings from an assembly of hard rods. Y axis is the intensity while X axis is the scattering vector 2qa. (a) Without structure factor and (b) with structure factor.

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